Modified Bateman solution for identical eigenvalues

International Nuclear Information System (INIS)

Dreher, Raymond

2013-01-01

Highlights: ► Solving indeterminacies due to identical eigenvalues in Bateman’s solution. ► Exact analytical solution of Bateman’s equations for identical eigenvalues. ► Algorithm calculating higher order derivatives appearing in this solution. ► Alternative evaluation of the derivatives through the Taylor polynomial. ► Implementation of an example program demonstrating the developed solution. - Abstract: In this paper we develop a general solution to the Bateman equations taking into account the special case of identical eigenvalues. A characteristic of this new solution is the presence of higher order derivatives. It is shown that the derivatives can be obtained analytically and also computed in an efficient manner

Eigenvalue Decomposition-Based Modified Newton Algorithm

Directory of Open Access Journals (Sweden)

Wen-jun Wang

2013-01-01

Full Text Available When the Hessian matrix is not positive, the Newton direction may not be the descending direction. A new method named eigenvalue decomposition-based modified Newton algorithm is presented, which first takes the eigenvalue decomposition of the Hessian matrix, then replaces the negative eigenvalues with their absolute values, and finally reconstructs the Hessian matrix and modifies the searching direction. The new searching direction is always the descending direction. The convergence of the algorithm is proven and the conclusion on convergence rate is presented qualitatively. Finally, a numerical experiment is given for comparing the convergence domains of the modified algorithm and the classical algorithm.

Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

Directory of Open Access Journals (Sweden)

Manfred Möller

2013-01-01

Full Text Available Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameter λ and which has separable boundary conditions depending linearly on λ. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.

Eigenvalue study of a chaotic resonator

Energy Technology Data Exchange (ETDEWEB)

Banova, Todorka [Technische Universitaet Darmstadt, Institut fuer Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstrasse 8, D-64289 Darmstadt (Germany); Technische Universitaet Darmstadt, Graduate School of Computational Engineering, Dolivostrasse 15, D-64293 Darmstadt (Germany); Ackermann, Wolfgang; Weiland, Thomas [Technische Universitaet Darmstadt, Institut fuer Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstrasse 8, D-64289 Darmstadt (Germany)

2013-07-01

The field of quantum chaos comprises the study of the manifestations of classical chaos in the properties of the corresponding quantum systems. Within this work, we compute the eigenfrequencies that are needed for the level spacing analysis of a microwave resonator with chaotic characteristics. The major challenges posed by our work are: first, the ability of the approaches to tackle the large scale eigenvalue problem and second, the capability to extract many, i.e. order of thousands, eigenfrequencies for the considered cavity. The first proposed approach for an accurate eigenfrequency extraction takes into consideration the evaluated electric field computations in time domain of a superconducting cavity and by means of signal-processing techniques extracts the eigenfrequencies. The second approach is based on the finite element method with curvilinear elements, which transforms the continuous eigenvalue problem to a discrete generalized eigenvalue problem. Afterwards, the Lanczos algorithm is used for the solution of the generalized eigenvalue problem. In the poster, a summary of the applied algorithms, as well as, critical implementation details together with the simulation results are provided.

The BR eigenvalue algorithm

Energy Technology Data Exchange (ETDEWEB)

Geist, G.A. [Oak Ridge National Lab., TN (United States). Computer Science and Mathematics Div.; Howell, G.W. [Florida Inst. of Tech., Melbourne, FL (United States). Dept. of Applied Mathematics; Watkins, D.S. [Washington State Univ., Pullman, WA (United States). Dept. of Pure and Applied Mathematics

1997-11-01

The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.

Application of collocation meshless method to eigenvalue problem

International Nuclear Information System (INIS)

Saitoh, Ayumu; Matsui, Nobuyuki; Itoh, Taku; Kamitani, Atsushi; Nakamura, Hiroaki

2012-01-01

The numerical method for solving the nonlinear eigenvalue problem has been developed by using the collocation Element-Free Galerkin Method (EFGM) and its performance has been numerically investigated. The results of computations show that the approximate solution of the nonlinear eigenvalue problem can be obtained stably by using the developed method. Therefore, it can be concluded that the developed method is useful for solving the nonlinear eigenvalue problem. (author)

Inequalities among eigenvalues of Sturm–Liouville problems

Directory of Open Access Journals (Sweden)

Kong Q

1999-01-01

Full Text Available There are well-known inequalities among the eigenvalues of Sturm–Liouville problems with periodic, semi-periodic, Dirichlet and Neumann boundary conditions. In this paper, for an arbitrary coupled self-adjoint boundary condition, we identify two separated boundary conditions corresponding to the Dirichlet and Neumann conditions in the classical case, and establish analogous inequalities. It is also well-known that the lowest periodic eigenvalue is simple; here we prove a similar result for the general case. Moreover, we show that the algebraic and geometric multiplicities of the eigenvalues of self-adjoint regular Sturm–Liouville problems with coupled boundary conditions are the same. An important step in our approach is to obtain a representation of the fundamental solutions for sufficiently negative values of the spectral parameter. Our approach yields the existence and boundedness from below of the eigenvalues of arbitrary self-adjoint regular Sturm–Liouville problems without using operator theory.

A method for the solution of the RPA eigenvalue

International Nuclear Information System (INIS)

Hoffman, M.J.H.; De Kock, P.R.

1986-01-01

The RPA eigenvalue problem requires the diagonalization of a 2nx2n matrix. In practical calculations, n (the number of particle-hole basis states) can be a few hundred and the diagonalization of such a large non-symmetric matrix may take quite a long time. In this report we firstly discuss sufficient conditions for real and non-zero RPA eigenvalues. The presence of zero or imaginary eigenvalues is related to the relative importance of the groundstate correlations to the total interaction energy. We then rewrite the RPA eigenvalue problem for the cases where these conditions are fulfilled in a form which only requires the diagonalization of two symmetric nxn matrices. The extend to which this method can be applied when zero eigenvalues occur, is also discussed

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements

Science.gov (United States)

Wu, Sheng-Jhih; Chu, Moody T.

2017-08-01

An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements

International Nuclear Information System (INIS)

Wu, Sheng-Jhih; Chu, Moody T

2017-01-01

An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing–Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations. (paper)

Accounting for Sampling Error in Genetic Eigenvalues Using Random Matrix Theory.

Science.gov (United States)

Sztepanacz, Jacqueline L; Blows, Mark W

2017-07-01

The distribution of genetic variance in multivariate phenotypes is characterized by the empirical spectral distribution of the eigenvalues of the genetic covariance matrix. Empirical estimates of genetic eigenvalues from random effects linear models are known to be overdispersed by sampling error, where large eigenvalues are biased upward, and small eigenvalues are biased downward. The overdispersion of the leading eigenvalues of sample covariance matrices have been demonstrated to conform to the Tracy-Widom (TW) distribution. Here we show that genetic eigenvalues estimated using restricted maximum likelihood (REML) in a multivariate random effects model with an unconstrained genetic covariance structure will also conform to the TW distribution after empirical scaling and centering. However, where estimation procedures using either REML or MCMC impose boundary constraints, the resulting genetic eigenvalues tend not be TW distributed. We show how using confidence intervals from sampling distributions of genetic eigenvalues without reference to the TW distribution is insufficient protection against mistaking sampling error as genetic variance, particularly when eigenvalues are small. By scaling such sampling distributions to the appropriate TW distribution, the critical value of the TW statistic can be used to determine if the magnitude of a genetic eigenvalue exceeds the sampling error for each eigenvalue in the spectral distribution of a given genetic covariance matrix. Copyright © 2017 by the Genetics Society of America.

The eigenvalue problem in phase space.

Science.gov (United States)

Cohen, Leon

2018-06-30

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

Extreme eigenvalues of sample covariance and correlation matrices

DEFF Research Database (Denmark)

Heiny, Johannes

This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... matrix of a p-dimensional heavy-tailed time series when p converges to infinity together with the sample size n. We generalize the growth rates of p existing in the literature. Assuming a regular variation condition with tail index ... eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables. The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high dimension...

Hessian eigenvalue distribution in a random Gaussian landscape

Science.gov (United States)

Yamada, Masaki; Vilenkin, Alexander

2018-03-01

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of 1/ N expansion, where N is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

An adjoint-based scheme for eigenvalue error improvement

International Nuclear Information System (INIS)

Merton, S.R.; Smedley-Stevenson, R.P.; Pain, C.C.; El-Sheikh, A.H.; Buchan, A.G.

2011-01-01

A scheme for improving the accuracy and reducing the error in eigenvalue calculations is presented. Using a rst order Taylor series expansion of both the eigenvalue solution and the residual of the governing equation, an approximation to the error in the eigenvalue is derived. This is done using a convolution of the equation residual and adjoint solution, which is calculated in-line with the primal solution. A defect correction on the solution is then performed in which the approximation to the error is used to apply a correction to the eigenvalue. The method is shown to dramatically improve convergence of the eigenvalue. The equation for the eigenvalue is shown to simplify when certain normalizations are applied to the eigenvector. Two such normalizations are considered; the rst of these is a fission-source type of normalisation and the second is an eigenvector normalisation. Results are demonstrated on a number of demanding elliptic problems using continuous Galerkin weighted nite elements. Moreover, the correction scheme may also be applied to hyperbolic problems and arbitrary discretization. This is not limited to spatial corrections and may be used throughout the phase space of the discrete equation. The applied correction not only improves fidelity of the calculation, it allows assessment of the reliability of numerical schemes to be made and could be used to guide mesh adaption algorithms or to automate mesh generation schemes. (author)

Efficient solutions to the NDA-NCA low-order eigenvalue problem

International Nuclear Information System (INIS)

Willert, J. A.; Kelley, C. T.

2013-01-01

Recent algorithmic advances combine moment-based acceleration and Jacobian-Free Newton-Krylov (JFNK) methods to accelerate the computation of the dominant eigenvalue in a k-eigenvalue calculation. In particular, NDA-NCA [1], builds a sequence of low-order (LO) diffusion-based eigenvalue problems in which the solution converges to the true eigenvalue solution. Within NDA-NCA, the solution to the LO k-eigenvalue problem is computed by solving a system of nonlinear equation using some variant of Newton's method. We show that we can speed up the solution to the LO problem dramatically by abandoning the JFNK method and exploiting the structure of the Jacobian matrix. (authors)

A robust multilevel simultaneous eigenvalue solver

Science.gov (United States)

Costiner, Sorin; Taasan, Shlomo

1993-01-01

Multilevel (ML) algorithms for eigenvalue problems are often faced with several types of difficulties such as: the mixing of approximated eigenvectors by the solution process, the approximation of incomplete clusters of eigenvectors, the poor representation of solution on coarse levels, and the existence of close or equal eigenvalues. Algorithms that do not treat appropriately these difficulties usually fail, or their performance degrades when facing them. These issues motivated the development of a robust adaptive ML algorithm which treats these difficulties, for the calculation of a few eigenvectors and their corresponding eigenvalues. The main techniques used in the new algorithm include: the adaptive completion and separation of the relevant clusters on different levels, the simultaneous treatment of solutions within each cluster, and the robustness tests which monitor the algorithm's efficiency and convergence. The eigenvectors' separation efficiency is based on a new ML projection technique generalizing the Rayleigh Ritz projection, combined with a technique, the backrotations. These separation techniques, when combined with an FMG formulation, in many cases lead to algorithms of O(qN) complexity, for q eigenvectors of size N on the finest level. Previously developed ML algorithms are less focused on the mentioned difficulties. Moreover, algorithms which employ fine level separation techniques are of O(q(sub 2)N) complexity and usually do not overcome all these difficulties. Computational examples are presented where Schrodinger type eigenvalue problems in 2-D and 3-D, having equal and closely clustered eigenvalues, are solved with the efficiency of the Poisson multigrid solver. A second order approximation is obtained in O(qN) work, where the total computational work is equivalent to only a few fine level relaxations per eigenvector.

Computing the eigenvalues and eigenvectors of a fuzzy matrix

Directory of Open Access Journals (Sweden)

A. Kumar

2012-08-01

Full Text Available Computation of fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix is a challenging problem. Determining the maximal and minimal symmetric solution can help to find the eigenvalues. So, we try to compute these eigenvalues by determining the maximal and minimal symmetric solution of the fully fuzzy linear system $widetilde{A}widetilde{X}= widetilde{lambda} widetilde{X}.$

Frequency response as a surrogate eigenvalue problem in topology optimization

DEFF Research Database (Denmark)

Andreassen, Erik; Ferrari, Federico; Sigmund, Ole

2018-01-01

This article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity as...

Generalization of Samuelson's inequality and location of eigenvalues

Indian Academy of Sciences (India)

We prove a generalization of Samuelson's inequality for higher order central moments. Bounds for the eigenvalues are obtained when a given complex × matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations.

EvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.

Science.gov (United States)

Tal-Ezer, Hillel

2016-05-19

Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen , D. C. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations ; Springer , 1997 ; Lehoucq , R. B. ; Sorensen , D. C. SIAM Journal on Matrix Analysis and Applications 1996 , 17 , 789 ; Calvetti , D. ; Reichel , L. ; Sorensen , D. C. Electronic Transactions on Numerical Analysis 1994 , 2 , 21 ) (or Eigs of Matlab) but significantly simpler.

Asymptotic Distribution of Eigenvalues of Weakly Dilute Wishart Matrices

Energy Technology Data Exchange (ETDEWEB)

Khorunzhy, A. [Institute for Low Temperature Physics (Ukraine)], E-mail: khorunjy@ilt.kharkov.ua; Rodgers, G. J. [Brunel University, Uxbridge, Department of Mathematics and Statistics (United Kingdom)], E-mail: g.j.rodgers@brunel.ac.uk

2000-03-15

We study the eigenvalue distribution of large random matrices that are randomly diluted. We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigenvalue distribution with a singular component at the origin. These include the Wishart random matrix ensemble and Gaussian random matrices with correlated entries. Our results show that the singularity in the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroys it.

Eigenvalues of the simplified ideal MHD ballooning equation

International Nuclear Information System (INIS)

Paris, R.B.; Auby, N.; Dagazian, R.Y.

1986-01-01

The investigation of the spectrum of the simplified differential equation describing the variation of the amplitude of the ideal MHD ballooning instability along magnetic field lines constitutes a multiparameter Schroedinger eigenvalue problem. An exact eigenvalue relation for the discrete part of the spectrum is obtained in terms of the oblate spheroidal functions. The dependence of the eigenvalues lambda on the two free parameters γ 2 and μ 2 of the equation is discussed, together with certain analytical approximations in the limits of small and large γ 2 . A brief review of the principal properties of the spheroidal functions is given in an appendix

Eigenvalue distributions of Wilson loops

International Nuclear Information System (INIS)

Lohmayer, Robert

2010-01-01

In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the ''symmetric'', the ''antisymmetric'', and the ''true'' eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. We study the transition by a detailed analysis of the average of the modulus square of the characteristic polynomial. Furthermore

Deflation of Eigenvalues for GMRES in Lattice QCD

International Nuclear Information System (INIS)

Morgan, Ronald B.; Wilcox, Walter

2002-01-01

Versions of GMRES with deflation of eigenvalues are applied to lattice QCD problems. Approximate eigenvectors corresponding to the smallest eigenvalues are generated at the same time that linear equations are solved. The eigenvectors improve convergence for the linear equations, and they help solve other right-hand sides

Jacobi-Davidson methods for generalized MHD-eigenvalue problems

NARCIS (Netherlands)

J.G.L. Booten; D.R. Fokkema; G.L.G. Sleijpen; H.A. van der Vorst (Henk)

1995-01-01

textabstractA Jacobi-Davidson algorithm for computing selected eigenvalues and associated eigenvectors of the generalized eigenvalue problem $Ax = lambda Bx$ is presented. In this paper the emphasis is put on the case where one of the matrices, say the B-matrix, is Hermitian positive definite. The

On the distribution of eigenvalues of certain matrix ensembles

International Nuclear Information System (INIS)

Bogomolny, E.; Bohigas, O.; Pato, M.P.

1995-01-01

Invariant random matrix ensembles with weak confinement potentials of the eigenvalues, corresponding to indeterminate moment problems, are investigated. These ensembles are characterized by the fact that the mean density of eigenvalues tends to a continuous function with increasing matrix dimension contrary to the usual cases where it grows indefinitely. It is demonstrated that the standard asymptotic formulae are not applicable in these cases and that the asymptotic distribution of eigenvalues can deviate from the classical ones. (author)

Eigenstructure of of singular systems. Perturbation analysis of simple eigenvalues

OpenAIRE

García Planas, María Isabel; Tarragona Romero, Sonia

2014-01-01

The problem to study small perturbations of simple eigenvalues with a change of parameters is of general interest in applied mathematics. After to introduce a systematic way to know if an eigenvalue of a singular system is simple or not, the aim of this work is to study the behavior of a simple eigenvalue of singular linear system family

The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile ...

Indian Academy of Sciences (India)

R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22

following universal inequalities for the λi's in the case when n = 2: λk+1 − λk ≤. 2 .... with V ≥ 0 on and eigenvalue problems with a weight (e.g., the fixed ...... [29] Protter M H, Universal inequalities for eigenvalues, Maximum Principles and Eigenvalue. Problems in ... minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl.

Efficient methods for time-absorption (α) eigenvalue calculations

International Nuclear Information System (INIS)

Hill, T.R.

1983-01-01

The time-absorption eigenvalue (α) calculation is one of the options found in most discrete-ordinates transport codes. Several methods have been developed at Los Alamos to improve the efficiency of this calculation. Two procedures, based on coarse-mesh rebalance, to accelerate the α eigenvalue search are derived. A hybrid scheme to automatically choose the more-effective rebalance method is described. The α rebalance scheme permits some simple modifications to the iteration strategy that eliminates many unnecessary calculations required in the standard search procedure. For several fast supercritical test problems, these methods resulted in convergence with one-fifth the number of iterations required for the conventional eigenvalue search procedure

An eigenvalue localization set for tensors and its applications.

Science.gov (United States)

Zhao, Jianxing; Sang, Caili

2017-01-01

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al . (Linear Algebra Appl. 481:36-53, 2015) and Huang et al . (J. Inequal. Appl. 2016:254, 2016). As an application of this set, new bounds for the minimum eigenvalue of [Formula: see text]-tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al ., the advantage of our results is that, without considering the selection of nonempty proper subsets S of [Formula: see text], we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of [Formula: see text]-tensors. Finally, numerical examples are given to verify the theoretical results.

Bound-state Dirac eigenvalues for scalar potentials

International Nuclear Information System (INIS)

Ram, B.; Arafah, M.

1981-01-01

The Dirac equation is solved with a linear and a quadratic scalar potential using an approach in which the Dirac equation is first transformed to a one-dimensional Schroedinger equation with an effective potential. The WKB method is used to obtain the energy eigenvalues. The eigenvalues for the quadratic scalar potential are real just as they are for the linear potential. The results with the linear potential agree well with those obtained by Critchfield. (author)

An eigenvalue localization set for tensors and its applications

Directory of Open Access Journals (Sweden)

Jianxing Zhao

2017-03-01

Full Text Available Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al. (Linear Algebra Appl. 481:36-53, 2015 and Huang et al. (J. Inequal. Appl. 2016:254, 2016. As an application of this set, new bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al., the advantage of our results is that, without considering the selection of nonempty proper subsets S of N = { 1 , 2 , … , n } $N=\\{1,2,\\ldots,n\\}$ , we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors. Finally, numerical examples are given to verify the theoretical results.

Eigenvalue distributions of Wilson loops

Energy Technology Data Exchange (ETDEWEB)

Lohmayer, Robert

2010-07-01

In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the ''symmetric'', the ''antisymmetric'', and the ''true'' eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. We study the transition by a detailed analysis of the average of the

Eigenvalues of PT-symmetric oscillators with polynomial potentials

International Nuclear Information System (INIS)

Shin, Kwang C

2005-01-01

We study the eigenvalue problem -u''(z) - [(iz) m + P m-1 (iz)]u(z) λu(z) with the boundary condition that u(z) decays to zero as z tends to infinity along the rays arg z = -π/2 ± 2π/(m+2) in the complex plane, where P m-1 (z) = a 1 z m-1 + a 2 z m-2 + . . . + a m-1 z is a polynomial and integers m ≥ 3. We provide an asymptotic expansion of the eigenvalues λ n as n → +∞, and prove that for each real polynomial P m-1 , the eigenvalues are all real and positive, with only finitely many exceptions

Oscillators and Eigenvalues

DEFF Research Database (Denmark)

Lindberg, Erik

1997-01-01

In order to obtain insight in the nature of nonlinear oscillators the eigenvalues of the linearized Jacobian of the differential equations describing the oscillator are found and displayed as functions of time. A number of oscillators are studied including Dewey's oscillator (piecewise linear wit...... with negative resistance), Kennedy's Colpitts-oscillator (with and without chaos) and a new 4'th order oscillator with hyper-chaos....

Cluster structure in the correlation coefficient matrix can be characterized by abnormal eigenvalues

Science.gov (United States)

Nie, Chun-Xiao

2018-02-01

In a large number of previous studies, the researchers found that some of the eigenvalues of the financial correlation matrix were greater than the predicted values of the random matrix theory (RMT). Here, we call these eigenvalues as abnormal eigenvalues. In order to reveal the hidden meaning of these abnormal eigenvalues, we study the toy model with cluster structure and find that these eigenvalues are related to the cluster structure of the correlation coefficient matrix. In this paper, model-based experiments show that in most cases, the number of abnormal eigenvalues of the correlation matrix is equal to the number of clusters. In addition, empirical studies show that the sum of the abnormal eigenvalues is related to the clarity of the cluster structure and is negatively correlated with the correlation dimension.

INDEFINITE COPOSITIVE MATRICES WITH EXACTLY ONE POSITIVE EIGENVALUE OR EXACTLY ONE NEGATIVE EIGENVALUE

NARCIS (Netherlands)

Jargalsaikhan, Bolor

Checking copositivity of a matrix is a co-NP-complete problem. This paper studies copositive matrices with certain spectral properties. It shows that an indefinite matrix with exactly one positive eigenvalue is copositive if and only if the matrix is nonnegative. Moreover, it shows that finding out

The nonconforming virtual element method for eigenvalue problems

Energy Technology Data Exchange (ETDEWEB)

Gardini, Francesca [Univ. of Pavia (Italy). Dept. of Mathematics; Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Vacca, Giuseppe [Univ. of Milano-Bicocca, Milan (Italy). Dept. of Mathematics and Applications

2018-02-05

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L²-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

Solving the RPA eigenvalue equation in real-space

CERN Document Server

Muta, A; Hashimoto, Y; Yabana, K

2002-01-01

We present a computational method to solve the RPA eigenvalue equation employing a uniform grid representation in three-dimensional Cartesian coordinates. The conjugate gradient method is used for this purpose as an interactive method for a generalized eigenvalue problem. No construction of unoccupied orbitals is required in the procedure. We expect this method to be useful for systems lacking spatial symmetry to calculate accurate eigenvalues and transition matrix elements of a few low-lying excitations. Some applications are presented to demonstrate the feasibility of the method, considering the simplified mean-field model as an example of a nuclear physics system and the electronic excitations in molecules with time-dependent density functional theory as an example of an electronic system. (author)

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Science.gov (United States)

Böttcher, A.; Bogoya, J. M.; Grudsky, S. M.; Maximenko, E. A.

2017-11-01

Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegő's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegő theorem on the asymptotic behaviour of the determinants of Toeplitz matrices are known. Starting in the 1950s, the asymptotics of the maximum and minimum eigenvalues were actively investigated. However, investigation of the individual asymptotics of all the eigenvalues and eigenvectors of Toeplitz matrices started only quite recently: the first papers on this subject were published in 2009-2010. A survey of this new field is presented here. Bibliography: 55 titles.

Perturbation of eigenvalues of preconditioned Navier-Stokes operators

Energy Technology Data Exchange (ETDEWEB)

Elman, H.C. [Univ. of Maryland, College Park, MD (United States)

1996-12-31

We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier-Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations, we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size, and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.

MAIA, Eigenvalues for MHD Equation of Tokamak Plasma Stability Problems

International Nuclear Information System (INIS)

Tanaka, Y.; Azumi, M.; Kurita, G.; Tsunematsu, T.; Takeda, T.

1986-01-01

1 - Description of program or function: This program solves an eigenvalue problem zBx=Ax where A and B are real block tri-diagonal matrices. This eigenvalue problem is derived from a reduced set of linear resistive MHD equations which is often employed to study tokamak plasma stability problem. 2 - Method of solution: Both the determinant and inverse iteration methods are employed. 3 - Restrictions on the complexity of the problem: The eigenvalue z must be real

Smallest eigenvalue distribution of the fixed-trace Laguerre beta-ensemble

International Nuclear Information System (INIS)

Chen Yang; Liu Dangzheng; Zhou Dasheng

2010-01-01

In this paper we study the entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed-trace Laguerre ensemble of N x N random matrices. We showed that for finite N the smallest eigenvalue distribution may be expressed in terms of Jack polynomials. Furthermore, based on the exact results, we found a limiting distribution when the smallest eigenvalue is suitably scaled with N followed by a large N limit. Our results turn out to be the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed-trace constraint. This suggests in a broad sense, the global constraint does not influence local correlations, at least, in the large N limit. Consequently, we have solved an open problem: the determination of the smallest eigenvalue distribution of the reduced density matrix-obtained by tracing out the environmental degrees of freedom-for a bipartite quantum system of unequal dimensions.

Fourier convergence analysis applied to neutron diffusion Eigenvalue problem

International Nuclear Information System (INIS)

Lee, Hyun Chul; Noh, Jae Man; Joo, Hyung Kook

2004-01-01

Fourier error analysis has been a standard technique for the stability and convergence analysis of linear and nonlinear iterative methods. Though the methods can be applied to Eigenvalue problems too, all the Fourier convergence analyses have been performed only for fixed source problems and a Fourier convergence analysis for Eigenvalue problem has never been reported. Lee et al proposed new 2-D/1-D coupling methods and they showed that the new ones are unconditionally stable while one of the two existing ones is unstable at a small mesh size and that the new ones are better than the existing ones in terms of the convergence rate. In this paper the convergence of method A in reference 4 for the diffusion Eigenvalue problem was analyzed by the Fourier analysis. The Fourier convergence analysis presented in this paper is the first one applied to a neutronics eigenvalue problem to the best of our knowledge

Modern algorithms for large sparse eigenvalue problems

International Nuclear Information System (INIS)

Meyer, A.

1987-01-01

The volume is written for mathematicians interested in (numerical) linear algebra and in the solution of large sparse eigenvalue problems, as well as for specialists in engineering, who use the considered algorithms in the investigation of eigenoscillations of structures, in reactor physics, etc. Some variants of the algorithms based on the idea of a gradient-type direction of movement are presented and their convergence properties are discussed. From this, a general strategy for the direct use of preconditionings for the eigenvalue problem is derived. In this new approach the necessity of the solution of large linear systems is entirely avoided. Hence, these methods represent a new alternative to some other modern eigenvalue algorithms, as they show a slightly slower convergence on the one hand but essentially lower numerical and data processing problems on the other hand. A brief description and comparison of some well-known methods (i.e. simultaneous iteration, Lanczos algorithm) completes this volume. (author)

The Schrodinger Eigenvalue March

Science.gov (United States)

Tannous, C.; Langlois, J.

2011-01-01

A simple numerical method for the determination of Schrodinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric…

Periodic Solutions, Eigenvalue Curves, and Degeneracy of the Fractional Mathieu Equation

International Nuclear Information System (INIS)

Parra-Hinojosa, A; Gutiérrez-Vega, J C

2016-01-01

We investigate the eigenvalue curves, the behavior of the periodic solutions, and the orthogonality properties of the Mathieu equation with an additional fractional derivative term using the method of harmonic balance. The addition of the fractional derivative term breaks the hermiticity of the equation in such a way that its eigenvalues need not be real nor its eigenfunctions orthogonal. We show that for a certain choice of parameters the eigenvalue curves reveal the appearance of degenerate eigenvalues. We offer a detailed discussion of the matrix representation of the differential operator corresponding to the fractional Mathieu equation, as well as some numerical examples of its periodic solutions. (paper)

Existence of solutions for a fourth order eigenvalue problem ] {Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions

Directory of Open Access Journals (Sweden)

Khalil Ben Haddouch

2016-04-01

Full Text Available In this work we will study the eigenvalues for a fourth order elliptic equation with $p(x$-growth conditions $\\Delta^2_{p(x} u=\\lambda |u|^{p(x-2} u$, under Neumann boundary conditions, where $p(x$ is a continuous function defined on the bounded domain with $p(x>1$. Through the Ljusternik-Schnireleman theory on $C^1$-manifold, we prove the existence of infinitely many eigenvalue sequences and $\\sup \\Lambda =+\\infty$, where $\\Lambda$ is the set of all eigenvalues.

TWO-DIMENSIONAL APPROXIMATION OF EIGENVALUE PROBLEMS IN SHELL THEORY: FLEXURAL SHELLS

Institute of Scientific and Technical Information of China (English)

无

2000-01-01

The eigenvalue problem for a thin linearly elastic shell, of thickness 2e, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as ε→0,the eigenvalue problem for the two-dimensional"flexural shell"model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.

Estimates for lower order eigenvalues of a clamped plate problem

OpenAIRE

Cheng, Qing-Ming; Huang, Guangyue; Wei, Guoxin

2009-01-01

For a bounded domain $\\Omega$ in a complete Riemannian manifold $M^n$, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.

Eigenvalue treatment of cosmological models

International Nuclear Information System (INIS)

Novello, M.; Soares, D.

1976-08-01

From the decomposition of Weyl tensor into its electric and magnetic parts, it is formulated the eigenvalue problem for cosmological models, and is used quasi-maxwellian form of Einstein's equation to propagate it along a time-like congruence. Three related theorems are presented

Eigenvalues of Words in Two Positive Definite Letters

OpenAIRE

Hillar, Christopher J; Johnson, Charles R

2005-01-01

The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical physics. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do. In the process, a fundamental question about solvability of symmetric word equations is encoun...

Eigenvalues of the Transferences of Gaussian Optical Systems

Directory of Open Access Journals (Sweden)

W.F. Harris

2005-12-01

Full Text Available The  problem  of  how  to  define  an  average eye leads to the question of what eigenvalues are  possible  for  ray  transferences.  This  paper examines the set of possible eigenvalues in the simplest possible case, that of optical systems consisting  of  elements  that  are  stigmatic  and centred on a common axis.

Solving eigenvalue response matrix equations with nonlinear techniques

International Nuclear Information System (INIS)

Roberts, Jeremy A.; Forget, Benoit

2014-01-01

Highlights: • High performance solvers were applied within ERMM for the first time. • Accelerated fixed-point methods were developed that reduce computational times by 2–3. • A nonlinear, Newton-based ERMM led to similar improvement and more robustness. • A 3-D, SN-based ERMM shows how ERMM can apply fine-mesh methods to full-core analysis. - Abstract: This paper presents new algorithms for use in the eigenvalue response matrix method (ERMM) for reactor eigenvalue problems. ERMM spatially decomposes a domain into independent nodes linked via boundary conditions approximated as truncated orthogonal expansions, the coefficients of which are response functions. In its simplest form, ERMM consists of a two-level eigenproblem: an outer Picard iteration updates the k-eigenvalue via balance, while the inner λ-eigenproblem imposes neutron balance between nodes. Efficient methods are developed for solving the inner λ-eigenvalue problem within the outer Picard iteration. Based on results from several diffusion and transport benchmark models, it was found that the Krylov–Schur method applied to the λ-eigenvalue problem reduces Picard solver times (excluding response generation) by a factor of 2–5. Furthermore, alternative methods, including Picard acceleration schemes, Steffensen’s method, and Newton’s method, are developed in this paper. These approaches often yield faster k-convergence and a need for fewer k-dependent response function evaluations, which is important because response generation is often the primary cost for problems using responses computed online (i.e., not from a precomputed database). Accelerated Picard iteration was found to reduce total computational times by 2–3 compared to the unaccelerated case for problems dominated by response generation. In addition, Newton’s method was found to provide nearly the same performance with improved robustness

A scheme for the evaluation of dominant time-eigenvalues of a nuclear reactor

International Nuclear Information System (INIS)

Modak, R.S.; Gupta, Anurag

2007-01-01

This paper presents a scheme to obtain the fundamental and few dominant solutions of the prompt time eigenvalue problem (referred to as α-eigenvalue problem) for a nuclear reactor using multi-group neutron diffusion theory. The scheme is based on the use of an algorithm called Orthomin(1). This algorithm was originally proposed by Suetomi and Sekimoto [Suetomi, E., Sekimoto, H., 1991. Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations. Ann. Nucl. Energy 18 (4), 205-227] to obtain the fundamental K-eigenvalue (K-effective) of nuclear reactors. Recently, it has been shown that the algorithm can be used to obtain the further dominant K-modes also. Since α-eigenvalue problem is usually more difficult to solve than the K-eigenvalue problem, an attempt has been made here to use Orthomin(1) for its solution. Numerical results are given for realistic 3-D test case

Energy eigenvalues of helium-like atoms in dense plasmas

International Nuclear Information System (INIS)

Hashino, Tasuke; Nakazaki, Shinobu; Kato, Takako; Kashiwabara, Hiromichi.

1987-04-01

Calculations based on a variational method with wave functions including the correlation of electrons are carried out to obtain energy eigenvalues of Schroedinger's equation for helium-like atoms embedded in dense plasmas, taking the Debye-Hueckel approximation. Energy eigenvalues for the 1 1 S, 2 1 S, and 2 3 S states are obtained as a function of Debye screening length. (author)

Eigenvalues of the volume operator in loop quantum gravity

International Nuclear Information System (INIS)

Meissner, Krzysztof A

2006-01-01

We present a simple method to calculate certain sums of the eigenvalues of the volume operator in loop quantum gravity. We derive the asymptotic distribution of the eigenvalues in the classical limit of very large spins, which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity

Absence of positive eigenvalues for hard-core N-body systems

DEFF Research Database (Denmark)

Ito, K.; Skibsted, Erik

We show absence of positive eigenvalues for generalized 2-body hard-core Schrödinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N-body hard-core Schrödinger operators, N≥ 2, is presented. This scheme inv...

Evaluation of Eigenvalue Routines for Large Scale Applications

Directory of Open Access Journals (Sweden)

V.A. Tischler

1994-01-01

Full Text Available The NASA structural analysis (NASTRAN∗ program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this article is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the block Lanczos aLgorithm. This eigenvalue routine is now availabLe in several mathematics libraries as well as in severaL commerciaL versions of NASTRAN. In addition, the eRA Y library maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The block Lanczos algorithm was found to be extremely efficient, particularly for very large problems.

Solving complex band structure problems with the FEAST eigenvalue algorithm

Science.gov (United States)

Laux, S. E.

2012-08-01

With straightforward extension, the FEAST eigenvalue algorithm [Polizzi, Phys. Rev. B 79, 115112 (2009)] is capable of solving the generalized eigenvalue problems representing traveling-wave problems—as exemplified by the complex band-structure problem—even though the matrices involved are complex, non-Hermitian, and singular, and hence outside the originally stated range of applicability of the algorithm. The obtained eigenvalues/eigenvectors, however, contain spurious solutions which must be detected and removed. The efficiency and parallel structure of the original algorithm are unaltered. The complex band structures of Si layers of varying thicknesses and InAs nanowires of varying radii are computed as test problems.

Joint density of eigenvalues in spiked multivariate models.

Science.gov (United States)

Dharmawansa, Prathapasinghe; Johnstone, Iain M

2014-01-01

The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and 'linear' statistics used in testing.

An algorithm of α-and γ-mode eigenvalue calculations by Monte Carlo method

International Nuclear Information System (INIS)

Yamamoto, Toshihiro; Miyoshi, Yoshinori

2003-01-01

A new algorithm for Monte Carlo calculation was developed to obtain α- and γ-mode eigenvalues. The α is a prompt neutron time decay constant measured in subcritical experiments, and the γ is a spatial decay constant measured in an exponential method for determining the subcriticality. This algorithm can be implemented into existing Monte Carlo eigenvalue calculation codes with minimum modifications. The algorithm was implemented into MCNP code and the performance of calculating the both mode eigenvalues were verified through comparison of the calculated eigenvalues with the ones obtained by fixed source calculations. (author)

Complex energy eigenvalues of a linear potential with a parabolical barrier

International Nuclear Information System (INIS)

Malherbe, J.B.

1978-01-01

The physical meaning and restrictions of complex energy eigenvalues are briefly discussed. It is indicated that a quasi-stationary phase describes an idealised disintegration system. Approximate resonance-eigenvalues of the one dimensional Schrodinger equation with a linear potential and parabolic barrier are calculated by means of Connor's semiclassical method. This method is based on the generalized WKB-method of Miller and Good. The results obtained confirm the correctness of a model representation which explains the unusual distribution of eigenvalues by certain other linear potentials in a complex energy level [af

Simultaneous multigrid techniques for nonlinear eigenvalue problems: Solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions

Science.gov (United States)

Costiner, Sorin; Ta'asan, Shlomo

1995-07-01

Algorithms for nonlinear eigenvalue problems (EP's) often require solving self-consistently a large number of EP's. Convergence difficulties may occur if the solution is not sought in an appropriate region, if global constraints have to be satisfied, or if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP's obtained from discretizations of partial differential EP have often been shown to be more efficient than single level algorithms. This paper presents MG techniques and a MG algorithm for nonlinear Schrödinger Poisson EP's. The algorithm overcomes the above mentioned difficulties combining the following techniques: a MG simultaneous treatment of the eigenvectors and nonlinearity, and with the global constrains; MG stable subspace continuation techniques for the treatment of nonlinearity; and a MG projection coupled with backrotations for separation of solutions. These techniques keep the solutions in an appropriate region, where the algorithm converges fast, and reduce the large number of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues. Computational examples for the nonlinear Schrödinger-Poisson EP in two and three dimensions, presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N and for the corresponding eigenvalues. One MG simultaneous cycle per fine level was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic convergence rate of 0.15 per MG cycle is attained.

2nd International Workshop on Eigenvalue Problems : Algorithms, Software and Applications in Petascale Computing

CERN Document Server

Zhang, Shao-Liang; Imamura, Toshiyuki; Yamamoto, Yusaku; Kuramashi, Yoshinobu; Hoshi, Takeo

2017-01-01

This book provides state-of-the-art and interdisciplinary topics on solving matrix eigenvalue problems, particularly by using recent petascale and upcoming post-petascale supercomputers. It gathers selected topics presented at the International Workshops on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2014 and EPASA2015), which brought together leading researchers working on the numerical solution of matrix eigenvalue problems to discuss and exchange ideas – and in so doing helped to create a community for researchers in eigenvalue problems. The topics presented in the book, including novel numerical algorithms, high-performance implementation techniques, software developments and sample applications, will contribute to various fields that involve solving large-scale eigenvalue problems.

Recurrence quantity analysis based on matrix eigenvalues

Science.gov (United States)

Yang, Pengbo; Shang, Pengjian

2018-06-01

Recurrence plots is a powerful tool for visualization and analysis of dynamical systems. Recurrence quantification analysis (RQA), based on point density and diagonal and vertical line structures in the recurrence plots, is considered to be alternative measures to quantify the complexity of dynamical systems. In this paper, we present a new measure based on recurrence matrix to quantify the dynamical properties of a given system. Matrix eigenvalues can reflect the basic characteristics of the complex systems, so we show the properties of the system by exploring the eigenvalues of the recurrence matrix. Considering that Shannon entropy has been defined as a complexity measure, we propose the definition of entropy of matrix eigenvalues (EOME) as a new RQA measure. We confirm that EOME can be used as a metric to quantify the behavior changes of the system. As a given dynamical system changes from a non-chaotic to a chaotic regime, the EOME will increase as well. The bigger EOME values imply higher complexity and lower predictability. We also study the effect of some factors on EOME,including data length, recurrence threshold, the embedding dimension, and additional noise. Finally, we demonstrate an application in physiology. The advantage of this measure lies in a high sensitivity and simple computation.

Multigrid techniques for nonlinear eigenvalue probems: Solutions of a nonlinear Schroedinger eigenvalue problem in 2D and 3D

Science.gov (United States)

Costiner, Sorin; Taasan, Shlomo

1994-01-01

This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.

Accurate high-lying eigenvalues of Schroedinger and Sturm-Liouville problems

International Nuclear Information System (INIS)

Vanden Berghe, G.; Van Daele, M.; De Meyer, H.

1994-01-01

A modified difference and a Numerov-like scheme have been introduced in a shooting algorithm for the determination of the (higher-lying) eigenvalues of Schroedinger equations and Sturm-Liouville problems. Some numerical experiments are introduced. Time measurements have been performed. The proposed algorithms are compared with other previously introduced shooting schemes. The structure of the eigenvalue error is discussed. ((orig.))

Bounds and estimates for the linearly perturbed eigenvalue problem

International Nuclear Information System (INIS)

Raddatz, W.D.

1983-01-01

This thesis considers the problem of bounding and estimating the discrete portion of the spectrum of a linearly perturbed self-adjoint operator, M(x). It is supposed that one knows an incomplete set of data consisting in the first few coefficients of the Taylor series expansions of one or more of the eigenvalues of M(x) about x = 0. The foundations of the variational study of eigen-values are first presented. These are then used to construct the best possible upper bounds and estimates using various sets of given information. Lower bounds are obtained by estimating the error in the upper bounds. The extension of these bounds and estimates to the eigenvalues of the doubly-perturbed operator M(x,y) is discussed. The results presented have numerous practical application in the physical sciences, including problems in atomic physics and the theory of vibrations of acoustical and mechanical systems

An algebraic substructuring using multiple shifts for eigenvalue computations

International Nuclear Information System (INIS)

Ko, Jin Hwan; Jung, Sung Nam; Byun, Do Young; Bai, Zhaojun

2008-01-01

Algebraic substructuring (AS) is a state-of-the-art method in eigenvalue computations, especially for large-sized problems, but originally it was designed to calculate only the smallest eigenvalues. Recently, an updated version of AS has been introduced to calculate the interior eigenvalues over a specified range by using a shift concept that is referred to as the shifted AS. In this work, we propose a combined method of both AS and the shifted AS by using multiple shifts for solving a considerable number of eigensolutions in a large-sized problem, which is an emerging computational issue of noise or vibration analysis in vehicle design. In addition, we investigated the accuracy of the shifted AS by presenting an error criterion. The proposed method has been applied to the FE model of an automobile body. The combined method yielded a higher efficiency without loss of accuracy in comparison to the original AS

On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations

Directory of Open Access Journals (Sweden)

Heinrich Voss

2016-06-01

Full Text Available Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

DEFF Research Database (Denmark)

Hurtado, Ana; Markvorsen, Steen; Palmer, Vicente

2013-01-01

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball...

Stratified source-sampling techniques for Monte Carlo eigenvalue analysis

International Nuclear Information System (INIS)

Mohamed, A.

1998-01-01

In 1995, at a conference on criticality safety, a special session was devoted to the Monte Carlo ''Eigenvalue of the World'' problem. Argonne presented a paper, at that session, in which the anomalies originally observed in that problem were reproduced in a much simplified model-problem configuration, and removed by a version of stratified source-sampling. In this paper, stratified source-sampling techniques are generalized and applied to three different Eigenvalue of the World configurations which take into account real-world statistical noise sources not included in the model problem, but which differ in the amount of neutronic coupling among the constituents of each configuration. It is concluded that, in Monte Carlo eigenvalue analysis of loosely-coupled arrays, the use of stratified source-sampling reduces the probability of encountering an anomalous result over that if conventional source-sampling methods are used. However, this gain in reliability is substantially less than that observed in the model-problem results

An Optimal Lower Eigenvalue System

Directory of Open Access Journals (Sweden)

Yingfan Liu

2011-01-01

Full Text Available An optimal lower eigenvalue system is studied, and main theorems including a series of necessary and suffcient conditions concerning existence and a Lipschitz continuity result concerning stability are obtained. As applications, solvability results to some von-Neumann-type input-output inequalities, growth, and optimal growth factors, as well as Leontief-type balanced and optimal balanced growth paths, are also gotten.

Complex eigenvalues for neutron transport equation with quadratically anisotropic scattering

International Nuclear Information System (INIS)

Sjoestrand, N.G.

1981-01-01

Complex eigenvalues for the monoenergetic neutron transport equation in the buckling approximation have been calculated for various combinations of linearly and quadratically anisotropic scattering. The results are discussed in terms of the time-dependent case. Tables are given of complex bucklings for real decay constants and of complex decay constants for real bucklings. The results fit nicely into the pattern of real and purely imaginary eigenvalues obtained earlier. (author)

On the solution of two-point linear differential eigenvalue problems. [numerical technique with application to Orr-Sommerfeld equation

Science.gov (United States)

Antar, B. N.

1976-01-01

A numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems. The technique is designed to search for complex eigenvalues belonging to complex operators. With this method, any domain of the complex eigenvalue plane could be scanned and the eigenvalues within it, if any, located. For an application of the method, the eigenvalues of the Orr-Sommerfeld equation of the plane Poiseuille flow are determined within a specified portion of the c-plane. The eigenvalues for alpha = 1 and R = 10,000 are tabulated and compared for accuracy with existing solutions.

A subspace preconditioning algorithm for eigenvector/eigenvalue computation

Energy Technology Data Exchange (ETDEWEB)

Bramble, J.H.; Knyazev, A.V.; Pasciak, J.E.

1996-12-31

We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigen-spaces of a symmetric positive definite matrix. In our applications, the dimension of a matrix is large and the cost of its inverting is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning. Estimates will be provided which show that the preconditioned method converges linearly and uniformly in the matrix dimension when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.

On a quadratic inverse eigenvalue problem

International Nuclear Information System (INIS)

Cai, Yunfeng; Xu, Shufang

2009-01-01

This paper concerns the quadratic inverse eigenvalue problem (QIEP) of constructing real symmetric matrices M, C and K of size n × n, with M nonsingular, so that the quadratic matrix polynomial Q(λ) ≡ λ 2 M + λC + K has a completely prescribed set of eigenvalues and eigenvectors. It is shown via construction that the QIEP has a solution if and only if r 0, where r and δ are computable from the prescribed spectral data. A necessary and sufficient condition for the existence of a solution to the QIEP with M being positive definite is also established in a constructive way. Furthermore, two algorithms are developed: one is to solve the QIEP; another is to find a particular solution to the QIEP with the leading coefficient matrix being positive definite, which also provides us an approach to a simultaneous reduction of real symmetric matrix triple (M, C, K) by real congruence. Numerical results show that the two algorithms are feasible and numerically reliable

The method of fundamental solutions for computing acoustic interior transmission eigenvalues

Science.gov (United States)

Kleefeld, Andreas; Pieronek, Lukas

2018-03-01

We analyze the method of fundamental solutions (MFS) in two different versions with focus on the computation of approximate acoustic interior transmission eigenvalues in 2D for homogeneous media. Our approach is mesh- and integration free, but suffers in general from the ill-conditioning effects of the discretized eigenoperator, which we could then successfully balance using an approved stabilization scheme. Our numerical examples cover many of the common scattering objects and prove to be very competitive in accuracy with the standard methods for PDE-related eigenvalue problems. We finally give an approximation analysis for our framework and provide error estimates, which bound interior transmission eigenvalue deviations in terms of some generalized MFS output.

Eigenvalues and expansion of bipartite graphs

DEFF Research Database (Denmark)

Høholdt, Tom; Janwa, Heeralal

2012-01-01

We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes that are optimal with respect to the bouns. We prove that BIBD-graphs are characterized by ...

Multi-level nonlinear diffusion acceleration method for multigroup transport k-Eigenvalue problems

International Nuclear Information System (INIS)

Anistratov, Dmitriy Y.

2011-01-01

The nonlinear diffusion acceleration (NDA) method is an efficient and flexible transport iterative scheme for solving reactor-physics problems. This paper presents a fast iterative algorithm for solving multigroup neutron transport eigenvalue problems in 1D slab geometry. The proposed method is defined by a multi-level system of equations that includes multigroup and effective one-group low-order NDA equations. The Eigenvalue is evaluated in the exact projected solution space of smallest dimensionality, namely, by solving the effective one- group eigenvalue transport problem. Numerical results that illustrate performance of the new algorithm are demonstrated. (author)

A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

Directory of Open Access Journals (Sweden)

Wang Xiaoxiao

2018-04-01

Full Text Available A set in the complex plane which involves n parameters in [0, 1] is given to localize all eigenvalues different from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we fix n parameters in [0, 1] to give a new set including all eigenvalues different from 1, which is tighter than those provided by Shen et al. (Linear Algebra Appl. 447 (2014 74-87 and Li et al. (Linear and Multilinear Algebra 63(11 (2015 2159-2170 for estimating the moduli of subdominant eigenvalues.

Accurate Valence Ionization Energies from Kohn-Sham Eigenvalues with the Help of Potential Adjustors.

Science.gov (United States)

Thierbach, Adrian; Neiss, Christian; Gallandi, Lukas; Marom, Noa; Körzdörfer, Thomas; Görling, Andreas

2017-10-10

An accurate yet computationally very efficient and formally well justified approach to calculate molecular ionization potentials is presented and tested. The first as well as higher ionization potentials are obtained as the negatives of the Kohn-Sham eigenvalues of the neutral molecule after adjusting the eigenvalues by a recently [ Görling Phys. Rev. B 2015 , 91 , 245120 ] introduced potential adjustor for exchange-correlation potentials. Technically the method is very simple. Besides a Kohn-Sham calculation of the neutral molecule, only a second Kohn-Sham calculation of the cation is required. The eigenvalue spectrum of the neutral molecule is shifted such that the negative of the eigenvalue of the highest occupied molecular orbital equals the energy difference of the total electronic energies of the cation minus the neutral molecule. For the first ionization potential this simply amounts to a ΔSCF calculation. Then, the higher ionization potentials are obtained as the negatives of the correspondingly shifted Kohn-Sham eigenvalues. Importantly, this shift of the Kohn-Sham eigenvalue spectrum is not just ad hoc. In fact, it is formally necessary for the physically correct energetic adjustment of the eigenvalue spectrum as it results from ensemble density-functional theory. An analogous approach for electron affinities is equally well obtained and justified. To illustrate the practical benefits of the approach, we calculate the valence ionization energies of test sets of small- and medium-sized molecules and photoelectron spectra of medium-sized electron acceptor molecules using a typical semilocal (PBE) and two typical global hybrid functionals (B3LYP and PBE0). The potential adjusted B3LYP and PBE0 eigenvalues yield valence ionization potentials that are in very good agreement with experimental values, reaching an accuracy that is as good as the best G 0 W 0 methods, however, at much lower computational costs. The potential adjusted PBE eigenvalues result in

EISPACK-J: subprogram package for solving eigenvalue problems

International Nuclear Information System (INIS)

Fujimura, Toichiro; Tsutsui, Tsuneo

1979-05-01

EISPACK-J, a subprogram package for solving eigenvalue problems, has been developed and subprograms with a variety of functions have been prepared. These subprograms can solve standard problems of complex matrices, general problems of real matrices and special problems in which only the required eigenvalues and eigenvectors are calculated. They are compared to existing subprograms, showing their features through benchmark tests. Many test problems, including realistic scale problems, are provided for the benchmark tests. Discussions are made on computer core storage and computing time required for each subprogram, and accuracy of the solution. The results show that the subprograms of EISPACK-J, based on Householder, QR and inverse iteration methods, are the best in computing time and accuracy. (author)

Generalized Eigenvalues for pairs on heritian matrices

Science.gov (United States)

Rublein, George

1988-01-01

A study was made of certain special cases of a generalized eigenvalue problem. Let A and B be nxn matrics. One may construct a certain polynomial, P(A,B, lambda) which specializes to the characteristic polynomial of B when A equals I. In particular, when B is hermitian, that characteristic polynomial, P(I,B, lambda) has real roots, and one can ask: are the roots of P(A,B, lambda) real when B is hermitian. We consider the case where A is positive definite and show that when N equals 3, the roots are indeed real. The basic tools needed in the proof are Shur's theorem on majorization for eigenvalues of hermitian matrices and the interlacing theorem for the eigenvalues of a positive definite hermitian matrix and one of its principal (n-1)x(n-1) minors. The method of proof first reduces the general problem to one where the diagonal of B has a certain structure: either diag (B) = diag (1,1,1) or diag (1,1,-1), or else the 2 x 2 principal minors of B are all 1. According as B has one of these three structures, we use an appropriate method to replace A by a positive diagonal matrix. Since it can be easily verified that P(D,B, lambda) has real roots, the result follows. For other configurations of B, a scaling and a continuity argument are used to prove the result in general.

The total Hartree-Fock energy-eigenvalue sum relationship in atoms

International Nuclear Information System (INIS)

Sen, K.D.

1979-01-01

Using the well known relationships for the isoelectronic changes in the total Hartree-Fock energy, nucleus-electron attraction energy and electron-electron repulsion energy in atoms a simple polynomial expansion in Z is obtained for the sum of the eigenvalues which can be used to calculate the total Hartree-Fock energy. Numerical results are presented for 2-10 electron series to show that the present relationship is a better approximation than the other available energy-eigenvalue relationships. (author)

Photonic band structure calculations using nonlinear eigenvalue techniques

International Nuclear Information System (INIS)

Spence, Alastair; Poulton, Chris

2005-01-01

This paper considers the numerical computation of the photonic band structure of periodic materials such as photonic crystals. This calculation involves the solution of a Hermitian nonlinear eigenvalue problem. Numerical methods for nonlinear eigenvalue problems are usually based on Newton's method or are extensions of techniques for the standard eigenvalue problem. We present a new variation on existing methods which has its derivation in methods for bifurcation problems, where bordered matrices are used to compute critical points in singular systems. This new approach has several advantages over the current methods. First, in our numerical calculations the new variation is more robust than existing techniques, having a larger domain of convergence. Second, the linear systems remain Hermitian and are nonsingular as the method converges. Third, the approach provides an elegant and efficient way of both thinking about the problem and organising the computer solution so that only one linear system needs to be factorised at each stage in the solution process. Finally, first- and higher-order derivatives are calculated as a natural extension of the basic method, and this has advantages in the electromagnetic problem discussed here, where the band structure is plotted as a set of paths in the (ω,k) plane

A new localization set for generalized eigenvalues

Directory of Open Access Journals (Sweden)

Jing Gao

2017-05-01

Full Text Available Abstract A new localization set for generalized eigenvalues is obtained. It is shown that the new set is tighter than that in (Numer. Linear Algebra Appl. 16:883-898, 2009. Numerical examples are given to verify the corresponding results.

High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform.

Science.gov (United States)

Gui, Tao; Lu, Chao; Lau, Alan Pak Tao; Wai, P K A

2017-08-21

In this paper, we experimentally investigate high-order modulation over a single discrete eigenvalue under the nonlinear Fourier transform (NFT) framework and exploit all degrees of freedom for encoding information. For a fixed eigenvalue, we compare different 4 bit/symbol modulation formats on the spectral amplitude and show that a 2-ring 16-APSK constellation achieves optimal performance. We then study joint spectral phase, spectral magnitude and eigenvalue modulation and found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation on the imaginary part of the eigenvalue in practical settings. We propose a spectral amplitude scaling method to mitigate such nonlinear ISI and demonstrate a record 4 GBaud 16-APSK on the spectral amplitude plus 2-bit eigenvalue modulation (total 6 bit/symbol at 24 Gb/s) transmission over 1000 km.

On a minimization of the eigenvalues of Schroedinger operator relatively domains

International Nuclear Information System (INIS)

Gasymov, Yu.S.; Niftiev, A.A.

2001-01-01

Minimization of the eigenvalues plays an important role in the operators spectral theory. The problem on the minimization of the eigenvalues of the Schroedinger operator by areas is considered in this work. The algorithm, analogous to the conditional gradient method, is proposed for the numerical solution of this problem in the common case. The result is generalized for the case of the positively determined completely continuous operator [ru

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science.

Science.gov (United States)

Marek, A; Blum, V; Johanni, R; Havu, V; Lang, B; Auckenthaler, T; Heinecke, A; Bungartz, H-J; Lederer, H

2014-05-28

Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem

Recent developments in semiclassical mechanics: eigenvalues and reaction rate constants

International Nuclear Information System (INIS)

Miller, W.H.

1976-04-01

A semiclassical treatment of eigenvalues for a multidimensional non-separable potential function and of the rate constant for a chemical reaction with an activation barrier is presented. Both phenomena are seen to be described by essentially the same semiclassical formalism, which is based on a construction of the total Hamiltonian in terms of the complete set of ''good'' action variables (or adiabatic invariants) associated with the minimum in the potential energy surface for the eigenvalue case, or the saddle point in the potential energy surface for the case of chemical reaction

Eigenvalue pinching on spinc manifolds

Science.gov (United States)

Roos, Saskia

2017-02-01

We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.

Eigenvalues of the -Laplacian and disconjugacy criteria

Directory of Open Access Journals (Sweden)

Pinasco Juan P

2006-01-01

Full Text Available We derive oscillation and nonoscillation criteria for the one-dimensional -Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.

Analysis of eigenvalue correction applied to biometrics

NARCIS (Netherlands)

Hendrikse, A.J.; Veldhuis, Raymond N.J.; Spreeuwers, Lieuwe Jan; Bazen, A.M.

Eigenvalue estimation plays an important role in biometrics. However, if the number of samples is limited, estimates are significantly biased. In this article we analyse the influence of this bias on the error rates of PCA/LDA based verification systems, using both synthetic data with realistic

First-order optical systems with unimodular eigenvalues

NARCIS (Netherlands)

Bastiaans, M.J.; Alieva, T.

2006-01-01

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues, is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the

A numerical study of the eigenvalues in the neutron diffusion theory

International Nuclear Information System (INIS)

Lima Bezerra, J. de.

1982-12-01

A systematic numerical study for the eigenvalue problem in one dimension was carried out. A computer code RED2G was developed to obtain and to discuss a number of numerical solutions concerning eigenvalues problems originating from the discretization of the two groups neutron diffusion equation in one dimension and steady state. The problem of eigenvalues was created from the discretization by the method of finite differences. The solutions were obtained by four different iterative methods, i.e. Power, Wielandt-1, Wielandt-2 and accelerated Power with the Chebyshev polinomials. The numerical results given by the solution of the two test-problems indicate that the RED2G code is fast and efficient in these calculations and the Wielandt-2 method has been found to be the best both in respect of rapidity of calculations as well as programation effort required. (E.G.) [pt

Ab initio nuclear structure - the large sparse matrix eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Vary, James P; Maris, Pieter [Department of Physics, Iowa State University, Ames, IA, 50011 (United States); Ng, Esmond; Yang, Chao [Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 (United States); Sosonkina, Masha, E-mail: jvary@iastate.ed [Scalable Computing Laboratory, Ames Laboratory, Iowa State University, Ames, IA, 50011 (United States)

2009-07-01

The structure and reactions of light nuclei represent fundamental and formidable challenges for microscopic theory based on realistic strong interaction potentials. Several ab initio methods have now emerged that provide nearly exact solutions for some nuclear properties. The ab initio no core shell model (NCSM) and the no core full configuration (NCFC) method, frame this quantum many-particle problem as a large sparse matrix eigenvalue problem where one evaluates the Hamiltonian matrix in a basis space consisting of many-fermion Slater determinants and then solves for a set of the lowest eigenvalues and their associated eigenvectors. The resulting eigenvectors are employed to evaluate a set of experimental quantities to test the underlying potential. For fundamental problems of interest, the matrix dimension often exceeds 10{sup 10} and the number of nonzero matrix elements may saturate available storage on present-day leadership class facilities. We survey recent results and advances in solving this large sparse matrix eigenvalue problem. We also outline the challenges that lie ahead for achieving further breakthroughs in fundamental nuclear theory using these ab initio approaches.

Ab initio nuclear structure - the large sparse matrix eigenvalue problem

International Nuclear Information System (INIS)

Vary, James P; Maris, Pieter; Ng, Esmond; Yang, Chao; Sosonkina, Masha

2009-01-01

The structure and reactions of light nuclei represent fundamental and formidable challenges for microscopic theory based on realistic strong interaction potentials. Several ab initio methods have now emerged that provide nearly exact solutions for some nuclear properties. The ab initio no core shell model (NCSM) and the no core full configuration (NCFC) method, frame this quantum many-particle problem as a large sparse matrix eigenvalue problem where one evaluates the Hamiltonian matrix in a basis space consisting of many-fermion Slater determinants and then solves for a set of the lowest eigenvalues and their associated eigenvectors. The resulting eigenvectors are employed to evaluate a set of experimental quantities to test the underlying potential. For fundamental problems of interest, the matrix dimension often exceeds 10 10 and the number of nonzero matrix elements may saturate available storage on present-day leadership class facilities. We survey recent results and advances in solving this large sparse matrix eigenvalue problem. We also outline the challenges that lie ahead for achieving further breakthroughs in fundamental nuclear theory using these ab initio approaches.

Large deviations of the maximum eigenvalue in Wishart random matrices

International Nuclear Information System (INIS)

Vivo, Pierpaolo; Majumdar, Satya N; Bohigas, Oriol

2007-01-01

We analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W = X T X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value (λ) = N/c decreases for large N as ∼exp[-β/2 N 2 Φ - (2√c + 1: c)], where β = 1, 2 corresponds respectively to real and complex Wishart matrices, c = N/M ≤ 1 and Φ - (x; c) is a rate (sometimes also called large deviation) function that we compute explicitly. The result for the anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of Wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier. Numerical simulations are in excellent agreement with the analytical predictions

Large deviations of the maximum eigenvalue in Wishart random matrices

Energy Technology Data Exchange (ETDEWEB)

Vivo, Pierpaolo [School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH (United Kingdom) ; Majumdar, Satya N [Laboratoire de Physique Theorique et Modeles Statistiques (UMR 8626 du CNRS), Universite Paris-Sud, Batiment 100, 91405 Orsay Cedex (France); Bohigas, Oriol [Laboratoire de Physique Theorique et Modeles Statistiques (UMR 8626 du CNRS), Universite Paris-Sud, Batiment 100, 91405 Orsay Cedex (France)

2007-04-20

We analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W = X{sup T}X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value ({lambda}) = N/c decreases for large N as {approx}exp[-{beta}/2 N{sup 2}{phi}{sub -} (2{radical}c + 1: c)], where {beta} = 1, 2 corresponds respectively to real and complex Wishart matrices, c = N/M {<=} 1 and {phi}{sub -}(x; c) is a rate (sometimes also called large deviation) function that we compute explicitly. The result for the anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of Wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier. Numerical simulations are in excellent agreement with the analytical predictions.

A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential

Directory of Open Access Journals (Sweden)

Craig Haile

2000-01-01

Full Text Available We study the time-independent Schrodinger equation with radially symmetric potential $k|x|^alpha$, $k ge 0$, $k in mathbb{R}, alpha ge 2$ on a bounded domain $Omega$ in $mathbb{R}^n$, $(n ge 2$ with Dirichlet boundary conditions. In particular, we compare the eigenvalue $lambda_2(Omega$ of the operator $-Delta + k |x|^alpha $ on $Omega$ with the eigenvalue $lambda_2(S_1$ of the same operator $-Delta +kr^alpha$ on a ball $S_1$, where $S_1$ has radius such that the first eigenvalues are the same ($lambda_1(Omega = lambda_1(S_1$. The main result is to show $lambda_2(Omega le lambda_2(S_1$. We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain $Omega$ with Dirichlet boundary conditions.

Higher-order relationship between eigen-value separation and static flux tilts

International Nuclear Information System (INIS)

Beckner, W.D.

1975-01-01

Spatial kinetics phenomena in nuclear reactors, such as xenon-induced spatial flux oscillations, are currently being analyzed using the higher harmonic solutions to the static reactor balance equation. An important parameter in such an analysis is a global quantity called eigenvalue separation. It is desirable to be able to experimentally measure this parameter in power reactors in order to confirm design calculations. Since spatial distortions in the flux shape depend on the eigenvalue separation of the reactor, an attempt has been made previously to use this fact as a means of measuring the parameter. It was postulated that an induced flux distortion or ''static flux tilt'' could be measured and theoretically related to eigenvalue separation. Unfortunately, the behavior of experimental data did not exactly agree with theoretical predictions, and values of the parameter found using the original static flux tilt technique were consistently low. The theory has been re-evaluated here and the previously observed discrepancy eliminated. Techniques have been also developed to allow for more accurate interpretation of experimental data. In order to make the method applicable to real systems, the theory has been extended to two spatial dimensions; extension to three dimensions follows directly. Possible trouble areas have been investigated, and experimental procedures for use of the technique to measure the eigenvalue separation in power reactors are presented

A note on quasilinear elliptic eigenvalue problems

Directory of Open Access Journals (Sweden)

Gianni Arioli

1999-11-01

Full Text Available We study an eigenvalue problem by a non-smooth critical point theory. Under general assumptions, we prove the existence of at least one solution as a minimum of a constrained energy functional. We apply some results on critical point theory with symmetry to provide a multiplicity result.

Complex eigenvalue analysis of railway wheel/rail squeal

African Journals Online (AJOL)

DR OKE

Squeal noise from wheel/rail and brake disc/pad frictional contact is typical in railways. ... squeal noise by multibody simulation of a rail car running on rigid rails. ... system, traditional complex eigenvalue analysis by finite element was used.

A teaching proposal for the study of Eigenvectors and Eigenvalues

Directory of Open Access Journals (Sweden)

María José Beltrán Meneu

2017-03-01

Full Text Available In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was motivated after observing students’ difficulties when treating eigenvectors and eigenvalues from a geometric point of view. It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.

Numerical computations of interior transmission eigenvalues for scattering objects with cavities

International Nuclear Information System (INIS)

Peters, Stefan; Kleefeld, Andreas

2016-01-01

In this article we extend the inside-outside duality for acoustic transmission eigenvalue problems by allowing scattering objects that may contain cavities. In this context we provide the functional analytical framework necessary to transfer the techniques that have been used in Kirsch and Lechleiter (2013 Inverse Problems, 29 104011) to derive the inside-outside duality. Additionally, extensive numerical results are presented to show that we are able to successfully detect interior transmission eigenvalues with the inside-outside duality approach for a variety of obstacles with and without cavities in three dimensions. In this context, we also discuss the advantages and disadvantages of the inside-outside duality approach from a numerical point of view. Furthermore we derive the integral equations necessary to extend the algorithm in Kleefeld (2013 Inverse Problems, 29 104012) to compute highly accurate interior transmission eigenvalues for scattering objects with cavities, which we will then use as reference values to examine the accuracy of the inside-outside duality algorithm. (paper)

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

Science.gov (United States)

Tischler, V. A.; Venkayya, Vipperla B.

1993-01-01

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems

Directory of Open Access Journals (Sweden)

Muhammed I. Syam

2017-11-01

Full Text Available This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional $2\\alpha $-order Sturm-Liouville problem where $\\frac{1}{2}< \\alpha \\leq 1$. In this paper, we implement the reproducing kernel method RKM to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at $x=1$. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

Eigenvalue translation method for mode calculations

International Nuclear Information System (INIS)

Gerck, E.; Cruz, C.H.B.

1978-11-01

A new method is described for the first few modes calculations in a interferometer that has several advantages over the ALLMAT subroutine, the Prony Method and the Fox and Li Method. In the illustrative results shown for the same cases it can be seen that the eigenvalue translation method is typically 100 fold times faster than the usual Fox and Li Method and 10 times faster than ALLMAT [pt

New exact approaches to the nuclear eigenvalue problem

International Nuclear Information System (INIS)

Andreozzi, F.; Lo Iudice, N.; Porrino, A.; Knapp, F.; Kvasil, J.

2005-01-01

In a recent past some of us have developed a new algorithm for diagonalizing the shell model Hamiltonian which consists of an iterative sequence of diagonalization of sub-matrices of small dimensions. The method, apart from being easy to implement, is robust, yielding always stable numerical solutions, and free of ghost eigenvalues. Subsequently, we have endowed the algorithm with an importance sampling, which leads to a drastic truncation of the shell model space, while keeping the accuracy of the solutions under control. Applications to typical nuclei show that the sampling yields also an extrapolation law to the exact eigenvalues. Complementary to the shell model algorithm is a method we are developing for studying collective and non collective excitations. To this purpose we solve the nuclear eigenvalue problem in a space which is the direct sum of Tamm-Dancoff n-phonon subspaces (n=0,1, ...N). The multiphonon basis is constructed by an iterative equation of motion method, which generates an over complete set of n-phonon states from the (n-1)-phonon basis. The redundancy is removed completely and exactly by a method based on the Choleski decomposition. The full Hamiltonian matrix comes out to have a simple structure and, therefore, can be drastically truncated before diagonalization by the mentioned importance sampling method. The phonon composition of the basis states allows removing naturally and maximally the spurious admixtures induced by the centre of mass motion. An application of the method to 16 O will be given for illustrative purposes. (authors)

Eigenvalue solutions in finite element thermal transient problems

International Nuclear Information System (INIS)

Stoker, J.R.

1975-01-01

The eigenvalue economiser concept can be useful in solving large finite element transient heat flow problems in which the boundary heat transfer coefficients are constant. The usual economiser theory is equivalent to applying a unit thermal 'force' to each of a small sub-set of nodes on the finite element mesh, and then calculating sets of resulting steady state temperatures. Subsequently it is assumed that the required transient temperature distributions can be approximated by a linear combination of this comparatively small set of master temperatures. The accuracy of a reduced eigenvalue calculation depends upon a good choice of master nodes, which presupposes at least a little knowledge about what sort of shape is expected in the unknown temperature distributions. There are some instances, however, where a reasonably good idea exists of the required shapes, permitting a modification to the economiser process which leads to greater economy in the number of master temperatures. The suggested new approach is to use manually prescribed temperature distributions as the master distributions, rather than using temperatures resulting from unit thermal forces. Thus, with a little pre-knowledge one may write down a set of master distributions which, as a linear combination, can represent the required solution over the range of interest to a reasonable engineering accuracy, and using the minimum number of variables. The proposed modified eigenvalue economiser technique then uses the master distributions in an automatic way to arrive at the required solution. The technique is illustrated by some simple finite element examples

Two-group k-eigenvalue benchmark calculations for planar geometry transport in a binary stochastic medium

International Nuclear Information System (INIS)

Davis, I.M.; Palmer, T.S.

2005-01-01

Benchmark calculations are performed for neutron transport in a two material (binary) stochastic multiplying medium. Spatial, angular, and energy dependence are included. The problem considered is based on a fuel assembly of a common pressurized water reactor. The mean chord length through the assembly is determined and used as the planar geometry system length. According to assumed or calculated material distributions, this system length is populated with alternating fuel and moderator segments of random size. Neutron flux distributions are numerically computed using a discretized form of the Boltzmann transport equation employing diffusion synthetic acceleration. Average quantities (group fluxes and k-eigenvalue) and variances are calculated from an ensemble of realizations of the mixing statistics. The effects of varying two parameters in the fuel, two different boundary conditions, and three different sets of mixing statistics are assessed. A probability distribution function (PDF) of the k-eigenvalue is generated and compared with previous research. Atomic mix solutions are compared with these benchmark ensemble average flux and k-eigenvalue solutions. Mixing statistics with large standard deviations give the most widely varying ensemble solutions of the flux and k-eigenvalue. The shape of the k-eigenvalue PDF qualitatively agrees with previous work. Its overall shape is independent of variations in fuel cross-sections for the problems considered, but its width is impacted by these variations. Statistical distributions with smaller standard deviations alter the shape of this PDF toward a normal distribution. The atomic mix approximation yields large over-predictions of the ensemble average k-eigenvalue and under-predictions of the flux. Qualitatively correct flux shapes are obtained in some cases. These benchmark calculations indicate that a model which includes higher statistical moments of the mixing statistics is needed for accurate predictions of binary

Lagrangian Differentiation, Integration and Eigenvalues Problems

International Nuclear Information System (INIS)

Durand, L.

1983-01-01

Calogero recently proposed a new and very powerful method for the solution of Sturm-Liouville eigenvalue problems based on Lagrangian differentiation. In this paper, some results of a numerical investigation of Calogero's method for physical interesting problems are presented. It is then shown that one can 'invert' his differentiation technique to obtain a flexible, factorially convergent Lagrangian integration scheme which should be useful in a variety of problems, e.g. solution of integral equations

An asymptotic expression for the eigenvalues of the normalization kernel of the resonating group method

International Nuclear Information System (INIS)

Lomnitz-Adler, J.; Brink, D.M.

1976-01-01

A generating function for the eigenvalues of the RGM Normalization Kernel is expressed in terms of the diagonal matrix elements of thw GCM Overlap Kernel. An asymptotic expression for the eigenvalues is obtained by using the Method of Steepest Descent. (Auth.)

A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus

Science.gov (United States)

Mohammadi, Jafar; Limmer, Steffen; Stańczak, Sławomir

2016-07-01

This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.

Methods for computing SN eigenvalues and eigenvectors of slab geometry transport problems

International Nuclear Information System (INIS)

Yavuz, Musa

1998-01-01

We discuss computational methods for computing the eigenvalues and eigenvectors of single energy-group neutral particle transport (S N ) problems in homogeneous slab geometry, with an arbitrary scattering anisotropy of order L. These eigensolutions are important when exact (or very accurate) solutions are desired for coarse spatial cell problems demanding rapid execution times. Three methods, one of which is 'new', are presented for determining the eigenvalues and eigenvectors of such S N problems. In the first method, separation of variables is directly applied to the S N equations. In the second method, common characteristics of the S N and P N-1 equations are used. In the new method, the eigenvalues and eigenvectors can be computed provided that the cell-interface Green's functions (transmission and reflection factors) are known. Numerical results for S 4 test problems are given to compare the new method with the existing methods

Methods for computing SN eigenvalues and eigenvectors of slab geometry transport problems

International Nuclear Information System (INIS)

Yavuz, M.

1997-01-01

We discuss computational methods for computing the eigenvalues and eigenvectors of single energy-group neutral particle transport (S N ) problems in homogeneous slab geometry, with an arbitrary scattering anisotropy of order L. These eigensolutions are important when exact (or very accurate) solutions are desired for coarse spatial cell problems demanding rapid execution times. Three methods, one of which is 'new', are presented for determining the eigenvalues and eigenvectors of such S N problems. In the first method, separation of variables is directly applied to the S N equations. In the second method, common characteristics of the S N and P N-1 equations are used. In the new method, the eigenvalues and eigenvectors can be computed provided that the cell-interface Green's functions (transmission and reflection factors) are known. Numerical results for S 4 test problems are given to compare the new method with the existing methods. (author)

Hybrid subgroup decomposition method for solving fine-group eigenvalue transport problems

International Nuclear Information System (INIS)

Yasseri, Saam; Rahnema, Farzad

2014-01-01

Highlights: • An acceleration technique for solving fine-group eigenvalue transport problems. • Coarse-group quasi transport theory to solve coarse-group eigenvalue transport problems. • Consistent and inconsistent formulations for coarse-group quasi transport theory. • Computational efficiency amplified by a factor of 2 using hybrid SGD for 1D BWR problem. - Abstract: In this paper, a new hybrid method for solving fine-group eigenvalue transport problems is developed. This method extends the subgroup decomposition method to efficiently couple a new coarse-group quasi transport theory with a set of fixed-source transport decomposition sweeps to obtain the fine-group transport solution. The advantages of the quasi transport theory are its high accuracy, straight-forward implementation and numerical stability. The hybrid method is analyzed for a 1D benchmark problem characteristic of boiling water reactors (BWR). It is shown that the method reproduces the fine-group transport solution with high accuracy while increasing the computational efficiency up to 12 times compared to direct fine-group transport calculations

A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

International Nuclear Information System (INIS)

Hwang, F-N; Wei, Z-H; Huang, T-M; Wang Weichung

2010-01-01

We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc's efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schroedinger's equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Directory of Open Access Journals (Sweden)

Ben C. Yee

2017-09-01

Full Text Available In this paper, we present a new multilevel in space and energy diffusion (MSED method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1 a grey (one-group diffusion equation used to efficiently converge the fission source and eigenvalue, (2 a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3 a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

Parallelization of mathematical library for generalized eigenvalue problem for real band matrices

International Nuclear Information System (INIS)

Tanaka, Yasuhisa.

1997-05-01

This research has focused on a parallelization of the mathematical library for a generalized eigenvalue problem for real band matrices on IBM SP and Hitachi SR2201. The origin of the library is LASO (Lanczos Algorithm with Selective Orthogonalization), which was developed on the basis of Block Lanczos method for standard eigenvalue problem for real band matrices at Texas University. We adopted D.O.F. (Degree Of Freedom) decomposition method for a parallelization of this library, and evaluated its parallel performance. (author)

On the number of eigenvalues of the discrete one-dimensional Dirac operator with a complex potential

Science.gov (United States)

Hulko, Artem

2018-03-01

In this paper we define a one-dimensional discrete Dirac operator on Z . We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. We also estimate the number of eigenvalues for the discrete Schrödinger operator with complex potential on Z . That is we extend the result obtained by Hulko (Bull Math Sci, to appear) to the whole Z.

Numerical Investigations on Several Stabilized Finite Element Methods for the Stokes Eigenvalue Problem

Directory of Open Access Journals (Sweden)

Pengzhan Huang

2011-01-01

Full Text Available Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and it is a favorite method among these methods for the Stokes eigenvalue problem.

AMDLIBF, IBM 360 Subroutine Library, Eigenvalues, Eigenvectors, Matrix Inversion

International Nuclear Information System (INIS)

Wang, Jesse Y.

1980-01-01

Description of problem or function: AMDLIBF is a subset of the IBM 360 Subroutine Library at the Applied Mathematics Division at Argonne. This subset includes library category F: Identification/Description: F152S F SYMINV: Invert sym. matrices, solve lin. systems; F154S A DOTP: Double plus precision accum. inner prod.; F156S F RAYCOR: Rayleigh corrections for eigenvalues; F161S F XTRADP: A fast extended precision inner product; F162S A XTRADP: Inner product of two DP real vectors; F202S F1 EIGEN: Eigen-system for real symmetric matrix; F203S F: Driver for F202S; F248S F RITZIT: Largest eigenvalue and vec. real sym. matrix; F261S F EIGINV: Inverse eigenvalue problem; F313S F CQZHES: Reduce cmplx matrices to upper Hess and tri; F314S F CQZVAL: Reduce complex matrix to upper Hess. form; F315S F CQZVEC: Eigenvectors of cmplx upper triang. syst.; F316S F CGG: Driver for complex general Eigen-problem; F402S F MATINV: Matrix inversion and sol. of linear eqns.; F403S F: Driver for F402S; F452S F CHOLLU,CHOLEQ: Sym. decomp. of pos. def. band matrices; F453S F MATINC: Inversion of complex matrices; F454S F CROUT: Solution of simultaneous linear equations; F455S F CROUTC: Sol. of simultaneous complex linear eqns.; F456S F1 DIAG: Integer preserving Gaussian elimination

New algorithms for the symmetric tridiagonal eigenvalue computation

Energy Technology Data Exchange (ETDEWEB)

Pan, V. [City Univ. of New York, Bronx, NY (United States)]|[International Computer Sciences Institute, Berkeley, CA (United States)

1994-12-31

The author presents new algorithms that accelerate the bisection method for the symmetric eigenvalue problem. The algorithms rely on some new techniques, which include acceleration of Newton`s iteration and can also be further applied to acceleration of some other iterative processes, in particular, of iterative algorithms for approximating polynomial zeros.

Eigenvalues of relaxed toroidal plasmas of arbitrary sharp edged cross sections. Vol. 2

Energy Technology Data Exchange (ETDEWEB)

Khalil, Sh M [Plasma Physics and Nuclear Fusion Department, Nuclear Research Center, Atomic Energy Authority, Cairo, (Egypt)

1996-03-01

Relaxed (force-free) toroidal plasmas described by the equations cur 1 B={mu}B, and grad {mu}=O (B is the magnetic field) induces interest in nuclear fusion. Its solution is perceived to describe the gross of the reversed field pinch (RFP), spheromak configuration, current limitation in toroidal plasmas, and others. The parameter {mu} plays an important roll in relaxed states. It cannot exceed the smallest eigenvalue {mu} (min), and that for a toroidal discharge there is a maximum toroidal current which is connected to this value. The values of{mu} were calculated numerically, using the methods of collocation points, for toroids of arbitrary aspect ratio {alpha} ({alpha} = R/a, ratio of major/minor radii of tokamak) and arbitrary curved cross-sections (circle, ellipse, cassini, and D-shaped). The aim of present work is to prove the applicability of the numerical methods for calculating the eigenvalues for toroidal plasmas having sharp edged cross sections and arbitrary aspect ratio. The lowest eigenvalue {mu} (min), satisfy the boundary condition {beta} .n = O (or RB. = O) for which the toroidal flux are calculated. These are the zero field eigenvalues of the equation cur 1 b={mu}B. The poloidal magnetic field lines corresponding to different cross sections are shown by plotting the boundary condition B.n=O. The plots showed good fulfillment of the boundary condition along the whole boundaries of different cross sections. Dependence of eigenvalues {mu}a on aspect ratio {alpha} is also obtained. Several runs of the programme with various wave numbers K showed that {mu}a is very insensitive to the choice of K. 8 figs.

A new formulation for the eigenvalue and the eigenfunction in the perturbation theory

International Nuclear Information System (INIS)

Korek, Mahmoud

1999-01-01

Full text.In infrared transitions, the problem of the ro vibrational eigenvalue and eigenfunction of a diatomic molecule is considered. It is shown that, for the transitions vJ↔v'J' the eigenvalues and the eigenfunctions of the two considered states can be expressed respectively in terms of one variable m (transition number), relating these two states, as E vm =Σ i=o e v (i) m i , Ψ vm =Σ i=0 φ v (i) m i and E v'm =Σ i=0 e v' (i) m i , Ψ v'm =Σ i=0 φ v' (i) m i , where m=[J'(J'+1)-J(J+1)]/2, and the coefficients e v (i) , φ v (i) , e v (i) , and φ v (i) , are given by analytical expressions. This m-representation of the eigenvalues and the eigenfunctions is more advantageous for the calculation of many factors in spectroscopy that are given in terms of m as the line intensities, the wave number of a transition, the Herman-Wallis coefficients,...etc. The numerical application to the ground state of the molecule CO shows that the present formulation provides a simple and accurate method for the calculation of the eigenvalues and the eigenfunctions for the two considered states

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

Science.gov (United States)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

Ground eigenvalue and eigenfunction of a spin-weighted spheroidal wave equation in low frequencies

Institute of Scientific and Technical Information of China (English)

Tang Wen-Lin; Tian Gui-Hua

2011-01-01

Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their analytic ground eigenvalues and eigenfunctions are obtained by means of a series in low frequency. The ground eigenvalue and eigenfunction for small complex frequencies are numerically determined.

Algorithm 589. SICEDR: a FORTRAN subroutine for improving the accuracy of computed matrix eigenvalues

International Nuclear Information System (INIS)

Dongarra, J.J.

1982-01-01

SICEDR is a FORTRAN subroutine for improving the accuracy of a computed real eigenvalue and improving or computing the associated eigenvector. It is first used to generate information during the determination of the eigenvalues by the Schur decomposition technique. In particular, the Schur decomposition technique results in an orthogonal matrix Q and an upper quasi-triangular matrix T, such that A = QTQ/sup T/. Matrices A, Q, and T and the approximate eigenvalue, say lambda, are then used in the improvement phase. SICEDR uses an iterative method similar to iterative improvement for linear systems to improve the accuracy of lambda and improve or compute the eigenvector x in O(n 2 ) work, where n is the order of the matrix A

Two new eigenvalue localization sets for tensors and theirs applications

Directory of Open Access Journals (Sweden)

Zhao Jianxing

2017-10-01

Full Text Available A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324 and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50. As an application, a weaker checkable sufficient condition for the positive (semi-definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1, 187-198. As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems

Science.gov (United States)

Bäcker, A.

Summary: We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two-dimensional area-preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area-preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigenvectors and compare them with the corresponding random matrix distributions. The Husimi representation allows for a direct comparison of the localization of the eigenstates in phase space with the corresponding classical structures. Examples for a perturbed cat map and the standard map with different parameters are shown. Billiard systems and the corresponding quantum billiards are another important class of systems (which are also relevant to applications, for example in mesoscopic physics). We provide a detailed exposition of the boundary integral method, which is one important method to determine the eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several methods to determine the eigenvalues from the Fredholm equation and illustrate them for the stadium billiard. The occurrence of spurious solutions is discussed in detail and illustrated for the circular billiard, the stadium billiard, and the annular sector billiard. We emphasize the role of the normal derivative function to compute the normalization of eigenfunctions, momentum representations or autocorrelation functions in a very efficient and direct way. Some examples for these quantities are given and discussed.

Oscillatory Stability and Eigenvalue Sensitivity Analysis of A DFIG Wind Turbine System

DEFF Research Database (Denmark)

Yang, Lihui; Xu, Zhao; Østergaard, Jacob

2011-01-01

This paper focuses on modeling and oscillatory stability analysis of a wind turbine with doubly fed induction generator (DFIG). A detailed mathematical model of DFIG wind turbine with vector-control loops is developed, based on which the loci of the system Jacobian's eigenvalues have been analyzed......, showing that, without appropriate controller tuning a Hopf bifurcation can occur in such a system due to various factors, such as wind speed. Subsequently, eigenvalue sensitivity with respect to machine and control parameters is performed to assess their impacts on system stability. Moreover, the Hopf...

The solution of a chiral random matrix model with complex eigenvalues

International Nuclear Information System (INIS)

Akemann, G

2003-01-01

We describe in detail the solution of the extension of the chiral Gaussian unitary ensemble (chGUE) into the complex plane. The correlation functions of the model are first calculated for a finite number of N complex eigenvalues, where we exploit the existence of orthogonal Laguerre polynomials in the complex plane. When taking the large-N limit we derive new correlation functions in the case of weak and strong non-Hermiticity, thus describing the transition from the chGUE to a generalized Ginibre ensemble. We briefly discuss applications to the Dirac operator eigenvalue spectrum in quantum chromodynamics with non-vanishing chemical potential. This is an extended version of hep-th/0204068

The cosmological constant as an eigenvalue of the Hamiltonian constraint in a varying speed of light theory

Energy Technology Data Exchange (ETDEWEB)

Garattini, Remo [Univ. degli Studi di Bergamo, Dalmine (Italy). Dept. of Engineering and Applied Sciences; I.N.F.N., Sezione di Milano, Milan (Italy); De Laurentis, Mariafelicia [Tomsk State Pedagogical Univ. (Russian Federation). Dept. of Theoretical Physics; INFN, Sezione di Napoli (Italy); Complutense Univ. di Monte S. Angelo, Napoli (Italy)

2017-01-15

In the framework of a Varying Speed of Light theory, we study the eigenvalues associated with the Wheeler-DeWitt equation representing the vacuum expectation values associated with the cosmological constant. We find that the Wheeler-DeWitt equation for the Friedmann-Lemaitre-Robertson-Walker metric is completely equivalent to a Sturm-Liouville problem provided that the related eigenvalue and the cosmological constant be identified. The explicit calculation is performed with the help of a variational procedure with trial wave functionals related to the Bessel function of the second kind K{sub ν}(x). After having verified that in ordinary General Relativity no eigenvalue appears, we find that in a Varying Speed of Light theory this is not the case. Nevertheless, instead of a single eigenvalue, we discover the existence of a family of eigenvalues associated to a negative power of the scale. A brief comment on what happens at the inflationary scale is also included. (copyright 2016 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)

A Universal Quantum Circuit Scheme For Finding Complex Eigenvalues

OpenAIRE

Daskin, Anmer; Grama, Ananth; Kais, Sabre

2013-01-01

We present a general quantum circuit design for finding eigenvalues of non-unitary matrices on quantum computers using the iterative phase estimation algorithm. In particular, we show how the method can be used for the simulation of resonance states for quantum systems.

Singular perturbation of simple eigenvalues

International Nuclear Information System (INIS)

Greenlee, W.M.

1976-01-01

Two operator theoretic theorems which generalize those of asymptotic regular perturbation theory and which apply to singular perturbation problems are proved. Application of these theorems to concrete problems is involved, but the perturbation expansions for eigenvalues and eigenvectors are developed in terms of solutions of linear operator equations. The method of correctors, as well as traditional boundary layer techniques, can be used to apply these theorems. The current formulation should be applicable to highly singular ''hard core'' potential perturbations of the radial equation of quantum mechanics. The theorems are applied to a comparatively simple model problem whose analysis is basic to that of the quantum mechanical problem

On the eigenvalues of S.Π for arbitrary spin in a constant magnetic field

International Nuclear Information System (INIS)

Jayaraman, J.; Oliveira, M.A.B. de.

1985-01-01

Utilizing the intimate connection of a charged particle in a nomogeneous magnetic field to that of a harmonic oscillator, it was established in a recent communication that the eigenvalue spectrum of the matrix operator S.Π for spin 1 is purely real for any intensity of the external magnetic field thereby removing a false impression to the contrary in the recent literature. Here these results are extended to arbitrary spin the reality of the eigenvalue spectrum. The case of spin 3/2 is discussed in some details and it is demonstrated that the complex eigenvalues implied the spectrum by a recent analysis of Weaver, for sufficiently intense magnetic field, when the particle number n assumes values 0 and 1 do not in fact appear at all. (Author) [pt

Eigenvalue estimates of positive integral operators with analytic ...

Indian Academy of Sciences (India)

Eigenvalue estimates of positive integral operators. 337 will be used to denote, respectively, the complex line integral of f along γ and the integral of f with respect to arc-length measure. In the first case we assume γ has an orientation. The notation Lp(γ ) will denote the Lp space of normalized arc length measure on γ with.

Escape rate from strange sets as an eigenvalue

International Nuclear Information System (INIS)

Tel, T.

1986-06-01

A new method is applied for calculating the escape rate from chaotic repellers or semi-attractors, based on the eigenvalue problem of the master equation of discrete dynamical systems. The corresponding eigenfunction is found to be smooth along unstable directions and to be, in general, a fractal measure. Examples of one and two dimensional maps are investigated. (author)

Random matrices, Frobenius eigenvalues, and monodromy

CERN Document Server

Katz, Nicholas M

1998-01-01

The main topic of this book is the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and L-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinit

Maximal imaginery eigenvalues in optimal systems

Directory of Open Access Journals (Sweden)

David Di Ruscio

1991-07-01

Full Text Available In this note we present equations that uniquely determine the maximum possible imaginary value of the closed loop eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen, provided a real symmetric solution of the algebraic Riccati equation exists. In addition, the corresponding state weight matrix and the solution to the algebraic Riccati equation are derived for a class of linear systems. A fundamental lemma for the existence of a real symmetric solution to the algebraic Riccati equation is derived for this class of linear systems.

Solving Large Scale Nonlinear Eigenvalue Problem in Next-Generation Accelerator Design

Energy Technology Data Exchange (ETDEWEB)

Liao, Ben-Shan; Bai, Zhaojun; /UC, Davis; Lee, Lie-Quan; Ko, Kwok; /SLAC

2006-09-28

A number of numerical methods, including inverse iteration, method of successive linear problem and nonlinear Arnoldi algorithm, are studied in this paper to solve a large scale nonlinear eigenvalue problem arising from finite element analysis of resonant frequencies and external Q{sub e} values of a waveguide loaded cavity in the next-generation accelerator design. They present a nonlinear Rayleigh-Ritz iterative projection algorithm, NRRIT in short and demonstrate that it is the most promising approach for a model scale cavity design. The NRRIT algorithm is an extension of the nonlinear Arnoldi algorithm due to Voss. Computational challenges of solving such a nonlinear eigenvalue problem for a full scale cavity design are outlined.

New approach to calculate bound state eigenvalues

International Nuclear Information System (INIS)

Gerck, E.; Gallas, J.A.C.

1983-01-01

A method of solving the radial Schrodinger equation for bound states is discussed. The method is based on a new piecewise representation of the second derivative operator on a set of functions that obey the boundary conditions. This representation is trivially diagonalised and leads to closed form expressions of the type E sub(n)=E(ab+b+c/n+...) for the eigenvalues. Examples are given for the power-law and logarithmic potentials. (Author) [pt

Eigenvalue estimates for submanifolds with bounded f-mean curvature

Indian Academy of Sciences (India)

GUANGYUE HUANG

1College of Mathematics and Information Science, Henan Normal University,. Xinxiang 453007 ... submanifolds in a hyperbolic space with the norm of their mean curvature vector bounded above by a constant. ..... [2] Batista M, Cavalcante M P and Pyo J, Some isoperimetric inequalities and eigenvalue estimates in ...

Genetic Algorithm Applied to the Eigenvalue Equalization Filtered-x LMS Algorithm (EE-FXLMS

Directory of Open Access Journals (Sweden)

Stephan P. Lovstedt

2008-01-01

Full Text Available The FXLMS algorithm, used extensively in active noise control (ANC, exhibits frequency-dependent convergence behavior. This leads to degraded performance for time-varying tonal noise and noise with multiple stationary tones. Previous work by the authors proposed the eigenvalue equalization filtered-x least mean squares (EE-FXLMS algorithm. For that algorithm, magnitude coefficients of the secondary path transfer function are modified to decrease variation in the eigenvalues of the filtered-x autocorrelation matrix, while preserving the phase, giving faster convergence and increasing overall attenuation. This paper revisits the EE-FXLMS algorithm, using a genetic algorithm to find magnitude coefficients that give the least variation in eigenvalues. This method overcomes some of the problems with implementing the EE-FXLMS algorithm arising from finite resolution of sampled systems. Experimental control results using the original secondary path model, and a modified secondary path model for both the previous implementation of EE-FXLMS and the genetic algorithm implementation are compared.

The numerical analysis of eigenvalue problem solutions in multigroup neutron diffusion theory

International Nuclear Information System (INIS)

Woznicki, Z.I.

1995-01-01

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iterations within global iterations. Particular iterative strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 35 figs, 16 tabs

On the calculation of the eigenvalues of the Faddeev equation kernel on the nonphysical sheet of energy

International Nuclear Information System (INIS)

Moeller, K.

1978-01-01

A system of three particles is considered which interacts by rank-1 separable potential. For the Faddeev equation kernel of this system a method is proposed for calculating the eigenvalues on the nonphysical sheet of the three-particle cms-energy. From the consideration of the analytical structure of the eigenvalues in the energy plane it follows that the analytical continuations of the eigenvalues from the physical to the nonphysical region are different above and below the three-particle threshold. In this paper the continuation below the threshold is discussed. (author)

Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations

Directory of Open Access Journals (Sweden)

Winfried Auzinger

2006-01-01

Full Text Available We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.

A method for eigenvalues of sparse lambda-matrices

International Nuclear Information System (INIS)

Yang, W.H.

1982-01-01

The matrix N(lambda) whose elements are functions of a parameter lambda is called the lambda-matrix. Those values of lambda that make the matrix singular are of great interest in many applied fields. An efficient method for those eigenvalues of a lambda-matrix is presented. A simple explicit convergence criterion is given as well as the algorithm and two numerical examples

Fundaments of transport equation splitting and the eigenvalue problem

International Nuclear Information System (INIS)

Stancic, V.

2000-01-01

In order to remove some singularities concerning the boundary conditions of one dimensional transport equation, a split form of transport equation describing the forward i.e. μ≥0, and a backward μ<0 directed neutrons is being proposed here. The eigenvalue problem has also been considered here (author)

Elementary Baecklund transformations for a discrete Ablowitz-Ladik eigenvalue problem

International Nuclear Information System (INIS)

Rourke, David E

2004-01-01

Elementary Baecklund transformations (BTs) are described for a discretization of the Zakharov-Shabat eigenvalue problem (a special case of the Ablowitz-Ladik eigenvalue problem). Elementary BTs allow the process of adding bound states to a system (i.e., the add-one-soliton BT) to be 'factorized' to solving two simpler sub-problems. They are used to determine the effect on the scattering data when bound states are added. They are shown to provide a method of calculating discrete solitons-this is achieved by constructing a lattice of intermediate potentials, with the parameters used in the calculation of the lattice simply related to the soliton scattering data. When the potentials, S n , T n , in the system are related by S n = -T n , they enable simple derivations to be obtained of the add-one-soliton BT and the nonlinear superposition formula

Monotonicity of energy eigenvalues for Coulomb systems

International Nuclear Information System (INIS)

Englisch, R.

1983-01-01

Generalising results by earlier workers for a large class of Hamiltonians (among others, Hamiltonians of Coulomb systems) which can be written in the form H(α) = H 0 + αH' the present works shows that their eigenvalues decrease with increasing α. This result is applied to Coulomb systems in which the distances between the infinitely heavy particles are varying and also is used to obtain a completion and simplification of proof for the stability of the biexciton. (author)

Intergenerational Correlation in Monte Carlo k-Eigenvalue Calculation

International Nuclear Information System (INIS)

Ueki, Taro

2002-01-01

This paper investigates intergenerational correlation in the Monte Carlo k-eigenvalue calculation of a neutron effective multiplicative factor. To this end, the exponential transform for path stretching has been applied to large fissionable media with localized highly multiplying regions because in such media an exponentially decaying shape is a rough representation of the importance of source particles. The numerical results show that the difference between real and apparent variances virtually vanishes for an appropriate value of the exponential transform parameter. This indicates that the intergenerational correlation of k-eigenvalue samples could be eliminated by the adjoint biasing of particle transport. The relation between the biasing of particle transport and the intergenerational correlation is therefore investigated in the framework of collision estimators, and the following conclusion has been obtained: Within the leading order approximation with respect to the number of histories per generation, the intergenerational correlation vanishes when immediate importance is constant, and the immediate importance under simulation can be made constant by the biasing of particle transport with a function adjoint to the source neutron's distribution, i.e., the importance over all future generations

Topological derivatives of eigenvalues and neural networks in identification of imperfections

International Nuclear Information System (INIS)

Grzanek, M; Nowakowski, A; Sokolowski, J

2008-01-01

Numerical method for identification of imperfections is devised for elliptic spectral problems. The neural networks are employed for numerical solution. The topological derivatives of eigenvalues are used in the learning procedure of the neural networks. The topological derivatives of eigenvalues are determined by the methods of asymptotic analysis in singularly perturbed geometrical domains. The convergence of the numerical method in a probabilistic setting is analysed. The method is presented for the identification of small singular perturbations of the boundary of geometrical domain, however the framework is general and can be used for numerical solutions of inverse problems in the presence of small imperfections in the interior of the domain. Some numerical results are given for elliptic spectral problem in two spatial dimensions.

Solving eigenvalue problems on curved surfaces using the Closest Point Method

KAUST Repository

Macdonald, Colin B.

2011-06-01

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace-Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach. © 2011 Elsevier Inc.

On the Solution of the Eigenvalue Assignment Problem for Discrete-Time Systems

Directory of Open Access Journals (Sweden)

El-Sayed M. E. Mostafa

2017-01-01

Full Text Available The output feedback eigenvalue assignment problem for discrete-time systems is considered. The problem is formulated first as an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to find a local solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a logarithmic barrier method is proposed for finding the local solution. The conjugate gradient method is further extended to tackle the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. The performance of the methods is illustrated through various test examples.

Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

International Nuclear Information System (INIS)

Gene Golub; Kwok Ko

2009-01-01

The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.

p-Norm SDD tensors and eigenvalue localization

Directory of Open Access Journals (Sweden)

Qilong Liu

2016-07-01

Full Text Available Abstract We present a new class of nonsingular tensors (p-norm strictly diagonally dominant tensors, which is a subclass of strong H $\\mathcal{H}$ -tensors. As applications of the results, we give a new eigenvalue inclusion set, which is tighter than those provided by Li et al. (Linear Multilinear Algebra 64:727-736, 2016 in some case. Based on this set, we give a checkable sufficient condition for the positive (semidefiniteness of an even-order symmetric tensor.

Hardy inequality, compact embeddings and properties of certain eigenvalue problems

Czech Academy of Sciences Publication Activity Database

Drábek, P.; Kufner, Alois

2017-01-01

Roč. 49, č. 1 (2017), s. 5-17 ISSN 0049-4704 Institutional support: RVO:67985840 Keywords : BD-property * compact embeddings * degenerate and singular eigenvalue problem Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics https://www.openstarts.units.it/handle/10077/16201

Bonnesen-style inequality for the first eigenvalue on a complete surface of constant curvature

Directory of Open Access Journals (Sweden)

Niufa Fang

2017-08-01

Full Text Available Abstract By Cheeger’s isoperimetric constants, some lower bounds and upper bounds of λ 1 $\\lambda_{1}$ , the first eigenvalue on a complete surface of constant curvature, are given. Some Bonnesen-style inequalities and reverse Bonnesen-style inequalities for the first eigenvalue are obtained. Those Bonnesen-style inequalities obtained are stronger than the well-known Osserman’s results and the upper bound is stronger than Osserman’s results (Osserman in Proceedings of the International Congress of Mathematicians, Helsinki, 1978.

Discontinuous Sturm-Liouville Problems with Eigenvalue Dependent Boundary Condition

Energy Technology Data Exchange (ETDEWEB)

Amirov, R. Kh., E-mail: emirov@cumhuriyet.edu.tr; Ozkan, A. S., E-mail: sozkan@cumhuriyet.edu.tr [Cumhuriyet University, Department of Mathematics Faculty of Art and Science (Turkey)

2014-12-15

In this study, an inverse problem for Sturm-Liouville differential operators with discontinuities is studied when an eigenparameter appears not only in the differential equation but it also appears in the boundary condition. Uniqueness theorems of inverse problems according to the Prüfer angle, the Weyl function and two different eigenvalues sets are proved.

Reflectance variability of surface coatings reveals characteristic eigenvalue spectra

Science.gov (United States)

Medina, José M.; Díaz, José A.; Barros, Rui

2012-10-01

We have examined the trial-to-trial variability of the reflectance spectra of surface coatings containing effect pigments. Principal component analysis of reflectances was done at each detection angle separately. A method for classification of principal components is applied based on the eigenvalue spectra. It was found that the eigenvalue spectra follow characteristic power laws and depend on the detection angle. Three different subsets of principal components were examined to separate the relevant spectral features related to the pigments from other noise sources. Reconstruction of the reflectance spectra by taking only the first subset indicated that reflectance variability was higher at near-specular reflection, suggesting a correlation with the trial-to-trial deposition of effect pigments. Reconstruction by using the second subset indicates that variability was higher at short wavelengths. Finally, reconstruction by using only the third subset indicates that reflectance variability was not totally random as a function of the wavelength. The methods employed can be useful in the evaluation of color variability in industrial paint application processes.

Solving the generalized symmetric eigenvalue problem using tile algorithms on multicore architectures

KAUST Repository

Ltaief, Hatem

2012-01-01

This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric eigenvalue problem by computing the Cholesky factorization of the right hand side symmetric definite positive matrix (first stage), and applying the inverse of the freshly computed triangular Cholesky factors to the original dense symmetric matrix of the problem (second stage). Calculating the eigenpairs of the resulting problem is then equivalent to the eigenpairs of the original problem. The computation proceeds by reducing the updated dense symmetric matrix to symmetric band form (third stage). The band structure is further reduced by applying a bulge chasing procedure, which annihilates the extra off-diagonal entries using orthogonal transformations (fourth stage). More details on the third and fourth stage can be found in Haidar et al. [Accepted at SC\\'11, November 2011]. The eigenvalues are then calculated from the tridiagonal form using the standard LAPACK QR algorithm (i.e., DTSEQR routine), while the complex and challenging eigenvector computations will be addressed in a companion paper. The tasks from the various stages can concurrently run in an out-of-order fashion. The data dependencies are cautiously tracked by the dynamic runtime system environment QUARK, which ensures the dependencies are not violated for numerical correctness purposes. The obtained tile four-stage generalized symmetric eigenvalue solver significantly outperforms the state-of-the-art numerical libraries (up to 21-fold speed up against multithreaded LAPACK with optimized multithreaded MKL BLAS and up to 4-fold speed up against the corresponding routine from the commercial numerical software Intel MKL) on four sockets twelve cores AMD system with a 24000×24000 matrix size. © 2012 The authors and IOS Press. All rights reserved.

Advanced Variance Reduction for Global k-Eigenvalue Simulations in MCNP

Energy Technology Data Exchange (ETDEWEB)

Edward W. Larsen

2008-06-01

The "criticality" or k-eigenvalue of a nuclear system determines whether the system is critical (k=1), or the extent to which it is subcritical (k<1) or supercritical (k>1). Calculations of k are frequently performed at nuclear facilities to determine the criticality of nuclear reactor cores, spent nuclear fuel storage casks, and other fissile systems. These calculations can be expensive, and current Monte Carlo methods have certain well-known deficiencies. In this project, we have developed and tested a new "functional Monte Carlo" (FMC) method that overcomes several of these deficiencies. The current state-of-the-art Monte Carlo k-eigenvalue method estimates the fission source for a sequence of fission generations (cycles), during each of which M particles per cycle are processed. After a series of "inactive" cycles during which the fission source "converges," a series of "active" cycles are performed. For each active cycle, the eigenvalue and eigenfunction are estimated; after N >> 1 active cycles are performed, the results are averaged to obtain estimates of the eigenvalue and eigenfunction and their standard deviations. This method has several disadvantages: (i) the estimate of k depends on the number M of particles per cycle, (iii) for optically thick systems, the eigenfunction estimate may not converge due to undersampling of the fission source, and (iii) since the fission source in any cycle depends on the estimated fission source from the previous cycle (the fission sources in different cycles are correlated), the estimated variance in k is smaller than the real variance. For an acceptably large number M of particles per cycle, the estimate of k is nearly independent of M; this essentially takes care of item (i). Item (ii) can be addressed by taking M sufficiently large, but for optically thick systems a sufficiently large M can easily be unrealistic. Item (iii) cannot be accounted for by taking M or N sufficiently large; it is an inherent deficiency due

The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory

International Nuclear Information System (INIS)

Woznicki, Z.I.

1994-01-01

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs

The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory

Energy Technology Data Exchange (ETDEWEB)

Woznicki, Z I [Institute of Atomic Energy, Otwock-Swierk (Poland)

1994-12-31

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs.

Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue

Directory of Open Access Journals (Sweden)

Shu-Guang Guo

2016-05-01

Full Text Available Abstract A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013 and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013 determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.

Investigation, development and application of optimal output feedback theory. Vol. 4: Measures of eigenvalue/eigenvector sensitivity to system parameters and unmodeled dynamics

Science.gov (United States)

Halyo, Nesim

1987-01-01

Some measures of eigenvalue and eigenvector sensitivity applicable to both continuous and discrete linear systems are developed and investigated. An infinite series representation is developed for the eigenvalues and eigenvectors of a system. The coefficients of the series are coupled, but can be obtained recursively using a nonlinear coupled vector difference equation. A new sensitivity measure is developed by considering the effects of unmodeled dynamics. It is shown that the sensitivity is high when any unmodeled eigenvalue is near a modeled eigenvalue. Using a simple example where the sensor dynamics have been neglected, it is shown that high feedback gains produce high eigenvalue/eigenvector sensitivity. The smallest singular value of the return difference is shown not to reflect eigenvalue sensitivity since it increases with the feedback gains. Using an upper bound obtained from the infinite series, a procedure to evaluate whether the sensitivity to parameter variations is within given acceptable bounds is developed and demonstrated by an example.

Distribution of the Largest Eigenvalues of the Levi-Smirnov Ensemble

International Nuclear Information System (INIS)

Wieczorek, W.

2004-01-01

We calculate the distribution of the k-th largest eigenvalue in the random matrix Levi - Smirnov Ensemble (LSE), using the spectral dualism between LSE and chiral Gaussian Unitary Ensemble (GUE). Then we reconstruct universal spectral oscillations and we investigate an asymptotic behavior of the spectral distribution. (author)

Multi-level methods for solving multigroup transport eigenvalue problems in 1D slab geometry

International Nuclear Information System (INIS)

Anistratov, D. Y.; Gol'din, V. Y.

2009-01-01

A methodology for solving eigenvalue problems for the multigroup neutron transport equation in 1D slab geometry is presented. In this paper we formulate and compare different variants of nonlinear multi-level iteration methods. They are defined by means of multigroup and effective one-group low-order quasi diffusion (LOQD) equations. We analyze the effects of utilization of the effective one-group LOQD problem for estimating the eigenvalue. We present numerical results to demonstrate the performance of the iteration algorithms in different types of reactor-physics problems. (authors)

Distribution of Schmidt-like eigenvalues for Gaussian ensembles of the random matrix theory

Science.gov (United States)

Pato, Mauricio P.; Oshanin, Gleb

2013-03-01

We study the probability distribution function P(β)n(w) of the Schmidt-like random variable w = x21/(∑j = 1nx2j/n), where xj, (j = 1, 2, …, n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P(β)n(w) converges to the Marčenko-Pastur form, i.e. is defined as P_{n}^{( \\beta )}(w) \\sim \\sqrt{(4 - w)/w} for w ∈ [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (β = 2) we present exact explicit expressions for P(β = 2)n(w) which are valid for arbitrary n and analyse their behaviour.

Distribution of Schmidt-like eigenvalues for Gaussian ensembles of the random matrix theory

International Nuclear Information System (INIS)

Pato, Mauricio P; Oshanin, Gleb

2013-01-01

We study the probability distribution function P (β) n (w) of the Schmidt-like random variable w = x 2 1 /(∑ j=1 n x 2 j /n), where x j , (j = 1, 2, …, n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P (β) n (w) converges to the Marčenko–Pastur form, i.e. is defined as P n (β) (w)∼√((4 - w)/w) for w ∈ [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (β = 2) we present exact explicit expressions for P (β=2) n (w) which are valid for arbitrary n and analyse their behaviour. (paper)

A comparison of maximum likelihood and other estimators of eigenvalues from several correlated Monte Carlo samples

International Nuclear Information System (INIS)

Beer, M.

1980-01-01

The maximum likelihood method for the multivariate normal distribution is applied to the case of several individual eigenvalues. Correlated Monte Carlo estimates of the eigenvalue are assumed to follow this prescription and aspects of the assumption are examined. Monte Carlo cell calculations using the SAM-CE and VIM codes for the TRX-1 and TRX-2 benchmark reactors, and SAM-CE full core results are analyzed with this method. Variance reductions of a few percent to a factor of 2 are obtained from maximum likelihood estimation as compared with the simple average and the minimum variance individual eigenvalue. The numerical results verify that the use of sample variances and correlation coefficients in place of the corresponding population statistics still leads to nearly minimum variance estimation for a sufficient number of histories and aggregates

Eigenvalue-dependent neutron energy spectra: Definitions, analyses, and applications

International Nuclear Information System (INIS)

Cacuci, D.G.; Ronen, Y.; Shayer, Z.; Wagschal, J.J.; Yeivin, Y.

1982-01-01

A general qualitative analysis of spectral effects that arise from solving the kappa-, α-, γ-, and sigma-eigenvalue formulations of the neutron transport equation for nuclear systems that deviate (to first order) from criticality is presented. Hierarchies of neutron spectra softness are established and expressed concisely in terms of the newly introduced spatialdependent local spectral indices for the core and for the reflector. It is shown that each hierarchy is preserved, regardless of the nature of the specific physical mechanism that cause the system to deviate from criticality. Qualitative conclusions regarding the general behavior of the spectrum-dependent integral spectral indices and ICRs corresponding to the kappa-, α-, γ-, and sigma-eigenvalue formalisms are also presented. By defining spectral indices separately for the core and for the reflector, it is possible to account for the characteristics of neutron spectra in both the core and the reflector. The distinctions between the spectra in the core and in the reflector could not have been accounted for by using a single type of spectral index (e.g., a spectral index for the entire system or a spectral index solely for the core)

A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer.

Science.gov (United States)

Hwang, Seong Jae; Collins, Maxwell D; Ravi, Sathya N; Ithapu, Vamsi K; Adluru, Nagesh; Johnson, Sterling C; Singh, Vikas

2015-12-01

Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a "black box" can often become restrictive. Many 'human in the loop' settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other 'views' of the disease pathology, involving clinical measurements and other image-derived representations.

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

International Nuclear Information System (INIS)

Kabashima, Yoshiyuki; Takahashi, Hisanao; Watanabe, Osamu

2010-01-01

A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.

MIMO Channel Model with Propagation Mechanism and the Properties of Correlation and Eigenvalue in Mobile Environments

Directory of Open Access Journals (Sweden)

Yuuki Kanemiyo

2012-01-01

Full Text Available This paper described a spatial correlation and eigenvalue in a multiple-input multiple-output (MIMO channel. A MIMO channel model with a multipath propagation mechanism was proposed and showed the channel matrix. The spatial correlation coefficient formula −,′−′( between MIMO channel matrix elements was derived for the model and was expressed as a directive wave term added to the product of mobile site correlation −′( and base site correlation −′( without LOS path, which are calculated independently of each other. By using −,′−′(, it is possible to create the channel matrix element with a fixed correlation value estimated by −,′−′( for a given multipath condition and a given antenna configuration. Furthermore, the correlation and the channel matrix eigenvalue were simulated, and the simulated and theoretical correlation values agreed well. The simulated eigenvalue showed that the average of the first eigenvalue λ1 hardly depends on the correlation −,′−′(, but the others do depend on −,′−′( and approach 1 as −,′−′( decreases. Moreover, as the path moves into LOS, the 1 state with mobile movement becomes more stable than the 1 of NLOS path.

Gravitational lensing by eigenvalue distributions of random matrix models

Science.gov (United States)

Martínez Alonso, Luis; Medina, Elena

2018-05-01

We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles.

Cessna Citation X Business Aircraft Eigenvalue Stability – Part2: Flight Envelope Analysis

Directory of Open Access Journals (Sweden)

Yamina BOUGHARI

2017-12-01

Full Text Available Civil aircraft flight control clearance is a time consuming, thus an expensive process in the aerospace industry. This process has to be investigated and proved to be safe for thousands of combinations in terms of speeds, altitudes, gross weights, Xcg / weight configurations and angles of attack. Even in this case, a worst-case condition that could lead to a critical situation might be missed. To address this problem, models that are able to describe an aircraft’s dynamics by taking into account all uncertainties over a region within a flight envelope have been developed using Linear Fractional Representation. In order to investigate the Cessna Citation X aircraft Eigenvalue Stability envelope, the Linear Fractional Representation models are implemented using the speeds and the altitudes as varying parameters. In this paper Part 2, the aircraft longitudinal eigenvalue stability is analyzed in a continuous range of flight envelope with varying parameter of True airspeed and altitude, instead of a single point, like classical methods. This is known as the aeroelastic stability envelope, required for civil aircraft certification as given by the Circular Advisory “Aeroelastic Stability Substantiation of Transport Category Airplanes AC No: 25.629-18”. In this new methodology the analysis is performed in time domain based on Lyapunov stability and solved by convex optimization algorithms by using the linear matrix inequalities to evaluate the eigenvalue stability, which is reduced to search for the negative eigenvalues in a region of flight envelope. It can also be used to study the stability of a system during an arbitrary motion from one point to another in the flight envelope. A whole aircraft analysis results’ for its entire envelope are presented in the form of graphs, thus offering good readability, and making them easily exploitable.

Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions

OpenAIRE

Guliyev, Namig J.

2008-01-01

International audience; Inverse problems of recovering the coefficients of Sturm–Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: 1) from the sequences of eigenvalues and norming constants; 2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

Solving the generalized symmetric eigenvalue problem using tile algorithms on multicore architectures

KAUST Repository

Ltaief, Hatem; Luszczek, Piotr R.; Haidar, Azzam; Dongarra, Jack

2012-01-01

This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric

A new S-type eigenvalue inclusion set for tensors and its applications.

Science.gov (United States)

Huang, Zheng-Ge; Wang, Li-Gong; Xu, Zhong; Cui, Jing-Jing

2016-01-01

In this paper, a new S -type eigenvalue localization set for a tensor is derived by dividing [Formula: see text] into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H -eigenvalue of strong M -tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).

Improved simple graphical solution for the eigenvalues of the finite square well potential

International Nuclear Information System (INIS)

Burge, E.J.

1985-01-01

The three principal graphical methods for obtaining the energy eigenvalues of the finite square well potential are presented. The forms of the wavefunctions within the well, and the corresponding linear probability densities, are derived directly from the method. A simple extension of the method allows the energy level spectrum to be obtained directly on a linear energy scale. The variations of the energy eigenvalues with well depth and width are separately and jointly displayed, and explicit corresponding functional relationships are derived. Two universal graphs are deduced which allow the rapid appreciation and calculation of the dependence of the energy levels on the depth and width of the well and on the mass of the particle. (author)

Asymptotic eigenvalue estimates for a Robin problem with a large parameter

Czech Academy of Sciences Publication Activity Database

Exner, Pavel; Minakov, A.; Parnovski, L.

2014-01-01

Roč. 71, č. 2 (2014), s. 141-156 ISSN 0032-5155 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : Laplacian * Robin problem * eigenvalue asymptotics Subject RIV: BE - Theoretical Physics Impact factor: 0.250, year: 2014

Dynamic Eigenvalue Problem of Concrete Slab Road Surface

Science.gov (United States)

Pawlak, Urszula; Szczecina, Michał

2017-10-01

The paper presents an analysis of the dynamic eigenvalue problem of concrete slab road surface. A sample concrete slab was modelled using Autodesk Robot Structural Analysis software and calculated with Finite Element Method. The slab was set on a one-parameter elastic subsoil, for which the modulus of elasticity was separately calculated. The eigen frequencies and eigenvectors (as maximal vertical nodal displacements) were presented. On the basis of the results of calculations, some basic recommendations for designers of concrete road surfaces were offered.

On a residual freedom of the next-to-leading BFKL eigenvalue in color adjoint representation in planar N=4 SYM

Energy Technology Data Exchange (ETDEWEB)

Bondarenko, Sergey; Prygarin, Alex [Physics Department, Ariel University,Ariel 40700, territories administered by (Israel)

2016-07-15

We discuss a residual freedom of the next-to-leading BFKL eigenvalue that originates from ambiguity in redistributing the next-to-leading (NLO) corrections between the adjoint BFKL eigenvalue and eigenfunctions in planar N=4 super-Yang-Mills (SYM) Theory. In terms of the remainder function of the Bern-Dixon-Smirnov (BDS) amplitude this freedom is translated to reshuffling correction between the eigenvalue and the impact factors in the multi-Regge kinematics (MRK) in the next-to-leading logarithm approximation (NLA). We show that the modified NLO BFKL eigenvalue suggested by the authors in ref. http://arxiv.org/abs/1510.00589 can be introduced in the MRK expression for the remainder function by shifting the anomalous dimension in the impact factor in such a way that the two and three loop remainder function is left unchanged to the NLA accuracy.

Numerical method for the eigenvalue problem and the singular equation by using the multi-grid method and application to ordinary differential equation

International Nuclear Information System (INIS)

Kanki, Takashi; Uyama, Tadao; Tokuda, Shinji.

1995-07-01

In the numerical method to compute the matching data which are necessary for resistive MHD stability analyses, it is required to solve the eigenvalue problem and the associated singular equation. An iterative method is developed to solve the eigenvalue problem and the singular equation. In this method, the eigenvalue problem is replaced with an equivalent nonlinear equation and a singular equation is derived from Newton's method for the nonlinear equation. The multi-grid method (MGM), a high speed iterative method, can be applied to this method. The convergence of the eigenvalue and the eigenvector, and the CPU time in this method are investigated for a model equation. It is confirmed from the numerical results that this method is effective for solving the eigenvalue problem and the singular equation with numerical stability and high accuracy. It is shown by improving the MGM that the CPU time for this method is 50 times shorter than that of the direct method. (author)

Schiffer's Conjecture, Interior Transmission Eigenvalues and Invisibility Cloaking: Singular Problem vs. Nonsingular Problem

OpenAIRE

Liu, Hongyu

2012-01-01

In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

Institute of Scientific and Technical Information of China (English)

Tang Wen-Lin; Tian Gui-Hua

2011-01-01

The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.

Imaginary eigenvalue solution in RPA and phase transition

International Nuclear Information System (INIS)

Yao Yujie; Jing Xiaogong; Zhao Guoquan; Wu Shishu

1993-01-01

The phase transition (PT) of a many-particle system with a close-shell configuration, the stability of the Hartree-Fock (HF) solution and the random phase approximation (RPA) are studied by means of a generalized three-level solvable model. The question whether the occurrence of an imaginary eigenvalue solution in RPA (OISA) may be considered as a signature of PT is explored in some detail. It is found that there is no close relation between OISA and PT. Generally, OISA shows that RPA becomes poor

Eigenfunctions and Eigenvalues for a Scalar Riemann-Hilbert Problem Associated to Inverse Scattering

Science.gov (United States)

Pelinovsky, Dmitry E.; Sulem, Catherine

A complete set of eigenfunctions is introduced within the Riemann-Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schrödinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation.

Asymptotics of the Perron-Frobenius eigenvalue of nonnegative Hessenberg-Toeplitz matrices

NARCIS (Netherlands)

Janssen, A.J.E.M.

1989-01-01

Asymptotic results for the Perron-Frobenius eigenvalue of a nonnegative Hessenberg-Toeplitz matrix as the dimension of the matrix tends to 8 are given. The results are used and interpreted in terms of source entropies in the case where the Hessenberg-Toeplitz matrix arises as the transition matrix

Analytic approximation to the largest eigenvalue distribution of a white Wishart matrix

CSIR Research Space (South Africa)

Vlok, JD

2012-08-14

Full Text Available offers largely simplified computation and provides statistics such as the mean value and region of support of the largest eigenvalue distribution. Numeric results from the literature are compared with the approximation and Monte Carlo simulation results...

Uncertainty Estimates in Cold Critical Eigenvalue Predictions

International Nuclear Information System (INIS)

Karve, Atul A.; Moore, Brian R.; Mills, Vernon W.; Marrotte, Gary N.

2005-01-01

A recent cycle of a General Electric boiling water reactor performed two beginning-of-cycle local cold criticals. The eigenvalues estimated by the core simulator were 0.99826 and 1.00610. The large spread in them (= 0.00784) is a source of concern, and it is studied here. An analysis process is developed using statistical techniques, where first a transfer function relating the core observable Y (eigenvalue) to various factors (X's) is established. Engineering judgment is used to recognize the best candidates for X's. They are identified as power-weighted assembly k ∞ 's of selected assemblies around the withdrawn rods. These are a small subset of many X's that could potentially influence Y. However, the intention here is not to do a comprehensive study by accounting for all the X's. Rather, the scope is to demonstrate that the process developed is reasonable and to show its applicability to performing detailed studies. Variability in X's is obtained by perturbing nodal k ∞ 's since they directly influence the buckling term in the quasi-two-group diffusion equation model of the core simulator. Any perturbations introduced in them are bounded by standard well-established uncertainties. The resulting perturbations in the X's may not necessarily be directly correlated to physical attributes, but they encompass numerous biases and uncertainties credited to input and modeling uncertainties. The 'vital few' from the 'unimportant many' X's are determined, and then they are subgrouped according to assembly type, location, exposure, and control rod insertion. The goal is to study how the subgroups influence Y in order to have a better understanding of the variability observed in it

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

Czech Academy of Sciences Publication Activity Database

Lotoreichik, Vladimir; Rohleder, J.

2017-01-01

Roč. 263, č. 1 (2017), s. 491-508 ISSN 0022-0396 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : Laplace operator * mixed boundary conditions * eigenvalue inequality * polyhedral domain * Lipschitz domain Subject RIV: BE - Theoretical Physics OBOR OECD: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect) Impact factor: 1.988, year: 2016

Solution of the multigroup neutron diffusion Eigenvalue problem in slab geometry by modified power method

Energy Technology Data Exchange (ETDEWEB)

Zanette, Rodrigo [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pós-Graduação em Matemática Aplicada; Petersen, Claudio Z.; Tavares, Matheus G., E-mail: rodrigozanette@hotmail.com, E-mail: claudiopetersen@yahoo.com.br, E-mail: matheus.gulartetavares@gmail.com [Universidade Federal de Pelotas (UFPEL), RS (Brazil). Programa de Pós-Graduação em Modelagem Matemática

2017-07-01

We describe in this work the application of the modified power method for solve the multigroup neutron diffusion eigenvalue problem in slab geometry considering two-dimensions for nuclear reactor global calculations. It is well known that criticality calculations can often be best approached by solving eigenvalue problems. The criticality in nuclear reactors physics plays a relevant role since establishes the ratio between the numbers of neutrons generated in successive fission reactions. In order to solve the eigenvalue problem, a modified power method is used to obtain the dominant eigenvalue (effective multiplication factor (K{sub eff})) and its corresponding eigenfunction (scalar neutron flux), which is non-negative in every domain, that is, physically relevant. The innovation of this work is solving the neutron diffusion equation in analytical form for each new iteration of the power method. For solve this problem we propose to apply the Finite Fourier Sine Transform on one of the spatial variables obtaining a transformed problem which is resolved by well-established methods for ordinary differential equations. The inverse Fourier transform is used to reconstruct the solution for the original problem. It is known that the power method is an iterative source method in which is updated by the neutron flux expression of previous iteration. Thus, for each new iteration, the neutron flux expression becomes larger and more complex due to analytical solution what makes propose that it be reconstructed through an polynomial interpolation. The methodology is implemented to solve a homogeneous problem and the results are compared with works presents in the literature. (author)

Generalization of the Fourier Convergence Analysis in the Neutron Diffusion Eigenvalue Problem

International Nuclear Information System (INIS)

Lee, Hyun Chul; Noh, Jae Man; Joo, Hyung Kook

2005-01-01

Fourier error analysis has been a standard technique for the stability and convergence analysis of linear and nonlinear iterative methods. Lee et al proposed new 2- D/1-D coupling methods and demonstrated several advantages of the new methods by performing a Fourier convergence analysis of the methods as well as two existing methods for a fixed source problem. We demonstrated the Fourier convergence analysis of one of the 2-D/1-D coupling methods applied to a neutron diffusion eigenvalue problem. However, the technique cannot be used directly to analyze the convergence of the other 2-D/1-D coupling methods since some algorithm-specific features were used in our previous study. In this paper we generalized the Fourier convergence analysis technique proposed and analyzed the convergence of the 2-D/1-D coupling methods applied to a neutron diffusion Eigenvalue problem using the generalized technique

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

Czech Academy of Sciences Publication Activity Database

Antunes, P. R. S.; Freitas, P.; Krejčiřík, David

2017-01-01

Roč. 10, č. 4 (2017), s. 357-379 ISSN 1864-8258 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : Eigenvalue optimisation * Robin Laplacian * negative boundary parameter * Bareket's conjecture Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.182, year: 2016

Inversion for Eigenvalues and Modes Using Sierra-SD and ROL.

Energy Technology Data Exchange (ETDEWEB)

Walsh, Timothy [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Aquino, Wilkins [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Ridzal, Denis [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Kouri, Drew Philip [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

2015-12-01

In this report we formulate eigenvalue-based methods for model calibration using a PDE-constrained optimization framework. We derive the abstract optimization operators from first principles and implement these methods using Sierra-SD and the Rapid Optimization Library (ROL). To demon- strate this approach, we use experimental measurements and an inverse solution to compute the joint and elastic foam properties of a low-fidelity unit (LFU) model.

A Schur Method for Designing LQ-optimal Systems with Prescribed Eigenvalues

Directory of Open Access Journals (Sweden)

David Di Ruscio

1990-01-01

Full Text Available In this paper a new algorithm for solving the LQ-optimal pole placement problem is presented. The method studied is a variant of the classical eigenvector approach and instead uses a set of Schur vectors, thereby gaining substantial numerical advantages. An important task in this method is the LQ-optimal pole placement problem for a second order (sub system. The paper presents a detailed analytical solution to this problem. This part is not only important for solving the general n-dimensional problem but also provides an understanding of the behaviour of an optimal system: The paper shows that in some cases it is an infinite number; in others a finite number, and in still others, non state weighting matrices Q that give the system a set of prescribed eigenvalues. Equations are presented that uniquely determine these state weight matrices as a function of the new prescribed eigcnvalues. From this result we have been able to derive the maximum possible imaginary part of the eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen.

On eigenvalue asymptotics for strong delta-interactions supported by surfaces with boundaries

Czech Academy of Sciences Publication Activity Database

Dittrich, Jaroslav; Exner, Pavel; Kuhn, C.; Pankrashkin, K.

2016-01-01

Roč. 97, 1-2 (2016), s. 1-25 ISSN 0921-7134 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : singular Schrodinger operator * delta-interaction * strong coupling * eigenvalue Subject RIV: BE - Theoretical Physics Impact factor: 0.933, year: 2016

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

Science.gov (United States)

Lakshtanov, E.; Vainberg, B.

2013-10-01

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on a possible location of the transmission eigenvalues. If the index of refraction \\sqrt{n(x)} is real, then we obtain a result on the existence of infinitely many positive ITEs and the Weyl-type lower bound on its counting function. All the results are obtained under the assumption that n(x) - 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).

NESTLE: Few-group neutron diffusion equation solver utilizing the nodal expansion method for eigenvalue, adjoint, fixed-source steady-state and transient problems

International Nuclear Information System (INIS)

Turinsky, P.J.; Al-Chalabi, R.M.K.; Engrand, P.; Sarsour, H.N.; Faure, F.X.; Guo, W.

1994-06-01

NESTLE is a FORTRAN77 code that solves the few-group neutron diffusion equation utilizing the Nodal Expansion Method (NEM). NESTLE can solve the eigenvalue (criticality); eigenvalue adjoint; external fixed-source steady-state; or external fixed-source. or eigenvalue initiated transient problems. The code name NESTLE originates from the multi-problem solution capability, abbreviating Nodal Eigenvalue, Steady-state, Transient, Le core Evaluator. The eigenvalue problem allows criticality searches to be completed, and the external fixed-source steady-state problem can search to achieve a specified power level. Transient problems model delayed neutrons via precursor groups. Several core properties can be input as time dependent. Two or four energy groups can be utilized, with all energy groups being thermal groups (i.e. upscatter exits) if desired. Core geometries modelled include Cartesian and Hexagonal. Three, two and one dimensional models can be utilized with various symmetries. The non-linear iterative strategy associated with the NEM method is employed. An advantage of the non-linear iterative strategy is that NSTLE can be utilized to solve either the nodal or Finite Difference Method representation of the few-group neutron diffusion equation

Prolongation structure and linear eigenvalue equations for Einstein-Maxwell fields

International Nuclear Information System (INIS)

Kramer, D.; Neugebauer, G.

1981-01-01

The Einstein-Maxwell equations for stationary axisymmetric exterior fields are shown to be the integrability conditions of a set of linear eigenvalue equations for pseudopotentials. Using the method of Wahlquist and Estabrook (J. Math Phys.; 16:1 (1975)) it is shown that the prolongation structure of the Einstein-Maxwell equations contains the SU(2,1) Lie algebra. A new mapping of known solutions to other solutions has been found. (author)

Inequalities among partial traces of hermitian operators and partial sums of their eigenvalues

International Nuclear Information System (INIS)

Daboul, J.

1990-01-01

Two different proofs of the following inequality are given: Tr sup(k)(H):= sup(k)Σ sub(i=1) h sub(i) :sup(k)Σ sub(i=1)(X sub(i), Hx sub(i))≥ sup(k)Σ sub(i=1)E sub(i), for k = 1,-,N, where H is a Hermitian matrix, the {X sub(i), i = 1,2-,k } are any k orthonormal vectors and the e sub(i) are the eigenvalues of H, ordered according to increasing values. This result is a generalization of the well-known fact, that ground state of a Hamiltonian is given by its lowest eigenvalue, E sub(i). It can also be regarded as a generalization, for Hermitian operators, of the invariance of the trace under unitary transformation. A few consequences of the above result are also derived. (author)

On the numerical solution of coupled eigenvalue differential equations arising in molecular spectroscopy

International Nuclear Information System (INIS)

Friedman, R.S.; Jamieson, M.J.; Preston, S.C.

1990-01-01

A method for solving coupled eigenvalue differential equations is given and its relation to an existing technique is shown. Use of the Gram-Schmidt process to overcome the severe instabilities arising in molecular problems is described in detail. (orig.)

Some remarks on the optimization of eigenvalue problems involving the p-Laplacian

Directory of Open Access Journals (Sweden)

Wacław Pielichowski

2008-01-01

Full Text Available Given a bounded domain \\(\\Omega \\subset \\mathbb{R}^n\\, numbers \\(p \\gt 1\\, \\(\\alpha \\geq 0\\ and \\(A \\in [0,|\\Omega |]\\, consider the optimization problem: find a subset \\(D \\subset \\Omega \\, of measure \\(A\\, for which the first eigenvalue of the operator \\(u\\mapsto -\\text{div} (|\

Approximative analytic eigenvalues for orbital excitations in the case of a coulomb potential plus linear and quadratic radial terms

International Nuclear Information System (INIS)

Rekab, S.; Zenine, N.

2006-01-01

We consider the three dimensional non relativistic eigenvalue problem in the case of a Coulomb potential plus linear and quadratic radial terms. In the framework of the Rayleigh-Schrodinger Perturbation Theory, using a specific choice of the unperturbed Hamiltonian, we obtain approximate analytic expressions for the eigenvalues of orbital excitations. The implications and the range of validity of the obtained analytic expression are discussed

On the number of negative eigenvalues of the Laplacian on a metric graph

International Nuclear Information System (INIS)

Behrndt, Jussi; Luger, Annemarie

2010-01-01

The number of negative eigenvalues of self-adjoint Laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure. This extends and complements earlier results by Kostrykin and Schrader (2006 Contemp. Math. 415 201-25).

On the number of negative eigenvalues of the Laplacian on a metric graph

Energy Technology Data Exchange (ETDEWEB)

Behrndt, Jussi [Institut fuer Mathematik, MA 6-4, Technische Universitaet Berlin, Strasse des 17. Juni 136, 10623 Berlin (Germany); Luger, Annemarie, E-mail: behrndt@math.tu-berlin.d, E-mail: luger@maths.lth.s [Center for Mathematical Sciences, Lund Institute of Technology/Lund University, Box 118, SE-221 00 Lund (Sweden)

2010-11-26

The number of negative eigenvalues of self-adjoint Laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure. This extends and complements earlier results by Kostrykin and Schrader (2006 Contemp. Math. 415 201-25).

An algebraic sub-structuring method for large-scale eigenvalue calculation

International Nuclear Information System (INIS)

Yang, C.; Gao, W.; Bai, Z.; Li, X.; Lee, L.; Husbands, P.; Ng, E.

2004-01-01

We examine sub-structuring methods for solving large-scale generalized eigenvalue problems from a purely algebraic point of view. We use the term 'algebraic sub-structuring' to refer to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to provide approximate solutions to the original problem. We are interested in the question of which spectral components one should extract from each sub-structure in order to produce an approximate solution to the original problem with a desired level of accuracy. Error estimate for the approximation to the smallest eigenpair is developed. The estimate leads to a simple heuristic for choosing spectral components (modes) from each sub-structure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic sub-structuring can be effectively used to solve a generalized eigenvalue problem arising from the simulation of an accelerator structure. One interesting characteristic of this application is that the stiffness matrix produced by a hierarchical vector finite elements scheme contains a null space of large dimension. We present an efficient scheme to deflate this null space in the algebraic sub-structuring process

Parallel algorithms for 2-D cylindrical transport equations of Eigenvalue problem

International Nuclear Information System (INIS)

Wei, J.; Yang, S.

2013-01-01

In this paper, aimed at the neutron transport equations of eigenvalue problem under 2-D cylindrical geometry on unstructured grid, the discrete scheme of Sn discrete ordinate and discontinuous finite is built, and the parallel computation for the scheme is realized on MPI systems. Numerical experiments indicate that the designed parallel algorithm can reach perfect speedup, it has good practicality and scalability. (authors)

Lq-perturbations of leading coefficients of elliptic operators: Asymptotics of eigenvalues

Directory of Open Access Journals (Sweden)

Vladimir Kozlov

2006-01-01

Full Text Available We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L∞-norm.

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

KAUST Repository

Haidar, Azzam

2012-01-01

Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel\\'s Math Kernel Library.

Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Directory of Open Access Journals (Sweden)

Ruyun Ma

2018-01-01

Full Text Available Let T>2 be an integer and T={1,2,…,T}. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem -Δ2u(j-1+q(ju(j=λg(ju(j,  j∈T, u(0=u(T,  u(1=u(T+1, and we determine the sign of the corresponding eigenfunctions, where λ is a parameter, q(j≥0 and q(j≢0 in T, and the weight function g changes its sign in T. As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem.

On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue

OpenAIRE

Berg, M. van den; Ferone, V.; Nitsch, C.; Trombetti, C.

2016-01-01

Let $\\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\\Omega|$. We obtain some properties of the set function $F:\\Omega\\mapsto \\R^+$ defined by $$ F(\\Omega)=\\frac{T(\\Omega)\\lambda_1(\\Omega)}{|\\Omega|} ,$$ where $T(\\Omega)$ and $\\lambda_1(\\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\\'olya bound $F(\\Omega)\\le 1,$ and show that $$F(\\Omega)\\le 1- \

A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

Directory of Open Access Journals (Sweden)

Fatemeh Mohammad

2014-05-01

Full Text Available In this paper‎, ‎we represent an inexact inverse‎ ‎subspace iteration method for computing a few eigenpairs of the‎ ‎generalized eigenvalue problem $Ax = \\lambda Bx$[Q.~Ye and P.~Zhang‎, ‎Inexact inverse subspace iteration for generalized eigenvalue‎ ‎problems‎, ‎Linear Algebra and its Application‎, ‎434 (2011 1697-1715‎‎]‎. ‎In particular‎, ‎the linear convergence property of the inverse‎ ‎subspace iteration is preserved‎.

Density profiles of small Dirac operator eigenvalues for two color QCD at nonzero chemical potential compared to matrix models

OpenAIRE

Akemann, G; Bittner, E; Lombardo, M; Markum, H; Pullirsch, R

2004-01-01

We investigate the eigenvalue spectrum of the staggered Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis.

Density profiles of small Dirac operator eigenvalues for two color QCD at nonzero chemical potential compared to matrix models

International Nuclear Information System (INIS)

Akemann, Gernot; Bittner, Elmar; Lombardo, Maria-Paola; Markum, Harald; Pullirsch, Rainer

2005-01-01

We investigate the eigenvalue spectrum of the staggered Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis

Solving large nonlinear generalized eigenvalue problems from Density Functional Theory calculations in parallel

DEFF Research Database (Denmark)

Bendtsen, Claus; Nielsen, Ole Holm; Hansen, Lars Bruno

2001-01-01

The quantum mechanical ground state of electrons is described by Density Functional Theory, which leads to large minimization problems. An efficient minimization method uses a self-consistent field (SCF) solution of large eigenvalue problems. The iterative Davidson algorithm is often used, and we...

Coarse-mesh rebalancing acceleration for eigenvalue problems

International Nuclear Information System (INIS)

Asaoka, T.; Nakahara, Y.; Miyasaka, S.

1974-01-01

The coarse-mesh rebalance method is adopted for Monte Carlo schemes for aiming at accelerating the convergence of a source iteration process. At every completion of the Monte Carlo game for one batch of neutron histories, the scaling factor for the neutron flux is calculated to achieve the neutron balance in each coarse-mesh zone into which the total system is divided. This rebalance factor is multiplied to the weight of each fission source neutron in the coarse-mesh zone for playing the next Monte Carlo game. The numerical examples have shown that the coarse-mesh rebalance Monte Carlo calculation gives a good estimate of the eigenvalue already after several batches with a negligible extra computer time compared to the standard Monte Carlo. 5 references. (U.S.)

On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems

International Nuclear Information System (INIS)

Dambrine, M.; Kateb, D.

2011-01-01

We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivity σ 1 are layered in a body of conductivity σ 2 . We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173-184, 2009) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311-321, EDP Sci., Les Ulis, 2009).

Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

Directory of Open Access Journals (Sweden)

Serguei I. Iakovlev

2013-01-01

Full Text Available It is shown that any \\(\\mu \\in \\mathbb{C}\\ is an infinite multiplicity eigenvalue of the Steklov smoothing operator \\(S_h\\ acting on the space \\(L^1_{loc}(\\mathbb{R}\\. For \\(\\mu \

Eigenvalues calculation algorithms for {lambda}-modes determination. Parallelization approach

Energy Technology Data Exchange (ETDEWEB)

Vidal, V. [Universidad Politecnica de Valencia (Spain). Departamento de Sistemas Informaticos y Computacion; Verdu, G.; Munoz-Cobo, J.L. [Universidad Politecnica de Valencia (Spain). Departamento de Ingenieria Quimica y Nuclear; Ginestart, D. [Universidad Politecnica de Valencia (Spain). Departamento de Matematica Aplicada

1997-03-01

In this paper, we review two methods to obtain the {lambda}-modes of a nuclear reactor, Subspace Iteration method and Arnoldi`s method, which are popular methods to solve the partial eigenvalue problem for a given matrix. In the developed application for the neutron diffusion equation we include improved acceleration techniques for both methods. Also, we propose two parallelization approaches for these methods, a coarse grain parallelization and a fine grain one. We have tested the developed algorithms with two realistic problems, focusing on the efficiency of the methods according to the CPU times. (author).

A comparison of the eigenvalue equations in kappa, α, lambda and γ in reactor theory. Application to fast and thermal systems in unreflected and reflected configurations

International Nuclear Information System (INIS)

Velarde, G.; Ahnert, C.; Aragones, J.M.

1977-01-01

A comparative study of the eigenvalue transport in kappa, lambda, γ and α is made. The neutronic fluxes obtained by solving the transport equation in the four eigenvalue types are compared numerically for fast and thermal systems in unreflected and reflected configurations. Important conclusions will be obtained about the appropiate use of each eigenvalue depending on the calculation type to be performed. (author)

Dimensionality of social networks using motifs and eigenvalues.

Directory of Open Access Journals (Sweden)

Anthony Bonato

Full Text Available We consider the dimensionality of social networks, and develop experiments aimed at predicting that dimension. We find that a social network model with nodes and links sampled from an m-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when m scales logarithmically with the number of nodes of the network. This supports a logarithmic dimension hypothesis, and we provide evidence with two different social networks, Facebook and LinkedIn. Further, we employ two different methods for confirming the hypothesis: the first uses the distribution of motif counts, and the second exploits the eigenvalue distribution.

Effect of continuous eigenvalue spectrum on plasma transport in toroidal systems

International Nuclear Information System (INIS)

Yamagishi, Tomejiro

1993-03-01

The effect of the continuous eigenvalue of the Vlasov equation on the cross field ion thermal flux is investigated. The continuum contribution due to the toroidal drift resonance is found to play an important role in ion transport particularly near the edge, which may apply to the interpretation of the sharp increase of ion heat conductivity near the periphery observed in large tokamaks. (author)

Evaluation of upper and lower bounds to energy eigenvalues in Shoenberg's perturbation-theory ground state by means of partitioning technique

International Nuclear Information System (INIS)

Logrado, P.G.; Vianna, J.D.M.

Upper and lower bounds for the energy eigenvalues is Schoenberg's perturbation-theory ground state are studied. After a review of the characteristic features of the partitioning techniques the perturbative expansion proposed by Schoenberg is generated from an exact operator equation. The upper and lower bounds for the ground state eigenvalue are derived by using reaction and wave operators concepts, the bracketing function and operator inequalities. (Author) [pt

Fourth-order Perturbed Eigenvalue Equation for Stepwise Damage Detection of Aeroplane Wing

Directory of Open Access Journals (Sweden)

Wong Chun Nam

2016-01-01

Full Text Available Perturbed eigenvalue equations up to fourth-order are established to detect structural damage in aeroplane wing. Complete set of perturbation terms including orthogonal and non-orthogonal coefficients are computed using perturbed eigenvalue and orthonormal equations. Then the perturbed eigenparameters are optimized using BFGS approach. Finite element model with small to large stepwise damage is used to represent actual aeroplane wing. In small damaged level, termination number is the same for both approaches, while rms errors and termination d-norms are very close. For medium damaged level, termination number is larger for third-order perturbation with lower d-norm and smaller rms error. In large damaged level, termination number is much larger for third-order perturbation with same d-norm and larger rms error. These trends are more significant as the damaged level increases. As the stepwise damage effect increases with damage level, the increase in stepwise effect leads to the increase in model order. Hence, fourth-order perturbation is more accurate to estimate the model solution.

Heuristic geometric ''eigenvalue universality'' in a one-dimensional neutron transport problem with anisotropic scattering

International Nuclear Information System (INIS)

Goncalves, G.A.; Vilhena, M.T. de; Bodmann, B.E.J.

2010-01-01

In the present work we propose a heuristic construction of a transport equation for neutrons with anisotropic scattering considering only the radial cylinder dimension. The eigenvalues of the solutions of the equation correspond to the positive values for the one dimensional case. The central idea of the procedure is the application of the S N method for the discretisation of the angular variable followed by the application of the zero order Hankel transformation. The basis the construction of the scattering terms in form of an integro-differential equation for stationary transport resides in the hypothesis that the eigenvalues that compose the elementary solutions are independent of geometry for a homogeneous medium. We compare the solutions for the cartesian one dimensional problem for an infinite cylinder with azimuthal symmetry and linear anisotropic scattering for two cases. (orig.)

Three-dimensional multiple reciprocity boundary element method for one-group neutron diffusion eigenvalue computations

International Nuclear Information System (INIS)

Itagaki, Masafumi; Sahashi, Naoki.

1996-01-01

The multiple reciprocity method (MRM) in conjunction with the boundary element method has been employed to solve one-group eigenvalue problems described by the three-dimensional (3-D) neutron diffusion equation. The domain integral related to the fission source is transformed into a series of boundary-only integrals, with the aid of the higher order fundamental solutions based on the spherical and the modified spherical Bessel functions. Since each degree of the higher order fundamental solutions in the 3-D cases has a singularity of order (1/r), the above series of boundary integrals requires additional terms which do not appear in the 2-D MRM formulation. The critical eigenvalue itself can be also described using only boundary integrals. Test calculations show that Wielandt's spectral shift technique guarantees rapid and stable convergence of 3-D MRM computations. (author)

The Second Eigenvalue of the p-Laplacian as p Goes to 1

Directory of Open Access Journals (Sweden)

Enea Parini

2010-01-01

Full Text Available The asymptotic behaviour of the second eigenvalue of the p-Laplacian operator as p goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if p is close enough to 1.

Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem

KAUST Repository

Raman, Aaswath

2010-02-26

We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.

Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem

KAUST Repository

Raman, Aaswath; Fan, Shanhui

2010-01-01

We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.

Super-quantum curves from super-eigenvalue models

Energy Technology Data Exchange (ETDEWEB)

Ciosmak, Paweł [Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,ul. Banacha 2, 02-097 Warsaw (Poland); Hadasz, Leszek [M. Smoluchowski Institute of Physics, Jagiellonian University,ul. Łojasiewicza 11, 30-348 Kraków (Poland); Manabe, Masahide [Faculty of Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw (Poland); Sułkowski, Piotr [Faculty of Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw (Poland); Walter Burke Institute for Theoretical Physics, California Institute of Technology,1200 E. California Blvd, Pasadena, CA 91125 (United States)

2016-10-10

In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce β-deformed version of those models, and derive differential equations for associated α/β-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.

Super-quantum curves from super-eigenvalue models

International Nuclear Information System (INIS)

Ciosmak, Paweł; Hadasz, Leszek; Manabe, Masahide; Sułkowski, Piotr

2016-01-01

In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce β-deformed version of those models, and derive differential equations for associated α/β-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.

Super-quantum curves from super-eigenvalue models

Science.gov (United States)

Ciosmak, Paweł; Hadasz, Leszek; Manabe, Masahide; Sułkowski, Piotr

2016-10-01

In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce β-deformed version of those models, and derive differential equations for associated α/ β-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.

Algorithm-Eigenvalue Estimation of Hyperspectral Wishart Covariance Matrices from a Limited Number of Samples

Science.gov (United States)

2015-03-01

biometrics for physiological characteristics, hyperspectral remote sensing for detecting signals buried in noise and clutter, and medical genetics for...s); %approximate bounds on the eigenvalues used in Eq. (8) that are derived %from the Marcenko- Pastur law k=p./n; %band

Perturbative stability of the approximate Killing field eigenvalue problem

International Nuclear Information System (INIS)

Beetle, Christopher; Wilder, Shawn

2014-01-01

An approximate Killing field may be defined on a compact, Riemannian geometry by solving an eigenvalue problem for a certain elliptic operator. This paper studies the effect of small perturbations in the Riemannian metric on the resulting vector field. It shows that small metric perturbations, as measured using a Sobolev-type supremum norm on the space of Riemannian geometries on a fixed manifold, yield small perturbations in the approximate Killing field, as measured using a Hilbert-type square integral norm. It also discusses applications to the problem of computing the spin of a generic black hole in general relativity. (paper)

A parallel algorithm for the non-symmetric eigenvalue problem

International Nuclear Information System (INIS)

Sidani, M.M.

1991-01-01

An algorithm is presented for the solution of the non-symmetric eigenvalue problem. The algorithm is based on a divide-and-conquer procedure that provides initial approximations to the eigenpairs, which are then refined using Newton iterations. Since the smaller subproblems can be solved independently, and since Newton iterations with different initial guesses can be started simultaneously, the algorithm - unlike the standard QR method - is ideal for parallel computers. The author also reports on his investigation of deflation methods designed to obtain further eigenpairs if needed. Numerical results from implementations on a host of parallel machines (distributed and shared-memory) are presented

Correlation between eigenvalues and sorted diagonal matrix elements of a large dimensional matrix

International Nuclear Information System (INIS)

Arima, A.

2008-01-01

Functional dependences of eigenvalues as functions of sorted diagonal elements are given for realistic nuclear shell model (NSM) hamiltonian, the uniform distribution hamiltonian and the GOE hamiltonian. In the NSM case, the dependence is found to be linear. We discuss extrapolation methods for more accurate predictions for low-lying states. (author)

Some algorithms for the solution of the symmetric eigenvalue problem on a multiprocessor electronic computer

International Nuclear Information System (INIS)

Molchanov, I.N.; Khimich, A.N.

1984-01-01

This article shows how a reflection method can be used to find the eigenvalues of a matrix by transforming the matrix to tridiagonal form. The method of conjugate gradients is used to find the smallest eigenvalue and the corresponding eigenvector of symmetric positive-definite band matrices. Topics considered include the computational scheme of the reflection method, the organization of parallel calculations by the reflection method, the computational scheme of the conjugate gradient method, the organization of parallel calculations by the conjugate gradient method, and the effectiveness of parallel algorithms. It is concluded that it is possible to increase the overall effectiveness of the multiprocessor electronic computers by either letting the newly available processors of a new problem operate in the multiprocessor mode, or by improving the coefficient of uniform partition of the original information

Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator

International Nuclear Information System (INIS)

Macfarlane, A J; Pfeiffer, Hendryk

2003-01-01

The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is now well known for powers up to four. The paper describes an extension of this uniformity for the totally antisymmetrized nth powers up to n = 9, identifying families of representations with integer eigenvalues 5, ..., 9 for the quadratic Casimir operator, in each case providing a formula for the dimensions of the representations in the family as a function of D = dim g. This generalizes previous results for powers j and Casimir eigenvalues j, j ≤ 4. Many intriguing, perhaps puzzling, features of the dimension formulae are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered

The discontinuous finite element method for solving Eigenvalue problems of transport equations

International Nuclear Information System (INIS)

Yang, Shulin; Wang, Ruihong

2011-01-01

In this paper, the multigroup transport equations for solving the eigenvalues λ and K_e_f_f under two dimensional cylindrical coordinate are discussed. Aimed at the equations, the discretizing way combining discontinuous finite element method (DFE) with discrete ordinate method (SN) is developed, and the iterative algorithms and steps are studied. The numerical results show that the algorithms are efficient. (author)

Eigenvalue condition for the Weinberg angle and possible new leptons and quarks

International Nuclear Information System (INIS)

Ma, E.

1977-01-01

In a given SU(2) x U(1) gauge model of the weak and electro-magnetic interactions, if one assumes that an eigenvalue condition exists for the mixing (Weinberg) angle, then its value can be computed in lower-order perturbation theory. This idea is illustrated with several examples, including two which are in agreement with all the present available data. (auth.)

A note on the gap between the first two eigenvalues for the Schroedinger operator

International Nuclear Information System (INIS)

Benguria, R.

1986-01-01

By means of a commutation formula, the author gives a simple proof of the upper bound of Wong et al on the gap between the first two eigenvalues in the Schrodinger operator. Unfortunately this proof does not seem to generalise into higher dimensions. (author)

Extending the subspace hybrid method for eigenvalue problems in reactor physics calculation

International Nuclear Information System (INIS)

Zhang, Q.; Abdel-Khalik, H. S.

2013-01-01

This paper presents an innovative hybrid Monte-Carlo-Deterministic method denoted by the SUBSPACE method designed for improving the efficiency of hybrid methods for reactor analysis applications. The SUBSPACE method achieves its high computational efficiency by taking advantage of the existing correlations between desired responses. Recently, significant gains in computational efficiency have been demonstrated using this method for source driven problems. Within this work the mathematical theory behind the SUBSPACE method is introduced and extended to address core wide level k-eigenvalue problems. The method's efficiency is demonstrated based on a three-dimensional quarter-core problem, where responses are sought on the pin cell level. The SUBSPACE method is compared to the FW-CADIS method and is found to be more efficient for the utilized test problem because of the reason that the FW-CADIS method solves a forward eigenvalue problem and an adjoint fixed-source problem while the SUBSPACE method only solves an adjoint fixed-source problem. Based on the favorable results obtained here, we are confident that the applicability of Monte Carlo for large scale reactor analysis could be realized in the near future. (authors)

Efficient eigenvalue determination for arbitrary Pauli products based on generalized spin-spin interactions

Science.gov (United States)

Leibfried, D.; Wineland, D. J.

2018-03-01

Effective spin-spin interactions between ? qubits enable the determination of the eigenvalue of an arbitrary Pauli product of dimension N with a constant, small number of multi-qubit gates that is independent of N and encodes the eigenvalue in the measurement basis states of an extra ancilla qubit. Such interactions are available whenever qubits can be coupled to a shared harmonic oscillator, a situation that can be realized in many physical qubit implementations. For example, suitable interactions have already been realized for up to 14 qubits in ion traps. It should be possible to implement stabilizer codes for quantum error correction with a constant number of multi-qubit gates, in contrast to typical constructions with a number of two-qubit gates that increases as a function of N. The special case of finding the parity of N qubits only requires a small number of operations that is independent of N. This compares favorably to algorithms for computing the parity on conventional machines, which implies a genuine quantum advantage.

Nonlinear Eigenvalue Problems in Elliptic Variational Inequalities: a local study

International Nuclear Information System (INIS)

Conrad, F.; Brauner, C.; Issard-Roch, F.; Nicolaenko, B.

1985-01-01

The authors consider a class of Nonlinear Eigenvalue Problems (N.L.E.P.) associated with Elliptic Variational Inequalities (E.V.I.). First the authors introduce the main tools for a local study of branches of solutions; the authors extend the linearization process required in the case of equations. Next the authors prove the existence of arcs of solutions close to regular vs singular points, and determine their local behavior up to the first order. Finally, the authors discuss the connection between their regularity condition and some stability concept. 37 references, 6 figures

Numerical method for multigroup one-dimensional SN eigenvalue problems with no spatial truncation error

International Nuclear Information System (INIS)

Abreu, M.P.; Filho, H.A.; Barros, R.C.

1993-01-01

The authors describe a new nodal method for multigroup slab-geometry discrete ordinates S N eigenvalue problems that is completely free from all spatial truncation errors. The unknowns in the method are the node-edge angular fluxes, the node-average angular fluxes, and the effective multiplication factor k eff . The numerical values obtained for these quantities are exactly those of the dominant analytic solution of the S N eigenvalue problem apart from finite arithmetic considerations. This method is based on the use of the standard balance equation and two nonstandard auxiliary equations. In the nonmultiplying regions, e.g., the reflector, we use the multigroup spectral Green's function (SGF) auxiliary equations. In the fuel regions, we use the multigroup spectral diamond (SD) auxiliary equations. The SD auxiliary equation is an extension of the conventional auxiliary equation used in the diamond difference (DD) method. This hybrid characteristic of the SD-SGF method improves both the numerical stability and the convergence rate

Wielandt acceleration for MCNP5 Monte Carlo eigenvalue calculations

International Nuclear Information System (INIS)

Brown, F.

2007-01-01

Monte Carlo criticality calculations use the power iteration method to determine the eigenvalue (k eff ) and eigenfunction (fission source distribution) of the fundamental mode. A recently proposed method for accelerating convergence of the Monte Carlo power iteration using Wielandt's method has been implemented in a test version of MCNP5. The method is shown to provide dramatic improvements in convergence rates and to greatly reduce the possibility of false convergence assessment. The method is effective and efficient, improving the Monte Carlo figure-of-merit for many problems. In addition, the method should eliminate most of the underprediction bias in confidence intervals for Monte Carlo criticality calculations. (authors)

Variational methods for eigenvalue problems an introduction to the weinstein method of intermediate problems

CERN Document Server

Gould, S H

1966-01-01

The first edition of this book gave a systematic exposition of the Weinstein method of calculating lower bounds of eigenvalues by means of intermediate problems. This second edition presents new developments in the framework of the material contained in the first edition, which is retained in somewhat modified form.

Dependence of the fundamental time eigenvalue of linear transport operator on the system size and other parameters - An application of the Perron-Frobenius theorem

International Nuclear Information System (INIS)

Sahni, D.C.

1991-01-01

Many papers have been devoted to the study of the spectral properties of the linear (neutron) transport equation. Most of the theoretical investigations have concentrated on the existence (or otherwise) of a continuous spectrum, point spectrum, a leading/dominant eigenvalue, and a corresponding positive eigenvector. It is shown that the fundamental time eigenvalue of the linear transport operator increases with the size of the system. This follows from the increase in the largest eigenvalue of a non-negative irreducible matrix whenever any matrix element his increased. This result of matrix analysis is generalized to more general Krein-Rutman operators that leave a cone of vectors invariant

Critical eigenvalue in LMFBRs: a physics assessment

International Nuclear Information System (INIS)

McKnight, R.D.; Collins, P.J.; Olsen, D.N.

1984-01-01

This paper summarizes recent work to put the analysis of past critical eigenvalue measurements from the US critical experiments program on a consistent basis. The integral data base includes 53 configurations built in 11 ZPPR assemblies which simulate mixed oxide LMFBRs. Both conventional and heterogeneous designs representing 350, 700, and 900 MWe sizes and with and without simulated control rods and/or control rod positions have been studied. The review of the integral data base includes quantitative assessment of experimental uncertainties in the measured excess reactivity. Analyses have been done with design level and higher-order methods using ENDF/B-IV data. Comparisons of these analyses with the experiments are used to generate recommended bias factors for criticality predictions. Recommended methods for analysis of LMFBR fast critical assemblies and LMFBR design calculations are presented. Unresolved issues and areas which require additional experimental or analytical study are identified

Positive Eigenvalues of Generalized Words in Two Hermitian Positive Definite Matrices

OpenAIRE

Hillar, Christopher; Johnson, Charles R.

2005-01-01

We define a word in two positive definite (complex Hermitian) matrices $A$ and $B$ as a finite product of real powers of $A$ and $B$. The question of which words have only positive eigenvalues is addressed. This question was raised some time ago in connection with a long-standing problem in theoretical physics, and it was previously approached by the authors for words in two real positive definite matrices with positive integral exponents. A large class of words that do guarantee positive eig...

Estimates of the eigenvalues of operator arising in swelling pressure model

International Nuclear Information System (INIS)

Kanguzhin, Baltabek; Zhapsarbayeva, Lyailya

2016-01-01

Swelling pressures from materials confined by structures can cause structural deformations and instability. Due to the complexity of interactions between expansive solid and solid-liquid equilibrium, the forces exerting on retaining structures from swelling are highly nonlinear. This work is our initial attempt to study a simplistic spectral problem based on the Euler-elastic beam theory and some simplistic swelling pressure model. In this work estimates of the eigenvalues of some initial/boundary value problem for nonlinear Euler-elastic beam equation are obtained.

Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

NARCIS (Netherlands)

Jarlebring, E.; Hochstenbach, M.E.

2009-01-01

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called

Absolute Monotonicity of Functions Related To Estimates of First Eigenvalue of Laplace Operator on Riemannian Manifolds

Directory of Open Access Journals (Sweden)

Feng Qi

2014-10-01

Full Text Available The authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of Laplace operator on Riemannian manifolds.

Ground-state energies and highest occupied eigenvalues of atoms in exchange-only density-functional theory

Science.gov (United States)

Li, Yan; Harbola, Manoj K.; Krieger, J. B.; Sahni, Viraht

1989-11-01

The exchange-correlation potential of the Kohn-Sham density-functional theory has recently been interpreted as the work required to move an electron against the electric field of its Fermi-Coulomb hole charge distribution. In this paper we present self-consistent results for ground-state total energies and highest occupied eigenvalues of closed subshell atoms as obtained by this formalism in the exchange-only approximation. The total energies, which are an upper bound, lie within 50 ppm of Hartree-Fock theory for atoms heavier than Be. The highest occupied eigenvalues, as a consequence of this interpretation, approximate well the experimental ionization potentials. In addition, the self-consistently calculated exchange potentials are very close to those of Talman and co-workers [J. D. Talman and W. F. Shadwick, Phys. Rev. A 14, 36 (1976); K. Aashamar, T. M. Luke, and J. D. Talman, At. Data Nucl. Data Tables 22, 443 (1978)].

Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold.

Science.gov (United States)

Palacios, Jonathan; Yeh, Harry; Wang, Wenping; Zhang, Yue; Laramee, Robert S; Sharma, Ritesh; Schultz, Thomas; Zhang, Eugene

2016-03-01

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.

Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media

Directory of Open Access Journals (Sweden)

Vicenţiu RăDulescu

2005-06-01

Full Text Available We study nonlinear eigenvalue problems of the type −div(a(x∇u=g(λ,x,u in ℝN, where a(x is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality. A specific minimax method is developed without making use of Palais-Smale condition.

Convergence diagnostics for Eigenvalue problems with linear regression model

International Nuclear Information System (INIS)

Shi, Bo; Petrovic, Bojan

2011-01-01

Although the Monte Carlo method has been extensively used for criticality/Eigenvalue problems, a reliable, robust, and efficient convergence diagnostics method is still desired. Most methods are based on integral parameters (multiplication factor, entropy) and either condense the local distribution information into a single value (e.g., entropy) or even disregard it. We propose to employ the detailed cycle-by-cycle local flux evolution obtained by using mesh tally mechanism to assess the source and flux convergence. By applying a linear regression model to each individual mesh in a mesh tally for convergence diagnostics, a global convergence criterion can be obtained. We exemplify this method on two problems and obtain promising diagnostics results. (author)

Computation of dominant eigenvalues and eigenvectors: A comparative study of algorithms

International Nuclear Information System (INIS)

Nightingale, M.P.; Viswanath, V.S.; Mueller, G.

1993-01-01

We investigate two widely used recursive algorithms for the computation of eigenvectors with extreme eigenvalues of large symmetric matrices---the modified Lanczoes method and the conjugate-gradient method. The goal is to establish a connection between their underlying principles and to evaluate their performance in applications to Hamiltonian and transfer matrices of selected model systems of interest in condensed matter physics and statistical mechanics. The conjugate-gradient method is found to converge more rapidly for understandable reasons, while storage requirements are the same for both methods

An eigenvalue approach to quantum plasmonics based on a self-consistent hydrodynamics method.

Science.gov (United States)

Ding, Kun; Chan, C T

2018-02-28

Plasmonics has attracted much attention not only because it has useful properties such as strong field enhancement, but also because it reveals the quantum nature of matter. To handle quantum plasmonics effects, ab initio packages or empirical Feibelman d-parameters have been used to explore the quantum correction of plasmonic resonances. However, most of these methods are formulated within the quasi-static framework. The self-consistent hydrodynamics model offers a reliable approach to study quantum plasmonics because it can incorporate the quantum effect of the electron gas into classical electrodynamics in a consistent manner. Instead of the standard scattering method, we formulate the self-consistent hydrodynamics method as an eigenvalue problem to study quantum plasmonics with electrons and photons treated on the same footing. We find that the eigenvalue approach must involve a global operator, which originates from the energy functional of the electron gas. This manifests the intrinsic nonlocality of the response of quantum plasmonic resonances. Our model gives the analytical forms of quantum corrections to plasmonic modes, incorporating quantum electron spill-out effects and electrodynamical retardation. We apply our method to study the quantum surface plasmon polariton for a single flat interface.

Eigenvalue for Densely Defined Perturbations of Multivalued Maximal Monotone Operators in Reflexive Banach Spaces

Directory of Open Access Journals (Sweden)

Boubakari Ibrahimou

2013-01-01

maximal monotone with and . Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem where the operator is a single-valued of class . The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator is multivalued and is investigated.

Code development for eigenvalue total sensitivity analysis and total uncertainty analysis

International Nuclear Information System (INIS)

Wan, Chenghui; Cao, Liangzhi; Wu, Hongchun; Zu, Tiejun; Shen, Wei

2015-01-01

Highlights: • We develop a new code for total sensitivity and uncertainty analysis. • The implicit effects of cross sections can be considered. • The results of our code agree well with TSUNAMI-1D. • Detailed analysis for origins of implicit effects is performed. - Abstract: The uncertainties of multigroup cross sections notably impact eigenvalue of neutron-transport equation. We report on a total sensitivity analysis and total uncertainty analysis code named UNICORN that has been developed by applying the direct numerical perturbation method and statistical sampling method. In order to consider the contributions of various basic cross sections and the implicit effects which are indirect results of multigroup cross sections through resonance self-shielding calculation, an improved multigroup cross-section perturbation model is developed. The DRAGON 4.0 code, with application of WIMSD-4 format library, is used by UNICORN to carry out the resonance self-shielding and neutron-transport calculations. In addition, the bootstrap technique has been applied to the statistical sampling method in UNICORN to obtain much steadier and more reliable uncertainty results. The UNICORN code has been verified against TSUNAMI-1D by analyzing the case of TMI-1 pin-cell. The numerical results show that the total uncertainty of eigenvalue caused by cross sections can reach up to be about 0.72%. Therefore the contributions of the basic cross sections and their implicit effects are not negligible

Solving Eigenvalue response matrix equations with Jacobian-Free Newton-Krylov methods

International Nuclear Information System (INIS)

Roberts, Jeremy A.; Forget, Benoit

2011-01-01

The response matrix method for reactor eigenvalue problems is motivated as a technique for solving coarse mesh transport equations, and the classical approach of power iteration (PI) for solution is described. The method is then reformulated as a nonlinear system of equations, and the associated Jacobian is derived. A Jacobian-Free Newton-Krylov (JFNK) method is employed to solve the system, using an approximate Jacobian coupled with incomplete factorization as a preconditioner. The unpreconditioned JFNK slightly outperforms PI, and preconditioned JFNK outperforms both PI and Steffensen-accelerated PI significantly. (author)

Third-order WKBJ eigenvalues for Lennard-Jones and Varshni V potentials

International Nuclear Information System (INIS)

Kesarwani, R.N.; Varshni, Y.P.

1978-01-01

The WKBJ method is applied to the third order for obtaining the eigenvalues for the fifth potential of Varshni, and the relevant integrals are analytically evaluated. Numerical results are obtained for the Lennard-Jones Potential, which is a special case of the Varshni V potential, and are compared to the results of Harrison and Bernstein obtained by a numerical integration of the wave equation. Error estimates are made. It is shown that for diatomic potentials, the Langer correction is not needed if the WKBJ approximation is carried to second and higher orders. (author)

Computation of Double Eigenvalues for Infinite Matrices of a Certain Class

OpenAIRE

宮崎, 佳典; Yoshinori, MIYAZAKI; 静岡産業大学 国際情報学部; Faculty of Communications and Informatics, Shizuoka Sangyo University

2001-01-01

It has been shown that a series of three-term recurrence relations of a certain class is a powerful tool for solving zeros of some special functions and eigenvalue problems (EVPs) of certain differential equations. Such cases include: the zeros of J_v(z); the zeros of zJ′_v(z)+HJ_v(z); the EVP of the Mathieu differential equation; and the EVP of the spheroidal wave equation. Previously by the author, it was demonstrated that the three-term recurrence relations of the class may be reformulated...

Asymptotics for the number of negative eigenvalues of a model operator related to a system of three-particles on lattices

International Nuclear Information System (INIS)

Rasulov, T.H.

2009-04-01

A model operator H μ , μ > 0 associated to a system of three particles on the three-dimensional lattice Z 3 . We study the case where the parameter function w has a special form with the nondegenerate minimum at the n, n > 1 points. If the associated Friedrichs model has a zero energy resonance, then we prove that the operator H μ has infinitely many negative eigenvalues accumulating at zero. Moreover, we obtain an asymptotic value for the number of negative eigenvalues of H μ lying below z < 0 with respect to the spectral parameter z → -0. (author)

A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

Directory of Open Access Journals (Sweden)

Gift Muchatibaya

2018-01-01

Full Text Available The asymptotic and threshold behaviour of the eigenvalues of a perturbed difference operator inside a spectral gap is investigated. In particular, applications of the Titchmarsh-Weyl m-function theory as well as the Birman-Schwinger principle is performed to investigate the existence and behaviour of the eigenvalues of the operator H0+λWn inside the spectral gap of H0 in the limits λ↑∞ and λ↓0.

Method for the determination of the dominant eigenvalue of the neutron transport equation in a slab using fractional derivative

International Nuclear Information System (INIS)

Sperotto, Fabiola Aiub; Segatto, Cynthia Feijo; Zabadal, Jorge

2002-01-01

In this work, we determine the dominant eigenvalue of the one-dimensional neutron transport equation in a slab constructing an integral form for the neutron transport equation which is the expressed in terms of fractional derivative of the angular flux. Equating the fractional derivative of the angular flux to the integrate equation, we determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of Riemann-Liouville definition of fractional derivative. Once known the angular flux the dominant eigenvalue is calculated solving a transcendental equation resulting from the application of the boundary conditions. We report the methodology applied, for comparison with available results in literature. (author)

Leak detection of complex pipelines based on the filter diagonalization method: robust technique for eigenvalue assessment

International Nuclear Information System (INIS)

Lay-Ekuakille, Aimé; Pariset, Carlo; Trotta, Amerigo

2010-01-01

The FDM (filter diagonalization method), an interesting technique used in nuclear magnetic resonance data processing for tackling FFT (fast Fourier transform) limitations, can be used by considering pipelines, especially complex configurations, as a vascular apparatus with arteries, veins, capillaries, etc. Thrombosis, which might occur in humans, can be considered as a leakage for the complex pipeline, the human vascular apparatus. The choice of eigenvalues in FDM or in spectra-based techniques is a key issue in recovering the solution of the main equation (for FDM) or frequency domain transformation (for FFT) in order to determine the accuracy in detecting leaks in pipelines. This paper deals with the possibility of improving the leak detection accuracy of the FDM technique thanks to a robust algorithm by assessing the problem of eigenvalues, making it less experimental and more analytical using Tikhonov-based regularization techniques. The paper starts from the results of previous experimental procedures carried out by the authors

The Application of Vector Fitting to Eigenvalue-based Harmonic Stability Analysis

DEFF Research Database (Denmark)

Dowlatabadi, Mohammadkazem Bakhshizadeh; Yoon, Changwoo; Hjerrild, Jesper

2017-01-01

Participation factor analysis is an interesting feature of the eigenvalue-based stability analysis in a power system, which enables the developers to identify the problematic elements in a multi-vendor project like in an offshore wind power plant. However, this method needs a full state space model...... of the elements that is not always possible to have in a competitive world due to confidentiality. In this paper, by using an identification method, the state space models for power converters are extracted from the provided data by the suppliers. Some uncertainties in the identification process are also...

Instability of the cored barotropic disc: the linear eigenvalue formulation

Science.gov (United States)

Polyachenko, E. V.

2018-05-01

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured. The paper suggests a new method for finding the spiral patterns based on an expansion of small amplitude perturbations over Lagrange polynomials in small radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to a galactic model with the cored exponential density profile.

A Monte Carlo neutron transport code for eigenvalue calculations on a dual-GPU system and CUDA environment

Energy Technology Data Exchange (ETDEWEB)

Liu, T.; Ding, A.; Ji, W.; Xu, X. G. [Nuclear Engineering and Engineering Physics, Rensselaer Polytechnic Inst., Troy, NY 12180 (United States); Carothers, C. D. [Dept. of Computer Science, Rensselaer Polytechnic Inst. RPI (United States); Brown, F. B. [Los Alamos National Laboratory (LANL) (United States)

2012-07-01

Monte Carlo (MC) method is able to accurately calculate eigenvalues in reactor analysis. Its lengthy computation time can be reduced by general-purpose computing on Graphics Processing Units (GPU), one of the latest parallel computing techniques under development. The method of porting a regular transport code to GPU is usually very straightforward due to the 'embarrassingly parallel' nature of MC code. However, the situation becomes different for eigenvalue calculation in that it will be performed on a generation-by-generation basis and the thread coordination should be explicitly taken care of. This paper presents our effort to develop such a GPU-based MC code in Compute Unified Device Architecture (CUDA) environment. The code is able to perform eigenvalue calculation under simple geometries on a multi-GPU system. The specifics of algorithm design, including thread organization and memory management were described in detail. The original CPU version of the code was tested on an Intel Xeon X5660 2.8 GHz CPU, and the adapted GPU version was tested on NVIDIA Tesla M2090 GPUs. Double-precision floating point format was used throughout the calculation. The result showed that a speedup of 7.0 and 33.3 were obtained for a bare spherical core and a binary slab system respectively. The speedup factor was further increased by a factor of {approx}2 on a dual GPU system. The upper limit of device-level parallelism was analyzed, and a possible method to enhance the thread-level parallelism was proposed. (authors)

Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

Science.gov (United States)

Ogilvie, Karen; Olde Daalhuis, Adri B.

2015-11-01

By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct.

A Monte Carlo neutron transport code for eigenvalue calculations on a dual-GPU system and CUDA environment

International Nuclear Information System (INIS)

Liu, T.; Ding, A.; Ji, W.; Xu, X. G.; Carothers, C. D.; Brown, F. B.

2012-01-01

Monte Carlo (MC) method is able to accurately calculate eigenvalues in reactor analysis. Its lengthy computation time can be reduced by general-purpose computing on Graphics Processing Units (GPU), one of the latest parallel computing techniques under development. The method of porting a regular transport code to GPU is usually very straightforward due to the 'embarrassingly parallel' nature of MC code. However, the situation becomes different for eigenvalue calculation in that it will be performed on a generation-by-generation basis and the thread coordination should be explicitly taken care of. This paper presents our effort to develop such a GPU-based MC code in Compute Unified Device Architecture (CUDA) environment. The code is able to perform eigenvalue calculation under simple geometries on a multi-GPU system. The specifics of algorithm design, including thread organization and memory management were described in detail. The original CPU version of the code was tested on an Intel Xeon X5660 2.8 GHz CPU, and the adapted GPU version was tested on NVIDIA Tesla M2090 GPUs. Double-precision floating point format was used throughout the calculation. The result showed that a speedup of 7.0 and 33.3 were obtained for a bare spherical core and a binary slab system respectively. The speedup factor was further increased by a factor of ∼2 on a dual GPU system. The upper limit of device-level parallelism was analyzed, and a possible method to enhance the thread-level parallelism was proposed. (authors)

Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

Science.gov (United States)

Huang, Tsung-Ming; Lin, Wen-Wei; Tian, Heng; Chen, Guan-Hua

2018-03-01

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 × 4000 at the density functional tight binding level, corresponding to a 8 × 8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction-diffusion systems with unilateral terms

Czech Academy of Sciences Publication Activity Database

Kučera, Milan; Navrátil, J.

2018-01-01

Roč. 166, January (2018), s. 154-180 ISSN 0362-546X Institutional support: RVO:67985840 Keywords : global bifurcation * maximal eigenvalue * positively homogeneous operators Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.192, year: 2016 http://www.sciencedirect.com/science/article/pii/S0362546X17302559?via%3Dihub

Perturbation of embedded eigenvalue by a near-lying resonance

Energy Technology Data Exchange (ETDEWEB)

Belyaev, V B; Motovilov, A K

1997-12-31

The case of quantum-mechanical system (including electronic molecules) is considered where Hamiltonian allows a separation, in particular by the Faddeev method, of a weakly coupled channel. Width (i.e. the imaginary part) of the resonance generated by a discrete spectrum eigenvalue of the separated channel is studied in the case where main part of the Hamiltonian gives itself another resonance. It is shown that if real parts of these resonances coincide and, at the same time, a coupling between the separated and main channels is sufficiently small then the width of the resonance generated by the separated (molecular) channel is inversely proportional to the width of the main (nuclear) channel resonance. This phenomenon being a kind of universal law, may play an important role increasing the `cold fusion` probability in electronic molecules whose nuclear constituents have narrow pre-threshold resonances. 21 refs.

Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem

Directory of Open Access Journals (Sweden)

Xuqing Zhang

2013-01-01

Full Text Available This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001, a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.

Asymmetric modes and complex time eigenvalues of the one-speed neutron transport equation in a homogeneous sphere

International Nuclear Information System (INIS)

Paranjape, S.D.; Kumar, V.; Sahni, D.C.

1993-01-01

The one-speed, time-dependent, isotropically scattering, integral transport equation in a homogeneous sphere has been converted into a criticality-like problem by considering exponential time behaviour of the scalar flux. This criticality problem has been converted into a matrix eigenvalue problem using the Fourier transform technique. The time eigenvalues λ, which are complex in general, have been determined for spherically symmetric as well as asymmetric modes. For the former case, the real decay constants and the real parts of complex decay constants decrease monotonically with increasing system size and form two distinct families of single-valued functions. For the spherically asymmetric modes, certain new features emerge. The real decay constants are found to be multi-valued functions of system size and they do not always decrease monotonically with increasing system size. As the system size increases from zero onwards, the decay constants alternate between complex and real values and the real and complex decay constant curves interlace. (Author)

Equi-frequency contour of photonic crystals with the extended Dirichlet-to-Neumann wave vector eigenvalue equation method

International Nuclear Information System (INIS)

Jiang Bin; Zhang Yejing; Wang Yufei; Liu Anjin; Zheng Wanhua

2012-01-01

We present the extended Dirichlet-to-Neumann wave vector eigenvalue equation (DtN-WVEE) method to calculate the equi-frequency contour (EFC) of square lattice photonic crystals (PhCs). With the extended DtN-WVEE method and Snell's law, the effective refractive index of the mode with a circular EFC can be obtained, which is further validated with the refractive index weighted by the electric field or magnetic field. To further verify the EFC calculated by the DtN-WVEE method, the finite-difference time-domain method is also used. Compared with other wave vector eigenvalue equation methods that calculate EFC directly, the size of the eigenmatrix used in the DtN-WVEE method is much smaller, and the computation time is significantly reduced. Since the DtN-WVEE method solves wave vectors for given arbitrary frequencies, it can also find applications in studying the optical properties of a PhC with dispersive, lossy and magnetic materials. (paper)

The eigenvalues of the SN transport matrix

International Nuclear Information System (INIS)

Ourique, L.E.; Vilhena, M.T. de

2005-01-01

In a recent paper, we analyze the dependence of the eigenvalues of the S N matrix transport, associated with the system of linear differential equations that corresponds to the S N approximations of the transport equation [1]. By considering a control parameter, we have shown that there exist some bifurcation points. This means that the solutions of S N approximations change from oscillatory to non-oscillatory behavior, a different approach of the study by [2]. Nowadays, the one-dimensional transport equation and related problems have been a source of new techniques for solving particular cases as well the development of analytical methods that search aspects of existence and uniqueness of the solutions [3], [4]. In this work, we generalize the results shown in [1], searching for a model of the distribution of the bifurcation points of the S N matrix transport, studying the one-dimensional case in a slab, with anisotropic differential cross section of order 3. The result indicates that the bifurcation points obey a certain rule of distribution. Beside that, the condition number of the matrix transport increases too much in the neighborhood of these points, as we have seen in [1]. (author)

Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction-diffusion systems with unilateral terms

Czech Academy of Sciences Publication Activity Database

Kučera, Milan; Navrátil, J.

2018-01-01

Roč. 166, January (2018), s. 154-180 ISSN 0362-546X Institutional support: RVO:67985840 Keywords : global bifurcation * maximal eigenvalue * positively homogeneous operators Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.192, year: 2016 http://www. science direct.com/ science /article/pii/S0362546X17302559?via%3Dihub

Eigenvalues and eigenvectors of the translation matrices of spherical waves of multiple-scattering theory

International Nuclear Information System (INIS)

Torrini, M.

1983-01-01

The exponential nature of the translation matrix G of spherical free waves has been set forth in a previous paper.The explicit expression of the exponential form of the translation matrix is given here, once the eigenvectros and the eigenvalues of G have been found. In addition, the eigenproblem relative to the matrix which transforms outgoing waves scattered by a centre in a set of spherical free waves centered at a different point is solved

Eigenvalues of Casimir operators for the general linear, the special linear, and the orthosymplectic Lie superalgebras

International Nuclear Information System (INIS)

Scheunert, M.

1982-10-01

The generators of the algebras under consideration can be written in a canonical two-index form and hence the associated standard seuqence of Casimir elements can be constructed. Following the classical approach by Perelomov and Popov, we obtain the eigenvalues of these Casimir elements in an arbitrary highest weight module by calculating the corresponding generating functions. (orig.)

Computing eigenvalue sensitivity coefficients to nuclear data based on the CLUTCH method with RMC code

International Nuclear Information System (INIS)

Qiu, Yishu; She, Ding; Tang, Xiao; Wang, Kan; Liang, Jingang

2016-01-01

Highlights: • A new algorithm is proposed to reduce memory consumption for sensitivity analysis. • The fission matrix method is used to generate adjoint fission source distributions. • Sensitivity analysis is performed on a detailed 3D full-core benchmark with RMC. - Abstract: Recently, there is a need to develop advanced methods of computing eigenvalue sensitivity coefficients to nuclear data in the continuous-energy Monte Carlo codes. One of these methods is the iterated fission probability (IFP) method, which is adopted by most of Monte Carlo codes of having the capabilities of computing sensitivity coefficients, including the Reactor Monte Carlo code RMC. Though it is accurate theoretically, the IFP method faces the challenge of huge memory consumption. Therefore, it may sometimes produce poor sensitivity coefficients since the number of particles in each active cycle is not sufficient enough due to the limitation of computer memory capacity. In this work, two algorithms of the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Tracklength importance CHaracterization (CLUTCH) method, namely, the collision-event-based algorithm (C-CLUTCH) which is also implemented in SCALE and the fission-event-based algorithm (F-CLUTCH) which is put forward in this work, are investigated and implemented in RMC to reduce memory requirements for computing eigenvalue sensitivity coefficients. While the C-CLUTCH algorithm requires to store concerning reaction rates of every collision, the F-CLUTCH algorithm only stores concerning reaction rates of every fission point. In addition, the fission matrix method is put forward to generate the adjoint fission source distribution for the CLUTCH method to compute sensitivity coefficients. These newly proposed approaches implemented in RMC code are verified by a SF96 lattice model and the MIT BEAVRS benchmark problem. The numerical results indicate the accuracy of the F-CLUTCH algorithm is the same as the C

Application of zero eigenvalue for solving the potential, heat, and wave equations using a sequence of special functions

Directory of Open Access Journals (Sweden)

2006-01-01

Full Text Available In the solution of boundary value problems, usually zero eigenvalue is ignored. This case also happens in calculating the eigenvalues of matrices, so that we would often like to find the nonzero solutions of the linear system A X = λ X when λ ≠ 0 . But λ = 0 implies that det A = 0 for X ≠ 0 and then the rank of matrix A is reduced at least one degree. This comment can similarly be stated for boundary value problems. In other words, if at least one of the eigens of equations related to the main problem is considered zero, then one of the solutions will be specified in advance. By using this note, first we study a class of special functions and then apply it for the potential, heat, and wave equations in spherical coordinate. In this way, some practical examples are also given.

Analysis of structural correlations in a model binary 3D liquid through the eigenvalues and eigenvectors of the atomic stress tensors

International Nuclear Information System (INIS)

Levashov, V. A.

2016-01-01

It is possible to associate with every atom or molecule in a liquid its own atomic stress tensor. These atomic stress tensors can be used to describe liquids’ structures and to investigate the connection between structural and dynamic properties. In particular, atomic stresses allow to address atomic scale correlations relevant to the Green-Kubo expression for viscosity. Previously correlations between the atomic stresses of different atoms were studied using the Cartesian representation of the stress tensors or the representation based on spherical harmonics. In this paper we address structural correlations in a 3D model binary liquid using the eigenvalues and eigenvectors of the atomic stress tensors. This approach allows to interpret correlations relevant to the Green-Kubo expression for viscosity in a simple geometric way. On decrease of temperature the changes in the relevant stress correlation function between different atoms are significantly more pronounced than the changes in the pair density function. We demonstrate that this behaviour originates from the orientational correlations between the eigenvectors of the atomic stress tensors. We also found correlations between the eigenvalues of the same atomic stress tensor. For the studied system, with purely repulsive interactions between the particles, the eigenvalues of every atomic stress tensor are positive and they can be ordered: λ 1 ≥ λ 2 ≥ λ 3 ≥ 0. We found that, for the particles of a given type, the probability distributions of the ratios (λ 2 /λ 1 ) and (λ 3 /λ 2 ) are essentially identical to each other in the liquids state. We also found that λ 2 tends to be equal to the geometric average of λ 1 and λ 3 . In our view, correlations between the eigenvalues may represent “the Poisson ratio effect” at the atomic scale.

Analysis of structural correlations in a model binary 3D liquid through the eigenvalues and eigenvectors of the atomic stress tensors

Energy Technology Data Exchange (ETDEWEB)

Levashov, V. A. [Technological Design Institute of Scientific Instrument Engineering, Novosibirsk 630058 (Russian Federation)

2016-03-07

It is possible to associate with every atom or molecule in a liquid its own atomic stress tensor. These atomic stress tensors can be used to describe liquids’ structures and to investigate the connection between structural and dynamic properties. In particular, atomic stresses allow to address atomic scale correlations relevant to the Green-Kubo expression for viscosity. Previously correlations between the atomic stresses of different atoms were studied using the Cartesian representation of the stress tensors or the representation based on spherical harmonics. In this paper we address structural correlations in a 3D model binary liquid using the eigenvalues and eigenvectors of the atomic stress tensors. This approach allows to interpret correlations relevant to the Green-Kubo expression for viscosity in a simple geometric way. On decrease of temperature the changes in the relevant stress correlation function between different atoms are significantly more pronounced than the changes in the pair density function. We demonstrate that this behaviour originates from the orientational correlations between the eigenvectors of the atomic stress tensors. We also found correlations between the eigenvalues of the same atomic stress tensor. For the studied system, with purely repulsive interactions between the particles, the eigenvalues of every atomic stress tensor are positive and they can be ordered: λ{sub 1} ≥ λ{sub 2} ≥ λ{sub 3} ≥ 0. We found that, for the particles of a given type, the probability distributions of the ratios (λ{sub 2}/λ{sub 1}) and (λ{sub 3}/λ{sub 2}) are essentially identical to each other in the liquids state. We also found that λ{sub 2} tends to be equal to the geometric average of λ{sub 1} and λ{sub 3}. In our view, correlations between the eigenvalues may represent “the Poisson ratio effect” at the atomic scale.

A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction Part II : analysis of convergence

NARCIS (Netherlands)

Bourne, D.P.; Elman, H.; Osborn, J.E.

2009-01-01

This paper is the second part of a two-part paper treating a non-self-adjoint quadratic eigenvalue problem for the linear stability of solutions to the Taylor-Couette problem for flow of a viscous liquid in a deformable cylinder, with the cylinder modelled as a membrane. The first part formulated

POSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT EIGENVALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS

Directory of Open Access Journals (Sweden)

FAOUZI HADDOUCHI

2015-11-01

Full Text Available In this paper, we study the existence of positive solutions of a three-point integral boundary value problem (BVP for the following second-order differential equation u''(t + \\lambda a(tf(u(t = 0; 0 0 is a parameter, 0 <\\eta < 1, 0 <\\alpha < 1/{\\eta}. . By using the properties of the Green's function and Krasnoselskii's fixed point theorem on cones, the eigenvalue intervals of the nonlinear boundary value problem are considered, some sufficient conditions for the existence of at least one positive solutions are established.

Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

KAUST Repository

Bonito, Andrea; Guermond, Jean-Luc

2011-01-01

We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space H-α with α ∈ (1/2, 1). The method is shown to be convergent and spectrally correct. © 2011 American Mathematical Society.

Experimental Validation of Model Updating and Damage Detection via Eigenvalue Sensitivity Methods with Artificial Boundary Conditions

Science.gov (United States)

2017-09-01

features can be evaluated such a welds and bolted connections. With these situations investigated, test articles with more complex geometry can be used...become more capable. Despite these advancements, the construction of physical prototypes remains an essential aspect of design and testing . FEM...prototype, there may be design deficiencies that cannot be identified until the prototype is tested . Using eigenvalue sensitivities, enhanced by

The strategy of spectral shifts and the sets of correct methods for calculating eigenvalues of general tridiagonal matrices

International Nuclear Information System (INIS)

Emel'yanenko, G.A.; Sek, I.E.

1988-01-01

Many correctable unknown methods for eigenvalue calculation of general tridiagonal matrices with real elements; criteria of singular tridiagonal matrices; necessary and sufficient conditions of tridiagonal matrix degeneracy; process with boundary conditions according to calculation processes of general upper and lower tridiagonal matrix minors are obtained. 6 refs

Computations of zeros of special functions and eigenvalues of differential equations by matrix method

OpenAIRE

Miyazaki, Yoshinori

2000-01-01

This paper is strongly based on two powerful general theorems proved by Ikebe, et. al in 1993[15] and 1996[13], which will be referred to as Theorem A and Theorem B in this paper. They were recently published and justify the approximate computations of simple eigenvalues of infinite matrices of certain types by truncation, giving an extremely accurate error estimates. So far, they have applied to some important problems in engineering, such as computing the zeros of some special functions, an...

Effective Perron-Frobenius eigenvalue for a correlated random map

Science.gov (United States)

Pool, Roman R.; Cáceres, Manuel O.

2010-09-01

We investigate the evolution of random positive linear maps with various type of disorder by analytic perturbation and direct simulation. Our theoretical result indicates that the statistics of a random linear map can be successfully described for long time by the mean-value vector state. The growth rate can be characterized by an effective Perron-Frobenius eigenvalue that strongly depends on the type of correlation between the elements of the projection matrix. We apply this approach to an age-structured population dynamics model. We show that the asymptotic mean-value vector state characterizes the population growth rate when the age-structured model has random vital parameters. In this case our approach reveals the nontrivial dependence of the effective growth rate with cross correlations. The problem was reduced to the calculation of the smallest positive root of a secular polynomial, which can be obtained by perturbations in terms of Green’s function diagrammatic technique built with noncommutative cumulants for arbitrary n -point correlations.

A multilevel, level-set method for optimizing eigenvalues in shape design problems

International Nuclear Information System (INIS)

Haber, E.

2004-01-01

In this paper, we consider optimal design problems that involve shape optimization. The goal is to determine the shape of a certain structure such that it is either as rigid or as soft as possible. To achieve this goal we combine two new ideas for an efficient solution of the problem. First, we replace the eigenvalue problem with an approximation by using inverse iteration. Second, we use a level set method but rather than propagating the front we use constrained optimization methods combined with multilevel continuation techniques. Combining these two ideas we obtain a robust and rapid method for the solution of the optimal design problem

The numerical analysis of eigenvalue problem solutions in the multigroup diffusion theory

International Nuclear Information System (INIS)

Woznick, Z.I.

1994-01-01

In this paper a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations is described. Usually the solution method is based on the system of inner and outer iterations. The presented matrix formalism allows us to visualize clearly, how the used matrix splitting influences the structure of the matrix in an eigenvalue problem to be solved as well as the independence between inner and outer iterations within global iterations. To keep the page limit, the present version of the paper consists only with first three of five sections given in the original paper under the same title (which will be published soon). (author). 13 refs

Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model

Energy Technology Data Exchange (ETDEWEB)

Shubov, Marianna A., E-mail: marianna.shubov@gmail.com [University of New Hampshire, Department of Mathematics and Statistics (United States)

2016-06-15

We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff’s law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.

Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model

International Nuclear Information System (INIS)

Shubov, Marianna A.

2016-01-01

We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff’s law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.

On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values

International Nuclear Information System (INIS)

Wei Yi; Fyodorov, Yan V

2008-01-01

Given any fixed N x N positive semi-definite diagonal matrix G ≥ 0 we derive the explicit formula for the density of complex eigenvalues for random matrices A of the form A=U√G where the random unitary matrices U are distributed on the group U(N) according to the Haar measure. (fast track communication)

A numerical method for eigenvalue problems in modeling liquid crystals

Energy Technology Data Exchange (ETDEWEB)

Baglama, J.; Farrell, P.A.; Reichel, L.; Ruttan, A. [Kent State Univ., OH (United States); Calvetti, D. [Stevens Inst. of Technology, Hoboken, NJ (United States)

1996-12-31

Equilibrium configurations of liquid crystals in finite containments are minimizers of the thermodynamic free energy of the system. It is important to be able to track the equilibrium configurations as the temperature of the liquid crystals decreases. The path of the minimal energy configuration at bifurcation points can be computed from the null space of a large sparse symmetric matrix. We describe a new variant of the implicitly restarted Lanczos method that is well suited for the computation of extreme eigenvalues of a large sparse symmetric matrix, and we use this method to determine the desired null space. Our implicitly restarted Lanczos method determines adoptively a polynomial filter by using Leja shifts, and does not require factorization of the matrix. The storage requirement of the method is small, and this makes it attractive to use for the present application.

No Eigenvalues Outside the Limiting Support of Generally Correlated Gaussian Matrices

KAUST Repository

Kammoun, Abla; Alouini, Mohamed-Slim

2016-01-01

This paper investigates the behaviour of the spectrum of generally correlated Gaussian random matrices whose columns are zero-mean independent vectors but have different correlations, under the specific regime where the number of their columns and that of their rows grow at infinity with the same pace. Following the approach proposed in [1], we prove that under some mild conditions, there is no eigenvalue outside the limiting support of generally correlated Gaussian matrices. As an outcome of this result, we establish that the smallest singular value of these matrices is almost surely greater than zero. From a practical perspective, this control of the smallest singular value is paramount to applications from statistical signal processing and wireless communication, in which this kind of matrices naturally arise.

No Eigenvalues Outside the Limiting Support of Generally Correlated Gaussian Matrices

KAUST Repository

Kammoun, Abla

2016-05-04

This paper investigates the behaviour of the spectrum of generally correlated Gaussian random matrices whose columns are zero-mean independent vectors but have different correlations, under the specific regime where the number of their columns and that of their rows grow at infinity with the same pace. Following the approach proposed in [1], we prove that under some mild conditions, there is no eigenvalue outside the limiting support of generally correlated Gaussian matrices. As an outcome of this result, we establish that the smallest singular value of these matrices is almost surely greater than zero. From a practical perspective, this control of the smallest singular value is paramount to applications from statistical signal processing and wireless communication, in which this kind of matrices naturally arise.

Movable geometry and eigenvalue search capability in the MC21 Monte Carlo code

International Nuclear Information System (INIS)

Gill, D. F.; Nease, B. R.; Griesheimer, D. P.

2013-01-01

A description of a robust and flexible movable geometry implementation in the Monte Carlo code MC21 is described along with a search algorithm that can be used in conjunction with the movable geometry capability to perform eigenvalue searches based on the position of some geometric component. The natural use of the combined movement and search capability is searching to critical through variation of control rod (or control drum) position. The movable geometry discussion provides the mathematical framework for moving surfaces in the MC21 combinatorial solid geometry description. A discussion of the interface between the movable geometry system and the user is also described, particularly the ability to create a hierarchy of movable groups. Combined with the hierarchical geometry description in MC21 the movable group framework provides a very powerful system for inline geometry modification. The eigenvalue search algorithm implemented in MC21 is also described. The foundations of this algorithm are a regula falsi search though several considerations are made in an effort to increase the efficiency of the algorithm for use with Monte Carlo. Specifically, criteria are developed to determine after each batch whether the Monte Carlo calculation should be continued, the search iteration can be rejected, or the search iteration has converged. These criteria seek to minimize the amount of time spent per iteration. Results for the regula falsi method are shown, illustrating that the method as implemented is indeed convergent and that the optimizations made ultimately reduce the total computational expense. (authors)

Movable geometry and eigenvalue search capability in the MC21 Monte Carlo code

Energy Technology Data Exchange (ETDEWEB)

Gill, D. F.; Nease, B. R.; Griesheimer, D. P. [Bettis Atomic Power Laboratory, PO Box 79, West Mifflin, PA 15122 (United States)

2013-07-01

A description of a robust and flexible movable geometry implementation in the Monte Carlo code MC21 is described along with a search algorithm that can be used in conjunction with the movable geometry capability to perform eigenvalue searches based on the position of some geometric component. The natural use of the combined movement and search capability is searching to critical through variation of control rod (or control drum) position. The movable geometry discussion provides the mathematical framework for moving surfaces in the MC21 combinatorial solid geometry description. A discussion of the interface between the movable geometry system and the user is also described, particularly the ability to create a hierarchy of movable groups. Combined with the hierarchical geometry description in MC21 the movable group framework provides a very powerful system for inline geometry modification. The eigenvalue search algorithm implemented in MC21 is also described. The foundations of this algorithm are a regula falsi search though several considerations are made in an effort to increase the efficiency of the algorithm for use with Monte Carlo. Specifically, criteria are developed to determine after each batch whether the Monte Carlo calculation should be continued, the search iteration can be rejected, or the search iteration has converged. These criteria seek to minimize the amount of time spent per iteration. Results for the regula falsi method are shown, illustrating that the method as implemented is indeed convergent and that the optimizations made ultimately reduce the total computational expense. (authors)

Quantum single oscillator with a (mod X) sup(#betta#) - type potential : energy eigenvalues and specific heat

International Nuclear Information System (INIS)

Mariz, A.M.; Rio Grande do Norte Univ., Natal; Tsallis, C.

1982-01-01

The quantum single one-dimensional oscillator associated with a potential proportional to /X/ sup(#betta#) (#betta# > 0) is discussed. The exact energy eigenvalues recently established by Turschner are further elaborated and convenient exact as well as asymptotic relations are exhibited. The exact T → 0 and T → infinite specific heat is discussed and numerical results for typical values of #betta# and intermediate temperature are presented. (Author) [pt

Employing Eigenvalue Ratios to Generate Prior Fracture-like Features for Stochastic Hydrogeophysical Characterization of a Fractured Aquifer System

Science.gov (United States)

Brewster, J.; Oware, E. K.

2017-12-01

Groundwater hosted in fractured rocks constitutes almost 65% of the principal aquifers in the US. The exploitation and contaminant management of fractured aquifers require fracture flow and transport modeling, which in turn requires a detailed understanding of the structure of the aquifer. The widely used equivalent porous medium approach to modeling fractured aquifer systems is inadequate to accurately predict fracture transport processes due to the averaging of the sharp lithological contrast between the matrix and the fractures. The potential of geophysical imaging (GI) to estimate spatially continuous subsurface profiles in a minimally invasive fashion is well proven. Conventional deterministic GI strategies, however, produce geologically unrealistic, smoothed-out results due to commonly enforced smoothing constraints. Stochastic GI of fractured aquifers is becoming increasing appealing due to its ability to recover realistic fracture features while providing multiple likely realizations that enable uncertainty assessment. Generating prior spatial features consistent with the expected target structures is crucial in stochastic imaging. We propose to utilize eigenvalue ratios to resolve the elongated fracture features expected in a fractured aquifer system. Eigenvalues capture the major and minor directions of variability in a region, which can be employed to evaluate shape descriptors, such as eccentricity (elongation) and orientation of features in the region. Eccentricity ranges from zero to one, representing a circularly sharped to a line feature, respectively. Here, we apply eigenvalue ratios to define a joint objective parameter consisting of eccentricity (shape) and direction terms to guide the generation of prior fracture-like features in some predefined principal directions for stochastic GI. Preliminary unconditional, synthetic experiments reveal the potential of the algorithm to simulate prior fracture-like features. We illustrate the strategy with a

The eigenvalue problem. Alpha, lambda and gamma modes and its applications

International Nuclear Information System (INIS)

Carreno, A.; Vidal-Ferrandiz, A.; Verdu, G.; Ginestar, D.

2017-01-01

Modal analysis has been efficiently used to study different problems in reactor physics. In this sense, several eigenvalue problems can be defined for neutron transport equation: the λ-modes, the γ-modes and the α-modes. However, for simplicity, the neutron diffusion equation is used as approximation of each one of these equations that they have been discretized by a high order finite elements. The obtained algebraic eigenproblems are large problems and have to be solved using iterative methods. In this work, we analyze two methods. The first one is the Krylov-Schur method and the second one is the modified block Newton method. The comparison of modes and the performance of these methods have been studied in two benchmark problems, a homogeneous 3D reactor and the 3D Langenbuch reactor. (author)

Overview of the ArbiTER edge plasma eigenvalue code

Science.gov (United States)

Baver, Derek; Myra, James; Umansky, Maxim

2011-10-01

The Arbitrary Topology Equation Reader, or ArbiTER, is a flexible eigenvalue solver that is currently under development for plasma physics applications. The ArbiTER code builds on the equation parser framework of the existing 2DX code, extending it to include a topology parser. This will give the code the capability to model problems with complicated geometries (such as multiple X-points and scrape-off layers) or model equations with arbitrary numbers of dimensions (e.g. for kinetic analysis). In the equation parser framework, model equations are not included in the program's source code. Instead, an input file contains instructions for building a matrix from profile functions and elementary differential operators. The program then executes these instructions in a sequential manner. These instructions may also be translated into analytic form, thus giving the code transparency as well as flexibility. We will present an overview of how the ArbiTER code is to work, as well as preliminary results from early versions of this code. Work supported by the U.S. DOE.

Multivariate analysis of eigenvalues and eigenvectors in tensor based morphometry

Science.gov (United States)

Rajagopalan, Vidya; Schwartzman, Armin; Hua, Xue; Leow, Alex; Thompson, Paul; Lepore, Natasha

2015-01-01

We develop a new algorithm to compute voxel-wise shape differences in tensor-based morphometry (TBM). As in standard TBM, we non-linearly register brain T1-weighed MRI data from a patient and control group to a template, and compute the Jacobian of the deformation fields. In standard TBM, the determinants of the Jacobian matrix at each voxel are statistically compared between the two groups. More recently, a multivariate extension of the statistical analysis involving the deformation tensors derived from the Jacobian matrices has been shown to improve statistical detection power.7 However, multivariate methods comprising large numbers of variables are computationally intensive and may be subject to noise. In addition, the anatomical interpretation of results is sometimes difficult. Here instead, we analyze the eigenvalues and the eigenvectors of the Jacobian matrices. Our method is validated on brain MRI data from Alzheimer's patients and healthy elderly controls from the Alzheimer's Disease Neuro Imaging Database.

Spectral/spatial optical CDMA code based on Diagonal Eigenvalue Unity

Science.gov (United States)

Najjar, Monia; Jellali, Nabiha; Ferchichi, Moez; Rezig, Houria

2017-11-01

A new two dimensional Diagonal Eigenvalue Unity (2D-DEU) code is developed for the spectral⧹spatial optical code division multiple access (OCDMA) system. It has a lower cross correlation value compared to two dimensional diluted perfect difference (2D-DPD), two dimensional Extended Enhanced Double Weight (2D-Extended-EDW) codes. Also, for the same code length, the number of users can be generated by the 2D-DEU code is higher than that provided by the others codes. The Bit Error Rate (BER) numerical analysis is developed by considering the effects of shot noise, phase induced intensity noise (PIIN), and thermal noise. The main result shows that BER is strongly affected by PIIN for the higher source power. The 2D-DEU code performance is compared with 2D-DPD, 2D-Extended-EDW and two dimensional multi-diagonals (2D-MD) codes. This comparison proves that the proposed 2D-DEU system outperforms the related codes.

Multigrid method applied to the solution of an elliptic, generalized eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Alchalabi, R.M. [BOC Group, Murray Hill, NJ (United States); Turinsky, P.J. [North Carolina State Univ., Raleigh, NC (United States)

1996-12-31

The work presented in this paper is concerned with the development of an efficient MG algorithm for the solution of an elliptic, generalized eigenvalue problem. The application is specifically applied to the multigroup neutron diffusion equation which is discretized by utilizing the Nodal Expansion Method (NEM). The underlying relaxation method is the Power Method, also known as the (Outer-Inner Method). The inner iterations are completed using Multi-color Line SOR, and the outer iterations are accelerated using Chebyshev Semi-iterative Method. Furthermore, the MG algorithm utilizes the consistent homogenization concept to construct the restriction operator, and a form function as a prolongation operator. The MG algorithm was integrated into the reactor neutronic analysis code NESTLE, and numerical results were obtained from solving production type benchmark problems.

Numerical study of Langevin equation in twisted Eguchi-Kawai model: distribution of eigenvalues of the plaquette matrix

International Nuclear Information System (INIS)

Migdal, A.A.; Polikarpov, M.I.; Veselov, A.I.; Yurov, V.P.

1983-01-01

The Langevin equation for the lattice theory with arbitrary gauge group is derived. The four-dimensional twisted Eguchi-Kawai model is investigated numerically. The results for the plaquette energy agree with those of the known Monte Carlo calculations. The new result is the distribution of eigenvalues of the plaquette matrix. In the strong coupling phase this distribution is smooth, whereas in the weak coupling phase a gap is clearly seen

Fast EEG spike detection via eigenvalue analysis and clustering of spatial amplitude distribution

Science.gov (United States)

Fukami, Tadanori; Shimada, Takamasa; Ishikawa, Bunnoshin

2018-06-01

Objective. In the current study, we tested a proposed method for fast spike detection in electroencephalography (EEG). Approach. We performed eigenvalue analysis in two-dimensional space spanned by gradients calculated from two neighboring samples to detect high-amplitude negative peaks. We extracted the spike candidates by imposing restrictions on parameters regarding spike shape and eigenvalues reflecting detection characteristics of individual medical doctors. We subsequently performed clustering, classifying detected peaks by considering the amplitude distribution at 19 scalp electrodes. Clusters with a small number of candidates were excluded. We then defined a score for eliminating spike candidates for which the pattern of detected electrodes differed from the overall pattern in a cluster. Spikes were detected by setting the score threshold. Main results. Based on visual inspection by a psychiatrist experienced in EEG, we evaluated the proposed method using two statistical measures of precision and recall with respect to detection performance. We found that precision and recall exhibited a trade-off relationship. The average recall value was 0.708 in eight subjects with the score threshold that maximized the F-measure, with 58.6  ±  36.2 spikes per subject. Under this condition, the average precision was 0.390, corresponding to a false positive rate 2.09 times higher than the true positive rate. Analysis of the required processing time revealed that, using a general-purpose computer, our method could be used to perform spike detection in 12.1% of the recording time. The process of narrowing down spike candidates based on shape occupied most of the processing time. Significance. Although the average recall value was comparable with that of other studies, the proposed method significantly shortened the processing time.

Eigenvalues of the Laplacian and of the Hecke operators for PSL(2,Z)

International Nuclear Information System (INIS)

Steil, G.

1994-03-01

A new method is described to compute with high accuracy a large number of eigenvalues and eigenfunctions (Maass wave forms) of the Laplacian and of the Hecke operators for the modular group. It relies essentially on the theory of Hecke operators. The results of the computations confirm some important conjectures from number theory, namely Ramanujan-Petersson, Sato-Tate, and the conjecture that the discrete spectrum of the Laplacian be simple. Examples of the numerical data are included as a reference. The algorithm can be generalized to other non-cocompact but cofinite arithmetic groups, like Picard group PSL(2, Z)[i]) and Hecke triangle groups Γ(√2) and Γ(√3). (orig.)

A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields

Energy Technology Data Exchange (ETDEWEB)

Lerche, I. [Institut für Geowissenschaften, Naturwissenschaftliche Fakultät III, Martin-Luther Universität, D-06099 Halle (Germany); Low, B. C. [High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colorado 80307 (United States)

2014-10-15

An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θB{sub φ}=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θB{sub φ}=aA|A|{sup 1/n}, n a positive constant, admits separable fields with A=(A{sub n}(θ))/(r{sup n}) , posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and A{sub n}(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=(H(θ,φ))/(r{sup n+2}) promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4

Use of exact albedo conditions in numerical methods for one-dimensional one-speed discrete ordinates eigenvalue problems

International Nuclear Information System (INIS)

Abreu, M.P. de

1994-01-01

The use of exact albedo boundary conditions in numerical methods applied to one-dimensional one-speed discrete ordinates (S n ) eigenvalue problems for nuclear reactor global calculations is described. An albedo operator that treats the reflector region around a nuclear reactor core implicitly is described and exactly was derived. To illustrate the method's efficiency and accuracy, it was used conventional linear diamond method with the albedo option to solve typical model problems. (author)

The discrete hungry Lotka Volterra system and a new algorithm for computing matrix eigenvalues

Science.gov (United States)

Fukuda, Akiko; Ishiwata, Emiko; Iwasaki, Masashi; Nakamura, Yoshimasa

2009-01-01

The discrete hungry Lotka-Volterra (dhLV) system is a generalization of the discrete Lotka-Volterra (dLV) system which stands for a prey-predator model in mathematical biology. In this paper, we show that (1) some invariants exist which are expressed by dhLV variables and are independent from the discrete time and (2) a dhLV variable converges to some positive constant or zero as the discrete time becomes sufficiently large. Some characteristic polynomial is then factorized with the help of the dhLV system. The asymptotic behaviour of the dhLV system enables us to design an algorithm for computing complex eigenvalues of a certain band matrix.

The discrete hungry Lotka–Volterra system and a new algorithm for computing matrix eigenvalues

International Nuclear Information System (INIS)

Fukuda, Akiko; Ishiwata, Emiko; Iwasaki, Masashi; Nakamura, Yoshimasa

2009-01-01

The discrete hungry Lotka–Volterra (dhLV) system is a generalization of the discrete Lotka–Volterra (dLV) system which stands for a prey–predator model in mathematical biology. In this paper, we show that (1) some invariants exist which are expressed by dhLV variables and are independent from the discrete time and (2) a dhLV variable converges to some positive constant or zero as the discrete time becomes sufficiently large. Some characteristic polynomial is then factorized with the help of the dhLV system. The asymptotic behaviour of the dhLV system enables us to design an algorithm for computing complex eigenvalues of a certain band matrix

A spectral nodal method for eigenvalue SN transport problems in two-dimensional rectangular geometry for energy multigroup nuclear reactor global calculations

International Nuclear Information System (INIS)

Silva, Davi Jose M.; Alves Filho, Hermes; Barros, Ricardo C.

2015-01-01

A spectral nodal method is developed for multigroup x,y-geometry discrete ordinates (S N ) eigenvalue problems for nuclear reactor global calculations. This method uses the conventional multigroup SN discretized spatial balance nodal equations with two non-standard auxiliary equations: the spectral diamond (SD) auxiliary equations for the discretization nodes inside the fuel regions, and the spectral Green's function (SGF) auxiliary equations for the non-multiplying regions, such as the baffle and the reactor. This spectral nodal method is derived from the analytical general solution of the SN transverse integrated nodal equations with constant approximations for the transverse leakage terms within each discretization node. The SD and SGF auxiliary equations have parameters, which are determined to preserve the homogeneous and the particular components of these local general solutions. Therefore, we refer to the offered method as the hybrid SD-SGF-Constant Nodal (SD-SGF-CN) method. The S N discretized spatial balance equations, together with the SD and the SGF auxiliary equations form the SD-SGF-CN equations. We solve the SD-SGF-CN equations by using the one-node block inversion inner iterations (NBI), wherein the most recent estimates for the incoming group node-edge average or prescribed boundary conditions are used to evaluate the outgoing group node-edge average fluxes in the directions of the S N transport sweeps, for each estimate of the dominant eigenvalue in the conventional Power outer iterations. We show in numerical calculations that the SD-SGF-CN method is very accurate for coarse-mesh multigroup S N eigenvalue problems, even though the transverse leakage terms are approximated rather simply. (author)

An Approximate Proximal Bundle Method to Minimize a Class of Maximum Eigenvalue Functions

Directory of Open Access Journals (Sweden)

Wei Wang

2014-01-01

Full Text Available We present an approximate nonsmooth algorithm to solve a minimization problem, in which the objective function is the sum of a maximum eigenvalue function of matrices and a convex function. The essential idea to solve the optimization problem in this paper is similar to the thought of proximal bundle method, but the difference is that we choose approximate subgradient and function value to construct approximate cutting-plane model to solve the above mentioned problem. An important advantage of the approximate cutting-plane model for objective function is that it is more stable than cutting-plane model. In addition, the approximate proximal bundle method algorithm can be given. Furthermore, the sequences generated by the algorithm converge to the optimal solution of the original problem.

On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble

Science.gov (United States)

Atkin, Max R.; Charlier, Christophe; Zohren, Stefan

2018-04-01

We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the Laguerre unitary ensemble. The probability that this ratio is greater than r  >  1 is expressed in terms of an Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as is found as an integral over containing two functions q 1(x) and q 2(x). These functions satisfy a system of two coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as and , as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x.

Eigenvalue-eigenvector decomposition (EED) analysis of dissimilarity and covariance matrix obtained from total synchronous fluorescence spectral (TSFS) data sets of herbal preparations: Optimizing the classification approach

Science.gov (United States)

Tarai, Madhumita; Kumar, Keshav; Divya, O.; Bairi, Partha; Mishra, Kishor Kumar; Mishra, Ashok Kumar

2017-09-01

The present work compares the dissimilarity and covariance based unsupervised chemometric classification approaches by taking the total synchronous fluorescence spectroscopy data sets acquired for the cumin and non-cumin based herbal preparations. The conventional decomposition method involves eigenvalue-eigenvector analysis of the covariance of the data set and finds the factors that can explain the overall major sources of variation present in the data set. The conventional approach does this irrespective of the fact that the samples belong to intrinsically different groups and hence leads to poor class separation. The present work shows that classification of such samples can be optimized by performing the eigenvalue-eigenvector decomposition on the pair-wise dissimilarity matrix.

Application of Higher Order Fission Matrix for Real Variance Estimation in McCARD Monte Carlo Eigenvalue Calculation

Energy Technology Data Exchange (ETDEWEB)

Park, Ho Jin [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of); Shim, Hyung Jin [Seoul National University, Seoul (Korea, Republic of)

2015-05-15

In a Monte Carlo (MC) eigenvalue calculation, it is well known that the apparent variance of a local tally such as pin power differs from the real variance considerably. The MC method in eigenvalue calculations uses a power iteration method. In the power iteration method, the fission matrix (FM) and fission source density (FSD) are used as the operator and the solution. The FM is useful to estimate a variance and covariance because the FM can be calculated by a few cycle calculations even at inactive cycle. Recently, S. Carney have implemented the higher order fission matrix (HOFM) capabilities into the MCNP6 MC code in order to apply to extend the perturbation theory to second order. In this study, the HOFM capability by the Hotelling deflation method was implemented into McCARD and used to predict the behavior of a real and apparent SD ratio. In the simple 1D slab problems, the Endo's theoretical model predicts well the real to apparent SD ratio. It was noted that the Endo's theoretical model with the McCARD higher mode FS solutions by the HOFM yields much better the real to apparent SD ratio than that with the analytic solutions. In the near future, the application for a high dominance ratio problem such as BEAVRS benchmark will be conducted.

Dependence of accuracy of ESPRIT estimates on signal eigenvalues: the case of a noisy sum of two real exponentials.

Science.gov (United States)

Alexandrov, Theodore; Golyandina, Nina; Timofeyev, Alexey

2009-02-26

This paper is devoted to estimation of parameters for a noisy sum of two real exponential functions. Singular Spectrum Analysis is used to extract the signal subspace and then the ESPRIT method exploiting signal subspace features is applied to obtain estimates of the desired exponential rates. Dependence of estimation quality on signal eigenvalues is investigated. The special design to test this relation is elaborated.

Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems

International Nuclear Information System (INIS)

Eaker, C.W.; Schatz, G.C.; De Leon, N.; Heller, E.J.

1984-01-01

Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie--Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon--Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method

A comparative study of history-based versus vectorized Monte Carlo methods in the GPU/CUDA environment for a simple neutron eigenvalue problem

International Nuclear Information System (INIS)

Liu, T.; Du, X.; Ji, W.; Xu, G.; Brown, F.B.

2013-01-01

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed. (authors)

A comparative study of history-based versus vectorized Monte Carlo methods in the GPU/CUDA environment for a simple neutron eigenvalue problem

Science.gov (United States)

Liu, Tianyu; Du, Xining; Ji, Wei; Xu, X. George; Brown, Forrest B.

2014-06-01

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.

On a Volume Constrained for the First Eigenvalue of the P-Laplacian Operator

International Nuclear Information System (INIS)

Ly, Idrissa

2009-10-01

In this paper, we are interested in a shape optimization problem which consists in minimizing the functional that associates to an open set the first eigenvalue for p-Laplacian operator with homogeneous boundary condition. The minimum is taken among all open subsets with prescribed measure of a given bounded domain. We study an existence result for the associate variational problem. Our technique consists in enlarging the class of admissible functions to the whole space W 0 1,p (D), penalizing those functions whose level sets have a measure which is less than those required. In fact, we study the minimizers of a family of penalized functionals J λ , λ > 0 showing they are Hoelder continuous. And we prove that such functions minimize the initial problem provided the penalization parameter λ is large enough. (author)

Eigenvalue-eigenvector decomposition (EED) analysis of dissimilarity and covariance matrix obtained from total synchronous fluorescence spectral (TSFS) data sets of herbal preparations: Optimizing the classification approach.

Science.gov (United States)

Tarai, Madhumita; Kumar, Keshav; Divya, O; Bairi, Partha; Mishra, Kishor Kumar; Mishra, Ashok Kumar

2017-09-05

The present work compares the dissimilarity and covariance based unsupervised chemometric classification approaches by taking the total synchronous fluorescence spectroscopy data sets acquired for the cumin and non-cumin based herbal preparations. The conventional decomposition method involves eigenvalue-eigenvector analysis of the covariance of the data set and finds the factors that can explain the overall major sources of variation present in the data set. The conventional approach does this irrespective of the fact that the samples belong to intrinsically different groups and hence leads to poor class separation. The present work shows that classification of such samples can be optimized by performing the eigenvalue-eigenvector decomposition on the pair-wise dissimilarity matrix. Copyright © 2017 Elsevier B.V. All rights reserved.

Eigenvalue sensitivity of sampled time systems operating in closed loop

Science.gov (United States)

Bernal, Dionisio

2018-05-01

The use of feedback to create closed-loop eigenstructures with high sensitivity has received some attention in the Structural Health Monitoring field. Although practical implementation is necessarily digital, and thus in sampled time, work thus far has center on the continuous time framework, both in design and in checking performance. It is shown in this paper that the performance in discrete time, at typical sampling rates, can differ notably from that anticipated in the continuous time formulation and that discrepancies can be particularly large on the real part of the eigenvalue sensitivities; a consequence being important error on the (linear estimate) of the level of damage at which closed-loop stability is lost. As one anticipates, explicit consideration of the sampling rate poses no special difficulties in the closed-loop eigenstructure design and the relevant expressions are developed in the paper, including a formula for the efficient evaluation of the derivative of the matrix exponential based on the theory of complex perturbations. The paper presents an easily reproduced numerical example showing the level of error that can result when the discrete time implementation of the controller is not considered.

Classical scattering theory of waves from the view point of an eigenvalue problem and application to target identification

International Nuclear Information System (INIS)

Bottcher, C.; Strayer, M.R.; Werby, M.F.

1993-01-01

The Helmholtz-Poincare Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWE's. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can by obtained in matrix form be expanding all relevant terms in partial wave expansions, including a biorthogonal expansion of the Green function. However some freedom of choice in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways to long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermition operator. The methodology will be explained in detail and examples will be presented

High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

Energy Technology Data Exchange (ETDEWEB)

Pieper, Andreas [Ernst-Moritz-Arndt-Universität Greifswald (Germany); Kreutzer, Moritz [Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany); Alvermann, Andreas, E-mail: alvermann@physik.uni-greifswald.de [Ernst-Moritz-Arndt-Universität Greifswald (Germany); Galgon, Martin [Bergische Universität Wuppertal (Germany); Fehske, Holger [Ernst-Moritz-Arndt-Universität Greifswald (Germany); Hager, Georg [Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany); Lang, Bruno [Bergische Universität Wuppertal (Germany); Wellein, Gerhard [Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

2016-11-15

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 10{sup 2} innermost eigenpairs of a topological insulator matrix with dimension 10{sup 9} derived from quantum physics applications.

A spectral nodal method for eigenvalue S{sub N} transport problems in two-dimensional rectangular geometry for energy multigroup nuclear reactor global calculations

Energy Technology Data Exchange (ETDEWEB)

Silva, Davi Jose M.; Alves Filho, Hermes; Barros, Ricardo C., E-mail: davijmsilva@yahoo.com.br, E-mail: halves@iprj.uerj.br, E-mail: rcbarros@pq.cnpq.br [Universidade do Estado do Rio de Janeiro (UERJ), Nova Friburgo, RJ (Brazil). Programa de Pos-Graduacao em Modelagem Computacional

2015-07-01

A spectral nodal method is developed for multigroup x,y-geometry discrete ordinates (S{sub N}) eigenvalue problems for nuclear reactor global calculations. This method uses the conventional multigroup SN discretized spatial balance nodal equations with two non-standard auxiliary equations: the spectral diamond (SD) auxiliary equations for the discretization nodes inside the fuel regions, and the spectral Green's function (SGF) auxiliary equations for the non-multiplying regions, such as the baffle and the reactor. This spectral nodal method is derived from the analytical general solution of the SN transverse integrated nodal equations with constant approximations for the transverse leakage terms within each discretization node. The SD and SGF auxiliary equations have parameters, which are determined to preserve the homogeneous and the particular components of these local general solutions. Therefore, we refer to the offered method as the hybrid SD-SGF-Constant Nodal (SD-SGF-CN) method. The S{sub N} discretized spatial balance equations, together with the SD and the SGF auxiliary equations form the SD-SGF-CN equations. We solve the SD-SGF-CN equations by using the one-node block inversion inner iterations (NBI), wherein the most recent estimates for the incoming group node-edge average or prescribed boundary conditions are used to evaluate the outgoing group node-edge average fluxes in the directions of the S{sub N} transport sweeps, for each estimate of the dominant eigenvalue in the conventional Power outer iterations. We show in numerical calculations that the SD-SGF-CN method is very accurate for coarse-mesh multigroup S{sub N} eigenvalue problems, even though the transverse leakage terms are approximated rather simply. (author)

Energy eigenvalues and squeezing properties of general systems of coupled quantum anharmonic oscillators

International Nuclear Information System (INIS)

Chung, N. N.; Chew, L. Y.

2007-01-01

We have generalized the two-step approach to the solution of systems of N coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems

The Path Resistance Method for Bounding the Smallest Nontrivial Eigenvalue of a Laplacian

Science.gov (United States)

Guattery, Stephen; Leighton, Tom; Miller, Gary L.

1997-01-01

We introduce the path resistance method for lower bounds on the smallest nontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits; it uses clique embeddings to produce lower bounds on lambda(sub 2) and star embeddings to produce lower bounds on the smallest Rayleigh quotient when there is a zero Dirichlet boundary condition. The method assigns priorities to the paths in the embedding; we show that, for an unweighted tree T, using uniform priorities for a clique embedding produces a lower bound on lambda(sub 2) that is off by at most an 0(log diameter(T)) factor. We show that the best bounds this method can produce for clique embeddings are the same as for a related method that uses clique embeddings and edge lengths to produce bounds.

Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Stathopoulos, A.; Fischer, C.F. [Vanderbilt Univ., Nashville, TN (United States); Saad, Y.

1994-12-31

The solution of the large, sparse, symmetric eigenvalue problem, Ax = {lambda}x, is central to many scientific applications. Among many iterative methods that attempt to solve this problem, the Lanczos and the Generalized Davidson (GD) are the most widely used methods. The Lanczos method builds an orthogonal basis for the Krylov subspace, from which the required eigenvectors are approximated through a Rayleigh-Ritz procedure. Each Lanczos iteration is economical to compute but the number of iterations may grow significantly for difficult problems. The GD method can be considered a preconditioned version of Lanczos. In each step the Rayleigh-Ritz procedure is solved and explicit orthogonalization of the preconditioned residual ((M {minus} {lambda}I){sup {minus}1}(A {minus} {lambda}I)x) is performed. Therefore, the GD method attempts to improve convergence and robustness at the expense of a more complicated step.

Topics in bound-state dynamical processes: semiclassical eigenvalues, reactive scattering kernels and gas-surface scattering models

International Nuclear Information System (INIS)

Adams, J.E.

1979-05-01

The difficulty of applying the WKB approximation to problems involving arbitrary potentials has been confronted. Recent work has produced a convenient expression for the potential correction term. However, this approach does not yield a unique correction term and hence cannot be used to construct the proper modification. An attempt is made to overcome the uniqueness difficulties by imposing a criterion which permits identification of the correct modification. Sections of this work are: semiclassical eigenvalues for potentials defined on a finite interval; reactive scattering exchange kernels; a unified model for elastic and inelastic scattering from a solid surface; and selective absorption on a solid surface

A self-interaction-free local hybrid functional: Accurate binding energies vis-à-vis accurate ionization potentials from Kohn-Sham eigenvalues

International Nuclear Information System (INIS)

Schmidt, Tobias; Kümmel, Stephan; Kraisler, Eli; Makmal, Adi; Kronik, Leeor

2014-01-01

We present and test a new approximation for the exchange-correlation (xc) energy of Kohn-Sham density functional theory. It combines exact exchange with a compatible non-local correlation functional. The functional is by construction free of one-electron self-interaction, respects constraints derived from uniform coordinate scaling, and has the correct asymptotic behavior of the xc energy density. It contains one parameter that is not determined ab initio. We investigate whether it is possible to construct a functional that yields accurate binding energies and affords other advantages, specifically Kohn-Sham eigenvalues that reliably reflect ionization potentials. Tests for a set of atoms and small molecules show that within our local-hybrid form accurate binding energies can be achieved by proper optimization of the free parameter in our functional, along with an improvement in dissociation energy curves and in Kohn-Sham eigenvalues. However, the correspondence of the latter to experimental ionization potentials is not yet satisfactory, and if we choose to optimize their prediction, a rather different value of the functional's parameter is obtained. We put this finding in a larger context by discussing similar observations for other functionals and possible directions for further functional development that our findings suggest

Self-diffusion in periodic porous media: a comparison of numerical simulation and eigenvalue methods.

Science.gov (United States)

Schwartz, L M; Bergman, D J; Dunn, K J; Mitra, P P

1996-01-01

Random walk computer simulations are an important tool in understanding magnetic resonance measurements in porous media. In this paper we focus on the description of pulsed field gradient spin echo (PGSE) experiments that measure the probability, P(R,t), that a diffusing water molecule will travel a distance R in a time t. Because PGSE simulations are often limited by statistical considerations, we will see that valuable insight can be gained by working with simple periodic geometries and comparing simulation data to the results of exact eigenvalue expansions. In this connection, our attention will be focused on (1) the wavevector, k, and time dependent magnetization, M(k, t); and (2) the normalized probability, Ps(delta R, t), that a diffusing particle will return to within delta R of the origin after time t.

On discontinuous Galerkin and discrete ordinates approximations for neutron transport equation and the critical eigenvalue

International Nuclear Information System (INIS)

Asadzadeh, M.; Thevenot, L.

2010-01-01

The objective of this paper is to give a mathematical framework for a fully discrete numerical approach for the study of the neutron transport equation in a cylindrical domain (container model,). More specifically, we consider the discontinuous Galerkin (D G) finite element method for spatial approximation of the mono-energetic, critical neutron transport equation in an infinite cylindrical domain ??in R3 with a polygonal convex cross-section ? The velocity discretization relies on a special quadrature rule developed to give optimal estimates in discrete ordinate parameters compatible with the quasi-uniform spatial mesh. We use interpolation spaces and derive optimal error estimates, up to maximal available regularity, for the fully discrete scalar flux. Finally we employ a duality argument and prove superconvergence estimates for the critical eigenvalue.

Depletion GPT-free sensitivity analysis for reactor eigenvalue problems

International Nuclear Information System (INIS)

Kennedy, C.; Abdel-Khalik, H.

2013-01-01

This manuscript introduces a novel approach to solving depletion perturbation theory problems without the need to set up or solve the generalized perturbation theory (GPT) equations. The approach, hereinafter denoted generalized perturbation theory free (GPT-Free), constructs a reduced order model (ROM) using methods based in perturbation theory and computes response sensitivity profiles in a manner that is independent of the number or type of responses, allowing for an efficient computation of sensitivities when many responses are required. Moreover, the reduction error from using the ROM is quantified in the GPT-Free approach by means of a Wilks' order statistics error metric denoted the K-metric. Traditional GPT has been recognized as the most computationally efficient approach for performing sensitivity analyses of models with many input parameters, e.g. when forward sensitivity analyses are computationally intractable. However, most neutronics codes that can solve the fundamental (homogenous) adjoint eigenvalue problem do not have GPT capabilities unless envisioned during code development. The GPT-Free approach addresses this limitation by requiring only the ability to compute the fundamental adjoint. This manuscript demonstrates the GPT-Free approach for depletion reactor calculations performed in SCALE6 using the 7x7 UAM assembly model. A ROM is developed for the assembly over a time horizon of 990 days. The approach both calculates the reduction error over the lifetime of the simulation using the K-metric and benchmarks the obtained sensitivities using sample calculations. (authors)

On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.

Science.gov (United States)

Semenov, Yuri S; Bratus, Alexander S; Novozhilov, Artem S

2014-12-01

We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.5, 1 and prove that the absolute minimum of w¯(q), which always exists, belongs to the interval (0, 0.5]. For the specific case of a single peaked landscape we also find lower and upper bounds on w¯(q), which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact. Copyright © 2014 Elsevier Inc. All rights reserved.

HERESY, 2-D Few-Group Static Eigenvalues Calculation for Thermal Reactor

International Nuclear Information System (INIS)

Finch, D.R.

1965-01-01

1 - Description of problem or function: HERESY3 solves the two- dimensional, few-group, static reactor eigenvalue problem using the heterogeneous (source-sink or Feinburg-Galanin) formalism. The solution yields the reactor k-effective and absorption reaction rates for each rod normalized to the most absorptive rod in the thermal level. Epithermal fissions are allowed at each resonance level, and lattice-averaged values of thermal utilization, resonance escape probability, thermal and resonance eta values, and the fast fission factor are calculated. Kernels in the calculation are based on age-diffusion theory. Both finite reactor lattices and infinitely repeating reactor super-cells may be calculated. Rod parameters may be calculated by several internal options, and a direct interface is provided to a HAMMER system (NESC Abstract 277) lattice library tape to obtain cell parameters. Criticality searches are provided on thermal utilization, thermal eta, and axial leakage buckling. 2 - Method of solution: Direct power iteration on matrix form of the heterogeneous critical equation is used. 3 - Restrictions on the complexity of the problem: Maxima of - 50 flux/geometry symmetry positions; 20 physically different assemblies; 9 resonance levels; 5000 rod coordinate positions

Application of the Laplace transform method for the albedo boundary conditions in multigroup neutron diffusion eigenvalue problems in slab geometry

International Nuclear Information System (INIS)

Petersen, Claudio Zen; Vilhena, Marco T.; Barros, Ricardo C.

2009-01-01

In this paper the application of the Laplace transform method is described in order to determine the energy-dependent albedo matrix that is used in the boundary conditions multigroup neutron diffusion eigenvalue problems in slab geometry for nuclear reactor global calculations. In slab geometry, the diffusion albedo substitutes without approximation the baffle-reflector system around the active domain. Numerical results to typical test problems are shown to illustrate the accuracy and the efficiency of the Chebysheff acceleration scheme. (orig.)

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Science.gov (United States)

Plestenjak, Bor; Gheorghiu, Călin I.; Hochstenbach, Michiel E.

2015-10-01

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

MARG2D code. 1. Eigenvalue problem for two dimensional Newcomb equation

Energy Technology Data Exchange (ETDEWEB)

Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko

1997-10-01

A new method and a code MARG2D have been developed to solve the 2-dimensional Newcomb equation which plays an important role in the magnetohydrodynamic (MHD) stability analysis in an axisymmetric toroidal plasma such as a tokamak. In the present formulation, an eigenvalue problem is posed for the 2-D Newcomb equation, where the weight function (the kinetic energy integral) and the boundary conditions at rational surfaces are chosen so that an eigenfunction correctly behaves as the linear combination of the small solution and the analytical solutions around each of the rational surfaces. Thus, the difficulty on solving the 2-D Newcomb equation has been resolved. By using the MARG2D code, the ideal MHD marginally stable state can be identified for a 2-D toroidal plasma. The code is indispensable on computing the outer-region matching data necessary for the resistive MHD stability analysis. Benchmark with ERATOJ, an ideal MHD stability code, has been carried out and the MARG2D code demonstrates that it indeed identifies both stable and marginally stable states against ideal MHD motion. (author)

Variational variance reduction for particle transport eigenvalue calculations using Monte Carlo adjoint simulation

International Nuclear Information System (INIS)

Densmore, Jeffery D.; Larsen, Edward W.

2003-01-01

The Variational Variance Reduction (VVR) method is an effective technique for increasing the efficiency of Monte Carlo simulations [Ann. Nucl. Energy 28 (2001) 457; Nucl. Sci. Eng., in press]. This method uses a variational functional, which employs first-order estimates of forward and adjoint fluxes, to yield a second-order estimate of a desired system characteristic - which, in this paper, is the criticality eigenvalue k. If Monte Carlo estimates of the forward and adjoint fluxes are used, each having global 'first-order' errors of O(1/√N), where N is the number of histories used in the Monte Carlo simulation, then the statistical error in the VVR estimation of k will in principle be O(1/N). In this paper, we develop this theoretical possibility and demonstrate with numerical examples that implementations of the VVR method for criticality problems can approximate O(1/N) convergence for significantly large values of N

Quadratic partial eigenvalue assignment in large-scale stochastic dynamic systems for resilient and economic design

Energy Technology Data Exchange (ETDEWEB)

Das, Sonjoy; Goswami, Kundan [University at Buffalo, NY (United States); Datta, Biswa N. [Northern Illinois University, IL (United States)

2014-12-10

Failure of structural systems under dynamic loading can be prevented via active vibration control which shifts the damped natural frequencies of the systems away from the dominant range of loading spectrum. The damped natural frequencies and the dynamic load typically show significant variations in practice. A computationally efficient methodology based on quadratic partial eigenvalue assignment technique and optimization under uncertainty has been formulated in the present work that will rigorously account for these variations and result in an economic and resilient design of structures. A novel scheme based on hierarchical clustering and importance sampling is also developed in this work for accurate and efficient estimation of probability of failure to guarantee the desired resilience level of the designed system. Numerical examples are presented to illustrate the proposed methodology.

Modification of the MORSE code for Monte Carlo eigenvalue problems by coarse-mesh rebalance acceleration

International Nuclear Information System (INIS)

Nishida, Takahiko; Horikami, Kunihiko; Suzuki, Tadakazu; Nakahara, Yasuaki; Taji, Yukichi

1975-09-01

The coarse-mesh rebalancing technique is introduced into the general-purpose neutron and gamma-ray Monte Carlo transport code MORSE, to accelerate the convergence rate of the iteration process for eigenvalue calculation in a nuclear reactor system. Two subroutines are thus attached to the code. One is bookkeeping routine 'COARSE' for obtaining the quantities related with the neutron balance in each coarse mesh cell, such as the number of neutrons absorbed in the cell, from random walks of neutrons in a batch. The other is rebalance factor calculation routine 'REBAL' for obtaining the scaling factor whereby the neutron flux in the cell is multiplied to attain the neutron balance. The two subroutines and algorithm of the coarse mesh rebalancing acceleration in a Monte Carlo game are described. (auth.)

Quadratic partial eigenvalue assignment in large-scale stochastic dynamic systems for resilient and economic design

International Nuclear Information System (INIS)

Das, Sonjoy; Goswami, Kundan; Datta, Biswa N.

2014-01-01

Failure of structural systems under dynamic loading can be prevented via active vibration control which shifts the damped natural frequencies of the systems away from the dominant range of loading spectrum. The damped natural frequencies and the dynamic load typically show significant variations in practice. A computationally efficient methodology based on quadratic partial eigenvalue assignment technique and optimization under uncertainty has been formulated in the present work that will rigorously account for these variations and result in an economic and resilient design of structures. A novel scheme based on hierarchical clustering and importance sampling is also developed in this work for accurate and efficient estimation of probability of failure to guarantee the desired resilience level of the designed system. Numerical examples are presented to illustrate the proposed methodology

Nature of complex time eigenvalues of the one speed transport equation in a homogeneous sphere

International Nuclear Information System (INIS)

Dahl, E.B.; Sahni, D.C.

1990-01-01

The complex time eigenvalues of the transport equation have been studied for one speed neutrons, scattered isotropically in a homogeneous sphere with vacuum boundary conditions. It is shown that the complex decay constants vary continuously with the radius of the sphere. Our earlier conjecture (Dahl and Sahni (1983-84)) regarding disjoint arcs is thus shown to be true. We also indicate that complex decay constants exist even for large assemblies, though with rapid oscillations in the corresponding eigenvectors. These modes cannot be predicted by the diffusion equation as this behaviour of the eigenvectors contradicts the assumption of 'slowly varying flux' needed to derive the diffusion approximation from the transport equation. For an infinite system, the existence of complex modes is related to the solution of a homogeneous equation. (author)

On the generalized eigenvalue method for energies and matrix elements in lattice field theory

Energy Technology Data Exchange (ETDEWEB)

Blossier, Benoit [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany)]|[Paris-XI Univ., 91 - Orsay (France). Lab. de Physique Theorique; Morte, Michele della [CERN, Geneva (Switzerland). Physics Dept.]|[Mainz Univ. (Germany). Inst. fuer Kernphysik; Hippel, Georg von; Sommer, Rainer [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Mendes, Tereza [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany)]|[Sao Paulo Univ. (Brazil). IFSC

2009-02-15

We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as exp(-(E{sub N+1}-E{sub n}) t). The gap E{sub N+1}-E{sub n} can be made large by increasing the number N of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order 1/m{sub b} in HQET. (orig.)

On the generalized eigenvalue method for energies and matrix elements in lattice field theory

International Nuclear Information System (INIS)

Blossier, Benoit; Mendes, Tereza; Sao Paulo Univ.

2009-02-01

We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as exp(-(E N+1 -E n ) t). The gap E N+1 -E n can be made large by increasing the number N of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order 1/m b in HQET. (orig.)

Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum

Science.gov (United States)

Guarnieri, F.; Moon, W.; Wettlaufer, J. S.

2017-09-01

Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with a negative constant drift, described by a Fokker-Planck equation with a potential V (x ) =-[b ln(x ) +a x ] , for b >0 and a finance. The Bessel-like process we consider can be solved by seeking solutions through an expansion into a complete set of eigenfunctions. The associated imaginary-time Schrödinger equation exhibits a mix of discrete and continuous eigenvalue spectra, corresponding to the quantum Coulomb potential describing the bound states of the hydrogen atom. We present a technique to evaluate the normalization factor of the continuous spectrum of eigenfunctions that relies solely upon their asymptotic behavior. We demonstrate the technique by solving the Brownian motion problem and the Bessel process both with a constant negative drift. We conclude with a comparison to other analytical methods and with numerical solutions.

Those Do What? Connecting Eigenvectors and Eigenvalues to the Rest of Linear Algebra: Using Visual Enhancements to Help Students Connect Eigenvectors to the Rest of Linear Algebra

Science.gov (United States)

Nyman, Melvin A.; Lapp, Douglas A.; St. John, Dennis; Berry, John S.

2010-01-01

This paper discusses student difficulties in grasping concepts from Linear Algebra--in particular, the connection of eigenvalues and eigenvectors to other important topics in linear algebra. Based on our prior observations from student interviews, we propose technology-enhanced instructional approaches that might positively impact student…

Quantum complex rotation and uniform semiclassical calculations of complex energy eigenvalues

International Nuclear Information System (INIS)

Connor, J.N.L.; Smith, A.D.

1983-01-01

Quantum and semiclassical calculations of complex energy eigenvalues have been carried out for an exponential potential of the form V 0 r 2 exp(-r) and Lennard-Jones (12,6) potential. A straightforward method, based on the complex coordinate rotation technique, is described for the quantum calculation of complex eigenenergies. For singular potentials, the method involves an inward and outward integration of the radial Schroedinger equation, followed by matching of the logarithmic derivatives of the wave functions at an intermediate point. For regular potentials, the method is simpler, as only an inward integration is required. Attention is drawn to the World War II researches of Hartree and co-workers who anticipated later quantum mechanical work on the complex rotation method. Complex eigenenergies are also calculated from a uniform semiclassical three turning point quantization formula, which allows for the proximity of the outer pair of complex turning points. Limiting cases of this formula, which are valid for very narrow or very broad widths, are also used in the calculations. We obtain good agreement between the semiclassical and quantum results. For the Lennard-Jones (12,6) potential, we compare resonance energies and widths from the complex energy definition of a resonance with those obtained from the time delay definition

The mass spectrum of high energy elementary particles via El Naschie's E(∞) golden mean nested oscillators, the Dunkerly-Southwell eigenvalue theorems and KAM

International Nuclear Information System (INIS)

Marek-Crnjac, L.

2003-01-01

In the present work we give a classical nested mechanical model and corresponding expressions for the theoretical masses of elementary particles, including the masses of quarks as being the joint eigenvalues of combined vibrating sets using the Southwell and the Dunkerly theorems. The role played by the golden mean in KAM theory and consequently our present model is also discussed

Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE

Science.gov (United States)

Forrester, Peter J.; Trinh, Allan K.

2018-05-01

The neighbourhood of the largest eigenvalue λmax in the Gaussian unitary ensemble (GUE) and Laguerre unitary ensemble (LUE) is referred to as the soft edge. It is known that there exists a particular centring and scaling such that the distribution of λmax tends to a universal form, with an error term bounded by 1/N2/3. We take up the problem of computing the exact functional form of the leading error term in a large N asymptotic expansion for both the GUE and LUE—two versions of the LUE are considered, one with the parameter a fixed and the other with a proportional to N. Both settings in the LUE case allow for an interpretation in terms of the distribution of a particular weighted path length in a model involving exponential variables on a rectangular grid, as the grid size gets large. We give operator theoretic forms of the corrections, which are corollaries of knowledge of the first two terms in the large N expansion of the scaled kernel and are readily computed using a method due to Bornemann. We also give expressions in terms of the solutions of particular systems of coupled differential equations, which provide an alternative method of computation. Both characterisations are well suited to a thinned generalisation of the original ensemble, whereby each eigenvalue is deleted independently with probability (1 - ξ). In Sec. V, we investigate using simulation the question of whether upon an appropriate centring and scaling a wider class of complex Hermitian random matrix ensembles have their leading correction to the distribution of λmax proportional to 1/N2/3.

Sorting waves and associated eigenvalues

Science.gov (United States)

Carbonari, Costanza; Colombini, Marco; Solari, Luca

2017-04-01

The presence of mixed sediment always characterizes gravel bed rivers. Sorting processes take place during bed load transport of heterogeneous sediment mixtures. The two main elements necessary to the occurrence of sorting are the heterogeneous character of sediments and the presence of an active sediment transport. When these two key ingredients are simultaneously present, the segregation of bed material is consistently detected both in the field [7] and in laboratory [3] observations. In heterogeneous sediment transport, bed altimetric variations and sorting always coexist and both mechanisms are independently capable of driving the formation of morphological patterns. Indeed, consistent patterns of longitudinal and transverse sorting are identified almost ubiquitously. In some cases, such as bar formation [2] and channel bends [5], sorting acts as a stabilizing effect and therefore the dominant mechanism driving pattern formation is associated with bed altimetric variations. In other cases, such as longitudinal streaks, sorting enhances system instability and can therefore be considered the prevailing mechanism. Bedload sheets, first observed by Khunle and Southard [1], represent another classic example of a morphological pattern essentially triggered by sorting, as theoretical [4] and experimental [3] results suggested. These sorting waves cause strong spatial and temporal fluctuations of bedload transport rate typical observed in gravel bed rivers. The problem of bed load transport of a sediment mixture is formulated in the framework of a 1D linear stability analysis. The base state consists of a uniform flow in an infinitely wide channel with active bed load transport. The behaviour of the eigenvalues associated with fluid motion, bed evolution and sorting processes in the space of the significant flow and sediment parameters is analysed. A comparison is attempted with the results of the theoretical analysis of Seminara Colombini and Parker [4] and Stecca

On the density of eigenvalues of a random matrix; Concernant la densite des racines caracteristiques d'une matrice stochastique

Energy Technology Data Exchange (ETDEWEB)

Mehta, M. L. [Institute of Fundamental Research Bombay (India); Gaudin, M. [Commissariat a l' energie atomique et aux energies alternatives - CEA, Centre d' Etudes Nucleaires de Saclay, Gif-sur-Yvette (France)

1960-07-01

An exact expression for the density of eigenvalues of a random- matrix is derived. When the order of the matrix becomes infinite, it can be seen very directly that it goes over to Wigner's 'semi-circle law'. Reprint of a paper published in 'Nuclear Physics' 18, 1960, p. 420-427 [French] On deduit une expression precise pour la densite des racines caracteristiques d'une matrice stochastique. Quand l'ordre de la matrice devient infini, on peut voir facilement qu'elle obeit a la loi dite 'semi-circulaire' de Wigner. Reproduction d'un article publie dans 'Nuclear Physics' 18, 1960, p. 420-427.

An algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group

International Nuclear Information System (INIS)

Wang, S.J.

1993-04-01

An algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group is formulated for the first time. One dimensional problem is treated explicitly in detail for both the finite dimensional and infinite dimensional Hilbert spaces. For the finite dimensional Hilbert space, the su(2) algebraic representation is used; while for the infinite dimensional Hilbert space, the Heisenberg-Weyl algebraic representation is employed. Fourier expansion technique is generalized to the generator space, which is suitable for analysis of irregular spectra. The polynormial operator basis is also used for complement, which is appropriate for analysis of some simple Hamiltonians. The proposed new approach is applied to solve the classical inverse Sturn-Liouville problem and to study the problems of quantum regular and irregular spectra. (orig.)

Construction of local boundary conditions for an eigenvalue problem using micro-local analysis: application to optical waveguide problems

International Nuclear Information System (INIS)

Barucq, Helene; Bekkey, Chokri; Djellouli, Rabia

2004-01-01

We present a general procedure based on the pseudo-differential calculus for deriving artificial boundary conditions for an eigenvalue problem that characterizes the propagation of guided modes in optical waveguides. This new approach allows the construction of local conditions that (a) are independent of the frequency regime, (b) preserve the sparsity pattern of the finite element discretization, and (c) are applicable to arbitrarily shaped convex artificial boundaries. The last feature has the potential for reducing the size of the computational domain. Numerical results are presented to highlight the potential of conditions of order 1/2 and 1, for improving significantly the computational efficiency of finite element methods for the solution of optical waveguide problems

The mass spectrum of high energy elementary particles via El Naschie's E sup ( supinfinity sup ) golden mean nested oscillators, the Dunkerly-Southwell eigenvalue theorems and KAM

CERN Document Server

Marek-Crnjac, L

2003-01-01

In the present work we give a classical nested mechanical model and corresponding expressions for the theoretical masses of elementary particles, including the masses of quarks as being the joint eigenvalues of combined vibrating sets using the Southwell and the Dunkerly theorems. The role played by the golden mean in KAM theory and consequently our present model is also discussed.

Eigenvector/eigenvalue analysis of a 3D current referential fault detection and diagnosis of an induction motor

International Nuclear Information System (INIS)

Pires, V. Fernao; Martins, J.F.; Pires, A.J.

2010-01-01

In this paper an integrated approach for on-line induction motor fault detection and diagnosis is presented. The need to insure a continuous and safety operation for induction motors involves preventive maintenance procedures combined with fault diagnosis techniques. The proposed approach uses an automatic three step algorithm. Firstly, the induction motor stator currents are measured which will give typical patterns that can be used to identify the fault. Secondly, the eigenvectors/eigenvalues of the 3D current referential are computed. Finally the proposed algorithm will discern if the motor is healthy or not and report the extent of the fault. Furthermore this algorithm is able to identify distinct faults (stator winding faults or broken bars). The proposed approach was experimentally implemented and its performance verified on various types of working conditions.

An Experiment of Robust Parallel Algorithm for the Eigenvalue problem of a Multigroup Neutron Diffusion based on modified FETI-DP : Part 2

International Nuclear Information System (INIS)

Chang, Jonghwa

2014-01-01

Today, we can use a computer cluster consist of a few hundreds CPUs with reasonable budget. Such computer system enables us to do detailed modeling of reactor core. The detailed modeling will improve the safety and the economics of a nuclear reactor by eliminating un-necessary conservatism or missing consideration. To take advantage of such a cluster computer, efficient parallel algorithms must be developed. Mechanical structure analysis community has studied the domain decomposition method to solve the stress-strain equation using the finite element methods. One of the most successful domain decomposition method in terms of robustness is FETI-DP. We have modified the original FETI-DP to solve the eigenvalue problem for the multi-group diffusion problem in previous study. In this study, we report the result of recent modification to handle the three-dimensional subdomain partitioning, and the sub-domain multi-group problem. Modified FETI-DP algorithm has been successfully applied for the eigenvalue problem of multi-group neutron diffusion equation. The overall CPU time is decreasing as number of sub-domains (partitions) is increasing. However, there may be a limit in decrement due to increment of the number of primal points will increase the CPU time spent by the solution of the global equation. Even distribution of computational load (criterion a) is important to achieve fast computation. The subdomain partition can be effectively performed using suitable graph theory partition package such as MeTIS

An Experiment of Robust Parallel Algorithm for the Eigenvalue problem of a Multigroup Neutron Diffusion based on modified FETI-DP : Part 2

Energy Technology Data Exchange (ETDEWEB)

Chang, Jonghwa [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of)

2014-10-15

Today, we can use a computer cluster consist of a few hundreds CPUs with reasonable budget. Such computer system enables us to do detailed modeling of reactor core. The detailed modeling will improve the safety and the economics of a nuclear reactor by eliminating un-necessary conservatism or missing consideration. To take advantage of such a cluster computer, efficient parallel algorithms must be developed. Mechanical structure analysis community has studied the domain decomposition method to solve the stress-strain equation using the finite element methods. One of the most successful domain decomposition method in terms of robustness is FETI-DP. We have modified the original FETI-DP to solve the eigenvalue problem for the multi-group diffusion problem in previous study. In this study, we report the result of recent modification to handle the three-dimensional subdomain partitioning, and the sub-domain multi-group problem. Modified FETI-DP algorithm has been successfully applied for the eigenvalue problem of multi-group neutron diffusion equation. The overall CPU time is decreasing as number of sub-domains (partitions) is increasing. However, there may be a limit in decrement due to increment of the number of primal points will increase the CPU time spent by the solution of the global equation. Even distribution of computational load (criterion a) is important to achieve fast computation. The subdomain partition can be effectively performed using suitable graph theory partition package such as MeTIS.

Application of bias factor method using random sampling technique for prediction accuracy improvement of critical eigenvalue of BWR

International Nuclear Information System (INIS)

Ito, Motohiro; Endo, Tomohiro; Yamamoto, Akio; Kuroda, Yusuke; Yoshii, Takashi

2017-01-01

The bias factor method based on the random sampling technique is applied to the benchmark problem of Peach Bottom Unit 2. Validity and availability of the present method, i.e. correction of calculation results and reduction of uncertainty, are confirmed in addition to features and performance of the present method. In the present study, core characteristics in cycle 3 are corrected with the proposed method using predicted and 'measured' critical eigenvalues in cycles 1 and 2. As the source of uncertainty, variance-covariance of cross sections is considered. The calculation results indicate that bias between predicted and measured results, and uncertainty owing to cross section can be reduced. Extension to other uncertainties such as thermal hydraulics properties will be a future task. (author)

Eigenvalue distributions for a class of covariance matrices with application to Bienenstock-Cooper-Munro neurons under noisy conditions.

Science.gov (United States)

Bazzani, Armando; Castellani, Gastone C; Cooper, Leon N

2010-05-01

We analyze the effects of noise correlations in the input to, or among, Bienenstock-Cooper-Munro neurons using the Wigner semicircular law to construct random, positive-definite symmetric correlation matrices and compute their eigenvalue distributions. In the finite dimensional case, we compare our analytic results with numerical simulations and show the effects of correlations on the lifetimes of synaptic strengths in various visual environments. These correlations can be due either to correlations in the noise from the input lateral geniculate nucleus neurons, or correlations in the variability of lateral connections in a network of neurons. In particular, we find that for fixed dimensionality, a large noise variance can give rise to long lifetimes of synaptic strengths. This may be of physiological significance.

Relativistic energy eigenvalues for the Dirac equation in the presence of vector and scalar potentials via the simple similarity transformation

International Nuclear Information System (INIS)

Barakat, T

2012-01-01

Based on the simple similarity transformation, we were able to transform the Dirac equation whose potential contains vector V (r) = -A/r + B 1 r and scalar S(r) = B 2 r types into a form nearly identical to the Schrödinger equation. The transformed equation is so simple that one can solve it by means of the asymptotic iteration method. Moreover, within the same framework we were able to obtain the relativistic energy eigenvalues for the Dirac equation with vector Coulomb plus scalar linear, and with pure scalar linear potentials; V (r) = -A/r, S(r) = B 2 r, and V (r) = 0, S(r) = B 2 r, respectively.

An accelerated conjugate gradient algorithm to compute low-lying eigenvalues - a study for the Dirac operator in SU(2) lattice QCD

International Nuclear Information System (INIS)

Kalkreuter, T.; Simma, H.

1995-07-01

The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with controlled numerical errors by a conjugate gradient (CG) method. This CG algorithm is accelerated by alternating it with exact diagonalizations in the subspace spanned by the numerically computed eigenvectors. We study this combined algorithm in case of the Dirac operator with (dynamical) Wilson fermions in four-dimensional SU(2) gauge fields. The algorithm is numerically very stable and can be parallelized in an efficient way. On lattices of sizes 4 4 - 16 4 an acceleration of the pure CG method by a factor of 4 - 8 is found. (orig.)

EISPACK, Subroutines for Eigenvalues, Eigenvectors, Matrix Operations

International Nuclear Information System (INIS)

Garbow, Burton S.; Cline, A.K.; Meyering, J.

1993-01-01

1 - Description of problem or function: EISPACK3 is a collection of 75 FORTRAN subroutines, both single- and double-precision, that compute the eigenvalues and eigenvectors of nine classes of matrices. The package can determine the Eigen-system of complex general, complex Hermitian, real general, real symmetric, real symmetric band, real symmetric tridiagonal, special real tridiagonal, generalized real, and generalized real symmetric matrices. In addition, there are two routines which use the singular value decomposition to solve certain least squares problem. The individual subroutines are - Identification/Description: BAKVEC: Back transform vectors of matrix formed by FIGI; BALANC: Balance a real general matrix; BALBAK: Back transform vectors of matrix formed by BALANC; BANDR: Reduce sym. band matrix to sym. tridiag. matrix; BANDV: Find some vectors of sym. band matrix; BISECT: Find some values of sym. tridiag. matrix; BQR: Find some values of sym. band matrix; CBABK2: Back transform vectors of matrix formed by CBAL; CBAL: Balance a complex general matrix; CDIV: Perform division of two complex quantities; CG: Driver subroutine for a complex general matrix; CH: Driver subroutine for a complex Hermitian matrix; CINVIT: Find some vectors of complex Hess. matrix; COMBAK: Back transform vectors of matrix formed by COMHES; COMHES: Reduce complex matrix to complex Hess. (elementary); COMLR: Find all values of complex Hess. matrix (LR); COMLR2: Find all values/vectors of cmplx Hess. matrix (LR); CCMQR: Find all values of complex Hessenberg matrix (QR); COMQR2: Find all values/vectors of cmplx Hess. matrix (QR); CORTB: Back transform vectors of matrix formed by CORTH; CORTH: Reduce complex matrix to complex Hess. (unitary); CSROOT: Find square root of complex quantity; ELMBAK: Back transform vectors of matrix formed by ELMHES; ELMHES: Reduce real matrix to real Hess. (elementary); ELTRAN: Accumulate transformations from ELMHES (for HQR2); EPSLON: Estimate unit roundoff

Evaluation of vectorized Monte Carlo algorithms on GPUs for a neutron Eigenvalue problem

International Nuclear Information System (INIS)

Du, X.; Liu, T.; Ji, W.; Xu, X. G.; Brown, F. B.

2013-01-01

Conventional Monte Carlo (MC) methods for radiation transport computations are 'history-based', which means that one particle history at a time is tracked. Simulations based on such methods suffer from thread divergence on the graphics processing unit (GPU), which severely affects the performance of GPUs. To circumvent this limitation, event-based vectorized MC algorithms can be utilized. A versatile software test-bed, called ARCHER - Accelerated Radiation-transport Computations in Heterogeneous Environments - was used for this study. ARCHER facilitates the development and testing of a MC code based on the vectorized MC algorithm implemented on GPUs by using NVIDIA's Compute Unified Device Architecture (CUDA). The ARCHER GPU code was designed to solve a neutron eigenvalue problem and was tested on a NVIDIA Tesla M2090 Fermi card. We found that although the vectorized MC method significantly reduces the occurrence of divergent branching and enhances the warp execution efficiency, the overall simulation speed is ten times slower than the conventional history-based MC method on GPUs. By analyzing detailed GPU profiling information from ARCHER, we discovered that the main reason was the large amount of global memory transactions, causing severe memory access latency. Several possible solutions to alleviate the memory latency issue are discussed. (authors)

Evaluation of vectorized Monte Carlo algorithms on GPUs for a neutron Eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Du, X.; Liu, T.; Ji, W.; Xu, X. G. [Nuclear Engineering Program, Rensselaer Polytechnic Institute, Troy, NY 12180 (United States); Brown, F. B. [Monte Carlo Codes Group, Los Alamos National Laboratory, Los Alamos, NM 87545 (United States)

2013-07-01

Conventional Monte Carlo (MC) methods for radiation transport computations are 'history-based', which means that one particle history at a time is tracked. Simulations based on such methods suffer from thread divergence on the graphics processing unit (GPU), which severely affects the performance of GPUs. To circumvent this limitation, event-based vectorized MC algorithms can be utilized. A versatile software test-bed, called ARCHER - Accelerated Radiation-transport Computations in Heterogeneous Environments - was used for this study. ARCHER facilitates the development and testing of a MC code based on the vectorized MC algorithm implemented on GPUs by using NVIDIA's Compute Unified Device Architecture (CUDA). The ARCHER{sub GPU} code was designed to solve a neutron eigenvalue problem and was tested on a NVIDIA Tesla M2090 Fermi card. We found that although the vectorized MC method significantly reduces the occurrence of divergent branching and enhances the warp execution efficiency, the overall simulation speed is ten times slower than the conventional history-based MC method on GPUs. By analyzing detailed GPU profiling information from ARCHER, we discovered that the main reason was the large amount of global memory transactions, causing severe memory access latency. Several possible solutions to alleviate the memory latency issue are discussed. (authors)

Problem on eigenfunctions and eigenvalues for effective Hamiltonians in pair channels of four-particle systems

International Nuclear Information System (INIS)

Gurbanovich, N.S.; Zelenskaya, I.N.

1976-01-01

The solution for eigenfunction and eigenvalue for effective Hamiltonians anti Hsub(p) in two-particle channels corresponding to division of four particles into groups (3.1) and (2.2) is very essential in the four-body problem as applied to nuclear reactions. The interaction of anti√sub(p) in each channel may be written in the form of an integral operator which takes account of the structure of a target nucleus or of an incident particle and satisfying the integral equation. While assuming the two-particle potentials to be central, it is possible to expand the effective interactions anti√sub(p) in partial waves and write the radial equation for anti Hsub(p). In the approximation on a mass shell the radial equations for the eigenfunctions of Hsub(p) are reduced to an algebraic equations system. The coefficients of the latter are expressed through the Fourier images for products of wave functions of bound clusters and the two-particle central potential which are localized in a momentum space

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Science.gov (United States)

Garmon, Savannah; Ordonez, Gonzalo

2017-06-01

It has been reported in the literature that the survival probability P(t) near an exceptional point where two eigenstates coalesce should generally exhibit an evolution P (t ) ˜t2e-Γ t, in which Γ is the decay rate of the coalesced eigenstate; this has been verified in a microwave billiard experiment [B. Dietz et al., Phys. Rev. E 75, 027201 (2007)]. However, the heuristic effective Hamiltonian that is usually employed to obtain this result ignores the possible influence of the continuum threshold on the dynamics. By contrast, in this work we employ an analytical approach starting from the microscopic Hamiltonian representing two simple models in order to show that the continuum threshold has a strong influence on the dynamics near exceptional points in a variety of circumstances. To report our results, we divide the exceptional points in Hermitian open quantum systems into two cases: at an EP2A two virtual bound states coalesce before forming a resonance, anti-resonance pair with complex conjugate eigenvalues, while at an EP2B two resonances coalesce before forming two different resonances. For the EP2B, which is the case studied in the microwave billiard experiment, we verify that the survival probability exhibits the previously reported modified exponential decay on intermediate time scales, but this is replaced with an inverse power law on very long time scales. Meanwhile, for the EP2A the influence from the continuum threshold is so strong that the evolution is non-exponential on all time scales and the heuristic approach fails completely. When the EP2A appears very near the threshold, we obtain the novel evolution P (t ) ˜1 -C1√{t } on intermediate time scales, while further away the parabolic decay (Zeno dynamics) on short time scales is enhanced.

An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

International Nuclear Information System (INIS)

Conca, Carlos; Mahadevan, Rajesh; Sanz, Leon

2009-01-01

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhaeuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101-117, 1996). Although, the problem in one dimension has a solution (cf. Krein in AMS Transl. Ser. 2(1):163-187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185-220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328-337, 1983)

Numerical method to calculate the quantum transmission, resonance and eigenvalue energies: application to a biased multibarrier systems

Energy Technology Data Exchange (ETDEWEB)

Maiz, F., E-mail: fethimaiz@gmail.com [University of Cartage, Nabeul Engineering Preparatory Institute, Merazka, 8000 Nabeul (Tunisia); King Khalid University, Faculty of Science, Physics Department, PO Box 9004, Abha 61413 (Saudi Arabia)

2015-04-15

A novel method to calculate the quantum transmission, resonance and eigenvalue energies forming the sub-bands structure of non-symmetrical, non-periodical semiconducting heterostructure potential has been proposed in this paper. The method can be applied on a multilayer system with varying thickness of the layer and effective mass of electrons and holes. Assuming an approximated effective mass and using Bastard's boundary conditions, Schrödinger equation at each media is solved and then using a confirmed recurrence method, the transmission and reflection coefficients and the energy quantification condition are expressed. They are simple combination of coupled equations. Schrödinger's equation solutions are Airy functions or plane waves, depending on the electrical potential energy slope. To illustrate the feasibility of the proposed method, the N barriers – (N−1) wells structure for N=3, 5, 8, 9, 17 and 35 are studied. All results show very good agreements with previously published results obtained from applying different methods on similar systems.

Diagonal Eigenvalue Unity (DEU) code for spectral amplitude coding-optical code division multiple access

Science.gov (United States)

Ahmed, Hassan Yousif; Nisar, K. S.

2013-08-01

Code with ideal in-phase cross correlation (CC) and practical code length to support high number of users are required in spectral amplitude coding-optical code division multiple access (SAC-OCDMA) systems. SAC systems are getting more attractive in the field of OCDMA because of its ability to eliminate the influence of multiple access interference (MAI) and also suppress the effect of phase induced intensity noise (PIIN). In this paper, we have proposed new Diagonal Eigenvalue Unity (DEU) code families with ideal in-phase CC based on Jordan block matrix with simple algebraic ways. Four sets of DEU code families based on the code weight W and number of users N for the combination (even, even), (even, odd), (odd, odd) and (odd, even) are constructed. This combination gives DEU code more flexibility in selection of code weight and number of users. These features made this code a compelling candidate for future optical communication systems. Numerical results show that the proposed DEU system outperforms reported codes. In addition, simulation results taken from a commercial optical systems simulator, Virtual Photonic Instrument (VPI™) shown that, using point to multipoint transmission in passive optical network (PON), DEU has better performance and could support long span with high data rate.

Individual eigenvalue distributions of crossover chiral random matrices and low-energy constants of SU(2) × U(1) lattice gauge theory

Science.gov (United States)

Yamamoto, Takuya; Nishigaki, Shinsuke M.

2018-02-01

We compute individual distributions of low-lying eigenvalues of a chiral random matrix ensemble interpolating symplectic and unitary symmetry classes by the Nyström-type method of evaluating the Fredholm Pfaffian and resolvents of the quaternion kernel. The one-parameter family of these distributions is shown to fit excellently the Dirac spectra of SU(2) lattice gauge theory with a constant U(1) background or dynamically fluctuating U(1) gauge field, which weakly breaks the pseudoreality of the unperturbed SU(2) Dirac operator. The observed linear dependence of the crossover parameter with the strength of the U(1) perturbations leads to precise determination of the pseudo-scalar decay constant, as well as the chiral condensate in the effective chiral Lagrangian of the AI class.

CoMFA, CoMSIA and Eigenvalue Analysis on Dibenzodioxepinone and Dibenzodioxocinone Derivatives as Cholesteryl Ester Transfer Protein Inhibitors

Directory of Open Access Journals (Sweden)

Mao-sheng Cheng

2008-08-01

Full Text Available Abstract: CoMFA, CoMSIA and eigenvalue analysis (EVA were performed to study the structural features of 61 diverse dibenzodioxepinone and dibenzodioxocinone analogues to probe cholesteryl ester transfer protein (CETP inhibitory activity. Three methods yielded statistically significant models upon assessment of cross-validation, bootstrapping, and progressive scrambling. This was further validated by an external set of 13 derivatives. Our results demonstrate that three models have a good interpolation as well as extrapolation. The hydrophobic features were confirmed to contribute significantly to inhibitor potencies, while a pre-oriented hydrogen bond provided by the hydroxyl group at the 3-position indicated a good correlation with previous SAR, and a hydrogen bond acceptor may play a crucial role in CETP inhibition. These derived models may help us to gain a deeper understanding of the binding interaction of these lactone-based compounds and aid in the design of new potent compounds against CETP.

Construction of accuracy-preserving surrogate for the eigenvalue radiation diffusion and/or transport problem

Energy Technology Data Exchange (ETDEWEB)

Wang, C.; Abdel-Khalik, H. S. [Dept. of Nuclear Engineering, North Caroline State Univ., Raleigh, NC 27695 (United States)

2012-07-01

The construction of surrogate models for high fidelity models is now considered an important objective in support of all engineering activities which require repeated execution of the simulation, such as verification studies, validation exercises, and uncertainty quantification. The surrogate must be computationally inexpensive to allow its repeated execution, and must be computationally accurate in order for its predictions to be credible. This manuscript introduces a new surrogate construction approach that reduces the dimensionality of the state solution via a range-finding algorithm from linear algebra. It then employs a proper orthogonal decomposition-like approach to solve for the reduced state. The algorithm provides an upper bound on the error resulting from the reduction. Different from the state-of-the-art, the new approach allows the user to define the desired accuracy a priori which controls the maximum allowable reduction. We demonstrate the utility of this approach using an eigenvalue radiation diffusion model, where the accuracy is selected to match machine precision. Results indicate that significant reduction is possible for typical reactor assembly models, which are currently considered expensive given the need to employ very fine mesh many group calculations to ensure the highest possible fidelity for the downstream core calculations. Given the potential for significant reduction in the computational cost, we believe it is possible to rethink the manner in which homogenization theory is currently employed in reactor design calculations. (authors)

Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels

KAUST Repository

Haidar, Azzam

2011-01-01

This paper introduces a novel implementation in reducing a symmetric dense matrix to tridiagonal form, which is the preprocessing step toward solving symmetric eigenvalue problems. Based on tile algorithms, the reduction follows a two-stage approach, where the tile matrix is first reduced to symmetric band form prior to the final condensed structure. The challenging trade-off between algorithmic performance and task granularity has been tackled through a grouping technique, which consists of aggregating fine-grained and memory-aware computational tasks during both stages, while sustaining the application\\'s overall high performance. A dynamic runtime environment system then schedules the different tasks in an out-of-order fashion. The performance for the tridiagonal reduction reported in this paper is unprecedented. Our implementation results in up to 50-fold and 12-fold improvement (130 Gflop/s) compared to the equivalent routines from LAPACK V3.2 and Intel MKL V10.3, respectively, on an eight socket hexa-core AMD Opteron multicore shared-memory system with a matrix size of 24000×24000. Copyright 2011 ACM.

The trace formula and the distribution of eigenvalues of Schroedinger operators on manifolds all of whose geodesics are closed

International Nuclear Information System (INIS)

Schubert, R.

1995-05-01

We investigate the behaviour of the remainder term R(E) in the Weyl formula {nvertical stroke E n ≤E}=Vol(M).E d/2 /[(4π) d/2 Γ(d/2+1)]+R(E) for the eigenvalues E n of a Schroedinger operator on a d-dimensional compact Riemannian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d-1)/2 Θ(√E), where Θ(x) is an almost periodic function of Besicovitch class B 2 which has a limit distribution whose density is a box-shaped function. Furthermore we derive a trace formula and study higher order terms in the asymptotics of the coefficients related to the periodic orbits. The periodicity of the geodesic flow leads to a very simple structure of the trace formula which is the reason why the limit distribution can be computed explicitly. (orig.)

An Experiment of Robust Parallel Algorithm for the Eigenvalue problem of a Multigroup Neutron Diffusion based on modified FETI-DP

Energy Technology Data Exchange (ETDEWEB)

Chang, Jonghwa [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of)

2014-05-15

Parallelization of Monte Carlo simulation is widely adpoted. There are also several parallel algorithms developed for the SN transport theory using the parallel wave sweeping algorithm and for the CPM using parallel ray tracing. For practical purpose of reactor physics application, the thermal feedback and burnup effects on the multigroup cross section should be considered. In this respect, the domain decomposition method(DDM) is suitable for distributing the expensive cross section calculation work. Parallel transport code and diffusion code based on the Raviart-Thomas mixed finite element method was developed. However most of the developed methods rely on the heuristic convergence of flux and current at the domain interfaces. Convergence was not attained in some cases. Mechanical stress computation community has also work on the DDM to solve the stress-strain equation using the finite element methods. The most successful domain decomposition method in terms of robustness is FETI-DP. We have modified the original FETI-DP to solve the eigenvalue problem for the multigroup diffusion problem in this study.

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

Energy Technology Data Exchange (ETDEWEB)

Menke, Lorenz Harry, E-mail: lnz2004@mindspring.com [University of Pittsburgh (United States)

2012-05-15

This paper derives all 36 analytical solutions of the energy eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor for polynomial degrees 1 through 4 using classical algebraic theory. By the use of double-parameterization the full general solution sets are illustrated in a compact, symmetric, structural, and usable form that is valid for asymmetry parameter {eta} is an element of (- {infinity}, + {infinity}). These results are useful for code developers in the area of Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR) and rotational spectroscopy who want to offer exact solutions whenever possible, rather that resorting to numerical solutions. In addition, by using standard linear algebra methods, the characteristic equations of all integer and half-integer spins I from 0 to 15, inclusive are represented in a compact and naturally parameterized form that illustrates structure and symmetries. This extends Nielson's listing of characteristic equations for integer spins out to I = 15, inclusive.

Structural analyses on piping systems of sodium reactors. 2. Eigenvalue analyses of hot-leg pipelines of large scale sodium reactors

International Nuclear Information System (INIS)

Furuhashi, Ichiro; Kasahara, Naoto

2002-01-01

Two types of finite element models analyzed eigenvalues of hot-leg pipelines of a large-scale sodium reactor. One is a beam element model, which is usual for pipe analyses. The other is a shell element model to evaluate particular modes in thin pipes with large diameters. Summary of analysis results: (1) A beam element model and a order natural frequency. A beam element model is available to get the first order vibration mode. (2) The maximum difference ratio of beam mode natural frequencies was 14% between a beam element model with no shear deformations and a shell element model. However, its difference becomes very small, when shear deformations are considered in beam element. (3) In the first order horizontal mode, the Y-piece acts like a pendulum, and the elbow acts like the hinge. The natural frequency is strongly affected by the bending and shear rigidities of the outer supporting pipe. (4) In the first order vertical mode, the vertical sections of the outer and inner pipes moves in the axial-directional piston mode, the horizontal section of inner pipe behaves like the cantilever, and the elbow acts like the hinge. The natural frequency is strongly affected by the axial rigidity of outer supporting pipe. (5) Both effective masses and participation factors were small for particular shell modes. (author)

Eigenvalue Problems.

Science.gov (United States)

1987-06-01

11.55), we get (11.56) 1 bu(u - ZhU)dx < bull - hUi{ !s Ch2V(a), where C = C(aOa 1lbo,blV 0 (a),X). Next we consider I(S-Sh)uIL and Ij(T-Th)uaL2 It is...Math. Ann. 97, 711-736. Courant, R. [1943]: Variational methods for the solution of prob- .-lems of equilibrium and vibrations, Bull . Amer. Math. Soc...fournir des bornes superieures ou inferieures, C.R. Acad. Sci., Paris 235, 995-997. .V Prodi, G. (1962]: Theoremi di tipo locale per il sistema de Navier

Fast matrix factorization algorithm for DOSY based on the eigenvalue decomposition and the difference approximation focusing on the size of observed matrix

International Nuclear Information System (INIS)

Tanaka, Yuho; Uruma, Kazunori; Furukawa, Toshihiro; Nakao, Tomoki; Izumi, Kenya; Utsumi, Hiroaki

2017-01-01

This paper deals with an analysis problem for diffusion-ordered NMR spectroscopy (DOSY). DOSY is formulated as a matrix factorization problem of a given observed matrix. In order to solve this problem, a direct exponential curve resolution algorithm (DECRA) is well known. DECRA is based on singular value decomposition; the advantage of this algorithm is that the initial value is not required. However, DECRA requires a long calculating time, depending on the size of the given observed matrix due to the singular value decomposition, and this is a serious problem in practical use. Thus, this paper proposes a new analysis algorithm for DOSY to achieve a short calculating time. In order to solve matrix factorization for DOSY without using singular value decomposition, this paper focuses on the size of the given observed matrix. The observed matrix in DOSY is also a rectangular matrix with more columns than rows, due to limitation of the measuring time; thus, the proposed algorithm transforms the given observed matrix into a small observed matrix. The proposed algorithm applies the eigenvalue decomposition and the difference approximation to the small observed matrix, and the matrix factorization problem for DOSY is solved. The simulation and a data analysis show that the proposed algorithm achieves a lower calculating time than DECRA as well as similar analysis result results to DECRA. (author)

Krein signature for instability of PT-symmetric states

Science.gov (United States)

Chernyavsky, Alexander; Pelinovsky, Dmitry E.

2018-05-01

Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT-symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.

An eigenvalue approach for the automatic scaling of unknowns in model-based reconstructions: Application to real-time phase-contrast flow MRI.

Science.gov (United States)

Tan, Zhengguo; Hohage, Thorsten; Kalentev, Oleksandr; Joseph, Arun A; Wang, Xiaoqing; Voit, Dirk; Merboldt, K Dietmar; Frahm, Jens

2017-12-01

The purpose of this work is to develop an automatic method for the scaling of unknowns in model-based nonlinear inverse reconstructions and to evaluate its application to real-time phase-contrast (RT-PC) flow magnetic resonance imaging (MRI). Model-based MRI reconstructions of parametric maps which describe a physical or physiological function require the solution of a nonlinear inverse problem, because the list of unknowns in the extended MRI signal equation comprises multiple functional parameters and all coil sensitivity profiles. Iterative solutions therefore rely on an appropriate scaling of unknowns to numerically balance partial derivatives and regularization terms. The scaling of unknowns emerges as a self-adjoint and positive-definite matrix which is expressible by its maximal eigenvalue and solved by power iterations. The proposed method is applied to RT-PC flow MRI based on highly undersampled acquisitions. Experimental validations include numerical phantoms providing ground truth and a wide range of human studies in the ascending aorta, carotid arteries, deep veins during muscular exercise and cerebrospinal fluid during deep respiration. For RT-PC flow MRI, model-based reconstructions with automatic scaling not only offer velocity maps with high spatiotemporal acuity and much reduced phase noise, but also ensure fast convergence as well as accurate and precise velocities for all conditions tested, i.e. for different velocity ranges, vessel sizes and the simultaneous presence of signals with velocity aliasing. In summary, the proposed automatic scaling of unknowns in model-based MRI reconstructions yields quantitatively reliable velocities for RT-PC flow MRI in various experimental scenarios. Copyright © 2017 John Wiley & Sons, Ltd.

Vibrations and Eigenvalues

Indian Academy of Sciences (India)

We make music by causing strings, membranes, or air columns to vibrate. Engineers design safe structures by control- ling vibrations. I will describe to you a very simple vibrating system and the mathematics needed to analyse it. The ideas were born in the work of Joseph-Louis Lagrange (1736–1813), and I begin by quot-.

Vibrations and Eigenvalues

Indian Academy of Sciences (India)

The vibrating string problem is the source of much mathe- matics and physics. ... ing this science [mechanics],and the art of solving the problems pertaining to it, to .... used tools for finding maxima and minima of functions of several variables.

Conditioning analysis of block incomplete factorization and its application to elliptic equations

NARCIS (Netherlands)

Lu, H.; Axelsson, Owe

1997-01-01

The paper deals with eigenvalue estimates for block incomplete fac- torization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the

Preclinical Diagnosis of Magnetic Resonance (MR Brain Images via Discrete Wavelet Packet Transform with Tsallis Entropy and Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM

Directory of Open Access Journals (Sweden)

Yudong Zhang

2015-03-01

Full Text Available Background: Developing an accurate computer-aided diagnosis (CAD system of MR brain images is essential for medical interpretation and analysis. In this study, we propose a novel automatic CAD system to distinguish abnormal brains from normal brains in MRI scanning. Methods: The proposed method simplifies the task to a binary classification problem. We used discrete wavelet packet transform (DWPT to extract wavelet packet coefficients from MR brain images. Next, Shannon entropy (SE and Tsallis entropy (TE were harnessed to obtain entropy features from DWPT coefficients. Finally, generalized eigenvalue proximate support vector machine (GEPSVM, and GEPSVM with radial basis function (RBF kernel, were employed as classifier. We tested the four proposed diagnosis methods (DWPT + SE + GEPSVM, DWPT + TE + GEPSVM, DWPT + SE + GEPSVM + RBF, and DWPT + TE + GEPSVM + RBF on three benchmark datasets of Dataset-66, Dataset-160, and Dataset-255. Results: The 10 repetition of K-fold stratified cross validation results showed the proposed DWPT + TE + GEPSVM + RBF method excelled not only other three proposed classifiers but also existing state-of-the-art methods in terms of classification accuracy. In addition, the DWPT + TE + GEPSVM + RBF method achieved accuracy of 100%, 100%, and 99.53% on Dataset-66, Dataset-160, and Dataset-255, respectively. For Dataset-255, the offline learning cost 8.4430s and online prediction cost merely 0.1059s. Conclusions: We have proved the effectiveness of the proposed method, which achieved nearly 100% accuracy over three benchmark datasets.

Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series

DEFF Research Database (Denmark)

Davis, Richard A.; Mikosch, Thomas Valentin; Pfaffel, Olivier

2016-01-01

In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. The time series constitutes a linear process across time and between components. The input noise of the linear process has regularly varying tails with index α∈(0,4) in...... particular, the time series has infinite fourth moment. We derive the limiting behavior for the largest eigenvalues of the sample covariance matrix and show point process convergence of the normalized eigenvalues. The limiting process has an explicit form involving points of a Poisson process and eigenvalues...... of a non-negative definite matrix. Based on this convergence we derive limit theory for a host of other continuous functionals of the eigenvalues, including the joint convergence of the largest eigenvalues, the joint convergence of the largest eigenvalue and the trace of the sample covariance matrix...

Iterative methods for the detection of Hopf bifurcations in finite element discretisation of incompressible flow problems

International Nuclear Information System (INIS)

Cliffe, K.A.; Garratt, T.J.; Spence, A.

1992-03-01

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalised eigenvalue problems arising from mixed finite element discretisations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and can be used in a scheme to determine the stability of steady state solutions and to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalised eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem to compute approximations to the eigenvalue of smallest real part. Numerical experiments are performed using a model of double diffusive convection. (author)

Random eigenvalue problems revisited

Indian Academy of Sciences (India)

statistical distributions; linear stochastic systems. 1. ... dimensional multivariate Gaussian random vector with mean µ ∈ Rm and covariance ... 5, the proposed analytical methods are applied to a three degree-of-freedom system and the ...... The joint pdf ofω1 andω3 is however close to a bivariate Gaussian density function.

Spectrum of the multigroup neutron transport operator for bounded spatial domains

International Nuclear Information System (INIS)

Larsen, E.W.

1979-01-01

The spectrum of the multigroup neutron transport operator A is studied for bounded spatial regions D which consist of a finite number of material subregions. Our main results provide simple conditions on the material cross sections which guarantee that (1) A possesses eigenvalues in the finite plane; (2) A possesses a ''leading'' eigenvalue lambda 0 which is real, not less than the real part of any other eigenvalue, and to which there corresponds at least one nonnegative eigenfunction psi/sub lambda/0; and (3) A possesses a ''dominant'' eigenvalue lambda 0 which is real, simple, greater than the real part of any other eigenvalue, and whose eigenfunction psi/sub lambda/0 satisfies psi/sub lambda/0> or =0 and ∫psi/sub lambda/0d 2 Ω>0. We give examples to illustrate the results and to show that a leading eigenvalue need not be simple, nor its eigenfunction(s) positive

Generalized mean detector for collaborative spectrum sensing

KAUST Repository

Shakir, Muhammad Zeeshan

2013-04-01

In this paper, a unified generalized eigenvalue based spectrum sensing framework referred to as Generalized mean detector (GMD) has been introduced. The generalization of the detectors namely (i) the eigenvalue ratio detector (ERD) involving the ratio of the largest and the smallest eigenvalues; (ii) the Geometric mean detector (GEMD) involving the ratio of the largest eigenvalue and the geometric mean of the eigenvalues and (iii) the Arithmetic mean detector (ARMD) involving the ratio of the largest and the arithmetic mean of the eigenvalues is explored. The foundation of the proposed unified framework is based on the calculation of exact analytical moments of the random variables of test statistics of the respective detectors. In this context, we approximate the probability density function (PDF) of the test statistics of the respective detectors by Gaussian/Gamma PDF using the moment matching method. Finally, we derive closed-form expressions to calculate the decision threshold of the eigenvalue based detectors by exchanging the derived exact moments of the random variables of test statistics with the moments of the Gaussian/Gamma distribution function. The performance of the eigenvalue based detectors is compared with the traditional detectors such as energy detector (ED) and cyclostationary detector (CSD) and validate the importance of the eigenvalue based detectors particularly over realistic wireless cognitive environments. Analytical and simulation results show that the GEMD and the ARMD yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, our results based on proposed simple and tractable approximation approaches are in perfect agreement with the empirical results. © 1972-2012 IEEE.

Phase diagram of the Dirac spectrum at nonzero chemical potential

International Nuclear Information System (INIS)

Osborn, J. C.; Splittorff, K.; Verbaarschot, J. J. M.

2008-01-01

The Dirac spectrum of QCD with dynamical fermions at nonzero chemical potential is characterized by three regions: a region with a constant eigenvalue density, a region where the eigenvalue density shows oscillations that grow exponentially with the volume and the remainder of the complex plane where the eigenvalue density is zero. In this paper we derive the phase diagram of the Dirac spectrum from a chiral Lagrangian. We show that the constant eigenvalue density corresponds to a pion condensed phase while the strongly oscillating region is given by a kaon condensed phase. The normal phase with nonzero chiral condensate but vanishing Bose condensates coincides with the region of the complex plane where there are no eigenvalues.

Design of a New Concentration Series for the Orthogonal Sample Design Approach and Estimation of the Number of Reactions in Chemical Systems.

Science.gov (United States)

Shi, Jiajia; Liu, Yuhai; Guo, Ran; Li, Xiaopei; He, Anqi; Gao, Yunlong; Wei, Yongju; Liu, Cuige; Zhao, Ying; Xu, Yizhuang; Noda, Isao; Wu, Jinguang

2015-11-01

A new concentration series is proposed for the construction of a two-dimensional (2D) synchronous spectrum for orthogonal sample design analysis to probe intermolecular interaction between solutes dissolved in the same solutions. The obtained 2D synchronous spectrum possesses the following two properties: (1) cross peaks in the 2D synchronous spectra can be used to reflect intermolecular interaction reliably, since interference portions that have nothing to do with intermolecular interaction are completely removed, and (2) the two-dimensional synchronous spectrum produced can effectively avoid accidental collinearity. Hence, the correct number of nonzero eigenvalues can be obtained so that the number of chemical reactions can be estimated. In a real chemical system, noise present in one-dimensional spectra may also produce nonzero eigenvalues. To get the correct number of chemical reactions, we classified nonzero eigenvalues into significant nonzero eigenvalues and insignificant nonzero eigenvalues. Significant nonzero eigenvalues can be identified by inspecting the pattern of the corresponding eigenvector with help of the Durbin-Watson statistic. As a result, the correct number of chemical reactions can be obtained from significant nonzero eigenvalues. This approach provides a solid basis to obtain insight into subtle spectral variations caused by intermolecular interaction.

Spectral asymmetry of the massless Dirac operator on a 3-torus

International Nuclear Information System (INIS)

Downes, Robert J.; Vassiliev, Dmitri; Levitin, Michael

2013-01-01

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant

Statistical properties of cross-correlation in the Korean stock market

Science.gov (United States)

Oh, G.; Eom, C.; Wang, F.; Jung, W.-S.; Stanley, H. E.; Kim, S.

2011-01-01

We investigate the statistical properties of the cross-correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the cross-correlation matrix is positively skewed and changes over time. We find that the eigenvalue distribution of original cross-correlation matrix deviates from the eigenvalues predicted by the RMT, and the largest eigenvalue is 52 times larger than the maximum value among the eigenvalues predicted by the RMT. The β_{473} coefficient, which reflect the largest eigenvalue property, is 0.8, while one of the eigenvalues in the RMT is approximately zero. Notably, we show that the entropy function E(σ) with the portfolio risk σ for the original and filtered cross-correlation matrices are consistent with a power-law function, E( σ) σ^{-γ}, with the exponent γ 2.92 and those for Asian currency crisis decreases significantly.

Application of Nondimensional Dynamic Influence Function Method for Eigenmode Analysis of Two-Dimensional Acoustic Cavities

Directory of Open Access Journals (Sweden)

S. W. Kang

2014-04-01

Full Text Available This paper establishes an improved NDIF method for the eigenvalue extraction of two-dimensional acoustic cavities with arbitrary shapes. The NDIF method, which was introduced by the authors in 1999, gives highly accurate eigenvalues despite employing a small number of nodes. However, it needs the inefficient procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues and mode shapes. The paper proposes a practical approach for overcoming the inefficient procedure by making the final system matrix equation of the NDIF method into a form of algebraic eigenvalue problem. The solution quality of the proposed method is investigated by obtaining the eigenvalues and mode shapes of a circular, a rectangular, and an arbitrarily shaped cavity.

Numerical solutions of the monoenergetic neutron transport equation with anisotropic scattering

International Nuclear Information System (INIS)

Dahl, B.

1985-01-01

The Boltzmann equation for monoenergetic neutrons has been solved numerically with high accuracy for homogeneous slabs and spheres with various degree of linear anisotropy. Vacuum boundary conditions are used. The numerical method is based on previous work by Carlvik. Benchmark values of the criticality factor and higher order eigenvalues are given for multiplying systems of thickness or diameter from 10 -5 to 20 mean free paths and with anisotropy coefficients from 0.0 to 0.3. For slab geometry, both even and odd mode eigenvalues are treated. With increasing anisotropy, an increasing number of complex eigenvalues is observer. The total flux is calculated from the eigenvector and tables of the fundamental mode flux are given. Accurate extrapolation distances are derived for various dimensions and anisotropy coefficients from our eigenvalue results on slabs and spheres and from the work by Sanchez on infinite cylinders.The time eigenvalue spectrum in subcritical systems has also been studied. First, the connection between the eigenvalues arising from the time dependent and stationary transport equation is established. Based on this, the spectrum of real time eigenvalues in slabs and spheres is calculated. For spheres, the existence of complex time eigenvalues in the region beyond the value corresponding to the Corngold limit is numerically established. The presence of such eigenvalues has earlier not been proved. It is further shown that the Boltzmann equation for a sphere is significantly simplified when the decay constant is at the Corngold limit. The spectrum of sphere diameters corresponding to this decay constant is calculated for various linear anisotropies, and detailed numerical results are given. (Author)

Multicomponent diffusion in basaltic melts at 1350 °C

Science.gov (United States)

Guo, Chenghuan; Zhang, Youxue

2018-05-01

Nine successful diffusion couple experiments were conducted in an 8-component SiO2-TiO2-Al2O3-FeO-MgO-CaO-Na2O-K2O system at ∼1350 °C and at 1 GPa, to study multicomponent diffusion in basaltic melts. At least 3 traverses were measured to obtain diffusion profiles for each experiment. Multicomponent diffusion matrix at 1350 °C was obtained by simultaneously fitting diffusion profiles of diffusion couple experiments. Furthermore, in order to better constrain the diffusion matrix and reconcile mineral dissolution data, mineral dissolution experiments in the literature and diffusion couple experiments from this study, were fit together. All features of diffusion profiles in both diffusion couple and mineral dissolution experiments were well reproduced by the diffusion matrix. Diffusion mechanism is inferred from eigenvectors of the diffusion matrix, and it shows that the diffusive exchange between network-formers SiO2 and Al2O3 is the slowest, the exchange of SiO2 with other oxide components is the second slowest with an eigenvalue that is only ∼10% larger, then the exchange between divalent oxide components and all the other oxide components is the third slowest with an eigenvalue that is twice the smallest eigenvalue, then the exchange of FeO + K2O with all the other oxide components is the fourth slowest with an eigenvalue that is 5 times the smallest eigenvalue, then the exchange of MgO with FeO + CaO is the third fastest with an eigenvalue that is 6.3 times the smallest eigenvalue, then the exchange of CaO + K2O with all the other oxide components is the second fastest with an eigenvalue that is 7.5 times the smallest eigenvalue, and the exchange of Na2O with all other oxide components is the fastest with an eigenvalue that is 31 times the smallest eigenvalue. The slowest and fastest eigenvectors are consistent with those for simpler systems in most literature. The obtained diffusion matrix was successfully applied to predict diffusion profiles during

Two-faced property of a market factor in asset pricing and diversification effect

Science.gov (United States)

Eom, Cheoljun

2017-04-01

This study empirically investigates the test hypothesis that a market factor acting as a representative common factor in the pricing models has a negative influence on constructing a well-diversified portfolio from the Markowitz mean-variance optimization function (MVOF). We use the comparative correlation matrix (C-CM) method to control a single eigenvalue among all eigenvalues included in the sample correlation matrix (S-CM), through the random matrix theory (RMT). In particular, this study observes the effect of the largest eigenvalue that has the property of the market factor. According to the results, the largest eigenvalue has the highest explanatory power on the stock return changes. The C-CM without the largest eigenvalue in the S-CM constructs a more diversified portfolio capable of improving the practical applicability of the MVOF. Moreover, the more diversified portfolio constructed from this C-CM has better out-of-sample performance in the future period. These results support the test hypothesis for the two-faced property of the market factor, defined by the largest eigenvalue.

Cucheb: A GPU implementation of the filtered Lanczos procedure

Science.gov (United States)

Aurentz, Jared L.; Kalantzis, Vassilis; Saad, Yousef

2017-11-01

This paper describes the software package Cucheb, a GPU implementation of the filtered Lanczos procedure for the solution of large sparse symmetric eigenvalue problems. The filtered Lanczos procedure uses a carefully chosen polynomial spectral transformation to accelerate convergence of the Lanczos method when computing eigenvalues within a desired interval. This method has proven particularly effective for eigenvalue problems that arise in electronic structure calculations and density functional theory. We compare our implementation against an equivalent CPU implementation and show that using the GPU can reduce the computation time by more than a factor of 10. Program Summary Program title: Cucheb Program Files doi:http://dx.doi.org/10.17632/rjr9tzchmh.1 Licensing provisions: MIT Programming language: CUDA C/C++ Nature of problem: Electronic structure calculations require the computation of all eigenvalue-eigenvector pairs of a symmetric matrix that lie inside a user-defined real interval. Solution method: To compute all the eigenvalues within a given interval a polynomial spectral transformation is constructed that maps the desired eigenvalues of the original matrix to the exterior of the spectrum of the transformed matrix. The Lanczos method is then used to compute the desired eigenvectors of the transformed matrix, which are then used to recover the desired eigenvalues of the original matrix. The bulk of the operations are executed in parallel using a graphics processing unit (GPU). Runtime: Variable, depending on the number of eigenvalues sought and the size and sparsity of the matrix. Additional comments: Cucheb is compatible with CUDA Toolkit v7.0 or greater.

Semi-analytical Karhunen-Loeve representation of irregular waves based on the prolate spheroidal wave functions

Science.gov (United States)

Lee, Gibbeum; Cho, Yeunwoo

2018-01-01

A new semi-analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in Karhunen-Loeve (K-L) representation of random data such as irregular ocean waves. Instead of direct numerical approach to this matrix eigenvalue problem, which may suffer from the computational inaccuracy for big data, a pair of integral and differential equations are considered, which are related to the so-called prolate spheroidal wave functions (PSWF). First, the PSWF is expressed as a summation of a small number of the analytical Legendre functions. After substituting them into the PSWF differential equation, a much smaller size matrix eigenvalue problem is obtained than the direct numerical K-L matrix eigenvalue problem. By solving this with a minimal numerical effort, the PSWF and the associated eigenvalue of the PSWF differential equation are obtained. Then, the eigenvalue of the PSWF integral equation is analytically expressed by the functional values of the PSWF and the eigenvalues obtained in the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K-L integral equation for the representation of exemplary wave data such as ordinary irregular waves. It is found that, with the same accuracy, the required memory size of the present method is smaller than that of the direct numerical K-L representation and the computation time of the present method is shorter than that of the semi-analytical method based on the sinusoidal functions.

The Green function in the secondary model of thermalization; La funcion de GREEN en el Modelo secundario de Termalizacion

Energy Technology Data Exchange (ETDEWEB)

Guasp, J

1972-07-01

The Green Function of the thermalization problem is studied in the secondary model case through the spatial Fourier transform. A relation between singularities and eigenvalues allows the determination of the analyticity dominion. The eigenvalue spectrum has a purely discrete part, laying on an interval of the imaginary axis of the K complex plane (the Fourier parameter), and another part, purely continuous, laying in the reminder of the imaginary axis. A correspondence between discrete eigenvalues and exponential modes of the Green Function, extemal properties for the eigenvalues and some remarkable properties of the eigenfunctions are established. (Author) 32 refs.

Spectral inversion of an indefinite Sturm-Liouville problem due to Richardson

International Nuclear Information System (INIS)

Shanley, Paul E

2009-01-01

We study an indefinite Sturm-Liouville problem due to Richardson whose complicated eigenvalue dependence on a parameter has been a puzzle for decades. In atomic physics a process exists that inverts the usual Schroedinger situation of an energy eigenvalue depending on a coupling parameter into the so-called Sturmian problem where the coupling parameter becomes the eigenvalue which then depends on the energy. We observe that the Richardson equation is of the Sturmian type. This means that the Richardson and its related Schroedinger eigenvalue functions are inverses of each other and that the Richardson spectrum is therefore no longer a puzzle

A new approach to the method of source-sink potentials for molecular conduction

Energy Technology Data Exchange (ETDEWEB)

Pickup, Barry T., E-mail: B.T.Pickup@sheffield.ac.uk, E-mail: P.W.Fowler@sheffield.ac.uk; Fowler, Patrick W., E-mail: B.T.Pickup@sheffield.ac.uk, E-mail: P.W.Fowler@sheffield.ac.uk; Borg, Martha [Department of Chemistry, University of Sheffield, Sheffield S3 7HF (United Kingdom); Sciriha, Irene [Department of Mathematics, University of Malta, Msida (Malta)

2015-11-21

We re-derive the tight-binding source-sink potential (SSP) equations for ballistic conduction through conjugated molecular structures in a form that avoids singularities. This enables derivation of new results for families of molecular devices in terms of eigenvectors and eigenvalues of the adjacency matrix of the molecular graph. In particular, we define the transmission of electrons through individual molecular orbitals (MO) and through MO shells. We make explicit the behaviour of the total current and individual MO and shell currents at molecular eigenvalues. A rich variety of behaviour is found. A SSP device has specific insulation or conduction at an eigenvalue of the molecular graph (a root of the characteristic polynomial) according to the multiplicities of that value in the spectra of four defined device polynomials. Conduction near eigenvalues is dominated by the transmission curves of nearby shells. A shell may be inert or active. An inert shell does not conduct at any energy, not even at its own eigenvalue. Conduction may occur at the eigenvalue of an inert shell, but is then carried entirely by other shells. If a shell is active, it carries all conduction at its own eigenvalue. For bipartite molecular graphs (alternant molecules), orbital conduction properties are governed by a pairing theorem. Inertness of shells for families such as chains and rings is predicted by selection rules based on node counting and degeneracy.

Black Hole Entropy with and without Log Correction in Loop Quantum Gravity

International Nuclear Information System (INIS)

Mitra, P.

2014-01-01

Earlier calculations of black hole entropy in loop quantum gravity have given a term proportional to the area with a correction involving the logarithm of the area when the area eigenvalue is close to the classical area. However the calculations yield an entropy proportional to the area eigenvalue with no such correction when the area eigenvalue is large compared to the classical area

Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian

Science.gov (United States)

Bender, Carl M.; Brody, Dorje C.

2018-04-01

The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.

Complex coordinate rotation and relativistic Hylleraas-CI: helium isoelectronic series

International Nuclear Information System (INIS)

Pestka, G; Bylicki, M; Karwowski, J

2007-01-01

A combination of the Hylleraas-CI (Hy-CI) and complex coordinate rotation (CCR) methods has been applied to study the nuclear charge dependence of the eigenvalues of the Dirac-Coulomb (DC) Hamiltonian corresponding to the ground states of helium isoelectronic series atoms. It has been shown that the CCR, due to the separation of the localized states from the unphysical Brown-Ravenhall continuum, removes the instabilities of the bound-state eigenvalues observed in large-basis set Hy-CI results. The Hy-CI-CCR results are in very good agreement with the most accurate ones available in the literature. Surprisingly, the difference between the DC Hy-CI-CCR eigenvalues and the eigenvalues of the positive-energy projected no-pair Hamiltonian is equal, up to the numerical accuracy of the results, to (Zα) 3 /6π, i.e. to (Zα) 3 relativistic many-body perturbation theory contribution for electron-electron Coulomb interaction operator. An excellent agreement between the Hy-CI-CCR eigenvalues shifted by (Zα) 3 /6π and the no-pair ones confirms the very high accuracy achieved in both approaches. The numerical accuracy of the Hy-CI-CCR DC eigenvalues is estimated to eight significant figures

Mixed-mode loading of the structural elements with defect

Directory of Open Access Journals (Sweden)

Larisa V. Stepanova

2015-06-01

Full Text Available In the article the problem of determining the stress-strain state near the mixed-mode crack tip in a power-law material under plane stress conditions is considered. The eigenfunction method is used for the mixed-mode crack tip problem. It is shown that the eigenfunction expansion method results in the nonlinear eigenvalue problem. The numeric solution of the nonlinear eigenvalue problem formulated is obtained. The power of the distance from the crack tip is the eigenvalue of the nonlinear eigenvalue problem considered whereas the angular distributions of the stress components are the eigenfunctions. The new eigenvalues different from the eigenvalues of the Hutchinson–Rice–Rosengren are found. It is shown that the new asymptotic solution can be interpreted as the self-similar intermediate asymptotics of the stress field in the vicinity of the crack tip at distances which are very small compared to the crack length or the size of the specimen and at distances which are large compared to the length of the completely damaged zone. The developed method allows us to construct the geometry of the completely damaged zone in vicinity of the crack tip.

The construction of partner potential from the general potential anharmonic in D-dimensional Schrodinger system

Science.gov (United States)

Suparmi; Cari, C.; Wea, K. N.; Wahyulianti

2018-03-01

The Schrodinger equation is the fundamental equation in quantum physics. The characteristic of the particle in physics potential field can be explained by using the Schrodinger equation. In this study, the solution of 4 dimensional Schrodinger equation for the anharmonic potential and the anharmonic partner potential have done. The method that used to solve the Schrodinger equation was the ansatz wave method, while to construction the partner potential was the supersymmetric method. The construction of partner potential used to explain the experiment result that cannot be explained by the original potential. The eigenvalue for anharmonic potential and the anharmonic partner potential have the same characteristic. Every increase of quantum orbital number the eigenvalue getting smaller. This result corresponds to Bohrn’s atomic theory that the eigenvalue is inversely proportional to the atomic shell. But the eigenvalue for the anharmonic partner potential higher than the eigenvalue for the anharmonic original potential.

Theoretical study of lithium clusters by electronic stress tensor

International Nuclear Information System (INIS)

Ichikawa, Kazuhide; Nozaki, Hiroo; Komazawa, Naoya; Tachibana, Akitomo

2012-01-01

We study the electronic structure of small lithium clusters Li_n (n = 2 ∼ 8) using the electronic stress tensor. We find that the three eigenvalues of the electronic stress tensor of the Li clusters are negative and degenerate, just like the stress tensor of liquid. This leads us to propose that we may characterize a metallic bond in terms of the electronic stress tensor. Our proposal is that in addition to the negativity of the three eigenvalues of the electronic stress tensor, their degeneracy characterizes some aspects of the metallic nature of chemical bonding. To quantify the degree of degeneracy, we use the differential eigenvalues of the electronic stress tensor. By comparing the Li clusters and hydrocarbon molecules, we show that the sign of the largest eigenvalue and the differential eigenvalues could be useful indices to evaluate the metallicity or covalency of a chemical bond.

Improved non-dimensional dynamic influence function method for vibration analysis of arbitrarily shaped plates with clamped edges

Directory of Open Access Journals (Sweden)

Sang-Wook Kang

2016-03-01

Full Text Available A new formulation for the non-dimensional dynamic influence function method, which was developed by the authors, is proposed to efficiently extract eigenvalues and mode shapes of clamped plates with arbitrary shapes. Compared with the finite element and boundary element methods, the non-dimensional dynamic influence function method yields highly accurate solutions in eigenvalue analysis problems of plates and membranes including acoustic cavities. However, the non-dimensional dynamic influence function method requires the uneconomic procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues because it produces a non-algebraic eigenvalue problem. This article describes a new approach that reduces the problem of free vibrations of clamped plates to an algebraic eigenvalue problem, the solution of which is straightforward. The validity and efficiency of the proposed method are illustrated through several numerical examples.

Diffusion escape through a cluster of small absorbing windows

Energy Technology Data Exchange (ETDEWEB)

Holcman, D [Department of Mathematics, Weizmann Institute of Science, Rehovot 76100 (Israel); Schuss, Z [Department of Mathematics, Tel-Aviv University, Tel-Aviv 69978 (Israel)

2008-04-18

We study the first eigenvalue of the Laplace equation in a bounded domain in R{sup d} (d=2,3) with mixed Neumann-Dirichlet (Zaremba) boundary conditions. The Neumann condition is imposed on most of the boundary and the Dirichlet boundary consists of a cluster of small windows. When the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problems with a single Dirichlet window. However, when two or more Dirichlet windows cluster tightly together they interact nonlinearly. We compare our asymptotic approximation of the eigenvalue to the escape rate of simulated Brownian particles through the small windows.

Spectral properties of the temporal evolution of brain network structure.

Science.gov (United States)

Wang, Rong; Zhang, Zhen-Zhen; Ma, Jun; Yang, Yong; Lin, Pan; Wu, Ying

2015-12-01

The temporal evolution properties of the brain network are crucial for complex brain processes. In this paper, we investigate the differences in the dynamic brain network during resting and visual stimulation states in a task-positive subnetwork, task-negative subnetwork, and whole-brain network. The dynamic brain network is first constructed from human functional magnetic resonance imaging data based on the sliding window method, and then the eigenvalues corresponding to the network are calculated. We use eigenvalue analysis to analyze the global properties of eigenvalues and the random matrix theory (RMT) method to measure the local properties. For global properties, the shifting of the eigenvalue distribution and the decrease in the largest eigenvalue are linked to visual stimulation in all networks. For local properties, the short-range correlation in eigenvalues as measured by the nearest neighbor spacing distribution is not always sensitive to visual stimulation. However, the long-range correlation in eigenvalues as evaluated by spectral rigidity and number variance not only predicts the universal behavior of the dynamic brain network but also suggests non-consistent changes in different networks. These results demonstrate that the dynamic brain network is more random for the task-positive subnetwork and whole-brain network under visual stimulation but is more regular for the task-negative subnetwork. Our findings provide deeper insight into the importance of spectral properties in the functional brain network, especially the incomparable role of RMT in revealing the intrinsic properties of complex systems.

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree-Fock.

Science.gov (United States)

Hesselmann, Andreas; Görling, Andreas

2011-01-21

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.

Universality, marginal operators, and limit cycles

International Nuclear Information System (INIS)

Glazek, Stanislaw D.; Wilson, Kenneth G.

2004-01-01

The universality of renormalization-group limit-cycle behavior is illustrated with a simple discrete Hamiltonian model. A nonperturbative renormalization-group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound-state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero--also a consequence of the limit cycle. Wegner's eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegner's eigenvalues are independent of location on the limit cycle, in contrast with Wegner's operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonian's bound-state eigenvalues converge much more quickly than otherwise to an exact geometric series

Improved non-dimensional dynamic influence function method based on tow-domain method for vibration analysis of membranes

Directory of Open Access Journals (Sweden)

SW Kang

2015-02-01

Full Text Available This article introduces an improved non-dimensional dynamic influence function method using a sub-domain method for efficiently extracting the eigenvalues and mode shapes of concave membranes with arbitrary shapes. The non-dimensional dynamic influence function method (non-dimensional dynamic influence function method, which was developed by the authors in 1999, gives highly accurate eigenvalues for membranes, plates, and acoustic cavities, compared with the finite element method. However, it needs the inefficient procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues and mode shapes. To overcome the inefficient procedure, this article proposes a practical approach to make the system matrix equation of the concave membrane of interest into a form of algebraic eigenvalue problem. It is shown by several case studies that the proposed method has a good convergence characteristics and yields very accurate eigenvalues, compared with an exact method and finite element method (ANSYS.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance.

Science.gov (United States)

Wang, Weichen; Fan, Jianqing

2017-06-01

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies.

Thick-Restart Lanczos Method for Electronic Structure Calculations

International Nuclear Information System (INIS)

Simon, Horst D.; Wang, L.-W.; Wu, Kesheng

1999-01-01

This paper describes two recent innovations related to the classic Lanczos method for eigenvalue problems, namely the thick-restart technique and dynamic restarting schemes. Combining these two new techniques we are able to implement an efficient eigenvalue problem solver. This paper will demonstrate its effectiveness on one particular class of problems for which this method is well suited: linear eigenvalue problems generated from non-self-consistent electronic structure calculations

Track 4: basic nuclear science variance reduction for Monte Carlo criticality simulations. 2. Assessment of MCNP Statistical Analysis of keff Eigenvalue Convergence with an Analytical Criticality Verification Test Set

International Nuclear Information System (INIS)

Sood, Avnet; Forster, R. Arthur; Parsons, D. Kent

2001-01-01

Monte Carlo simulations of nuclear criticality eigenvalue problems are often performed by general purpose radiation transport codes such as MCNP. MCNP performs detailed statistical analysis of the criticality calculation and provides feedback to the user with warning messages, tables, and graphs. The purpose of the analysis is to provide the user with sufficient information to assess spatial convergence of the eigenfunction and thus the validity of the criticality calculation. As a test of this statistical analysis package in MCNP, analytic criticality verification benchmark problems have been used for the first time to assess the performance of the criticality convergence tests in MCNP. The MCNP statistical analysis capability has been recently assessed using the 75 multigroup criticality verification analytic problem test set. MCNP was verified with these problems at the 10 -4 to 10 -5 statistical error level using 40 000 histories per cycle and 2000 active cycles. In all cases, the final boxed combined k eff answer was given with the standard deviation and three confidence intervals that contained the analytic k eff . To test the effectiveness of the statistical analysis checks in identifying poor eigenfunction convergence, ten problems from the test set were deliberately run incorrectly using 1000 histories per cycle, 200 active cycles, and 10 inactive cycles. Six problems with large dominance ratios were chosen from the test set because they do not achieve the normal spatial mode in the beginning of the calculation. To further stress the convergence tests, these problems were also started with an initial fission source point 1 cm from the boundary thus increasing the likelihood of a poorly converged initial fission source distribution. The final combined k eff confidence intervals for these deliberately ill-posed problems did not include the analytic k eff value. In no case did a bad confidence interval go undetected. Warning messages were given signaling that

Transmission eigenvalues and thermoacoustic tomography

International Nuclear Information System (INIS)

Finch, David; Hickmann, Kyle S

2013-01-01

The spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. Under a non-trapping hypothesis, we show that sparsity of the interior transmission spectrum implies a range separation condition for the thermoacoustic operator. In odd dimensions greater than or equal to 3, we prove that the interior transmission spectrum for a pair of radially symmetric non-trapping sound speeds is countable, and conclude that the ranges of the associated thermoacoustic maps have only trivial intersection. (paper)

Generalized eigenvalue based spectrum sensing

KAUST Repository

Shakir, Muhammad; Alouini, Mohamed-Slim

2012-01-01

of the decision threshold of the respective detectors. The decision threshold has been calculated in a closed form which is based on the approximation of Cumulative Distribution Functions (CDFs) of the respective test statistics. In this context, we exchange

Broadband Utilization of Eigenvalue Techniques

Directory of Open Access Journals (Sweden)

Z. Kus

1996-06-01

Full Text Available A large number of signal processing problems are concerned with estimating unknown parameters from sensor array measurement. This paper presents simple algorithm for estimating spatio-temporal spectrum of signals received by passive array derivated from MUSIC algorithm, based on eigenstructure of covariance matrices of received signals. Some simulation results are presented.

Influence of electron correlation and degeneracy on the Fukui matrix and extension of frontier molecular orbital theory to correlated quantum chemical methods.

Science.gov (United States)

Bultinck, Patrick; Van Neck, Dimitri; Acke, Guillaume; Ayers, Paul W

2012-02-21

The Fukui function is considered as the diagonal element of the Fukui matrix in position space, where the Fukui matrix is the derivative of the one particle density matrix (1DM) with respect to the number of electrons. Diagonalization of the Fukui matrix, expressed in an orthogonal orbital basis, explains why regions in space with negative Fukui functions exist. Using a test set of molecules, electron correlation is found to have a remarkable effect on the eigenvalues of the Fukui matrix. The Fukui matrices at the independent electron model level are mathematically proven to always have an eigenvalue equal to exactly unity while the rest of the eigenvalues possibly differ from zero but sum to zero. The loss of idempotency of the 1DM at correlated levels of theory causes the loss of these properties. The influence of electron correlation is examined in detail and the frontier molecular orbital concept is extended to correlated levels of theory by defining it as the eigenvector of the Fukui matrix with the largest eigenvalue. The effect of degeneracy on the Fukui matrix is examined in detail, revealing that this is another way by which the unity eigenvalue and perfect pairing of eigenvalues can disappear.

Eigenvectors determination of the ribosome dynamics model during mRNA translation using the Kleene Star algorithm

Science.gov (United States)

Ernawati; Carnia, E.; Supriatna, A. K.

2018-03-01

Eigenvalues and eigenvectors in max-plus algebra have the same important role as eigenvalues and eigenvectors in conventional algebra. In max-plus algebra, eigenvalues and eigenvectors are useful for knowing dynamics of the system such as in train system scheduling, scheduling production systems and scheduling learning activities in moving classes. In the translation of proteins in which the ribosome move uni-directionally along the mRNA strand to recruit the amino acids that make up the protein, eigenvalues and eigenvectors are used to calculate protein production rates and density of ribosomes on the mRNA. Based on this, it is important to examine the eigenvalues and eigenvectors in the process of protein translation. In this paper an eigenvector formula is given for a ribosome dynamics during mRNA translation by using the Kleene star algorithm in which the resulting eigenvector formula is simpler and easier to apply to the system than that introduced elsewhere. This paper also discusses the properties of the matrix {B}λ \\otimes n of model. Among the important properties, it always has the same elements in the first column for n = 1, 2,… if the eigenvalue is the time of initiation, λ = τin , and the column is the eigenvector of the model corresponding to λ.

Omega-mode perturbation theory and reactor kinetics for analyzing accelerator-driven subcritical systems

International Nuclear Information System (INIS)

Ren-Tai, Chiang

2003-01-01

An ω-mode first-order perturbation theory is developed for analyzing the time- and space-dependent neutron behavior in Accelerator-Driven Subcritical Systems (ADSS). The generalized point-kinetics equations are systematically derived using the ω-mode first-order perturbation theory and Fredholm Alternative Theorem. Seven sets of the ω-mode eigenvalues exist with using six groups of delayed neutrons and all ω eigenvalues are negative in ADSS. Seven ω-mode adjoint and forward eigenfunctions are employed to form the point-kinetic parameters. The neutron flux is expressed as a linear combination of the products of seven ω-eigenvalue-mode shape functions and their corresponding time functions up to the first order terms, and the lowest negative ω-eigenvalue mode is the dominant mode. (author)

Method for computing self-consistent solution in a gun code

Science.gov (United States)

Nelson, Eric M

2014-09-23

Complex gun code computations can be made to converge more quickly based on a selection of one or more relaxation parameters. An eigenvalue analysis is applied to error residuals to identify two error eigenvalues that are associated with respective error residuals. Relaxation values can be selected based on these eigenvalues so that error residuals associated with each can be alternately reduced in successive iterations. In some examples, relaxation values that would be unstable if used alone can be used.

Right-Hand Side Dependent Bounds for GMRES Applied to Ill-Posed Problems

KAUST Repository

Pestana, Jennifer

2014-01-01

© IFIP International Federation for Information Processing 2014. In this paper we apply simple GMRES bounds to the nearly singular systems that arise in ill-posed problems. Our bounds depend on the eigenvalues of the coefficient matrix, the right-hand side vector and the nonnormality of the system. The bounds show that GMRES residuals initially decrease, as residual components associated with large eigenvalues are reduced, after which semi-convergence can be expected because of the effects of small eigenvalues.

WKB analysis of PT-symmetric Sturm–Liouville problems

International Nuclear Information System (INIS)

Bender, Carl M; Jones, Hugh F

2012-01-01

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schrödinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm–Liouville problems, the eigenvalues of the PT-symmetric Sturm–Liouville problem grow like n 2 for large n. However, the novelty is that a PT eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’. (paper)

Studies of density functional methods for finite and extended electron systems

International Nuclear Information System (INIS)

Pedroza, A.C.

1984-11-01

In this thesis approximate density-functional (DF) schemes have been studied. The exact DF effective potential and orbital eigenvalues have been obtained for the first time for a number of atomic systems and compared with their approximate counterparts. A general finding is that the exact DF eigenvalues lie above the corresponding excitation energies for deeper shells and below the excitation energies for unoccupied shells. However, the differences between the true DF eigenvalues and the corresponding excitation energies were found to be smaller than the error in the local-density (LD) approximation to the DF eigenvalues. Non-local approximations by Langreth, Perdew, and Mehl (LPM) and by Gunnarsson and co-workers have been tested and were found to substantially improve the total energies compared to the LD approximation but to give less improvements for the DF potential and DF eigenvalues. As a further test of the LPM scheme the beryllium metal has been studied, and the calculated cohesive energy and structure factor have been compared with experiments and with results from the LD approximation. (author)

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

Energy Technology Data Exchange (ETDEWEB)

Vecharynski, Eugene [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Brabec, Jiri [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Shao, Meiyue [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Govind, Niranjan [Pacific Northwest National Lab. (PNNL), Richland, WA (United States). Environmental Molecular Sciences Lab.; Yang, Chao [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division

2017-12-01

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.

Spectral properties of Google matrix of Wikipedia and other networks

Science.gov (United States)

Ermann, Leonardo; Frahm, Klaus M.; Shepelyansky, Dima L.

2013-05-01

We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method, we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.

Infrared behavior of the Faddeev-Popov operator in Coulomb gauge QCD

International Nuclear Information System (INIS)

Nakagawa, Y.; Saito, T.; Toki, H.; Nakamura, A.

2007-01-01

We calculate the eigenvalue distribution of the Faddeev-Popov operator in Coulomb gauge QCD using quenched SU(3) lattice simulation. In the confinement phase, the density of the low-lying eigenvalues increases with lattice volume, and the confinement criterion is satisfied. Moreover, even in the deconfinement phase, the behavior of the FP eigenvalue density is qualitatively the same as in the confinement phase. This is consistent with the fact that the color-Coulomb potential is not screened in the deconfined phase

Spectral theory of Sturm-Liouville differential operators: proceedings of the 1984 workshop

Energy Technology Data Exchange (ETDEWEB)

Kaper, H.G.; Zettl, A. (eds.)

1984-12-01

This report contains the proceedings of the workshop which was held at Argonne during the period May 14 through June 15, 1984. The report contains 22 articles, authored or co-authored by the participants in the workshop. Topics covered at the workshop included the asymptotics of eigenvalues and eigenfunctions; qualitative and quantitative aspects of Sturm-Liouville eigenvalue problems with discrete and continuous spectra; polar, indefinite, and nonselfadjoint Sturm-Liouville eigenvalue problems; and systems of differential equations of Sturm-Liouville type.

The Schroedinger equation as a singular perturbation problem

International Nuclear Information System (INIS)

Jager, E.M. de; Kuepper, T.

1978-01-01

Comparisons are made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problem connected with the one dimensional Schroedinger equation in Hilbert space. The difference of the eigenvalues is estimated by applying Weyl's monotonicity principle and the minimum maximum principle. The difference of the eigenfunctions is estimated in L 2 norm and in maximum norm obtained by using simple tools from operator theory in Hilbert spaces. An application concerning perturbations of the Planck ideal linear oscillator is given. (author)

On the solution of the differential equation occurring in the problem of heat convection in laminar flow through a tube with slip—flow

Directory of Open Access Journals (Sweden)

Xanming Wang

1996-01-01

Full Text Available A technique is developed for evaluation of eigenvalues in solution of the differential equation d2y/dr2+(1/rdy/dr+λ2(β−r2y=0 which occurs in the problem of heat convection in laminar flow through a circular tube with silp-flow (β>1. A series solution requires the expansions of coeffecients involving extremely large numbers. No work has been reported in the case of β>1, because of its computational complexity in the evaluation of the eigenvalues. In this paper, a matrix was constructed and a computational algorithm was obtained to calculate the first four eigenvalues. Also, an asymptotic formula was developed to generate the full spectrum of eigenvalues. The computational results for various values of β were obtained.

Matrix calculus

CERN Document Server

Bodewig, E

1959-01-01

Matrix Calculus, Second Revised and Enlarged Edition focuses on systematic calculation with the building blocks of a matrix and rows and columns, shunning the use of individual elements. The publication first offers information on vectors, matrices, further applications, measures of the magnitude of a matrix, and forms. The text then examines eigenvalues and exact solutions, including the characteristic equation, eigenrows, extremum properties of the eigenvalues, bounds for the eigenvalues, elementary divisors, and bounds for the determinant. The text ponders on approximate solutions, as well

Ballooning Representation Approach to Low-Frequency Instabilities in Stellarators

International Nuclear Information System (INIS)

Dewar, R.L.; Gardner, H.J.; Lewandowski, J.; Persson, M.

1995-01-01

Local ideal MHD ballooning eigenvalues have been calculated on many field lines for heliac and torsatron cases using a parallel implementation of a ballooning code on a Thinking Machines Corporation CM-5 Global eigenvalues have been estimated for the torsatron test case using the ray tracing method of Dewar and Glasser and also by using the TERPSI-CHORE global eigenvalue code, with good agreement. As a preliminary to detailed study of H-1, 3-D visualizations of stability-related quantities have been produced. 6 refs

On the discrete Frobenius-Perron operator of the Bernoulli map

International Nuclear Information System (INIS)

Bai Zaiqiao

2006-01-01

We study the spectra of a finite-dimensional Frobenius-Perron operator (matrix) of the Bernoulli map derived from phase space discretization. The eigenvalues and (right and left) eigenvectors are analytically calculated, which are closely related to periodic orbits on the partition points. In the degenerate case, Jordan decomposition of the matrix is explicitly constructed. Except for the isolated eigenvalue 1, there is no definite limit with respect to eigenvalues when n → ∞. The behaviour of the eigenvectors is discussed in the limit of large n

Spectral asymptotic in the large coupling limit

CERN Document Server

Bruneau, V

2002-01-01

In this paper, we study a singular perturbation of an eigenvalues problem related to supra-conductor wave guides. Using boundary layer tools we perform a complete asymptotic expansion of the eigenvalues as the conductivity tends to $+\\infty$.

OPERATOR-RELATED FORMULATION OF THE EIGENVALUE PROBLEM FOR THE BOUNDARY PROBLEM OF ANALYSIS OF A THREE-DIMENSIONAL STRUCTURE WITH PIECEWISE-CONSTANT PHYSICAL AND GEOMETRICAL PARAMETERS ALONGSIDE THE BASIC DIRECTION WITHIN THE FRAMEWORK OF THE DISCRETE-CON

Directory of Open Access Journals (Sweden)

Akimov Pavel Alekseevich

2012-10-01

Full Text Available The proposed paper covers the operator-related formulation of the eigenvalue problem of analysis of a three-dimensional structure that has piecewise-constant physical and geometrical parameters alongside the so-called basic direction within the framework of a discrete-continual approach (a discrete-continual finite element method, a discrete-continual variation method. Generally, discrete-continual formulations represent contemporary mathematical models that become available for computer implementation. They make it possible for a researcher to consider the boundary effects whenever particular components of the solution represent rapidly varying functions. Another feature of discrete-continual methods is the absence of any limitations imposed on lengths of structures. The three-dimensional problem of elasticity is used as the design model of a structure. In accordance with the so-called method of extended domain, the domain in question is embordered by an extended one of an arbitrary shape. At the stage of numerical implementation, relative key features of discrete-continual methods include convenient mathematical formulas, effective computational patterns and algorithms, simple data processing, etc. The authors present their formulation of the problem in question for an isotropic medium with allowance for supports restrained by elastic elements while standard boundary conditions are also taken into consideration.

Activation and thermodynamic parameter study of the heteronuclear C=O···H-N hydrogen bonding of diphenylurethane isomeric structures by FT-IR spectroscopy using the regularized inversion of an eigenvalue problem.

Science.gov (United States)

Spegazzini, Nicolas; Siesler, Heinz W; Ozaki, Yukihiro

2012-08-02

The doublet of the ν(C=O) carbonyl band in isomeric urethane systems has been extensively discussed in qualitative terms on the basis of FT-IR spectroscopy of the macromolecular structures. Recently, a reaction extent model was proposed as an inverse kinetic problem for the synthesis of diphenylurethane for which hydrogen-bonded and non-hydrogen-bonded C=O functionalities were identified. In this article, the heteronuclear C=O···H-N hydrogen bonding in the isomeric structure of diphenylurethane synthesized from phenylisocyanate and phenol was investigated via FT-IR spectroscopy, using a methodology of regularization for the inverse reaction extent model through an eigenvalue problem. The kinetic and thermodynamic parameters of this system were derived directly from the spectroscopic data. The activation and thermodynamic parameters of the isomeric structures of diphenylurethane linked through a hydrogen bonding equilibrium were studied. The study determined the enthalpy (ΔH = 15.25 kJ/mol), entropy (TΔS = 14.61 kJ/mol), and free energy (ΔG = 0.6 kJ/mol) of heteronuclear C=O···H-N hydrogen bonding by FT-IR spectroscopy through direct calculation from the differences in the kinetic parameters (δΔ(‡)H, -TδΔ(‡)S, and δΔ(‡)G) at equilibrium in the chemical reaction system. The parameters obtained in this study may contribute toward a better understanding of the properties of, and interactions in, supramolecular systems, such as the switching behavior of hydrogen bonding.

Is space-time symmetry a suitable generalization of parity-time symmetry?

International Nuclear Information System (INIS)

Amore, Paolo; Fernández, Francisco M.; Garcia, Javier

2014-01-01

We discuss space-time symmetric Hamiltonian operators of the form H=H 0 +igH ′ , where H 0 is Hermitian and g real. H 0 is invariant under the unitary operations of a point group G while H ′ is invariant under transformation by elements of a subgroup G ′ of G. If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g. In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0eigenvalues for g>0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries. - Highlights: • Space-time symmetry is a generalization of PT symmetry. • The eigenvalues of a space-time Hamiltonian are either real or appear as pairs of complex conjugate numbers. • In some cases all the eigenvalues are real for some values of a potential-strength parameter g. • At some value of g space-time symmetry is broken and complex eigenvalues appear. • Some multidimensional oscillators exhibit broken space-time symmetry for all values of g

Korea-NEA Research Project for the Construction of Worldwide Reactor Physics Experiments Database and the Foundation of Sustainable Collaboration Environment

Energy Technology Data Exchange (ETDEWEB)

Kim, Sang Ji; Jang, Jin Wook

2004-07-15

740 documents on PWR, LMR and VHTR related critical experiments are procured by participating in the IRPhE Project under the auspices of the OECD/NEA Data Bank. These documents were reviewed in detail for classification, measurement technique study. Among the database, the SNEAK-7 critical benchmarks were selected for further analysis by modern analysis tool, which requires minimum amount of correction for the raw calculated value. The SNEAK-7A and 7B were selected due to their spectrum similarity with KALIMER-600. Core eigenvalue, spectral indices, central material worths were evaluated and compared with the measurement. JEF-2.2 predict core eigenvalues pretty well, while the conversion ratio was underestimated by about 3%. ENDF/B-VI overestimates core eigenvalue by about 500 pcm while the conversion ratio was underestimated by about 4%. Through the sensitivity analysis of core eigenvalue to the partial cross section, a problem in {sup 238}U capture cross section was identified and a necessity arose to reduce the measurement uncertainty of {sup 239}Pu due to the high high sensitivity of core eigenvalue to {sup 239}Pu fission cross section.

Computer code abstract: NESTLE

International Nuclear Information System (INIS)

Turinsky, P.J.; Al-Chalabi, R.M.K.; Engrand, P.; Sarsour, H.N.; Faure, F.X.; Guo, W.

1995-01-01

NESTLE is a few-group neutron diffusion equation solver utilizing the nodal expansion method (NEM) for eigenvalue, adjoint, and fixed-source steady-state and transient problems. The NESTLE code solve the eigenvalue (criticality), eigenvalue adjoint, external fixed-source steady-state, and external fixed-source or eigenvalue initiated transient problems. The eigenvalue problem allows criticality searches to be completed, and the external fixed-source steady-state problem can search to achieve a specified power level. Transient problems model delayed neutrons via precursor groups. Several core properties can be input as time dependent. Two- or four-energy groups can be utilized, with all energy groups being thermal groups (i.e., upscatter exits) is desired. Core geometries modeled include Cartesian and hexagonal. Three-, two-, and one-dimensional models can be utilized with various symmetries. The thermal conditions predicted by the thermal-hydraulic model of the core are used to correct cross sections for temperature and density effects. Cross sections for temperature and density effects. Cross sections are parameterized by color, control rod state (i.e., in or out), and burnup, allowing fuel depletion to be modeled. Either a macroscopic or microscopic model may be employed

Modified Poisson eigenfunctions for electrostatic Bernstein--Greene--Kruskal equilibria

International Nuclear Information System (INIS)

Ling, K.; Abraham-Shrauner, B.

1981-01-01

The stability of an electrostatic Bernstein--Greene--Kruskal equilibrium by Lewis and Symon's general linear stability analysis for spatially inhomogeneous Vlasov equilibria, which employs eigenfunctions and eigenvalues of the equilibrium Liouville operator and the modified Poisson operator, is considered. Analytic expressions for the Liouville eigenfuctions and eigenvalues have already been given; approximate analytic expressions for the dominant eigenfunction and eigenvalue of the modified Poisson operator are given. In the kinetic limit three methods are given: (i) the perturbation method, (ii) the Rayleigh--Ritz method, and (iii) a method based on a Hill's equation. In the fluid limit the Rayleigh--Ritz method is used. The dominant eigenfunction and eigenvalue are then substituted in the dispersion relation and the growth rate calculated. The growth rate agrees very well with previous results found by numerical simulation and by modified Poisson eigenfunctions calculated numerically

Spectral analysis of the SN approximations in a slab with quadratically anisotropic scattering

International Nuclear Information System (INIS)

Ourique, L.E.; Pazos, R.P.; Vilhena, M.T.; Barros, R.C.

2003-01-01

The spectral analysis of the S N approximations to the one-dimensional transport equation began with 3 and 4, following the studies of 1 and 2 about the discrete eigenvalues of the transport equation. In previous work about the influence of a parameter in the solutions of S N approximations, it was considered the total macroscopic cross section as a control parameter and was analyzed how its variation changes the nature of the eigenvalues of the S N transport matrix, in problems with linearly anisotropic scattering. It was showed the existence of bifurcations points, i.e., there exist some values of control parameters for which the S N transport matrix has only real eigenvalues while for other values the S N relation between the eigenvalues of S N transport matrix and control parameter, supposing quadratically anisotropic scattering. Numerical results are reported. (author)

Fine structure of spectral properties for random correlation matrices: an application to financial markets.

Science.gov (United States)

Livan, Giacomo; Alfarano, Simone; Scalas, Enrico

2011-07-01

We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain, and we analyze the consequences of such filtering on eigenvalue spectra. We show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge as a consequence of cross correlations between stocks. We interpret and corroborate these findings in terms of factor models, and we compare empirical spectra to those predicted by random matrix theory for such models.

Spectral Mixture Analysis: Linear and Semi-parametric Full and Iterated Partial Unmixing in Multi- and Hyperspectral Image Data

DEFF Research Database (Denmark)

Nielsen, Allan Aasbjerg

2001-01-01

) and non-negative least squares (NNLS), and the partial unmixing methods orthogonal subspace projection (OSP), constrained energy minimization (CEM) and an eigenvalue formulation alternative are dealt with. The solution to the eigenvalue formulation alternative proves to be identical to the CEM solution....... The matrix inversion involved in CEM can be avoided by working on (a subset of) orthogonally transformed data such as signal maximum autocorrelation factors, MAFs, or signal minimum noise fractions, MNFs. This will also cause the partial unmixing result to be independent of the noise isolated in the MAF....../MNFs not included in the analysis. CEM and the eigenvalue formulation alternative enable us to perform partial unmixing when we know one desired end-member spectrum only and not the full set of end-member spectra. This is an advantage over full unmixing and OSP. The eigenvalue formulation of CEM inspires us...

Telescopic projective methods for parabolic differential equations

CERN Document Server

Gear, C W

2003-01-01

Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components.

Telescopic projective methods for parabolic differential equations

International Nuclear Information System (INIS)

Gear, C.W.; Kevrekidis, Ioannis G.

2003-01-01

Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components

Fine structure of spectral properties for random correlation matrices: An application to financial markets

Science.gov (United States)

Livan, Giacomo; Alfarano, Simone; Scalas, Enrico

2011-07-01

We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain, and we analyze the consequences of such filtering on eigenvalue spectra. We show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge as a consequence of cross correlations between stocks. We interpret and corroborate these findings in terms of factor models, and we compare empirical spectra to those predicted by random matrix theory for such models.

Wave chaos in quantum systems with point interaction

International Nuclear Information System (INIS)

Albeverio, S.; Seba, P.

1991-01-01

The authors study perturbations H of the quantized version H 0 of integrable Hamiltonian systems by point interactions. They relate the eigenvalues of H to the zeros of a certain meromorphic function ξ. Assuming the eigenvalues of H 0 are Poisson distributed, they get detailed information on the joint distribution of the zeros of ξ and give bounds on the probability density for the spacings of eigenvalues of H. Their results confirm the wave chaos phenomenon, as different from the quantum chaos phenomenon predicted by random matrix theory

Dynamical stability in fluid-structure interaction

International Nuclear Information System (INIS)

Planchard, J.; Thomas, B.

1991-01-01

The aim of the paper is to investigate the dynamical stability of a group of elastic tubes placed in a cross-flow which obeys to the Navier-Stokes equations. The stability of this coupled system is deduced from the study of a quadratic eigenvalue problem arising in the linearized equations. The instability occurs when the real part of one of the eigenvalues becomes positive; the steady state is then replaced by a time-periodic state which is stable (Hopf bifurcation phenomenon). Some numerical methods for solving the quadratic eigenvalue problem are described [fr

On the spectrum of a periodic operator with a small localized perturbation

International Nuclear Information System (INIS)

Borisov, D I; Gadyl'shin, R R

2008-01-01

The paper deals with the spectrum of a periodic self-adjoint differential operator on the real axis perturbed by a small localized non-self-adjoint operator. We show that the continuous spectrum does not depend on the perturbation, the residual spectrum is empty, and the point spectrum has no finite accumulation points. We study the problem of the existence of eigenvalues embedded in the continuous spectrum, obtain necessary and sufficient conditions for the existence of eigenvalues, construct asymptotic expansions of the eigenvalues and corresponding eigenfunctions and consider some examples

Investigations in the Spectral Properties of Operators with Distant Perturbations (survey

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A. M. Golovina

2015-01-01

Full Text Available We propose a chronological overview of researches on operators with distant perturbations. Let us explain what "distant perturbations" mean. An elementary example of the operator with distant perturbations is a differential operator of the second order with two finite potentials. Supports of these operators are at a great distance from each other, i.e. they are \\distant".The study of such operators has been performed since the middle of the last century, mostly by foreign researchers see eg. R. Ahlrichs, T. Aktosun, M. Klaus, P. Aventini, P. Exner, E.B. Davies, V. Graffi, E.V. Harrell II, H.J. Silverstone, M. Mebkhout, R. Hoegh-Krohn, W. Hun ziker, V. Kostrykin, R. Schrader, J.D. Morgan (III, Y. Pinchover, O.K. Reity, H. Tamura, X. Wang, Y. Wang, S. Kondej, B. Simon, I. Veselic, D.I. Borisov, A.M. Golovina. The main objects of their investigation were the asymptotic behaviors of eigenvalues and corresponding eigenfunctions of perturbed operators. In several papers the research was focused on resolvents and eigenvalues of perturbed operator arising from the edge of the essential spectrum. The main results of the past century are the first members of the asymptotics of perturbed eigenvalues and the corresponding eigenfunctions and the first members of the asymptotics of resolvents of the perturbed operators. The main results of the last fifteen years are full asymptotic expansions for the eigenvalues and their corresponding functions and an explicit formula for the resolvent of the perturbed operator.In this paper, we also note that up to 2004 only different kind of potentials were considered as perturbing operators, and Laplace and Dirac operators were considered as unperturbed operators. Only since 2004, nonpotential perturbing operators appeared in the literature. Since 2012, an arbitrary elliptic differential operator is considered as an unperturbed operator.We propose a classification of investigations on distant perturbations, based on the

A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems

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Babaee, Hessam; Choi, Minseok; Sapsis, Themistoklis P.; Karniadakis, George Em

2017-09-01

We develop a new robust methodology for the stochastic Navier-Stokes equations based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) methods [1-3]. Both approaches are variants of a generalized Karhunen-Loève (KL) expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics, hence, capturing the low-dimensional structure of the solution. The DO and BO formulations are mathematically equivalent [3], but they exhibit computationally complimentary properties. Specifically, the BO formulation may fail due to crossing of the eigenvalues of the covariance matrix, while both BO and DO become unstable when there is a high condition number of the covariance matrix or zero eigenvalues. To this end, we combine the two methods into a robust hybrid framework and in addition we employ a pseudo-inverse technique to invert the covariance matrix. The robustness of the proposed method stems from addressing the following issues in the DO/BO formulation: (i) eigenvalue crossing: we resolve the issue of eigenvalue crossing in the BO formulation by switching to the DO near eigenvalue crossing using the equivalence theorem and switching back to BO when the distance between eigenvalues is larger than a threshold value; (ii) ill-conditioned covariance matrix: we utilize a pseudo-inverse strategy to invert the covariance matrix; (iii) adaptivity: we utilize an adaptive strategy to add/remove modes to resolve the covariance matrix up to a threshold value. In particular, we introduce a soft-threshold criterion to allow the system to adapt to the newly added/removed mode and therefore avoid repetitive and unnecessary mode addition/removal. When the total variance approaches zero, we show that the DO/BO formulation becomes equivalent to the evolution equation of the Optimally Time-Dependent modes [4]. We demonstrate the capability of the proposed methodology with several numerical examples, namely (i) stochastic Burgers equation: we

Dimensionality of genomic information and performance of the Algorithm for Proven and Young for different livestock species.

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Pocrnic, Ivan; Lourenco, Daniela A L; Masuda, Yutaka; Misztal, Ignacy

2016-10-31

A genomic relationship matrix (GRM) can be inverted efficiently with the Algorithm for Proven and Young (APY) through recursion on a small number of core animals. The number of core animals is theoretically linked to effective population size (N e ). In a simulation study, the optimal number of core animals was equal to the number of largest eigenvalues of GRM that explained 98% of its variation. The purpose of this study was to find the optimal number of core animals and estimate N e for different species. Datasets included phenotypes, pedigrees, and genotypes for populations of Holstein, Jersey, and Angus cattle, pigs, and broiler chickens. The number of genotyped animals varied from 15,000 for broiler chickens to 77,000 for Holsteins, and the number of single-nucleotide polymorphisms used for genomic prediction varied from 37,000 to 61,000. Eigenvalue decomposition of the GRM for each population determined numbers of largest eigenvalues corresponding to 90, 95, 98, and 99% of variation. The number of eigenvalues corresponding to 90% (98%) of variation was 4527 (14,026) for Holstein, 3325 (11,500) for Jersey, 3654 (10,605) for Angus, 1239 (4103) for pig, and 1655 (4171) for broiler chicken. Each trait in each species was analyzed using the APY inverse of the GRM with randomly selected core animals, and their number was equal to the number of largest eigenvalues. Realized accuracies peaked with the number of core animals corresponding to 98% of variation for Holstein and Jersey and closer to 99% for other breed/species. N e was estimated based on comparisons of eigenvalue decomposition in a simulation study. Assuming a genome length of 30 Morgan, N e was equal to 149 for Holsteins, 101 for Jerseys, 113 for Angus, 32 for pigs, and 44 for broilers. Eigenvalue profiles of GRM for common species are similar to those in simulation studies although they are affected by number of genotyped animals and genotyping quality. For all investigated species, the APY required

Input design for linear dynamic systems using maxmin criteria

DEFF Research Database (Denmark)

Sadegh, Payman; Hansen, Lars H.; Madsen, Henrik

1998-01-01

This paper considers the problem of input design for maximizing the smallest eigenvalue of the information matrix for linear dynamic systems. The optimization of the smallest eigenvalue is of interest in parameter estimation and parameter change detection problems. We describe a simple cutting...

On functional determinants of matrix differential operators with multiple zero modes

NARCIS (Netherlands)

Falco, G.M.; Fedorenko, Andrey A; Gruzberg, Ilya A

2017-01-01

We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of $r\\times r$ matrix second order differential

The Theory of Quantized Fields. III

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Schwinger, J.

1953-05-01

In this paper we discuss the electromagnetic field, as perturbed by a prescribed current. All quantities of physical interest in various situations, eigenvalues, eigenfunctions, and transformation probabilities, are derived from a general transformation function which is expressed in a non-Hermitian representation. The problems treated are: the determination of the energy-momentum eigenvalues and eigenfunctions for the isolated electromagnetic field, and the energy eigenvalues and eigenfunctions for the field perturbed by a time-independent current that departs from zero only within a finite time interval, and for a time-dependent current that assumes non-vanishing time-independent values initially and finally. The results are applied in a discussion of the intra-red catastrophe and of the adiabatic theorem. It is shown how the latter can be exploited to give a uniform formulation for all problems requiring the evaluation of transition probabilities or eigenvalue displacements.

Multidimensional periodic Schrödinger operator perturbation theory and applications

CERN Document Server

Veliev, Oktay

2015-01-01

The book describes the direct problems and the inverse problem of the multidimensional Schrödinger operator with a periodic potential. This concerns perturbation theory and constructive determination of the spectral invariants and finding the periodic potential from the given Bloch eigenvalues. The unique method of this book derives the asymptotic formulas for Bloch eigenvalues and Bloch functions for arbitrary dimension. Moreover, the measure of the iso-energetic surfaces in the high energy region is construct and estimated. It implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed in this book, the spectral invariants of the multidimensional operator from the given Bloch eigenvalues are determined. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential. This way the possibility to determine the potential constructively by using Bloch eigenvalues as input data is given. In the ...

Intrinsic character of Stokes matrices

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Gagnon, Jean-François; Rousseau, Christiane

2017-02-01

Two germs of linear analytic differential systems x k + 1Y‧ = A (x) Y with a non-resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in S allows supposing that R+ lies in the intersection of two sectors. Reordering of the coordinates of Y allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in x is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.

Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects

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Castellano, Claudio; Pastor-Satorras, Romualdo

2017-10-01

The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: the hub with its immediate neighbors and the densely connected set of nodes with maximum K -core index. We validate this formula by showing that it predicts, with good accuracy, the largest eigenvalue of a large set of synthetic and real-world topologies. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a by-product, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.

Feedback Loop Gains and Feedback Behavior (1996)

DEFF Research Database (Denmark)

Kampmann, Christian Erik

2012-01-01

Linking feedback loops and system behavior is part of the foundation of system dynamics, yet the lack of formal tools has so far prevented a systematic application of the concept, except for very simple systems. Having such tools at their disposal would be a great help to analysts in understanding...... large, complicated simulation models. The paper applies tools from graph theory formally linking individual feedback loop strengths to the system eigenvalues. The significance of a link or a loop gain and an eigenvalue can be expressed in the eigenvalue elasticity, i.e., the relative change...... of an eigenvalue resulting from a relative change in the gain. The elasticities of individual links and loops may be found through simple matrix operations on the linearized system. Even though the number of feedback loops can grow rapidly with system size, reaching astronomical proportions even for modest systems...