Soft Error Vulnerability of Iterative Linear Algebra Methods
Energy Technology Data Exchange (ETDEWEB)
Bronevetsky, G; de Supinski, B
2008-01-19
Devices are increasingly vulnerable to soft errors as their feature sizes shrink. Previously, soft error rates were significant primarily in space and high-atmospheric computing. Modern architectures now use features so small at sufficiently low voltages that soft errors are becoming important even at terrestrial altitudes. Due to their large number of components, supercomputers are particularly susceptible to soft errors. Since many large scale parallel scientific applications use iterative linear algebra methods, the soft error vulnerability of these methods constitutes a large fraction of the applications overall vulnerability. Many users consider these methods invulnerable to most soft errors since they converge from an imprecise solution to a precise one. However, we show in this paper that iterative methods are vulnerable to soft errors, exhibiting both silent data corruptions and poor ability to detect errors. Further, we evaluate a variety of soft error detection and tolerance techniques, including checkpointing, linear matrix encodings, and residual tracking techniques.
Soft Error Vulnerability of Iterative Linear Algebra Methods
Energy Technology Data Exchange (ETDEWEB)
Bronevetsky, G; de Supinski, B
2007-12-15
Devices become increasingly vulnerable to soft errors as their feature sizes shrink. Previously, soft errors primarily caused problems for space and high-atmospheric computing applications. Modern architectures now use features so small at sufficiently low voltages that soft errors are becoming significant even at terrestrial altitudes. The soft error vulnerability of iterative linear algebra methods, which many scientific applications use, is a critical aspect of the overall application vulnerability. These methods are often considered invulnerable to many soft errors because they converge from an imprecise solution to a precise one. However, we show that iterative methods can be vulnerable to soft errors, with a high rate of silent data corruptions. We quantify this vulnerability, with algorithms generating up to 8.5% erroneous results when subjected to a single bit-flip. Further, we show that detecting soft errors in an iterative method depends on its detailed convergence properties and requires more complex mechanisms than simply checking the residual. Finally, we explore inexpensive techniques to tolerate soft errors in these methods.
Voila: A visual object-oriented iterative linear algebra problem solving environment
Energy Technology Data Exchange (ETDEWEB)
Edwards, H.C.; Hayes, L.J. [Univ. of Texas, Austin, TX (United States)
1994-12-31
Application of iterative methods to solve a large linear system of equations currently involves writing a program which calls iterative method subprograms from a large software package. These subprograms have complex interfaces which are difficult to use and even more difficult to program. A problem solving environment specifically tailored to the development and application of iterative methods is needed. This need will be fulfilled by Voila, a problem solving environment which provides a visual programming interface to object-oriented iterative linear algebra kernels. Voila will provide several quantum improvements over current iterative method problem solving environments. First, programming and applying iterative methods is considerably simplified through Voila`s visual programming interface. Second, iterative method algorithm implementations are independent of any particular sparse matrix data structure through Voila`s object-oriented kernels. Third, the compile-link-debug process is eliminated as Voila operates as an interpreter.
On a modification of minimal iteration methods for solving systems of linear algebraic equations
Yukhno, L. F.
2010-04-01
Modifications of certain minimal iteration methods for solving systems of linear algebraic equations are proposed and examined. The modified methods are shown to be superior to the original versions with respect to the round-off error accumulation, which makes them applicable to solving ill-conditioned problems. Numerical results demonstrating the efficiency of the proposed modifications are given.
Leapfrog variants of iterative methods for linear algebra equations
Saylor, Paul E.
1988-01-01
Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.
Nordtvedt, Kenneth
2015-01-01
A method for constructing metric gravity's N-body Lagrangian is developed which uses iterative, liner algebraic euqations which enforce invariance properties of gravity --- exterior effacement, interior effacement, and the time dilation and Lorentz contraction of matter under boosts. The method is demonstrated by obtaining the full 1/c^4 order Lagrangian, and a combination of exterior and interior effacement enforcement permits construction of the full Schwarzschild temporal and spatial metric potentials.
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
It was proved numerically that the Domain Decomposition Method (DDM) with one layer overlapping grids is identical to the block iterative method of linear algebra equations. The results obtained using DDM could be in reasonable aggeement with the results of full-domain simulation. With the three dimensional solver developed by the authors, the flow field in a pipe was simulated using the full-domain DDM with one layer overlapping grids and with patched grids respectively. Both of the two cases led to the convergent solution. Further research shows the superiority of the DDM with one layer overlapping grids to the DDM with patched grids. A comparison between the numerical results obtained by the authors and the experimental results given by Enayet[3] shows that the numerical results are reasonable.
Liesen, Jörg
2015-01-01
This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...
Stoll, R R
1968-01-01
Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand
Edwards, Harold M
1995-01-01
In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject
Allenby, Reg
1995-01-01
As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.Solutions to the exercises are available onlin
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Shilov, Georgi E
1977-01-01
Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.
Berberian, Sterling K
2014-01-01
Introductory treatment covers basic theory of vector spaces and linear maps - dimension, determinants, eigenvalues, and eigenvectors - plus more advanced topics such as the study of canonical forms for matrices. 1992 edition.
Sahai, Vivek
2013-01-01
Beginning with the basic concepts of vector spaces such as linear independence, basis and dimension, quotient space, linear transformation and duality with an exposition of the theory of linear operators on a finite dimensional vector space, this book includes the concept of eigenvalues and eigenvectors, diagonalization, triangulation and Jordan and rational canonical forms. Inner product spaces which cover finite dimensional spectral theory and an elementary theory of bilinear forms are also discussed. This new edition of the book incorporates the rich feedback of its readers. We have added new subject matter in the text to make the book more comprehensive. Many new examples have been discussed to illustrate the text. More exercises have been included. We have taken care to arrange the exercises in increasing order of difficulty. There is now a new section of hints for almost all exercises, except those which are straightforward, to enhance their importance for individual study and for classroom use.
Institute of Scientific and Technical Information of China (English)
Shuang-suo Zhao; Zhang-hua Luo; Guo-feng Zhang
2000-01-01
This paper presents optimum an one-parameter iteration (OOPI) method and a multi-parameter iteration direct (MPID) method for efficiently solving linear algebraic systems with low order matrix A and high order matrix B: Y = (A B)Y +Ф. On parallel computers (also on serial computer) the former will be efficient, even very efficient under certain conditions, the latter will be universally very efficient.
Abian, Alexander
1973-01-01
Linear Associative Algebras focuses on finite dimensional linear associative algebras and the Wedderburn structure theorems.The publication first elaborates on semigroups and groups, rings and fields, direct sum and tensor product of rings, and polynomial and matrix rings. The text then ponders on vector spaces, including finite dimensional vector spaces and matrix representation of vectors. The book takes a look at linear associative algebras, as well as the idempotent and nilpotent elements of an algebra, ideals of an algebra, total matrix algebras and the canonical forms of matrices, matrix
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
Axler, Sheldon
2015-01-01
This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the ...
Directory of Open Access Journals (Sweden)
Sinan AYDIN
2009-04-01
Full Text Available Linear algebra is a basic course followed in mathematics, science, and engineering university departments.Generally, this course is taken in either the first or second year but there have been difficulties in teachingand learning. This type of active algebra has resulted in an increase in research by mathematics educationresearchers. But there is insufficient information on this subject in Turkish and therefore it has not beengiven any educational status. This paper aims to give a general overview of this subject in teaching andlearning. These education studies can be considered quadruple: a the history of linear algebra, b formalismobstacles of linear algebra and cognitive flexibility to improve teaching and learning, c the relation betweenlinear algebra and geometry, d using technology in the teaching and learning linear algebra.Mathematicseducation researchers cannot provide an absolute solution to overcome the teaching and learning difficultiesof linear algebra. Epistemological analyses and experimental teaching have shown the learning difficulties.Given these results, further advice and assistance can be offered locally.
Cooperstein, Bruce
2010-01-01
Vector SpacesFieldsThe Space FnVector Spaces over an Arbitrary Field Subspaces of Vector SpacesSpan and IndependenceBases and Finite Dimensional Vector SpacesBases and Infinite Dimensional Vector SpacesCoordinate VectorsLinear TransformationsIntroduction to Linear TransformationsThe Range and Kernel of a Linear TransformationThe Correspondence and Isomorphism TheoremsMatrix of a Linear TransformationThe Algebra of L(V, W) and Mmn(F)Invertible Transformations and MatricesPolynomialsThe Algebra of PolynomialsRoots of PolynomialsTheory of a Single Linear OperatorInvariant Subspaces of an Operator
Andrilli, Stephen
2010-01-01
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, expl
Blyth, T S
2002-01-01
Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers:...
Energy Technology Data Exchange (ETDEWEB)
Magnin, H.; Coulomb, J.L. (Laboratoire d' electrotechnique de Grenoble (UA CNRS 355) E.N.S.I.E.G. BP 46 38402 St. Martin d' Heres (FR))
1989-07-01
Electromagnetic field analysis by finite elements methods needs solving of large sparse systems of linear equations. Though no discernible structure for the distribution of non-zero elements can be found (e.g. multidiagonal structures,...), subsets of independent equations can be determined. Equations that are in a same subset are then solved in parallel. A good choice for the storage scheme of sparse matrices is also very important to speedup the resolution by vectorization. The modifications the authors made to data structures are presented, and the possibility to use some other schemes is discussed.
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
Multicore Performance of Block Algebraic Iterative Reconstruction Methods
DEFF Research Database (Denmark)
Sørensen, Hans Henrik B.; Hansen, Per Christian
2014-01-01
Algebraic iterative methods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction techniques (SIRT), both of which rely...... a fixed relaxation parameter in each method, namely, the one that leads to the fastest semiconvergence. Computational results show that for multicore computers, the sequential approach is preferable....
Hogben, Leslie
2013-01-01
With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.New to the Second EditionSeparate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of
Linear Algebra Thoroughly Explained
Vujičić, Milan
2008-01-01
Linear Algebra Thoroughly Explained provides a comprehensive introduction to the subject suitable for adoption as a self-contained text for courses at undergraduate and postgraduate level. The clear and comprehensive presentation of the basic theory is illustrated throughout with an abundance of worked examples. The book is written for teachers and students of linear algebra at all levels and across mathematics and the applied sciences, particularly physics and engineering. It will also be an invaluable addition to research libraries as a comprehensive resource book for the subject.
Linear Mappings of Quaternion Algebra
Kleyn, Aleks
2011-01-01
In the paper I considered linear and antilinear automorphisms of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of linear and antilinear automorphisms.
Blyth, T S
2002-01-01
Most of the introductory courses on linear algebra develop the basic theory of finite dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num ber of illustrative and worked examples, as well as many exercises that are strategi cally placed throughout the text. Solutions to the ex...
Computer Program For Linear Algebra
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Computer Program For Linear Algebra
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Redesigning linear algebra algorithms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.
1983-01-01
Many of the standard algorithms in linear algebra as implemented in FORTRAN do not achieve maximum performance on today's large-scale vector computers. The author examines the problem and constructs alternative formulations of algorithms that do not lose the clarity of the original algorithm or sacrifice the FORTRAN portable environment, but do gain the performance attainable on these supercomputers. The resulting implementation not only performs well on vector computers but also increases performance on conventional sequential computers. 13 references.
Redesigning linear algebra algorithms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.
1983-01-01
Many of the standard algorithms in linear algebra as implemented in FORTRAN do not achieve maximum performance on today's large-scale vector computers. In this paper we examine the problem and construct alternative formulations of algorithms that do not lose the clarity of the original algorithm or sacrifice the Fortran portable environment, but do gain the performance attainable on these supercomputers. The resulting implementation not only performs well on vector computers but also increases performance on conventional sequential computers.
Fundamentals of linear algebra
Dash, Rajani Ballav
2008-01-01
FUNDAMENTALS OF LINEAR ALGEBRA is a comprehensive Text Book, which can be used by students and teachers of All Indian Universities. The Text has easy, understandable form and covers all topics of UGC Curriculum. There are lots of worked out examples which helps the students in solving the problems without anybody's help. The Problem sets have been designed keeping in view of the questions asked in different examinations.
Cooperstein, Bruce
2015-01-01
Advanced Linear Algebra, Second Edition takes a gentle approach that starts with familiar concepts and then gradually builds to deeper results. Each section begins with an outline of previously introduced concepts and results necessary for mastering the new material. By reviewing what students need to know before moving forward, the text builds a solid foundation upon which to progress. The new edition of this successful text focuses on vector spaces and the maps between them that preserve their structure (linear transformations). Designed for advanced undergraduate and beginning graduate stud
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
Numerical linear algebra theory and applications
Beilina, Larisa; Karchevskii, Mikhail
2017-01-01
This book combines a solid theoretical background in linear algebra with practical algorithms for numerical solution of linear algebra problems. Developed from a number of courses taught repeatedly by the authors, the material covers topics like matrix algebra, theory for linear systems of equations, spectral theory, vector and matrix norms combined with main direct and iterative numerical methods, least squares problems, and eigen problems. Numerical algorithms illustrated by computer programs written in MATLAB® are also provided as supplementary material on SpringerLink to give the reader a better understanding of professional numerical software for the solution of real-life problems. Perfect for a one- or two-semester course on numerical linear algebra, matrix computation, and large sparse matrices, this text will interest students at the advanced undergraduate or graduate level.
Third SIAM conference on applied linear algebra and short course on linear algebra in statistics
Energy Technology Data Exchange (ETDEWEB)
1988-01-01
This report contains abstracts on the following themes: Large Scale Computing and Numerical Methods; Inverse Eigenvalue Problems; Qualitative and Combinatorial Analysis of Matrices; Linear Systems and Control; Parallel Matrix Computations; Signal Processing; Optimization; Multivariate Statistics; Core Linear Algebra; and Iterative Methods for Solving Linear Systems. (LSP)
Lopez, Cesar
2014-01-01
MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Linear Algebra introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving an introduction to
Energy Technology Data Exchange (ETDEWEB)
Casasent, D.; Ghosh, A.
1983-01-01
Many of the linear algebra operations and algorithms possible on optical matrix-vector processors are reviewed. Emphasis is given to the use of direct solutions and their realization on systolic optical processors. As an example, implicit and explicit solutions to partial differential equations are considered. The matrix-decomposition required is found to be the major operation recommended for optical realization. The pipelining and flow of data and operations are noted to be key issues in the realization of any algorithm on an optical systolic array processor. A realization of the direct solution by householder qr decomposition is provided as a specific case study. 19 references.
Linear algebra and projective geometry
Baer, Reinhold
2005-01-01
Geared toward upper-level undergraduates and graduate students, this text establishes that projective geometry and linear algebra are essentially identical. The supporting evidence consists of theorems offering an algebraic demonstration of certain geometric concepts. These focus on the representation of projective geometries by linear manifolds, of projectivities by semilinear transformations, of collineations by linear transformations, and of dualities by semilinear forms. These theorems lead to a reconstruction of the geometry that constituted the discussion's starting point, within algebra
Templates for Linear Algebra Problems
Bai, Z.; Day, D.; Demmel, J.; Dongarra, J.; Gu, M.; Ruhe, A.; Vorst, H.A. van der
2001-01-01
The increasing availability of advanced-architecture computers is having a very signicant eect on all spheres of scientic computation, including algorithm research and software development in numerical linear algebra. Linear algebra {in particular, the solution of linear systems of equations and eig
Linear Algebra and Image Processing
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS
Institute of Scientific and Technical Information of China (English)
Zhong-zhi Bai
2001-01-01
For the large sparse system of linear equations with symmetric positive definite block coefficient matrix resulted from suitable finite element discretization of the second-order self-adjoint elliptic boundary value problem, by making use of the algebraic multilevel iteration technique and the blocked preconditioning strategy, we construct preconditioning matrices having parallel computing function for the coefficient matrix and set up a class of parallel hybrid algebraic multilevel iteration methods for solving this kind of system of linear equations. Theoretical analyses show that, besides much suitable for implementing on the high-speed parallel multiprocessor systems, these new methods are optimal-order methods. That is to say, their convergence rates are independent of both the sizes and the levels of the constructed matrix sequence, and their computational workloads are bounded by linear functions in the order number of the considered system of linear equations,respectively.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Practical approach to linear algebra
Choudhary, Prabhat
2009-01-01
""Linear Algebra is the heart of applied science but there are divergent views concerning its meaning. The field of Linear Algebra is more beautiful and more fundamental than its rather dull name may suggest. More beautiful because it is full of powerful ideas that are quite unlike those normally emphasized in a linear algebra course in a mathematics department. Throughout the book the author follows the practice of first presenting required background material, which is then used to develop the results. The book is divided in ten chapters. Relevant material is included in each chapter from ot
Newton—Like Iteration Method for Solving Algebraic Equations
Institute of Scientific and Technical Information of China (English)
JihuanHE
1998-01-01
In this paper,a Newton-like iteration method is proposed to solve an approximate solution of an algebraic equation.The iteration formula obtained by homotopy perturbation method contains the well-known Newton iteration formulain logic.
Numerical linear algebra for reconstruction inverse problems
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Teaching Linear Algebra at University
Dorier, Jean-Luc
1997-01-01
Linear algebra represents, with calculus, the two main mathematical subjects taught in science universities. However this teaching has always been difficult. In the last two decades, it became an active area for research works in mathematics education in several countries. Our goal is to give a synthetic overview of the main results of these works focusing on the most recent developments. The main issues we will address concern: • the epistemological specificity of linear algebra and the inte...
Iterative solution of linear systems
Freund, Roland W.; Golub, Gene H.; Nachtigal, Noel M.
1992-01-01
Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for nonHermitian matrices.
Computational linear and commutative algebra
Kreuzer, Martin
2016-01-01
This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to prese...
Topics in quaternion linear algebra
Rodman, Leiba
2014-01-01
Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses...
Linear algebra, geometry and transformation
Solomon, Bruce
2014-01-01
Vectors, Mappings and Linearity Numeric Vectors Functions Mappings and Transformations Linearity The Matrix of a Linear Transformation Solving Linear Systems The Linear SystemThe Augmented Matrix and RRE Form Homogeneous Systems in RRE Form Inhomogeneous Systems in RRE Form The Gauss-Jordan Algorithm Two Mapping Answers Linear Geometry Geometric Vectors Geometric/Numeric Duality Dot-Product Geometry Lines, Planes, and Hyperplanes System Geometry and Row/Column Duality The Algebra of Matrices Matrix Operations Special Matrices Matrix Inversion A Logical Digression The Logic of the Inversion Alg
Introduction to computational linear algebra
Nassif, Nabil; Erhel, Jocelyne
2015-01-01
Introduction to Computational Linear Algebra introduces the reader with a background in basic mathematics and computer programming to the fundamentals of dense and sparse matrix computations with illustrating examples. The textbook is a synthesis of conceptual and practical topics in ""Matrix Computations."" The book's learning outcomes are twofold: to understand state-of-the-art computational tools to solve matrix computations problems (BLAS primitives, MATLAB® programming) as well as essential mathematical concepts needed to master the topics of numerical linear algebra. It is suitable for s
Overview of parallel algorithms in numerical linear algebra
Energy Technology Data Exchange (ETDEWEB)
Sameh, A.
1983-01-01
The author gives a brief survey of the development of multiprocessor algorithms for: (i) the direct solution of linear systems, (ii) the algebraic eigenvalue problem, and (iii) the direct and iterative methods for solving the finite-difference or finite-element linear systems of equations arising from the discretization of linear partial differential equations. 66 references.
Data Compression with Linear Algebra
Etler, David
2015-01-01
A presentation on the applications of linear algebra to image compression. Covers entropy, the discrete cosine transform, thresholding, quantization, and examples of images compressed with DCT. Given in Spring 2015 at Ocean County College as part of the honors program.
Principles of linear algebra with Mathematica
Shiskowski, Kenneth M
2013-01-01
A hands-on introduction to the theoretical and computational aspects of linear algebra using Mathematica® Many topics in linear algebra are simple, yet computationally intensive, and computer algebra systems such as Mathematica® are essential not only for learning to apply the concepts to computationally challenging problems, but also for visualizing many of the geometric aspects within this field of study. Principles of Linear Algebra with Mathematica uniquely bridges the gap between beginning linear algebra and computational linear algebra that is often encountered in applied settings,
A Linear Iterative Unfolding Method
Laszlo, Andras
2011-01-01
A frequently faced task in experimental physics is to measure the probability distribution of some quantity. Often this quantity to be measured is smeared by a non-ideal detector response or by some physical process. The procedure of removing this smearing effect from the measured distribution is called unfolding, and is a delicate problem in signal processing. Due to the numerical ill-posedness of this task, various methods were invented which, given some assumptions on the initial probability distribution, try to regularize the problem. Most of these methods definitely introduce bias on the estimate of the initial probability distribution. We propose a linear iterative method (motivated by the Neumann series / Landweber iteration known in functional analysis), which has the advantage that no assumptions on the initial probability distribution is needed, and the only regularization parameter is the stopping order of the iteration. Convergence is proved under certain quite general conditions, which hold for p...
Singh, Kuldeep
2013-01-01
Linear algebra is a fundamental area of mathematics, and is arguably the most powerful mathematical tool ever developed. It is a core topic of study within fields as diverse as: business, economics, engineering, physics, computer science, ecology, sociology, demography and genetics. For an example of linear algebra at work, one needs to look no further than the Google search engine, which relies upon linear algebra to rank the results of a search with respect to relevance. The strength of the text is in the large number of examples and the step-by-step explanation of each topic as it is introduced. It is compiled in a way that allows distance learning, with explicit solutions to set problems freely available online. The miscellaneous exercises at the end of each chapter comprise questions from past exam papers from various universities, helping to reinforce the reader's confidence. Also included, generally at the beginning of sections, are short historicalbiographies of the leading players in the field of lin...
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Matrix algebra for linear models
Gruber, Marvin H J
2013-01-01
Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. This evolution has made matrix methods a vital part of statistical education. Traditionally, matrix methods are taught in courses on everything from regression analysis to stochastic processes, thus creating a fractured view of the topic. Matrix Algebra for Linear Models offers readers a unique, unified view of matrix analysis theory (where and when necessary), methods, and their applications. Written f
Meromorphic functions and linear algebra
Nevanlinna, Olavi
2003-01-01
This volume describes for the first time in monograph form important applications in numerical methods of linear algebra. The author presents new material and extended results from recent papers in a very readable style. The main goal of the book is to study the behavior of the resolvent of a matrix under the perturbation by low rank matrices. Whereas the eigenvalues (the poles of the resolvent) and the pseudospectra (the sets where the resolvent takes large values) can move dramatically under such perturbations, the growth of the resolvent as a matrix-valued meromorphic function remains essen
On the Teaching of Linear Algebra
Dorier, Jean-Luc
2000-01-01
The book present the state-of-art research on the teaching and learning of linear algebra in the fist year of university, in an international perspective. It aims at giving to university teachers in charge of linear algebra courses a wide range of information from works including theoretical and experimental issues. These works try to better understand the meaning of linear algebra in an epistemological approach, as well as the constraints and the difficulties in its teaching and learning. Th...
An Application of Linear Algebra over Lattices
Directory of Open Access Journals (Sweden)
M. Hosseinyazdi
2008-03-01
Full Text Available In this paper, first we consider L n as a semimodule over a complete bounded distributive lattice L. Then we define the basic concepts of module theory for L n. After that, we proved many similar theorems in linear algebra for the space L n. An application of linear algebra over lattices for solving linear systems, was given
An Easy Method To Accelerate An Iterative Algebraic Equation Solver
Energy Technology Data Exchange (ETDEWEB)
Yao, Jin [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2014-01-06
This article proposes to add a simple term to an iterative algebraic equation solver with an order n convergence rate, and to raise the order of convergence to (2n - 1). In particular, a simple algebraic equation solver with the 5th order convergence but uses only 4 function values in each iteration, is described in details. When this scheme is applied to a Newton-Raphson method of the quadratic convergence for a system of algebraic equations, a cubic convergence can be achieved with an low overhead cost of function evaluation that can be ignored as the size of the system increases.
Gonzalez-Vega, Laureano
1999-01-01
Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
PC Basic Linear Algebra Subroutines
Energy Technology Data Exchange (ETDEWEB)
1992-03-09
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, and find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.
A modern introduction to linear algebra
Ricardo, Henry
2009-01-01
Useful Concepts and Results at the Heart of Linear AlgebraA one- or two-semester course for a wide variety of students at the sophomore/junior undergraduate levelA Modern Introduction to Linear Algebra provides a rigorous yet accessible matrix-oriented introduction to the essential concepts of linear algebra. Concrete, easy-to-understand examples motivate the theory.The book first discusses vectors, Gaussian elimination, and reduced row echelon forms. It then offers a thorough introduction to matrix algebra, including defining the determinant naturally from the PA=LU factorization of a matrix.
(Numerical algorithms for solving linear algebra problems). Final report
Energy Technology Data Exchange (ETDEWEB)
Golub, G.H.
1985-04-16
We have concentrated on developing and analyzing various numerical algorithms for solving problems arising in a linear algebra context. The papers and research fall into basically three categories: (1) iterative methods for solving linear equations arising from p.d.e.'s; (2) calculation of Gauss-type quadrature rules; and (3) solution of matrix and data problems arising in statistical computation. We summarize some of these results, highlighting those which are of most importance.
An Inquiry-Based Linear Algebra Class
Wang, Haohao; Posey, Lisa
2011-01-01
Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…
A Course on Applied Linear Algebra.
Wang, Tse-Wei
1989-01-01
Provides an overview of a course, "Applied Linear Algebra," for teaching the concepts and the physical and geometric interpretations of some linear algebra topics. Describes the philosophy of the course, the computer project assignments, and student feedback. Major topics of the course are listed. (YP)
Dispersion Operators Algebra and Linear Canonical Transformations
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-02-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Dispersion Operators Algebra and Linear Canonical Transformations
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-04-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Institute of Scientific and Technical Information of China (English)
Yao-lin Jiang
2003-01-01
In this paper we presented a convergence condition of parallel dynamic iteration methods for a nonlinear system of differential-algebraic equations with a periodic constraint.The convergence criterion is decided by the spectral expression of a linear operator derivedfrom system partitions. Numerical experiments given here confirm the theoretical work ofthe paper.
The Growing Importance of Linear Algebra in Undergraduate Mathematics.
Tucker, Alan
1993-01-01
Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum. (MDH)
The Growing Importance of Linear Algebra in Undergraduate Mathematics.
Tucker, Alan
1993-01-01
Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum. (MDH)
AIR Tools - A MATLAB Package of Algebraic Iterative Reconstruction Techniques
DEFF Research Database (Denmark)
Hansen, Per Christian; Saxild-Hansen, Maria
are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter...... and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a “training” algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP...... criterion; for the first two methods “training” can be used to find the optimal discrepancy parameter. The corresponding manuscript is: • P. C. Hansen and M. Saxild-Hansen, AIR Tools – A MATLAB Package of Algebraic Iterative Reconstruction Techniques, submitted to Journal of Computational and Applied...
Advanced linear algebra for engineers with Matlab
Dianat, Sohail A
2009-01-01
Matrices, Matrix Algebra, and Elementary Matrix OperationsBasic Concepts and NotationMatrix AlgebraElementary Row OperationsSolution of System of Linear EquationsMatrix PartitionsBlock MultiplicationInner, Outer, and Kronecker ProductsDeterminants, Matrix Inversion and Solutions to Systems of Linear EquationsDeterminant of a MatrixMatrix InversionSolution of Simultaneous Linear EquationsApplications: Circuit AnalysisHomogeneous Coordinates SystemRank, Nu
Linear Maps Preserving Idempotence on Nest Algebras
Institute of Scientific and Technical Information of China (English)
Jian Lian CUI; Jin Chuan HOU
2004-01-01
In this paper, we discuss the rank-1-preserving linear maps on nest algebras of Hilbertspace operators. We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear bijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism.
Numerical linear algebra with applications using Matlab
Ford, William
2014-01-01
Designed for those who want to gain a practical knowledge of modern computational techniques for the numerical solution of linear algebra problems, Numerical Linear Algebra with Applications contains all the material necessary for a first year graduate or advanced undergraduate course on numerical linear algebra with numerous applications to engineering and science. With a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions, this book is ideal for solving real-world problems. It provides necessary mathematical background information for
AIR Tools - A MATLAB package of algebraic iterative reconstruction methods
DEFF Research Database (Denmark)
Hansen, Per Christian; Saxild-Hansen, Maria
2012-01-01
are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter...... and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new ‘‘training’’ algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP...
Accelerating Dense Linear Algebra on the GPU
DEFF Research Database (Denmark)
Sørensen, Hans Henrik Brandenborg
and matrix-vector operations on GPUs. Such operations form the backbone of level 1 and level 2 routines in the Basic Linear Algebra Subroutines (BLAS) library and are therefore of great importance in many scientific applications. The target hardware is the most recent NVIDIA Tesla 20-series (Fermi...... architecture). Most of the techniques I discuss for accelerating dense linear algebra are applicable to memory-bound GPU algorithms in general....
Numerical linear algebra in data mining
Eldén, Lars
Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.
Meromorphic iterative roots of linear fractional functions
Institute of Scientific and Technical Information of China (English)
SHI YongGuo; CHEN Li
2009-01-01
Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more difficult in non-monotone cases. Therefore, it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions on R. In this paper, iterative roots of LFFs are studied on C. An equivalence between the iterative functional equation for non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications, we present all meromorphic iterative roots for functions z and 1/z.
Modeling digital switching circuits with linear algebra
Thornton, Mitchell A
2014-01-01
Modeling Digital Switching Circuits with Linear Algebra describes an approach for modeling digital information and circuitry that is an alternative to Boolean algebra. While the Boolean algebraic model has been wildly successful and is responsible for many advances in modern information technology, the approach described in this book offers new insight and different ways of solving problems. Modeling the bit as a vector instead of a scalar value in the set {0, 1} allows digital circuits to be characterized with transfer functions in the form of a linear transformation matrix. The use of transf
Stability of Linear Equations--Algebraic Approach
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
Stability of Linear Equations--Algebraic Approach
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
Linear Algebraic Method for Non-Linear Map Analysis
Energy Technology Data Exchange (ETDEWEB)
Yu,L.; Nash, B.
2009-05-04
We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.
Linear algebra a first course with applications
Knop, Larry E
2008-01-01
Linear Algebra: A First Course with Applications explores the fundamental ideas of linear algebra, including vector spaces, subspaces, basis, span, linear independence, linear transformation, eigenvalues, and eigenvectors, as well as a variety of applications, from inventories to graphics to Google's PageRank. Unlike other texts on the subject, this classroom-tested book gives students enough time to absorb the material by focusing on vector spaces early on and using computational sections as numerical interludes. It offers introductions to Maple™, MATLAB®, and TI-83 Plus for calculating matri
Finite-dimensional linear algebra
Gockenbach, Mark S
2010-01-01
Some Problems Posed on Vector SpacesLinear equationsBest approximationDiagonalizationSummaryFields and Vector SpacesFields Vector spaces Subspaces Linear combinations and spanning sets Linear independence Basis and dimension Properties of bases Polynomial interpolation and the Lagrange basis Continuous piecewise polynomial functionsLinear OperatorsLinear operatorsMore properties of linear operatorsIsomorphic vector spaces Linear operator equations Existence and uniqueness of solutions The fundamental theorem; inverse operatorsGaussian elimination Newton's method Linear ordinary differential eq
Lal, Ramji
2017-01-01
This is the second in a series of three volumes dealing with important topics in algebra. Volume 2 is an introduction to linear algebra (including linear algebra over rings), Galois theory, representation theory, and the theory of group extensions. The section on linear algebra (chapters 1–5) does not require any background material from Algebra 1, except an understanding of set theory. Linear algebra is the most applicable branch of mathematics, and it is essential for students of science and engineering As such, the text can be used for one-semester courses for these students. The remaining part of the volume discusses Jordan and rational forms, general linear algebra (linear algebra over rings), Galois theory, representation theory (linear algebra over group algebras), and the theory of extension of groups follow linear algebra, and is suitable as a text for the second and third year students specializing in mathematics. .
Lie algebras and linear differential equations.
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Linear-algebraic lambda-calculus
Arrighi, P; Arrighi, Pablo; Dowek, Gilles
2005-01-01
With a view towards models of quantum computation and/or the interpretation of linear logic, we define a functional language where all functions are linear operators by construction. A small step operational semantic (and hence an interpreter/simulator) is provided for this language in the form of a term rewrite system. The linear-algebraic lambda-calculus hereby constructed is linear in a different (yet related) sense to that, say, of the linear lambda-calculus. These various notions of linearity are discussed in the context of quantum programming languages. KEYWORDS: quantum lambda-calculus, linear lambda-calculus, $\\lambda$-calculus, quantum logics.
Accelerating sparse linear algebra using graphics processing units
Spagnoli, Kyle E.; Humphrey, John R.; Price, Daniel K.; Kelmelis, Eric J.
2011-06-01
The modern graphics processing unit (GPU) found in many standard personal computers is a highly parallel math processor capable of over 1 TFLOPS of peak computational throughput at a cost similar to a high-end CPU with excellent FLOPS-to-watt ratio. High-level sparse linear algebra operations are computationally intense, often requiring large amounts of parallel operations and would seem a natural fit for the processing power of the GPU. Our work is on a GPU accelerated implementation of sparse linear algebra routines. We present results from both direct and iterative sparse system solvers. The GPU execution model featured by NVIDIA GPUs based on CUDA demands very strong parallelism, requiring between hundreds and thousands of simultaneous operations to achieve high performance. Some constructs from linear algebra map extremely well to the GPU and others map poorly. CPUs, on the other hand, do well at smaller order parallelism and perform acceptably during low-parallelism code segments. Our work addresses this via hybrid a processing model, in which the CPU and GPU work simultaneously to produce results. In many cases, this is accomplished by allowing each platform to do the work it performs most naturally. For example, the CPU is responsible for graph theory portion of the direct solvers while the GPU simultaneously performs the low level linear algebra routines.
Optical systolic solutions of linear algebraic equations
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
Linear algebra for skew-polynomial matrices
Abramov, Sergei; Bronstein, Manuel
2002-01-01
We describe an algorithm for transforming skew-polynomial matrices over an Ore domain in row-reduced form, and show that this algorithm can be used to perform the standard calculations of linear algebra on such matrices (ranks, kernels, linear dependences, inhomogeneous solving). The main application of our algorithm is to desingularize recurrences and to compute the rational solutions of a large class of linear functional systems. It also turns out to be efficient when applied to ordinary co...
SLAPP: A systolic linear algebra parallel processor
Energy Technology Data Exchange (ETDEWEB)
Drake, B.L.; Luk, F.T.; Speiser, J.M.; Symanski, J.J. (Naval Ocean Systems Center and Cornell Univ.)
1987-07-01
Systolic array computer architectures provide a means for fast computation of the linear algebra algorithms that form the building blocks of many signal-processing algorithms, facilitating their real-time computation. For applications to signal processing, the systolic array operates on matrices, an inherently parallel view of the data, using numerical linear algebra algorithms that have been suitably parallelized to efficiently utilize the available hardware. This article describes work currently underway at the Naval Ocean Systems Center, San Diego, California, to build a two-dimensional systolic array, SLAPP, demonstrating efficient and modular parallelization of key matric computations for real-time signal- and image-processing problems.
Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver
Livne, Oren E
2011-01-01
Laplacian matrices of graphs arise in large-scale computational applications such as machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG) solver of the linear system Ax=b is presented, where A is a graph Laplacian. LAMG's run time and storage are linear in the number of graph edges. LAMG consists of a setup phase, in which a sequence of increasingly-coarser Laplacian systems is constructed, and an iterative solve phase using multigrid cycles. General graphs pose algorithmic challenges not encountered in traditional applications of algebraic multigrid. LAMG combines a lean piecewise-constant interpolation, judicious node aggregation based on a new node proximity definition, and an energy correction of the coarse-level systems. This results in fast convergence and substantial overhead and memory s...
Linear algebra and group theory for physicists
Rao, K N Srinivasa
2006-01-01
Professor Srinivasa Rao's text on Linear Algebra and Group Theory is directed to undergraduate and graduate students who wish to acquire a solid theoretical foundation in these mathematical topics which find extensive use in physics. Based on courses delivered during Professor Srinivasa Rao's long career at the University of Mysore, this text is remarkable for its clear exposition of the subject. Advanced students will find a range of topics such as the Representation theory of Linear Associative Algebras, a complete analysis of Dirac and Kemmer algebras, Representations of the Symmetric group via Young Tableaux, a systematic derivation of the Crystallographic point groups, a comprehensive and unified discussion of the Rotation and Lorentz groups and their representations, and an introduction to Dynkin diagrams in the classification of Lie groups. In addition, the first few chapters on Elementary Group Theory and Vector Spaces also provide useful instructional material even at an introductory level. An author...
KWIC Index for Numerical Linear Algebra
Energy Technology Data Exchange (ETDEWEB)
Carpenter, J.A.
1983-07-01
This report is a sequel to ORNL/CSD-106 in the ongoing supplements to Professor A.S. Householder's KWIC Index for Numerical Algebra. Beginning with the previous supplement, the subject has been restricted to Numerical Linear Algebra, roughly characterized by the American Mathematical Society's classification sections 15 and 65F but with little coverage of infinite matrices, matrices over fields of characteristics other than zero, operator theory, optimization and those parts of matrix theory primarily combinatorial in nature. Some consideration is given to the uses of graph theory in Numerical Linear Algebra, particularly with respect to algorithms for sparse matrix computations. The period covered by this report is roughly the calendar year 1982 as measured by the appearance of the articles in the American Mathematical Society's Contents of Mathematical Publications lagging actual appearance dates by up to nearly half a year. The review citations are limited to the Mathematical Reviews (MR).
An introduction to linear algebra
Mirsky, L
2003-01-01
Rigorous, self-contained coverage of determinants, vectors, matrices and linear equations, quadratic forms, more. Elementary, easily readable account with numerous examples and problems at the end of each chapter.
Journal Writing: Enlivening Elementary Linear Algebra.
Meel, David E.
1999-01-01
Examines the various issues surrounding the implementation of journal writing in an undergraduate linear algebra course. Identifies the benefits of incorporating journal writing into an undergraduate mathematics course, which are supported with students' comments from their journals and their reflections on the process. Contains 14 references.…
Two dimensional basic linear algebra communication subprograms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Whaley, R.C. [Univ. of Tennessee, Knoxville, TN (United States); Geijn, R.A. van de [Univ. of Texas, Austin, TX (United States)
1993-12-31
This paper describes a package of linear algebra communication routines for manipulating and communicating data structures that are distributed among the memories of a distributed memory MIMD computer. The motivation for the BLACS is to increase portability, efficiency and modularity at a high level. The audience of the BLACS are mathematical software experts and people with large scale scientific computation to perform.
Journal Writing: Enlivening Elementary Linear Algebra.
Meel, David E.
1999-01-01
Examines the various issues surrounding the implementation of journal writing in an undergraduate linear algebra course. Identifies the benefits of incorporating journal writing into an undergraduate mathematics course, which are supported with students' comments from their journals and their reflections on the process. Contains 14 references.…
Modules as Learning Tools in Linear Algebra
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio
2014-01-01
This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…
Noise limitations in optical linear algebra processors.
Batsell, S G; Jong, T L; Walkup, J F; Krile, T F
1990-05-10
A general statistical noise model is presented for optical linear algebra processors. A statistical analysis which includes device noise, the multiplication process, and the addition operation is undertaken. We focus on those processes which are architecturally independent. Finally, experimental results which verify the analytical predictions are also presented.
Constructive Learning in Undergraduate Linear Algebra
Chandler, Farrah Jackson; Taylor, Dewey T.
2008-01-01
In this article we describe a project that we used in our undergraduate linear algebra courses to help our students successfully master fundamental concepts and definitions and generate interest in the course. We describe our philosophy and discuss the projects overall success.
High performance linear algebra algorithms: An introduction
DEFF Research Database (Denmark)
Gustavson, F.G.; Wasniewski, Jerzy
2006-01-01
his Mini-Symposium consisted of two back to back sessions, each consisting of five presentations, held on the afternoon of Monday, June 21, 2004. A major theme of both sessions was novel data structures for the matrices of dense linear algebra, DLA. Talks one to four of session one all centered...
Parallel computations in linear algebra. II
Energy Technology Data Exchange (ETDEWEB)
Faddeeva, V.N.; Faddeev, D.K.
1982-05-01
For pt.I, see Kibernetika, vol.13, no.6, p.28 (1977). Considerable effort was devoted in the surveyed period to automatic decomposition of sequential algorithms, or rather of procedures or subprograms written in the algorithmic languages ALGOL or FORTRAN. The authors do not consider this body of research, they only note that, on the one hand, the available linear algebra subprograms included in Eispack provide convenient objects for testing various approaches to automatic construction of parallel programs and, on the other, an important state in this activity is the development of methods for fast and efficient solution of linear recurrences, which reduce to solving systems of linear equations with band-triangular matrix (in particular, of sufficiently small width). This article reflects the penetration of the parallelism ideas into the computational methods of linear algebra in recent years. 74 references.
LAPACKrc: Fast linear algebra kernels/solvers for FPGA accelerators
Energy Technology Data Exchange (ETDEWEB)
Gonzalez, Juan; Nunez, Rafael C, E-mail: juan.gonzalez@accelogic.co [Accelogic, 1830 Main Street, Suite 204, Weston, FL (United States)
2009-07-01
We present LAPACKrc, a family of FPGA-based linear algebra solvers able to achieve more than 100x speedup per commodity processor on certain problems. LAPACKrc subsumes some of the LAPACK and ScaLAPACK functionalities, and it also incorporates sparse direct and iterative matrix solvers. Current LAPACKrc prototypes demonstrate between 40x-150x speedup compared against top-of-the-line hardware/software systems. A technology roadmap is in place to validate current performance of LAPACKrc in HPC applications, and to increase the computational throughput by factors of hundreds within the next few years.
Exact linear modeling using Ore algebras
Schindelar, Kristina; Zerz, Eva
2010-01-01
Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.
Linear algebra algorithms for divisors on an algebraic curve
Khuri-Makdisi, Kamal
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
Iterative solution of large linear systems
Young, David M
2003-01-01
This self-contained treatment offers a systematic development of the theory of iterative methods. Its focal point resides in an analysis of the convergence properties of the successive overrelaxation (SOR) method, as applied to a linear system with a consistently ordered matrix. The text explores the convergence properties of the SOR method and related techniques in terms of the spectral radii of the associated matrices as well as in terms of certain matrix norms. Contents include a review of matrix theory and general properties of iterative methods; SOR method and stationary modified SOR meth
Linear algebra on high-performance computers
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Sorensen, D.C.
1986-01-01
This paper surveys work recently done at Argonne National Laboratory in an attempt to discover ways to construct numerical software for high-performance computers. The numerical algorithms are taken from several areas of numerical linear algebra. We discuss certain architectural features of advanced-computer architectures that will affect the design of algorithms. The technique of restructuring algorithms in terms of certain modules is reviewed. This technique has proved successful in obtaining a high level of transportability without severe loss of performance on a wide variety of both vector and parallel computers. The module technique is demonstrably effective for dense linear algebra problems. However, in the case of sparse and structured problems it may be difficult to identify general modules that will be as effective. New algorithms have been devised for certain problems in this category. We present examples in three important areas: banded systems, sparse QR factorization, and symmetric eigenvalue problems. 32 refs., 10 figs., 6 tabs.
Transformation of time dependence to linear algebra
Menšík, Miroslav
2005-10-01
Reduced density matrix and memory function in the Nakajima-Zwanzig equation are expanded in properly chosen basis of special functions. This trick completely transforms time dependence to linear algebra. Then, the master equation for memory function is constructed and expanded in the same basis functions. For the model of a simple harmonic oscillator it is shown that this trick introduces infinite partial summation of the memory function in the system-bath interaction.
CULA: hybrid GPU accelerated linear algebra routines
Humphrey, John R.; Price, Daniel K.; Spagnoli, Kyle E.; Paolini, Aaron L.; Kelmelis, Eric J.
2010-04-01
The modern graphics processing unit (GPU) found in many standard personal computers is a highly parallel math processor capable of nearly 1 TFLOPS peak throughput at a cost similar to a high-end CPU and an excellent FLOPS/watt ratio. High-level linear algebra operations are computationally intense, often requiring O(N3) operations and would seem a natural fit for the processing power of the GPU. Our work is on CULA, a GPU accelerated implementation of linear algebra routines. We present results from factorizations such as LU decomposition, singular value decomposition and QR decomposition along with applications like system solution and least squares. The GPU execution model featured by NVIDIA GPUs based on CUDA demands very strong parallelism, requiring between hundreds and thousands of simultaneous operations to achieve high performance. Some constructs from linear algebra map extremely well to the GPU and others map poorly. CPUs, on the other hand, do well at smaller order parallelism and perform acceptably during low-parallelism code segments. Our work addresses this via hybrid a processing model, in which the CPU and GPU work simultaneously to produce results. In many cases, this is accomplished by allowing each platform to do the work it performs most naturally.
KWIC index for numerical linear algebra. [Bibliography
Energy Technology Data Exchange (ETDEWEB)
Carpenter, J.A.
1982-07-01
This report is a sequel to ORNL/CSD-96 in the ongoing supplements to Professor A.S. Householder's KWIC Index for Numerical Algebra. With this supplement, the coverage has been restricted to Numerical Linear Algebra and is now roughly characterized by the American Mathematical Society's classification section 15 and 65F but with little coverage of inifinite matrices, matrices over fields of characteristics other than zero, operator theory, optimization and those parts of matrix theory primarily combinatorial in nature. Some recognition is made of the uses of graph theory in Numerical Linear Algebra, particularly as regards their use in algorithms for sparse matrix computations. The period covered by this report is roughly the calendar year 1981 as measured by the appearance of the articles in the American Mathematical Society's Contents of Mathematical Publications. The review citations are limited to the Mathematical Reviews (MR) and Das Zentralblatt fur Mathematik und Ihre Grenzgebiete (ZBL). Future reports will be made more timely by closer ovservation of the few journals which supply the bulk of the listings rather than what appears to be too much reliance on secondary sources. Some thought is being given to the physical appearance of these reports and the author welcomes comments concerning both their appearance and contents.
A SLAP (Sparse Linear Algebra Package) for the masses
Energy Technology Data Exchange (ETDEWEB)
Seager, M.K.
1988-12-12
A Sparse Linear Algebra Package (SLAP), written in FORTRAN77, for the iterative solution of large sparse symmetric and non-symmetric linear systems is presented. SLAP Version 2.0 consists of three levels of routines: ''high level'', ''core'' and ''utility.'' The ''core'' routines implement the following preconditioned iterative methods: iterative refinement, conjugate gradient, conjugate gradient on the normal equations, bi-conjugate gradient, bi-conjugate gradient squared, orthomin and generalized minimum residual. All of these methods do not require the data structure of the matrix being solved nor of the preconditioning matrix, but do require the ''user'' to supply a matrix vector product and preconditioning routines. The ''high level'' routines assume one of two specific data structures and provide the required ''user routines.'' The preconditioners supported are diagonal scaling and incomplete factorization. One of the SLAP data structures allows for the vectorization of the matrix multiply and the backsolve of the incomplete factorization operations on machines with hardware gather/scatter capabilities. We present results for SLAP on the Cray Y/MP and Alliant FX/8 machines. 10 refs., 3 figs., 10 tabs.
LAPACK: Linear algebra software for supercomputers
Energy Technology Data Exchange (ETDEWEB)
Bischof, C.H.
1991-01-01
This paper presents an overview of the LAPACK library, a portable, public-domain library to solve the most common linear algebra problems. This library provides a uniformly designed set of subroutines for solving systems of simultaneous linear equations, least-squares problems, and eigenvalue problems for dense and banded matrices. We elaborate on the design methodologies incorporated to make the LAPACK codes efficient on today's high-performance architectures. In particular, we discuss the use of block algorithms and the reliance on the Basic Linear Algebra Subprograms. We present performance results that show the suitability of the LAPACK approach for vector uniprocessors and shared-memory multiprocessors. We also address some issues that have to be dealt with in tuning LAPACK for specific architectures. Lastly, we present results that show that the LAPACK software can be adapted with little effort to distributed-memory environments, and we discuss future efforts resulting from this project. 31 refs., 10 figs., 2 tabs.
Parallel algorithms for numerical linear algebra
van der Vorst, H
1990-01-01
This is the first in a new series of books presenting research results and developments concerning the theory and applications of parallel computers, including vector, pipeline, array, fifth/future generation computers, and neural computers.All aspects of high-speed computing fall within the scope of the series, e.g. algorithm design, applications, software engineering, networking, taxonomy, models and architectural trends, performance, peripheral devices.Papers in Volume One cover the main streams of parallel linear algebra: systolic array algorithms, message-passing systems, algorithms for p
Optical linear algebra processors - Architectures and algorithms
Casasent, David
1986-01-01
Attention is given to the component design and optical configuration features of a generic optical linear algebra processor (OLAP) architecture, as well as the large number of OLAP architectures, number representations, algorithms and applications encountered in current literature. Number-representation issues associated with bipolar and complex-valued data representations, high-accuracy (including floating point) performance, and the base or radix to be employed, are discussed, together with case studies on a space-integrating frequency-multiplexed architecture and a hybrid space-integrating and time-integrating multichannel architecture.
Basic linear algebra subprograms for FORTRAN usage
Lawson, C. L.; Hanson, R. J.; Kincaid, D. R.; Krogh, F. T.
1977-01-01
A package of 38 low level subprograms for many of the basic operations of numerical linear algebra is presented. The package is intended to be used with FORTRAN. The operations in the package are dot products, elementary vector operations, Givens transformations, vector copy and swap, vector norms, vector scaling, and the indices of components of largest magnitude. The subprograms and a test driver are available in portable FORTRAN. Versions of the subprograms are also provided in assembly language for the IBM 360/67, the CDC 6600 and CDC 7600, and the Univac 1108.
Primitive parallel operations for computational linear algebra
Energy Technology Data Exchange (ETDEWEB)
Panetta, J.
1985-01-01
This work is a small step in the direction of code portability over parallel and vector machines. The proposal consists of a style of programming and a set of parallel operators built over abstract data types. Objects and operators are directed to the Computational Linear Algebra area, although the principles of the proposal can be applied to any other area. A subset of the operators was implemented on a 64-processor, distributed memory MIMD machine, and the results are that computationally intensive operators achieve asymptotically optimal speed-ups, but data movement operators are inefficient, some even intrinsically sequential.
Optical linear algebra processors - Architectures and algorithms
Casasent, David
1986-01-01
Attention is given to the component design and optical configuration features of a generic optical linear algebra processor (OLAP) architecture, as well as the large number of OLAP architectures, number representations, algorithms and applications encountered in current literature. Number-representation issues associated with bipolar and complex-valued data representations, high-accuracy (including floating point) performance, and the base or radix to be employed, are discussed, together with case studies on a space-integrating frequency-multiplexed architecture and a hybrid space-integrating and time-integrating multichannel architecture.
Mathematical methods linear algebra normed spaces distributions integration
Korevaar, Jacob
1968-01-01
Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector
Freely generated vertex algebras and non-linear Lie conformal algebras
De Sole, Alberto; Kac, Victor
2003-01-01
We introduce the notion of a non--linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a non--linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.
INPUT-OUTPUT STRUCTURE OF LINEAR-DIFFERENTIAL ALGEBRAIC SYSTEMS
KUIJPER, M; SCHUMACHER, JM
1993-01-01
Systems of linear differential and algebraic equations occur in various ways, for instance, as a result of automated modeling procedures and in problems involving algebraic constraints, such as zero dynamics and exact model matching. Differential/algebraic systems may represent an input-output relat
Computing Gröbner Bases within Linear Algebra
Suzuki, Akira
In this paper, we present an alternative algorithm to compute Gröbner bases, which is based on computations on sparse linear algebra. Both of S-polynomial computations and monomial reductions are computed in linear algebra simultaneously in this algorithm. So it can be implemented to any computational system which can handle linear algebra. For a given ideal in a polynomial ring, it calculates a Gröbner basis along with the corresponding term order appropriately.
Using Geometry to teach and learn Linear Algebra
Gueudet, Ghislaine
2006-01-01
International audience; Linear algebra is a difficult topic for undergraduate students. In France, the focus of beginning linear algebra courses is the study of abstract vector spaces, with or without an inner product, rather than matrix operations as is common in many other countries. This paper presents a study of the possible uses of geometry and "geometrical intuition" in the teaching and learning of linear algebra. Fischbein's work on intuition in science and mathematics is used to analy...
Linear algebra a first course with applications to differential equations
Apostol, Tom M
2014-01-01
Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.
Investigating Students' Modes of Thinking in Linear Algebra: The Case of Linear Independence
Çelik, Derya
2015-01-01
Linear algebra is one of the most challenging topics to learn and teach in many countries. To facilitate the teaching and learning of linear algebra, priority should be given to epistemologically analyze the concepts that the undergraduate students have difficulty in conceptualizing and to define their ways of reasoning in linear algebra. After…
Investigating Students' Modes of Thinking in Linear Algebra: The Case of Linear Independence
Çelik, Derya
2015-01-01
Linear algebra is one of the most challenging topics to learn and teach in many countries. To facilitate the teaching and learning of linear algebra, priority should be given to epistemologically analyze the concepts that the undergraduate students have difficulty in conceptualizing and to define their ways of reasoning in linear algebra. After…
New parallel algorithms in linear algebra
Energy Technology Data Exchange (ETDEWEB)
Evans, D.J.
1983-01-01
The well-known and established techniques for deriving the numerical solution of linear systems of equations of the form ax=b are based on the following two basic strategies: (a) the factorisation of the coefficient matrix into easily inverted factors l and u leading to the class of direct methods, i.e. Gaussian elimination, Lu decomposition, Choleski, and (b) the splitting of the matrix a into convenient forms l and u which result in the iterative methods of Jacobi, Gauss Seidel and Sor. However, both these strategies essentially lead to algorithms more suitable for sequential computers and the question of a more convenient factorisation or splitting strategy for parallel processing is discussed leading to the formulation of new techniques based on the factorisation and splitting of the coefficient matrix a into components which are essentially interlocking matrix quadrants. It is shown that such a proposition leads to new parallel algorithms for both direct and iterative methods of solving linear equations and eigenvalue analysis. 15 references.
Linear algebra applications using Matlab software
Directory of Open Access Journals (Sweden)
Cornelia Victoria Anghel
2005-10-01
Full Text Available The paper presents two ways of special matrix generating using some functions included in the MatLab software package. The MatLab software package contains a set of functions that generate special matrixes used in the linear algebra applications and the signal processing from different activity fields. The paper presents two tipes of special matrixes that can be generated using written sintaxes in the dialog window of the MatLab software and for the command validity we need to press the Enter task. The applications presented in the paper represent eamples of numerical calculus using the MatLab software and belong to the scientific field „Computer Assisted Mathematics” thus creating the symbiosis between mathematics and informatics.
Towards automatic synthesis of linear algebra programs
Energy Technology Data Exchange (ETDEWEB)
Boyle, J. M.
1979-01-01
Automating the writing of efficient computer programs from an abstract specification of the computation that they are to perform is discussed. Advantages offered by automatic synthesis of programs include economy, reliability, and improved service. The synthesis of simple linear algebra programs is considered in general and then illustrated for the usual matrix product, a column-oriented matrix product, a rank-one update matrix product, and a program to multiply three matrices. The accumulation of inner products and transformational implementation of program synthesis addressed. The discussion attempts to illustrate both the general strategy of the syntheses and how various tactics can be adapted to make the syntheses proceed deterministically to programs that are optimal with respect to certain criteria. (RWR)
Numerical stability in problems of linear algebra.
Babuska, I.
1972-01-01
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
Optical Linear Algebra for Computational Light Transport
O'Toole, Matthew
Active illumination refers to optical techniques that use controllable lights and cameras to analyze the way light propagates through the world. These techniques confer many unique imaging capabilities (e.g. high-precision 3D scanning, image-based relighting, imaging through scattering media), but at a significant cost; they often require long acquisition and processing times, rely on predictive models for light transport, and cease to function when exposed to bright ambient sunlight. We develop a mathematical framework for describing and analyzing such imaging techniques. This framework is deeply rooted in numerical linear algebra, and models the transfer of radiant energy through an unknown environment with the so-called light transport matrix. Performing active illumination on a scene equates to applying a numerical operator on this unknown matrix. The brute-force approach to active illumination follows a two-step procedure: (1) optically measure the light transport matrix and (2) evaluate the matrix operator numerically. This approach is infeasible in general, because the light transport matrix is often much too large to measure, store, and analyze directly. Using principles from optical linear algebra, we evaluate these matrix operators in the optical domain, without ever measuring the light transport matrix in the first place. Specifically, we explore numerical algorithms that can be implemented partially or fully with programmable optics. These optical algorithms provide solutions to many longstanding problems in computer vision and graphics, including the ability to (1) photo-realistically change the illumination conditions of a given photo with only a handful of measurements, (2) accurately capture the 3D shape of objects in the presence of complex transport properties and strong ambient illumination, and (3) overcome the multipath interference problem associated with time-of-flight cameras. Most importantly, we introduce an all-new imaging regime
Reed, D. A.; Patrick, M. L.
1985-01-01
The applicability of static data flow architectures to the iterative solution of sparse linear systems of equations is investigated. An analytic performance model of a static data flow computation is developed. This model includes both spatial parallelism, concurrent execution in multiple PE's, and pipelining, the streaming of data from array memories through the PE's. The performance model is used to analyze a row partitioned iterative algorithm for solving sparse linear systems of algebraic equations. Based on this analysis, design parameters for the static data flow architecture as a function of matrix sparsity and dimension are proposed.
Resources for Teaching Linear Algebra. MAA Notes Volume 42.
Carlson, David, Ed.; And Others
This book takes the position that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition, and pedagogy. It includes the recommendations of the Linear Algebra Curriculum Study Group with their core syllabus for the first course, and the thoughts of mathematics faculty who have taught linear…
Esparsidade, Estrutura, Escalamento e Estabilidade em Algebra Linear Computacional
Stern, Julio M
2011-01-01
Sparsity, Structure, Scaling and Stability in Computational Linear Algebra - Textbook from the IX School of Computer Science, held on July 24-31 of 1994 at Recife, Brazil. Esparsidade, Estrutura, Escalamento e Estabilidade em Algebra Linear Computacional - Livro texto da IX Escola de Computacao, realizada nos dias 24 a 31 de Julho de 1994 em Recife, Brasil. This textbook is written in Portuguese Language.
Emphasizing Language and Visualization in Teaching Linear Algebra
Hannah, John; Stewart, Sepideh; Thomas, Mike
2013-01-01
Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his…
Resources for Teaching Linear Algebra. MAA Notes Volume 42.
Carlson, David, Ed.; And Others
This book takes the position that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition, and pedagogy. It includes the recommendations of the Linear Algebra Curriculum Study Group with their core syllabus for the first course, and the thoughts of mathematics faculty who have taught linear…
Emphasizing Language and Visualization in Teaching Linear Algebra
Hannah, John; Stewart, Sepideh; Thomas, Mike
2013-01-01
Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his…
Linear Algebra Revisited: An Attempt to Understand Students' Conceptual Difficulties
Britton, Sandra; Henderson, Jenny
2009-01-01
This article looks at some of the conceptual difficulties that students have in a linear algebra course. An overview of previous research in this area is given, and the various theories that have been espoused regarding the reasons that students find linear algebra so difficult are discussed. Student responses to two questions testing the ability…
Schaum's outline of theory and problems of linear algebra
Lipschutz, Seymour
2001-01-01
This third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.
An Efficient Bayesian Iterative Method for Solving Linear Systems
Institute of Scientific and Technical Information of China (English)
Deng DING; Kin Sio FONG; Ka Hou CHAN
2012-01-01
This paper concerns with the statistical methods for solving general linear systems.After a brief review of Bayesian perspective for inverse problems,a new and efficient iterative method for general linear systems from a Bayesian perspective is proposed.The convergence of this iterative method is proved,and the corresponding error analysis is studied.Finally,numerical experiments are given to support the efficiency of this iterative method,and some conclusions are obtained.
A hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element
Indian Academy of Sciences (India)
SHWETABH SRIVASTAVA; DHARMENDRA K GUPTA; PREDRAG STANIMIROVIC; SUKHJIT SINGH; FALGUNI ROY
2017-05-01
A quadratically convergent Newton-type iterative scheme is proposed for approximating the generalized Drazin inverse bd of the Banach algebra element b. Further, its extension into the form of the hyperpower iterative method of arbitrary order p$\\leq$2 is presented. Convergence criteria along with the estimation of error bounds in the computation of bd are discussed. Convergence results confirms the high order convergence rate of the proposed iterative scheme.
The linear algebra survival guide illustrated with Mathematica
Szabo, Fred
2015-01-01
The Linear Algebra Survival Guide is a reference book with a free downloadable Mathematica notebook containing all of interactive code to make the content of the book playable in Mathematica and the Mathematica Player. It offers a concise introduction to the core topics of linear algebra which includes numerous exercises that will accompany a first or second course in linear algebra. This book will guide you through the powerful graphic displays and visualization of Mathematica that make the most abstract theories seem simple-- allowing you to tackle realistic problems using simple mathematic
3D Object Recognition Based on Linear Lie Algebra Model
Institute of Scientific and Technical Information of China (English)
LI Fang-xing; WU Ping-dong; SUN Hua-fei; PENG Lin-yu
2009-01-01
A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed.Then an algorithm of 3D object recognition using the linear Lie algebra models is presented.It is a convenient recognition method for the objects which are symmetric about some axis.By using the presented algorithm,the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained.At last some recognition results of practicalities are given.
On Graph C*-Algebras with a Linear Ideal Lattice
DEFF Research Database (Denmark)
Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
2010-01-01
At the cost of restricting the nature of the involved K-groups, we prove a classication result for a hitherto unexplored class of graph C-algebras, allowing us to classify all graph C-algebras on nitely many vertices with a nite linear ideal lattice if all pair of vertices are connected by innitely...
Quasi-lisse vertex algebras and modular linear differential equations
Arakawa, Tomoyuki
2016-01-01
We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level $-h^{\\vee}/6-1$, which expresses the homogeneous Schur limit of the superconformal index of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.
Vortex lattice theory: A linear algebra approach
Chamoun, George C.
Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.
Essential linear algebra with applications a problem-solving approach
Andreescu, Titu
2014-01-01
This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject’s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality. Key features include: • a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory; • over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance; • an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them. Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course. ...
Parallel and Scalable Sparse Basic Linear Algebra Subprograms
DEFF Research Database (Denmark)
Liu, Weifeng
Sparse basic linear algebra subprograms (BLAS) are fundamental building blocks for numerous scientific computations and graph applications. Compared with Dense BLAS, parallelization of Sparse BLAS routines entails extra challenges due to the irregularity of sparse data structures. This thesis...
The arithmetico-geometric sequence: an application of linear algebra
Orosi, Greg
2016-07-01
In this paper, we present a linear algebra-based derivation of the analytic formula for the sum of the first nth terms of the arithmetico-geometric sequence. Furthermore, the advantage of the derivation is briefly discussed.
Linear algebra on a Cray X-MP. Technical document
Energy Technology Data Exchange (ETDEWEB)
Freund, R.F.
1990-04-01
This paper discusses basic issues of vectorization as well as memory organization and contention for vector machines. There is an analysis of the implications of these issues for the performance of basic linear algebra operations, SAXPY and SDOT.
Parallel and Scalable Sparse Basic Linear Algebra Subprograms
DEFF Research Database (Denmark)
Liu, Weifeng
Sparse basic linear algebra subprograms (BLAS) are fundamental building blocks for numerous scientific computations and graph applications. Compared with Dense BLAS, parallelization of Sparse BLAS routines entails extra challenges due to the irregularity of sparse data structures. This thesis...
Positive Stabilization of Linear Differential Algebraic Equation System
Directory of Open Access Journals (Sweden)
Muhafzan
2016-01-01
Full Text Available We study in this paper the existence of a feedback for linear differential algebraic equation system such that the closed-loop system is positive and stable. A necessary and sufficient condition for such existence has been established. This result can be used to detect the existence of a state feedback law that makes the linear differential algebraic equation system in closed loop positive and stable.
Formalized Linear Algebra over Elementary Divisor Rings in Coq
Cano, Guillaume; Cohen, Cyril; Dénès, Maxime; Mörtberg, Anders; Siles, Vincent
2016-01-01
International audience; This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely pre-sented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms comput-in...
Performance modeling and prediction for linear algebra algorithms
Iakymchuk, Roman
2012-01-01
This dissertation incorporates two research projects: performance modeling and prediction for dense linear algebra algorithms, and high-performance computing on clouds. The first project is focused on dense matrix computations, which are often used as computational kernels for numerous scientific applications. To solve a particular mathematical operation, linear algebra libraries provide a variety of algorithms. The algorithm of choice depends, obviously, on its performance. Performance of su...
IDEALS GENERATED BY LINEAR FORMS AND SYMMETRIC ALGEBRAS
Directory of Open Access Journals (Sweden)
Gaetana Restuccia
2016-01-01
Full Text Available We consider ideals generated by linear forms in the variables X1 : : : ;Xn in the polynomial ring R[X1; : : : ;Xn], being R a commutative, Noetherian ring with identity. We investigate when a sequence a1; a2; : : : ; am of linear forms is an ssequence, in order to compute algebraic invariants of the symmetric algebra of the ideal I = (a1; a2; : : : ; am.
Accelerating R with high performance linear algebra libraries
Directory of Open Access Journals (Sweden)
Bogdan Oancea
2015-09-01
Full Text Available Linear algebra routines are basic building blocks for the statistical software. In this paper we analyzed how can we improve R performance for matrix computations. We benchmarked few matrix operations using the standard linear algebra libraries included in the R distribution and high performance libraries like OpenBLAS, GotoBLAS and MKL. Our tests showed the best results are obtained with the MKL library, the other two libraries having similar performances, but lower than MKL.
Formalized linear algebra over Elementary Divisor Rings in Coq
Cano, Guillaume; Cohen, Cyril; Dénès, Maxime; Mörtberg, Anders; Siles, Vincent
2016-01-01
International audience; This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely pre-sented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms comput-in...
MULTILEVEL ITERATION METHODS FOR SOLVING LINEAR ILL-POSED PROBLEMS
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.
Linear {GLP}-algebras and their elementary theories
Pakhomov, F. N.
2016-12-01
The polymodal provability logic {GLP} was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free {GLP}-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable [1]. For every positive integer n we solve the corresponding question for the logics {GLP}_n that are the fragments of {GLP} with n modalities. We prove that the elementary theory of the free {GLP}_n-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable for all n. We introduce the notion of a linear {GLP}_n-algebra and prove that all free {GLP}_n-algebras generated by the constants \\mathbf{0}, \\mathbf{1} are linear. We also consider the more general case of the logics {GLP}_α whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.
Convergence results on iteration algorithms to linear systems.
Wang, Zhuande; Yang, Chuansheng; Yuan, Yubo
2014-01-01
In order to solve the large scale linear systems, backward and Jacobi iteration algorithms are employed. The convergence is the most important issue. In this paper, a unified backward iterative matrix is proposed. It shows that some well-known iterative algorithms can be deduced with it. The most important result is that the convergence results have been proved. Firstly, the spectral radius of the Jacobi iterative matrix is positive and the one of backward iterative matrix is strongly positive (lager than a positive constant). Secondly, the mentioned two iterations have the same convergence results (convergence or divergence simultaneously). Finally, some numerical experiments show that the proposed algorithms are correct and have the merit of backward methods.
Symmetric linear systems - An application of algebraic systems theory
Hazewinkel, M.; Martin, C.
1983-01-01
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
Symmetric linear systems - An application of algebraic systems theory
Hazewinkel, M.; Martin, C.
1983-01-01
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
Linear algebra and probability for computer science applications
Davis, Ernest
2012-01-01
MATLABDesk calculator operations Booleans Nonstandard numbers Loops and conditionals Script file Functions Variable scope and parameter passingI: Linear Algebra Vectors Definition of vectors Applications of vectorsBasic operations on vectorsDot productVectors in MATLAB: Basic operationsPlotting vectors in MATLABVectors in other programming languagesMatrices Definition of matrices Applications of matrices Simple operations on matrices Multiplying a matrix times a vector Linear transformation Systems of linear equations Matrix multiplication Vectors as matrices Algebraic properties of matrix mul
The Linear Span of Projections in AH Algebras and for Inclusions of C*-Algebras
Directory of Open Access Journals (Sweden)
Dinh Trung Hoa
2013-01-01
Full Text Available In the first part of this paper, we show that an AH algebra A=lim→(Ai,ϕi has the LP property if and only if every element of the centre of Ai belongs to the closure of the linear span of projections in A. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital C*-algebras P⊂A with a finite Watatani index, if a faithful conditional expectation E:A→P has the Rokhlin property in the sense of Kodaka et al., then P has the LP property under the condition thatA has the LP property. As an application, let A be a simple unital C*-algebra with the LP property, α an action of a finite group G onto Aut(A. If α has the Rokhlin property in the sense of Izumi, then the fixed point algebra AG and the crossed product algebra A ⋊α G have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.
Embodied, Symbolic and Formal Thinking in Linear Algebra
Stewart, Sepideh; Thomas, Michael O. J.
2007-01-01
Students often find their first university linear algebra experience very challenging. While coping with procedural aspects of the subject, solving linear systems and manipulating matrices, they may struggle with crucial conceptual ideas underpinning them, making it very difficult to progress in more advanced courses. This research has sought to…
Embodied, Symbolic and Formal Thinking in Linear Algebra
Stewart, Sepideh; Thomas, Michael O. J.
2007-01-01
Students often find their first university linear algebra experience very challenging. While coping with procedural aspects of the subject, solving linear systems and manipulating matrices, they may struggle with crucial conceptual ideas underpinning them, making it very difficult to progress in more advanced courses. This research has sought to…
The N-Isometric Isomorphisms in Linear N-Normed C*-Algebras
Institute of Scientific and Technical Information of China (English)
Chun-Gil PARK; Themistocles M. RASSIAS
2006-01-01
We prove the Hyers-Ulam stability of linear N-isometries in linear N-normed Banach modules over a unital C*-algebra. The main purpose of this paper is to investigate N-isometric C*-algebra isomorphisms between linear N-normed C*-algebras, N-isometric Poisson C*-algebra isomorphisms between linear N-normed Poisson C*-algebras, N-isometric Lie C*-algebra isomorphisms between linear N-normed Lie C*-algebras, N-isometric Poisson JC*-algebra isomorphisms between linear N-normed Poisson JC*-algebras, and N-isometric Lie JC*-algebra isomorphisms between linear N-normed Lie JC*-algebras.Moreover, we prove the Hyers-Ulam stability of their N-isometric homomorphisms.
Introduction to linear algebra and differential equations
Dettman, John W
1986-01-01
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
An introduction to linear algebra and tensors
Akivis, M A; Silverman, Richard A
1978-01-01
Eminently readable, completely elementary treatment begins with linear spaces and ends with analytic geometry, covering multilinear forms, tensors, linear transformation, and more. 250 problems, most with hints and answers. 1972 edition.
Impact of hierarchical memory systems on linear algebra algorithm design
Energy Technology Data Exchange (ETDEWEB)
Gallivan, K.; Jalby, W.; Meier, U.; Sameh, A.H.
1988-01-01
Linear algebra algorithms based on the BLAS or extended BLAS do not achieve high performance on multivector processors with a hierarchical memory system because of a lack of data locality. For such machines, block linear algebra algorithms must be implemented in terms of matrix-matrix primitives (BLAS3). Designing efficient linear algebra algorithms for these architectures requires analysis of the behavior of the matrix-matrix primitives and the resulting block algorithms as a function of certain system parameters. The analysis must identify the limits of performance improvement possible via blocking and any contradictory trends that require trade-off consideration. The authors propose a methodology that facilitates such an analysis and use it to analyze the performance of the BLAS3 primitives used in block methods. A similar analysis of the block size-performance relationship is also performed at the algorithm level for block versions of the LU decomposition and the Gram-Schmidt orthogonalization procedures.
Scaling Linear Algebra Kernels using Remote Memory Access
Energy Technology Data Exchange (ETDEWEB)
Krishnan, Manoj Kumar; Lewis, Robert R.; Vishnu, Abhinav
2010-09-13
This paper describes the scalability of linear algebra kernels based on remote memory access approach. The current approach differs from the other linear algebra algorithms by the explicit use of shared memory and remote memory access (RMA) communication rather than message passing. It is suitable for clusters and scalable shared memory systems. The experimental results on large scale systems (Linux-Infiniband cluster, Cray XT) demonstrate consistent performance advantages over ScaLAPACK suite, the leading implementation of parallel linear algebra algorithms used today. For example, on a Cray XT4 for a matrix size of 102400, our RMA-based matrix multiplication achieved over 55 teraflops while ScaLAPACK’s pdgemm measured close to 42 teraflops on 10000 processes.
Emphasizing language and visualization in teaching linear algebra
Hannah, John; Stewart, Sepideh; Thomas, Mike
2013-06-01
Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his approach in both lectures and tutorials, and how he employed visualization and an emphasis on language to encourage conceptual thinking. We use Tall's framework of three worlds of mathematical thinking to reflect on the effect of these activities in students' learning. An analysis of students' attitudes to the course and their test and examination results help to answer questions about the value of such an approach, suggesting ways forward in teaching linear algebra.
RELAXED ASYNCHRONOUS ITERATIONS FOR THE LINEAR COMPLEMENTARITY PROBLEM
Institute of Scientific and Technical Information of China (English)
Zhong-zhi Bai; Yu-guang Huang
2002-01-01
We present a class of relaxed asynchronous parallel multisplitting iterative methods forsolving the linear complementarity problem on multiprocessor systems, and set up theirconvergence theories when the system matrix of the linear complementarity problem is anH-matrix with positive diagonal elements.
LDRD final report : autotuning for scalable linear algebra.
Energy Technology Data Exchange (ETDEWEB)
Heroux, Michael Allen; Marker, Bryan (University of Texas at Austin, Austin, TX)
2011-09-01
This report summarizes the progress made as part of a one year lab-directed research and development (LDRD) project to fund the research efforts of Bryan Marker at the University of Texas at Austin. The goal of the project was to develop new techniques for automatically tuning the performance of dense linear algebra kernels. These kernels often represent the majority of computational time in an application. The primary outcome from this work is a demonstration of the value of model driven engineering as an approach to accurately predict and study performance trade-offs for dense linear algebra computations.
CRPC research into linear algebra software for high performance computers
Energy Technology Data Exchange (ETDEWEB)
Choi, J.; Walker, D.W. [Oak Ridge National Lab., TN (United States). Mathematical Sciences Section; Dongarra, J.J. [Univ. of Tennessee, Knoxville, TN (United States). Dept. of Computer Science]|[Oak Ridge National Lab., TN (United States). Mathematical Sciences Section; Pozo, R. [Univ. of Tennessee, Knoxville, TN (United States). Dept. of Computer Science; Sorensen, D.C. [Rice Univ., Houston, TX (United States). Dept. of Computational and Applied Mathematics
1994-12-31
In this paper the authors look at a number of approaches being investigated in the Center for Research on Parallel Computation (CRPC) to develop linear algebra software for high-performance computers. These approaches are exemplified by the LAPACK, templates, and ARPACK projects. LAPACK is a software library for performing dense and banded linear algebra computations, and was designed to run efficiently on high-performance computers. The authors focus on the design of the distributed-memory version of LAPACK, and on an object-oriented interface to LAPACK.
Linear algebraic methods applied to intensity modulated radiation therapy.
Crooks, S M; Xing, L
2001-10-01
Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.
Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver (Journal Version)
Livne, Oren E
2011-01-01
Laplacian matrices of graphs arise in large-scale computational applications such as machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG) solver of the symmetric linear system Ax=b is presented, where A is a graph Laplacian. LAMG's run time and storage are linear in the number of graph edges. It is robust and requires no fine tuning. LAMG consists of a setup phase, in which a sequence of increasingly-coarser Laplacian systems is constructed, and an iterative solve phase using multigrid cycles. General graphs pose algorithmic challenges not encountered in traditional applications of algebraic multigrid. LAMG combines a lean piecewise-constant interpolation, judicious node aggregation based on a new node proximity definition, and a novel energy correction of the coarse-level systems. This results ...
A fast iterative scheme for the linearized Boltzmann equation
Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2017-06-01
Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L = (L + Nδh) - Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference
Linear algebra and matrix analysis for statistics
Banerjee, Sudipto
2014-01-01
Matrices, Vectors, and Their OperationsBasic definitions and notations Matrix addition and scalar-matrix multiplication Matrix multiplication Partitioned matricesThe ""trace"" of a square matrix Some special matricesSystems of Linear EquationsIntroduction Gaussian elimination Gauss-Jordan elimination Elementary matrices Homogeneous linear systems The inverse of a matrixMore on Linear EquationsThe LU decompositionCrout's Algorithm LU decomposition with row interchanges The LDU and Cholesky factorizations Inverse of partitioned matrices The LDU decomposition for partitioned matricesThe Sherman-W
Algebraic Framework for Linear and Morphological Scale-Spaces
Heijmans, H.J.A.M.; van den Boomgaard, R.
2002-01-01
This paper proposes a general algebraic construction technique for image scale-spaces. The basic idea is to first downscale the image by some factor using an invertible scaling, then apply an image operator (linear or morphological) at a unit scale, and finally resize the image to its original scale
A Choice Option between Proofs in Linear Algebra
Rensaa, Ragnhild Johanne
2007-01-01
At their final exam in linear algebra students at the author's university were given the possibility to choose between two types of proofs to be done. They could either prove two short statements by themselves or they could explain four steps in a given proof. This paper reports on investigations of students' responses to the choice option…
Clearing the Fog from the Undergraduate Course in Linear Algebra
Scott, Damon
2007-01-01
For over a decade it has been a common observation that a "fog" passes over the course in linear algebra once abstract vector spaces are presented. See [2, 3]. We show how this fog may be cleared by having the students translate "abstract" vector-space problems to isomorphic "concrete" settings, solve the "concrete" problem either by hand or with…
A Framework for Mathematical Thinking: The Case of Linear Algebra
Stewart, Sepideh; Thomas, Michael O. J.
2009-01-01
Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the…
Inverse Modelling Problems in Linear Algebra Undergraduate Courses
Martinez-Luaces, Victor E.
2013-01-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…
Strategies and Computer Projects for Teaching Linear Algebra.
Pecuch-Herrero, Marta
2000-01-01
Adopts several strategies for successful teaching and learning of linear algebra, which consist of a set of computer projects allowing students to explore new concepts, make conjectures, apply theorems, and work on applied projects of their choice. The resulting improvement in student learning has been remarkable. (Contains 12 references.)…
The IBM RISC System/6000 and linear algebra operations
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. (Tennessee Univ., Knoxville, TN (USA). Dept. of Computer Science Oak Ridge National Lab., TN (USA)); Mays, P. (Numerical Algorithms Group Ltd., Oxford (UK)); Radicati di Brozolo, G. (IBM European Center for Scientific and Engineering Computing, Rome (Italy))
1991-01-01
This paper discusses the IBM RISC System/6000 workstation and a set of experiments with blocked algorithms commonly used in solving problems in numerical linear algebra. We describe the performance of these algorithms and discuss the techniques used in achieving high performance on such an architecture. 10 refs., 11 figs., 6 tabs.
A set of Level 3 Basic Linear Algebra Subprograms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. (Univ. of Tennessee, Knoxville, TN (United States) Oak Ridge National Lab., TN (United States)); Du Croz, J.; Hammarling, S. (Numerical Algorithms Group Ltd., Oxford (United Kingdom)); Duff, I. (Harwell Lab., Oxfordshire (United Kingdom))
1990-03-01
This paper describes a set of Level 3 Basic Linear Algebra Subprograms (Level 3 BLAS). The Level 3 BLAS are targeted at matrix-matrix operations, with the aim of providing more efficient, but portable, implementations of algorithms on high-performance computers, especially those with hierarchical memory and parallel processing capability.
Creating Discussions with Classroom Voting in Linear Algebra
Cline, Kelly; Zullo, Holly; Duncan, Jonathan; Stewart, Ann; Snipes, Marie
2013-01-01
We present a study of classroom voting in linear algebra, in which the instructors posed multiple-choice questions to the class and then allowed a few minutes for consideration and small-group discussion. After each student in the class voted on the correct answer using a classroom response system, a set of clickers, the instructor then guided a…
Undergraduate Mathematics Students' Emotional Experiences in Linear Algebra Courses
Martínez-Sierra, Gustavo; García-González, María del Socorro
2016-01-01
Little is known about students' emotions in the field of Mathematics Education that go beyond students' emotions in problem solving. To start filling this gap this qualitative research has the aim to identify emotional experiences of undergraduate mathematics students in Linear Algebra courses. In order to obtain data, retrospective focus group…
Partially Flipped Linear Algebra: A Team-Based Approach
Carney, Debra; Ormes, Nicholas; Swanson, Rebecca
2015-01-01
In this article we describe a partially flipped Introductory Linear Algebra course developed by three faculty members at two different universities. We give motivation for our partially flipped design and describe our implementation in detail. Two main features of our course design are team-developed preview videos and related in-class activities.…
Inverse Modelling Problems in Linear Algebra Undergraduate Courses
Martinez-Luaces, Victor E.
2013-01-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…
Optical linear algebra processors: noise and error-source modeling.
Casasent, D; Ghosh, A
1985-06-01
The modeling of system and component noise and error sources in optical linear algebra processors (OLAP's) are considered, with attention to the frequency-multiplexed OLAP. General expressions are obtained for the output produced as a function of various component errors and noise. A digital simulator for this model is discussed.
A Framework for Mathematical Thinking: The Case of Linear Algebra
Stewart, Sepideh; Thomas, Michael O. J.
2009-01-01
Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the…
Definitions Are Important: The Case of Linear Algebra
Berman, Abraham; Shvartsman, Ludmila
2016-01-01
In this paper we describe an experiment in a linear algebra course. The aim of the experiment was to promote the students' understanding of the studied concepts focusing on their definitions. It seems to be a given that students should understand concepts' definitions before working substantially with them. Unfortunately, in many cases they do…
Using Cognitive Tutor Software in Learning Linear Algebra Word Concept
Yang, Kai-Ju
2015-01-01
This paper reports on a study of twelve 10th grade students using Cognitive Tutor, a math software program, to learn linear algebra word concept. The study's purpose was to examine whether students' mathematics performance as it is related to using Cognitive Tutor provided evidence to support Koedlinger's (2002) four instructional principles used…
Optical linear algebra processors - Noise and error-source modeling
Casasent, D.; Ghosh, A.
1985-01-01
The modeling of system and component noise and error sources in optical linear algebra processors (OLAPs) are considered, with attention to the frequency-multiplexed OLAP. General expressions are obtained for the output produced as a function of various component errors and noise. A digital simulator for this model is discussed.
Linear Algebra and the Experiences of a "Flipper"
Wright, Sarah E.
2015-01-01
This paper describes the linear algebra class I taught during Spring 2014 semester at Adelphi University. I discuss the details of how I flipped the class and incorporated elements of inquiry-based learning as well as the reasoning behind specific decisions I made. I give feedback from the students on the success of the course and provide my own…
Fifth SIAM conference on applied linear algebra. Final report
Energy Technology Data Exchange (ETDEWEB)
Lewis, J.G.; Gilbert, J.R.; Parlett, B.N.
1994-11-16
The SIAM Conferences on Applied Linear Algebra are the centerpiece of activities for the SIAG on Linear Algebra. They are held every three years and bring together a diverse group of applied linear algebraists, representing industry, government and academics in both matrix theory and matrix computations. This sequence of conferences has two related goals: (1) to be useful and interesting to linear algebraists of every area of specialization, and, (2) to develop and expose connections among problems in different areas. Many aspects of the 1994 conference were carefully chosen to enhance interchange between the various groups and yet still provide a solid focus on specialities. The organizing committee adopted a new meeting structure to resolve the conflict between these two goals at earlier meetings in the series. We have prepared this report for others who may wish to consider our structure as an alternative to more traditional arrangements.
Elements of Linear Algebra. Lecture Notes
Cotaescu, Ion I
2016-01-01
These pedagogical lecture notes address to the students in theoretical physics for helping them to understand the mechanisms of the linear operators defined on finite-dimensional vector spaces equipped with definite or indefinite inner products. The importance of the Dirac conjugation is pointed out presenting its general theory and a version of the Riesz theorem for the indefinite inner product spaces, based on the Dirac-Riesz map that combines the action of the Riesz map with that of the metric operator. The matrix representations of the linear operators on vector spaces with definite or indefinite inner products is also presented.
Reading between the Lines: Teaching Linear Algebra
Lewis, Jennifer M.; Blunk, Merrie L.
2012-01-01
This paper compares lessons on linear equations from the same curriculum materials taught by two teachers of different levels of mathematical knowledge for teaching (MKT). The analysis indicates that the mathematical quality of instruction in these two classrooms appears to be a function of differences in MKT. Although the two teachers were…
HOPF ALGEBRAIC APPROACH TO THE n LINEARLY RECURSIVE SEQUENCES
Institute of Scientific and Technical Information of China (English)
LIANGGUI
1994-01-01
It is proved that a linearly recursive sequence of n indicea over field F(n≥1) is autorntatically a product of n lioearly recurplve sequencea of 1-lndex over F by the theory of Hopf algebras.By the way,the correspondence between the set of linearly recursive sequenoes of 1-index and F[X]0 is generalised to the case of n-index.
Quasi-Linear Algebras and Integrability (the Heisenberg Picture
Directory of Open Access Journals (Sweden)
Alexei Zhedanov
2008-02-01
Full Text Available We study Poisson and operator algebras with the ''quasi-linear property'' from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators as functions of ''time'' t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution interpretation of the corresponding integrable systems.
GPU Linear algebra extensions for GNU/Octave
Bosi, L. B.; Mariotti, M.; Santocchia, A.
2012-06-01
Octave is one of the most widely used open source tools for numerical analysis and liner algebra. Our project aims to improve Octave by introducing support for GPU computing in order to speed up some linear algebra operations. The core of our work is a C library that executes some BLAS operations concerning vector- vector, vector matrix and matrix-matrix functions on the GPU. OpenCL functions are used to program GPU kernels, which are bound within the GNU/octave framework. We report the project implementation design and some preliminary results about performance.
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Institute of Scientific and Technical Information of China (English)
罗掌华
2001-01-01
讨论了针对求解线性问题Y=(A×B)Y+Φ1的OOPI算法和MPID算法的并行性能.在对算法的并行执行过程进行描述之后，文中给出了两种算法的存储性能要求和并行加速比.通过分析之后发现，OOPI算法的并行性能依赖于对m×m阶矩阵分解，而MPID算法却难以在处理机之间平均分配负载，这降低了并行机的有效利用率，为克服这两种算法在并行计算上的缺陷，提出了OMPID算法，并对该算法的收敛性进行了证明.%Parallelism of OOPI algorithm and MPID algorithm for so lving thelinear systems as Y=(AB)Y+Φ1 are discussed.After parallel pr ocess of algorithms is described,the memory requirements are analyzed and the ra tes of parallel acceleration are given.It is indicated by theoretical analysis t hat parallelism of OOPI algorithm depends on parallelism of decomposition of m matrix,and it is difficult for MPID algorithm to evenly allocate loads among processors.A kind of new optimum multi-parameter iteration algorithm is present ed,and converge of algorithm is proved.
Sectorial oscillation of linear differential equations and iterated order
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Dao-chun Sun
2004-10-01
Full Text Available In the present paper, we investigate higher order linear differential equations with entire coefficients of iterated order. Using value distribution theory of transcendental meromorphic functions and covering surface theory, we extend a result on the order of growth of solutions published by Bank and Langley [2].
Linear iterative technique for solution of nonlinear thermal network problems
Energy Technology Data Exchange (ETDEWEB)
Seabourn, C.M.
1976-11-01
A method for rapid and accurate solution of linear and/or nonlinear thermal network problems is described. It is a matrix iterative process that converges for nodal temperatures and variations of thermal conductivity with temperature. The method is computer oriented and can be changed easily for design studies.
The AOR Iterative Method for Preconditioned Linear Systems
Institute of Scientific and Technical Information of China (English)
WANG Zhuan-de; GAO Zhong-xi; HUANG Ting-zhu
2004-01-01
The preconditioned methods for solving linear system are discussed The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterative method does not converge. The results in corresponding references are improved and perfected.
Parallel linear algebra library for the Denelcor HEP
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Sorensen, D.C.
1984-10-01
This paper discusses the implementation of a library of algorithms for problems in linear algebra on the Denelcor HEP. The package includes some of the most heavily used subroutines from LINPACK, that is, solution of linear systems based on LU, Cholesky, and QR factorizations as well as the appropriate triangular solvers. The concept followed is to code these routines in terms of high-level modules, which provides a vehicle to achieve both transportability and efficiency across a wide range of architectures. We discuss this concept in the context of a numerical linear algebra software library which is adaptable to highly parallel computing systems. However, the concept is expected to applicable to other libraries as well.
Heterogenous Acceleration for Linear Algebra in Multi-coprocessor Environments
Energy Technology Data Exchange (ETDEWEB)
Luszczek, Piotr R [ORNL; Tomov, Stanimire Z [ORNL; Dongarra, Jack J [ORNL
2015-01-01
We present an efficient and scalable programming model for the development of linear algebra in heterogeneous multi-coprocessor environments. The model incorporates some of the current best design and implementation practices for the heterogeneous acceleration of dense linear algebra (DLA). Examples are given as the basis for solving linear systems' algorithms - the LU, QR, and Cholesky factorizations. To generate the extreme level of parallelism needed for the efficient use of coprocessors, algorithms of interest are redesigned and then split into well-chosen computational tasks. The tasks execution is scheduled over the computational components of a hybrid system of multi-core CPUs and coprocessors using a light-weight runtime system. The use of lightweight runtime systems keeps scheduling overhead low, while enabling the expression of parallelism through otherwise sequential code. This simplifies the development efforts and allows the exploration of the unique strengths of the various hardware components.
Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations
Seo, Jae Hong
Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.
Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability
Institute of Scientific and Technical Information of China (English)
CHEN ZHENG-XIN; CHEN QIONG
2012-01-01
Let Mn be the algebra of all n × n complex matrices and gl(n,C) be the general linear Lie algebra,where n ≥ 2.An invertible linear map ?:gl(n,C) →gl(n,C) preserves solvability in both directions if both ? and ?-1 map every solvable Lie subalgebra of gl(n,C) to some solvable Lie subalgebra.In this paper we classify the invertible linear maps preserving solvability on gl(n,C) in both directions.As a sequence,such maps coincide with the invertible linear maps preserving commutativity on Mn in both directions.
Near infrared reflectance analysis by Gauss-Jordan linear algebra
Honigs, D. E.; Freelin, J. M.; Hieftje, G. M.
1983-02-01
Near-infrared reflectance analysis (NIRA) is an analytical technique that uses the near-infrared diffuse reflectance of a sample at several discrete wavelengths to predict the concentration of one or more of the chemical species in that sample. However, because near-infrared bands from solid samples are both abundant and broad, the reflectance at a given wavelength usually contains contributions from several sample components, requiring extensive calculations on overlapped bands. In the present study, these calculations have been performed using an approach similar to that employed in multi-component spectrophotometry, but with Gauss-Jordan linear algebra serving as the computational vehicle. Using this approach, correlations for percent protein in wheat flour and percent benzene in hydrocarbons have been obtained and are evaluated. The advantages of a linear-algebra approach over the common one employing stepwise regression are explored.
[Relations between biomedical variables: mathematical analysis or linear algebra?].
Hucher, M; Berlie, J; Brunet, M
1977-01-01
The authors, after a short reminder of one pattern's structure, stress on the possible double approach of relations uniting the variables of this pattern: use of fonctions, what is within the mathematical analysis sphere, use of linear algebra profiting by matricial calculation's development and automatiosation. They precise the respective interests on these methods, their bounds and the imperatives for utilization, according to the kind of variables, of data, and the objective for work, understanding phenomenons or helping towards decision.
Error Control of Iterative Linear Solvers for Integrated Groundwater Models
Dixon, Matthew; Brush, Charles; Chung, Francis; Dogrul, Emin; Kadir, Tariq
2010-01-01
An open problem that arises when using modern iterative linear solvers, such as the preconditioned conjugate gradient (PCG) method or Generalized Minimum RESidual method (GMRES) is how to choose the residual tolerance in the linear solver to be consistent with the tolerance on the solution error. This problem is especially acute for integrated groundwater models which are implicitly coupled to another model, such as surface water models, and resolve both multiple scales of flow and temporal interaction terms, giving rise to linear systems with variable scaling. This article uses the theory of 'forward error bound estimation' to show how rescaling the linear system affects the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed using the USGS GSFLOW package and the California State Department of Water Resources' Integrated Water Flow Model (IWFM), we observe that this error bound guides the choice of a prac...
The design of linear algebra libraries for high performance computers
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. [Tennessee Univ., Knoxville, TN (United States). Dept. of Computer Science]|[Oak Ridge National Lab., TN (United States); Walker, D.W. [Oak Ridge National Lab., TN (United States)
1993-08-01
This paper discusses the design of linear algebra libraries for high performance computers. Particular emphasis is placed on the development of scalable algorithms for MIMD distributed memory concurrent computers. A brief description of the EISPACK, LINPACK, and LAPACK libraries is given, followed by an outline of ScaLAPACK, which is a distributed memory version of LAPACK currently under development. The importance of block-partitioned algorithms in reducing the frequency of data movement between different levels of hierarchical memory is stressed. The use of such algorithms helps reduce the message startup costs on distributed memory concurrent computers. Other key ideas in our approach are the use of distributed versions of the Level 3 Basic Linear Algebra Subprograms (BLAS) as computational building blocks, and the use of Basic Linear Algebra Communication Subprograms (BLACS) as communication building blocks. Together the distributed BLAS and the BLACS can be used to construct higher-level algorithms, and hide many details of the parallelism from the application developer. The block-cyclic data distribution is described, and adopted as a good way of distributing block-partitioned matrices. Block-partitioned versions of the Cholesky and LU factorizations are presented, and optimization issues associated with the implementation of the LU factorization algorithm on distributed memory concurrent computers are discussed, together with its performance on the Intel Delta system. Finally, approaches to the design of library interfaces are reviewed.
Nonlinear Alignment and Its Local Linear Iterative Solution
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Sumin Zhang
2016-01-01
Full Text Available In manifold learning, the aim of alignment is to derive the global coordinate of manifold from the local coordinates of manifold’s patches. At present, most of manifold learning algorithms assume that the relation between the global and local coordinates is locally linear and based on this linear relation align the local coordinates of manifold’s patches into the global coordinate of manifold. There are two contributions in this paper. First, the nonlinear relation between the manifold’s global and local coordinates is deduced by making use of the differentiation of local pullback functions defined on the differential manifold. Second, the method of local linear iterative alignment is used to align the manifold’s local coordinates into the manifold’s global coordinate. The experimental results presented in this paper show that the errors of noniterative alignment are considerably large and can be reduced to almost zero within the first two iterations. The large errors of noniterative/linear alignment verify the nonlinear nature of alignment and justify the necessity of iterative alignment.
Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics
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Alex J. Dragt; Filippo Neri; Govindan Rangarajan; David Douglas; Liam M. Healy; Robert D. Ryne
1988-12-01
The purpose of this paper is to present a summary of new methods, employing Lie algebraic tools, for characterizing beam dynamics in charged-particle optical systems. These methods are applicable to accelerator design, charged-particle beam transport, electron microscopes, and also light optics. The new methods represent the action of each separate element of a compound optical system, including all departures from paraxial optics, by a certain operator. The operators for the various elements can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. This paper deals mostly with accelerator design and charged-particle beam transport. The application of Lie algebraic methods to light optics and electron microscopes is described elsewhere (1, see also 44). To keep its scope within reasonable bounds, they restrict their treatment of accelerator design and charged-particle beam transport primarily to the use of Lie algebraic methods for the description of particle orbits in terms of transfer maps. There are other Lie algebraic or related approaches to accelerator problems that the reader may find of interest (2). For a general discussion of linear and nonlinear problems in accelerator physics see (3).
AZTEC: A parallel iterative package for the solving linear systems
Energy Technology Data Exchange (ETDEWEB)
Hutchinson, S.A.; Shadid, J.N.; Tuminaro, R.S. [Sandia National Labs., Albuquerque, NM (United States)
1996-12-31
We describe a parallel linear system package, AZTEC. The package incorporates a number of parallel iterative methods (e.g. GMRES, biCGSTAB, CGS, TFQMR) and preconditioners (e.g. Jacobi, Gauss-Seidel, polynomial, domain decomposition with LU or ILU within subdomains). Additionally, AZTEC allows for the reuse of previous preconditioning factorizations within Newton schemes for nonlinear methods. Currently, a number of different users are using this package to solve a variety of PDE applications.
Energy Technology Data Exchange (ETDEWEB)
Rice, J.R.
1984-12-31
On October 29 and 30, 1984 about 20 people met at Purdue University to consider extensions to the Basic Linear Algebra Subroutines (BLAS) and linear algebra software modules in general. The need for these extensions and new sets of modules is largely due to the advent of new supercomputer architectures which make it difficult for ordinary coding techniques to achieve even a significant fraction of the potential computing power. The workshop format was one of informal presentations with ample discussions followed by sessions of general discussions of the issues raised. This report is a summary of the presentations, the issues raised, the conclusions reached and the open issue discussions. Each participant had an opportunity to comment on this report, but it also clearly reflects the author's filtering of the extensive discussions. Section 2 describes seven proposals for linear algebra software modules and Section 3 describes four presentations on the use of such modules. Discussion summaries are given next; Section 4 for those where near concensus was reached and Section 5 where the issues were left open.
Kim, Joshua; Guan, Huaiqun; Gersten, David; Zhang, Tiezhi
2013-01-01
Tetrahedron beam computed tomography (TBCT) performs volumetric imaging using a stack of fan beams generated by a multiple pixel X-ray source. While the TBCT system was designed to overcome the scatter and detector issues faced by cone beam computed tomography (CBCT), it still suffers the same large cone angle artifacts as CBCT due to the use of approximate reconstruction algorithms. It has been shown that iterative reconstruction algorithms are better able to model irregular system geometries and that algebraic iterative algorithms in particular have been able to reduce cone artifacts appearing at large cone angles. In this paper, the SART algorithm is modified for the use with the different TBCT geometries and is tested using both simulated projection data and data acquired using the TBCT benchtop system. The modified SART reconstruction algorithms were able to mitigate the effects of using data generated at large cone angles and were also able to reconstruct CT images without the introduction of artifacts due to either the longitudinal or transverse truncation in the data sets. Algebraic iterative reconstruction can be especially useful for dual-source dual-detector TBCT, wherein the cone angle is the largest in the center of the field of view.
Directory of Open Access Journals (Sweden)
Joshua Kim
2013-01-01
Full Text Available Tetrahedron beam computed tomography (TBCT performs volumetric imaging using a stack of fan beams generated by a multiple pixel X-ray source. While the TBCT system was designed to overcome the scatter and detector issues faced by cone beam computed tomography (CBCT, it still suffers the same large cone angle artifacts as CBCT due to the use of approximate reconstruction algorithms. It has been shown that iterative reconstruction algorithms are better able to model irregular system geometries and that algebraic iterative algorithms in particular have been able to reduce cone artifacts appearing at large cone angles. In this paper, the SART algorithm is modified for the use with the different TBCT geometries and is tested using both simulated projection data and data acquired using the TBCT benchtop system. The modified SART reconstruction algorithms were able to mitigate the effects of using data generated at large cone angles and were also able to reconstruct CT images without the introduction of artifacts due to either the longitudinal or transverse truncation in the data sets. Algebraic iterative reconstruction can be especially useful for dual-source dual-detector TBCT, wherein the cone angle is the largest in the center of the field of view.
AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S
Klumpp, A. R.
1994-01-01
This package extends the Ada programming language to include linear algebra capabilities similar to those of the HAL/S programming language. The package is designed for avionics applications such as Space Station flight software. In addition to the HAL/S built-in functions, the package incorporates the quaternion functions used in the Shuttle and Galileo projects, and routines from LINPAK that solve systems of equations involving general square matrices. Language conventions in this package follow those of HAL/S to the maximum extent practical and minimize the effort required for writing new avionics software and translating existent software into Ada. Valid numeric types in this package include scalar, vector, matrix, and quaternion declarations. (Quaternions are fourcomponent vectors used in representing motion between two coordinate frames). Single precision and double precision floating point arithmetic is available in addition to the standard double precision integer manipulation. Infix operators are used instead of function calls to define dot products, cross products, quaternion products, and mixed scalar-vector, scalar-matrix, and vector-matrix products. The package contains two generic programs: one for floating point, and one for integer. The actual component type is passed as a formal parameter to the generic linear algebra package. The procedures for solving systems of linear equations defined by general matrices include GEFA, GECO, GESL, and GIDI. The HAL/S functions include ABVAL, UNIT, TRACE, DET, INVERSE, TRANSPOSE, GET, PUT, FETCH, PLACE, and IDENTITY. This package is written in Ada (Version 1.2) for batch execution and is machine independent. The linear algebra software depends on nothing outside the Ada language except for a call to a square root function for floating point scalars (such as SQRT in the DEC VAX MATHLIB library). This program was developed in 1989, and is a copyrighted work with all copyright vested in NASA.
AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S
Klumpp, A. R.
1994-01-01
This package extends the Ada programming language to include linear algebra capabilities similar to those of the HAL/S programming language. The package is designed for avionics applications such as Space Station flight software. In addition to the HAL/S built-in functions, the package incorporates the quaternion functions used in the Shuttle and Galileo projects, and routines from LINPAK that solve systems of equations involving general square matrices. Language conventions in this package follow those of HAL/S to the maximum extent practical and minimize the effort required for writing new avionics software and translating existent software into Ada. Valid numeric types in this package include scalar, vector, matrix, and quaternion declarations. (Quaternions are fourcomponent vectors used in representing motion between two coordinate frames). Single precision and double precision floating point arithmetic is available in addition to the standard double precision integer manipulation. Infix operators are used instead of function calls to define dot products, cross products, quaternion products, and mixed scalar-vector, scalar-matrix, and vector-matrix products. The package contains two generic programs: one for floating point, and one for integer. The actual component type is passed as a formal parameter to the generic linear algebra package. The procedures for solving systems of linear equations defined by general matrices include GEFA, GECO, GESL, and GIDI. The HAL/S functions include ABVAL, UNIT, TRACE, DET, INVERSE, TRANSPOSE, GET, PUT, FETCH, PLACE, and IDENTITY. This package is written in Ada (Version 1.2) for batch execution and is machine independent. The linear algebra software depends on nothing outside the Ada language except for a call to a square root function for floating point scalars (such as SQRT in the DEC VAX MATHLIB library). This program was developed in 1989, and is a copyrighted work with all copyright vested in NASA.
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…
Mechanical Analogy-based Iterative Method for Solving a System of Linear Equations
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Yu. V. Berchun
2015-01-01
Full Text Available The paper reviews prerequisites to creating a variety of the iterative methods to solve a system of linear equations (SLE. It considers the splitting methods, variation-type methods, projection-type methods, and the methods of relaxation.A new iterative method based on mechanical analogy (the movement without resistance of a material point, that is connected by ideal elastically-linear constraints with unending guides defined by equations of solved SLE. The mechanical system has the unique position of stable equilibrium, the coordinates of which correspond to the solution of linear algebraic equation. The model of the mechanical system is a system of ordinary differential equations of the second order, integration of which allows you to define the point trajectory. In contrast to the classical methods of relaxation the proposed method does not ensure a trajectory passage through the equilibrium position. Thus the convergence of the method is achieved through the iterative stop of a material point at the moment it passes through the next (from the beginning of the given iteration minimum of potential energy. After that the next iteration (with changed initial coordinates starts.A resource-intensive process of numerical integration of differential equations in order to obtain a precise law of motion (at each iteration is replaced by defining its approximation. The coefficients of the approximating polynomial of the fourth order are calculated from the initial conditions, including higher-order derivatives. The resulting approximation enables you to evaluate the kinetic energy of a material point to calculate approximately the moment of time to reach the maximum kinetic energy (and minimum of the potential one, i.e. the end of the iteration.The software implementation is done. The problems with symmetric positive definite matrix, generated as a result of using finite element method, allowed us to examine a convergence rate of the proposed method
Aydin, Sinan
2014-01-01
Linear algebra is a basic mathematical subject taught in mathematics and science depar-tments of universities. The teaching and learning of this course has always been difficult. This study aims to contribute to the research in linear algebra education, focusing on linear dependence and independence concepts. This was done by introducing…
Aydin, Sinan
2014-01-01
Linear algebra is a basic mathematical subject taught in mathematics and science depar-tments of universities. The teaching and learning of this course has always been difficult. This study aims to contribute to the research in linear algebra education, focusing on linear dependence and independence concepts. This was done by introducing…
Aydin, Sinan
2014-01-01
Linear algebra is a basic mathematical subject taught in mathematics and science depar-tments of universities. The teaching and learning of this course has always been difficult. This study aims to contribute to the research in linear algebra education, focusing on linear dependence and independence concepts. This was done by introducing…
Matrix preconditioning: a robust operation for optical linear algebra processors.
Ghosh, A; Paparao, P
1987-07-15
Analog electrooptical processors are best suited for applications demanding high computational throughput with tolerance for inaccuracies. Matrix preconditioning is one such application. Matrix preconditioning is a preprocessing step for reducing the condition number of a matrix and is used extensively with gradient algorithms for increasing the rate of convergence and improving the accuracy of the solution. In this paper, we describe a simple parallel algorithm for matrix preconditioning, which can be implemented efficiently on a pipelined optical linear algebra processor. From the results of our numerical experiments we show that the efficacy of the preconditioning algorithm is affected very little by the errors of the optical system.
Polynomial system solving for decoding linear codes and algebraic cryptanalysis
2009-01-01
This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The ...
Negative base encoding in optical linear algebra processors
Perlee, C.; Casasent, D.
1986-01-01
In the digital multiplication by analog convolution algorithm, the bits of two encoded numbers are convolved to form the product of the two numbers in mixed binary representation; this output can be easily converted to binary. Attention is presently given to negative base encoding, treating base -2 initially, and then showing that the negative base system can be readily extended to any radix. In general, negative base encoding in optical linear algebra processors represents a more efficient technique than either sign magnitude or 2's complement encoding, when the additions of digitally encoded products are performed in parallel.
First order linear ordinary differential equations in associative algebras
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Gordon Erlebacher
2004-01-01
Full Text Available In this paper, we study the linear differential equation $$ frac{dx}{dt}=sum_{i=1}^n a_i(t x b_i(t + f(t $$ in an associative but non-commutative algebra $mathcal{A}$, where the $b_i(t$ form a set of commuting $mathcal{A}$-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.
GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS
Institute of Scientific and Technical Information of China (English)
陈正新
2014-01-01
Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) 6= 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.
Linear $r$-matrix algebra for classical separable systems
Eilbeck, J C; Kuznetsov, V B; Tsiganov, A V; Kuznetsov, Vadim B.
1994-01-01
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\\times 2$ matrices for the whole hierarchy and construct the associated linear $r$-matrix algebra with the $r$-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Using the method of variable separation we provide the integration of the systems in classical mechanics conctructing the separation equations and, hence, the explicit form of action variables. The quantisation problem is discussed with the help of the separation variables.
Relating Reasoning Methodologies in Linear Logic and Process Algebra
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Yuxin Deng
2012-11-01
Full Text Available We show that the proof-theoretic notion of logical preorder coincides with the process-theoretic notion of contextual preorder for a CCS-like calculus obtained from the formula-as-process interpretation of a fragment of linear logic. The argument makes use of other standard notions in process algebra, namely a labeled transition system and a coinductively defined simulation relation. This result establishes a connection between an approach to reason about process specifications and a method to reason about logic specifications.
Student Learning of Basis, Span and Linear Independence in Linear Algebra
Stewart, Sepideh; Thomas, Michael O. J.
2010-01-01
One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have…
Comparison of approaches based on optimization and algebraic iteration for binary tomography
Cai, Weiwei; Ma, Lin
2010-12-01
Binary tomography represents a special category of tomographic problems, in which only two values are possible for the sought image pixels. The binary nature of the problems can potentially lead to a significant reduction in the number of view angles required for a satisfactory reconstruction, thusly enabling many interesting applications. However, the limited view angles result in a severely underdetermined system of equations, which is challenging to solve. Various approaches have been proposed to address such a challenge, and two categories of approaches include those based on optimization and those based on algebraic iteration. However, the relative strengths, limitations, and applicable ranges of these approaches have not been clearly defined in the past. Therefore, it is the main objective of this work to conduct a systematic comparison of approaches from each category. This comparison suggested that the approaches based on algebraic iteration offered both superior reconstruction fidelity and computation efficiency at low (two or three) view angles, and these advantages diminished at high view angles. Meanwhile, this work also investigated the application of regularization techniques, the selection of optimal regularization parameter, and the use of a local search technique for binary problems. We expect the results and conclusions reported in this work to provide valuable guidance for the design and development of algorithms for binary tomography problems.
Lighthouse: A User-Centered Web Service for Linear Algebra Software
Norris, Boyana; Bernstein, Sa-Lin; Nair, Ramya; Jessup, Elizabeth
2014-01-01
Various fields of science and engineering rely on linear algebra for large scale data analysis, modeling and simulation, machine learning, and other applied problems. Linear algebra computations often dominate the execution time of such applications. Meanwhile, experts in these domains typically lack the training or time required to develop efficient, high-performance implementations of linear algebra algorithms. In the Lighthouse project, we enable developers with varied backgrounds to readi...
Hageman, Louis A
2004-01-01
This graduate-level text examines the practical use of iterative methods in solving large, sparse systems of linear algebraic equations and in resolving multidimensional boundary-value problems. Assuming minimal mathematical background, it profiles the relative merits of several general iterative procedures. Topics include polynomial acceleration of basic iterative methods, Chebyshev and conjugate gradient acceleration procedures applicable to partitioning the linear system into a "red/black" block form, adaptive computational algorithms for the successive overrelaxation (SOR) method, and comp
Linear response theory an analytic-algebraic approach
De Nittis, Giuseppe
2017-01-01
This book presents a modern and systematic approach to Linear Response Theory (LRT) by combining analytic and algebraic ideas. LRT is a tool to study systems that are driven out of equilibrium by external perturbations. In particular the reader is provided with a new and robust tool to implement LRT for a wide array of systems. The proposed formalism in fact applies to periodic and random systems in the discrete and the continuum. After a short introduction describing the structure of the book, its aim and motivation, the basic elements of the theory are presented in chapter 2. The mathematical framework of the theory is outlined in chapters 3–5: the relevant von Neumann algebras, noncommutative $L^p$- and Sobolev spaces are introduced; their construction is then made explicit for common physical systems; the notion of isopectral perturbations and the associated dynamics are studied. Chapter 6 is dedicated to the main results, proofs of the Kubo and Kubo-Streda formulas. The book closes with a chapter about...
Robust parallel iterative solvers for linear and least-squares problems, Final Technical Report
Energy Technology Data Exchange (ETDEWEB)
Saad, Yousef
2014-01-16
The primary goal of this project is to study and develop robust iterative methods for solving linear systems of equations and least squares systems. The focus of the Minnesota team is on algorithms development, robustness issues, and on tests and validation of the methods on realistic problems. 1. The project begun with an investigation on how to practically update a preconditioner obtained from an ILU-type factorization, when the coefficient matrix changes. 2. We investigated strategies to improve robustness in parallel preconditioners in a specific case of a PDE with discontinuous coefficients. 3. We explored ways to adapt standard preconditioners for solving linear systems arising from the Helmholtz equation. These are often difficult linear systems to solve by iterative methods. 4. We have also worked on purely theoretical issues related to the analysis of Krylov subspace methods for linear systems. 5. We developed an effective strategy for performing ILU factorizations for the case when the matrix is highly indefinite. The strategy uses shifting in some optimal way. The method was extended to the solution of Helmholtz equations by using complex shifts, yielding very good results in many cases. 6. We addressed the difficult problem of preconditioning sparse systems of equations on GPUs. 7. A by-product of the above work is a software package consisting of an iterative solver library for GPUs based on CUDA. This was made publicly available. It was the first such library that offers complete iterative solvers for GPUs. 8. We considered another form of ILU which blends coarsening techniques from Multigrid with algebraic multilevel methods. 9. We have released a new version on our parallel solver - called pARMS [new version is version 3]. As part of this we have tested the code in complex settings - including the solution of Maxwell and Helmholtz equations and for a problem of crystal growth.10. As an application of polynomial preconditioning we considered the
Acoustooptic linear algebra processors - Architectures, algorithms, and applications
Casasent, D.
1984-01-01
Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.
Using linear algebra for protein structural comparison and classification
Directory of Open Access Journals (Sweden)
Janaína Gomide
2009-01-01
Full Text Available In this article, we describe a novel methodology to extract semantic characteristics from protein structures using linear algebra in order to compose structural signature vectors which may be used efficiently to compare and classify protein structures into fold families. These signatures are built from the pattern of hydrophobic intrachain interactions using Singular Value Decomposition (SVD and Latent Semantic Indexing (LSI techniques. Considering proteins as documents and contacts as terms, we have built a retrieval system which is able to find conserved contacts in samples of myoglobin fold family and to retrieve these proteins among proteins of varied folds with precision of up to 80%. The classifier is a web tool available at our laboratory website. Users can search for similar chains from a specific PDB, view and compare their contact maps and browse their structures using a JMol plug-in.
PC-BLAS. PC Basic Linear Algebra Subroutines
Energy Technology Data Exchange (ETDEWEB)
Hanson, R.J. [Sandia National Labs., Albuquerque, NM (United States)
1986-11-01
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, and find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.
A look at scalable dense linear algebra libraries
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. [Tennessee Univ., Knoxville, TN (United States). Dept. of Computer Science]|[Oak Ridge National Lab., TN (United States); Van de Geijn, R.A. [Texas Univ., Austin, TX (United States). Dept. of Computer Sciences; Walker, D.W. [Oak Ridge National Lab., TN (United States)
1992-08-01
We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization are presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 GFLOPS (double precision) for the largest problem considered.
A look at scalable dense linear algebra libraries
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. (Tennessee Univ., Knoxville, TN (United States) Dept. of Computer Science Oak Ridge National Lab., TN (United States)); van de Geijn, R. (Texas Univ., Austin, TX (United States). Dept. of Computer Sciences); Walker, D.W. (Oak Ridge National Lab., TN (United States))
1992-07-01
We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization are presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 Gflop/s (double precision) for the largest problem considered.
Linear algebra for dense matrices on a hypercube
Energy Technology Data Exchange (ETDEWEB)
Sears, M.P.
1990-01-01
A set of routines has been written for dense matrix operations optimized for the NCUBE/6400 parallel processor. This work was motivated by a Sandia effort to parallelize certain electronic structure calculations. Routines are included for matrix transpose, multiply, Cholesky decomposition, triangular inversion, and Householder tridiagonalization. The library is written in C and is callable from Fortran. Matrices up to order 1600 can be handled on 128 processors. For each operation, the algorithm used is presented along with typical timings and estimates of performance. Performance for order 1600 on 128 processors varies from 42 MFLOPs (Householder tridiagonalization, triangular inverse) up to 126 MFLOPs (matrix multiply). We also present performance results for communications and basic linear algebra operations (saxpy and dot products). 8 refs., 6 tabs.
A look at scalable dense linear algebra libraries
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. (Tennessee Univ., Knoxville, TN (United States). Dept. of Computer Science Oak Ridge National Lab., TN (United States)); Van de Geijn, R.A. (Texas Univ., Austin, TX (United States). Dept. of Computer Sciences); Walker, D.W. (Oak Ridge National Lab., TN (United States))
1992-01-01
We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization are presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 GFLOPS (double precision) for the largest problem considered.
Acoustooptic linear algebra processors - Architectures, algorithms, and applications
Casasent, D.
1984-01-01
Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.
Parallel Implementation of Linear Algebra Problems on Dawning—1000
Institute of Scientific and Technical Information of China (English)
迟学斌
1998-01-01
In this paper,some parallel algorithms are described for solving numerical linear algebra problems on Dawning-1000.They include matrix multiplication,LU factorization of a dense matrix,Cholesky factorization of a symmetric matrix,and eigendecomposition of symmetric matrix for real and complex data types.These programs are constructed based on fast BLAS library of Dawning-1000 under NX environment.Some comparison results under different parallel environments and implementing methods are also given for Cholesky factorization.The execution time,measured performance and speedup for each problem on Dawning-1000 are shown.For matrix multiplication and IU factorization,1.86GFLOPS and 1.53GFLOPS are reached.
Using linear algebra for protein structural comparison and classification.
Gomide, Janaína; Melo-Minardi, Raquel; Dos Santos, Marcos Augusto; Neshich, Goran; Meira, Wagner; Lopes, Júlio César; Santoro, Marcelo
2009-07-01
In this article, we describe a novel methodology to extract semantic characteristics from protein structures using linear algebra in order to compose structural signature vectors which may be used efficiently to compare and classify protein structures into fold families. These signatures are built from the pattern of hydrophobic intrachain interactions using Singular Value Decomposition (SVD) and Latent Semantic Indexing (LSI) techniques. Considering proteins as documents and contacts as terms, we have built a retrieval system which is able to find conserved contacts in samples of myoglobin fold family and to retrieve these proteins among proteins of varied folds with precision of up to 80%. The classifier is a web tool available at our laboratory website. Users can search for similar chains from a specific PDB, view and compare their contact maps and browse their structures using a JMol plug-in.
ON THE ITERATED ORDER OF MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
CaoTingbin; ChenZongxuan; ZhengXiumin; TuJin
2005-01-01
In this paper, we investigate complexhigher order linear differential equationshomogeneous and non-homogeneous with meromorphic coefficients. We obtain several results concerning the iterated order of meromorphic solutionsand the iterated convergence exponent of the zeros of meromorphic solutions.
Directory of Open Access Journals (Sweden)
Cheng-yi Zhang
2016-06-01
Full Text Available Abstract Some convergence conditions on successive over-relaxed (SOR iterative method and symmetric SOR (SSOR iterative method are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.
Optimization techniques for OpenCL-based linear algebra routines
Kozacik, Stephen; Fox, Paul; Humphrey, John; Kuller, Aryeh; Kelmelis, Eric; Prather, Dennis W.
2014-06-01
The OpenCL standard for general-purpose parallel programming allows a developer to target highly parallel computations towards graphics processing units (GPUs), CPUs, co-processing devices, and field programmable gate arrays (FPGAs). The computationally intense domains of linear algebra and image processing have shown significant speedups when implemented in the OpenCL environment. A major benefit of OpenCL is that a routine written for one device can be run across many different devices and architectures; however, a kernel optimized for one device may not exhibit high performance when executed on a different device. For this reason kernels must typically be hand-optimized for every target device family. Due to the large number of parameters that can affect performance, hand tuning for every possible device is impractical and often produces suboptimal results. For this work, we focused on optimizing the general matrix multiplication routine. General matrix multiplication is used as a building block for many linear algebra routines and often comprises a large portion of the run-time. Prior work has shown this routine to be a good candidate for high-performance implementation in OpenCL. We selected several candidate algorithms from the literature that are suitable for parameterization. We then developed parameterized kernels implementing these algorithms using only portable OpenCL features. Our implementation queries device information supplied by the OpenCL runtime and utilizes this as well as user input to generate a search space that satisfies device and algorithmic constraints. Preliminary results from our work confirm that optimizations are not portable from one device to the next, and show the benefits of automatic tuning. Using a standard set of tuning parameters seen in the literature for the NVIDIA Fermi architecture achieves a performance of 1.6 TFLOPS on an AMD 7970 device, while automatically tuning achieves a peak of 2.7 TFLOPS
Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences
2016-01-01
International audience; Sakata generalized the Berlekamp -- Massey algorithm to $n$ dimensions in~1988. The Berlekamp -- Massey -- Sakata (BMS)algorithm can be used for finding a Gröbner basis of a $0$-dimensionalideal of relations verified by a table. We investigate this problem usinglinear algebra techniques, with motivations such as accelerating change ofbasis algorithms (FGLM) or improving their complexity.We first define and characterize multidimensional linear recursive sequencesfor $0$...
A linear process-algebraic format with data for probabilistic automata
Katoen, Joost-Pieter; Pol, van de Jaco; Stoelinga, Mariëlle; Timmer, Mark
2011-01-01
This paper presents a novel linear process-algebraic format for probabilistic automata. The key ingredient is a symbolic transformation of probabilistic process algebra terms that incorporate data into this linear format while preserving strong probabilistic bisimulation. This generalises similar te
-Orthomorphisms and -Linear Operators on the Order Dual of an -Algebra
Directory of Open Access Journals (Sweden)
Ying Feng
2012-01-01
Full Text Available We consider the -orthomorphisms and -linear operators on the order dual of an -algebra. In particular, when the -algebra has the factorization property (not necessarily unital, we prove that the orthomorphisms, -orthomorphisms, and -linear operators on the order dual are precisely the same class of operators.
Montiel, Mariana; Bhatti, Uzma
2010-01-01
This article presents an overview of some issues that were confronted when delivering an online second Linear Algebra course (assuming a previous Introductory Linear Algebra course) to graduate students enrolled in a Secondary Mathematics Education program. The focus is on performance in one particular aspect of the course: "change of basis" and…
Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles
Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim
2016-01-01
This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…
The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.
Uhlig, Frank
2002-01-01
Describes how elementary linear algebra can be taught successfully while introducing students to the concept and practice of mathematical proof. Suggests exploring the concept of solvability of linear systems first via the row echelon form (REF). (Author/KHR)
The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.
Uhlig, Frank
2002-01-01
Describes how elementary linear algebra can be taught successfully while introducing students to the concept and practice of mathematical proof. Suggests exploring the concept of solvability of linear systems first via the row echelon form (REF). (Author/KHR)
Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report
Energy Technology Data Exchange (ETDEWEB)
NONE
1997-12-31
Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that people from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.
Probing the Locality of Excited States with Linear Algebra.
Etienne, Thibaud
2015-04-14
This article reports a novel theoretical approach related to the analysis of molecular excited states. The strategy introduced here involves gathering two pieces of physical information, coming from Hilbert and direct space operations, into a general, unique quantum mechanical descriptor of electronic transitions' locality. Moreover, the projection of Hilbert and direct space-derived indices in an Argand plane delivers a straightforward way to visually probe the ability of a dye to undergo a long- or short-range charge-transfer. This information can be applied, for instance, to the analysis of the electronic response of families of dyes to light absorption by unveiling the trend of a given push-pull chromophore to increase the electronic cloud polarization magnitude of its main transition with respect to the size extension of its conjugated spacer. We finally demonstrate that all the quantities reported in this article can be reliably approximated by a linear algebraic derivation, based on the contraction of detachment/attachment density matrices from canonical to atomic space. This alternative derivation has the remarkable advantage of a very low computational cost with respect to the previously used numerical integrations, making fast and accurate characterization of large molecular systems' excited states easily affordable.
Linear Commuting Maps on Parab olic Subalgebras of Finite-dimensional Simple Lie Algebras
Institute of Scientific and Technical Information of China (English)
CHEN Zheng-xin; WANG Bing
2014-01-01
A map ϕ on a Lie algebra g is called to be commuting if [ϕ(x), x] = 0 for all x∈g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapϕon P is commuting if and only ifϕis a scalar multiplication map on P .
Fast Linear Algebra Applications in Stochastic Inversion and Data Assimilation
Kitanidis, P. K.; Ambikasaran, S.; Saibaba, A.; Li, J. Y.; Darve, E. F.
2012-12-01
Inverse problems and data assimilation problems arise frequently in earth-science applications, such as hydraulic tomography, cross-well seismic travel-time tomography, electrical resistivity tomography, contaminant source identification, assimilation of weather data, etc. A common feature amongst inverse problems is that the parameters we are interested in estimating are hard to measure directly, and a crucial component of inverse modeling is using sparse data to evaluate many model parameters. To quantify uncertainty, stochastic methods such as the geostatistical approach to inverse problems and Kalman filtering are often used. The algorithms for the implementation of these methods were originally developed for small-size problems and their cost of implementation increases quickly with the size of the problem, which is usually defined by the number of observations and the number of unknowns. From a practical standpoint, it is critical to develop computational algorithms in linear algebra for which the computational effort, both in terms of storage and computational time, increases roughly linearly with the size of the problem. This is in contrast, for example, with matrix-vector products (resp. LU factorization) that scale quadratically (resp. cubically). This objective is achieved by tailoring methods to the structure of problems. We present an overview of the challenges and general approaches available for reducing computational cost and then present applications focusing on algorithms that use the hierarchical matrix approach. The hierarchical method reduces matrix vector products involving the dense covariance matrix from O(m2) to O(m log m), where m is the number of unknowns. We illustrate the performance of our algorithm on a few applications, such as monitoring CO2 concentrations using crosswell seismic tomography.
CONTINUITY AND LINEARITY OF ADDITIVE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES
Institute of Scientific and Technical Information of China (English)
HANDEGUANG
1996-01-01
This paper discusses the problem concerning the continuity and linearity of additive derivation of nest algebras on normed spaces. It is proved that erevy linear derivation of a nest algebra algN is continuous provided that one of the following conditions is satisfied:(1)0+（包含）0.(2)X-（包含于）X.(3)there exists a non-trivial idempotnet p in algN such that the range of p belongs to N, It is also proved that every additive derivation of a nest algebra is automatically linear if the underlying normed space is infinite dimemnsional.
Iterative solution of linear systems in the 20th century
Saad, Y.; Vorst, H.A. van der
2001-01-01
This paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century. Although iterative methods for solving linear systems find their origin in the early nineteenth century (work by Gauss), the field has seen an explosion of activi
Yildiz Ulus, Aysegul
2013-01-01
This paper examines experimental and algorithmic contributions of advanced calculators (graphing and computer algebra system, CAS) in teaching the concept of "diagonalization," one of the key topics in Linear Algebra courses taught at the undergraduate level. Specifically, the proposed hypothesis of this study is to assess the effective…
An Example of Competence-Based Learning: Use of Maxima in Linear Algebra for Engineers
Diaz, Ana; Garcia, Alfonsa; de la Villa, Agustin
2011-01-01
This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…
Dongarra, Jack
2012-11-01
We propose to study the impact on the energy footprint of two advanced algorithmic strategies in the context of high performance dense linear algebra libraries: (1) mixed precision algorithms with iterative refinement allow to run at the peak performance of single precision floating-point arithmetic while achieving double precision accuracy and (2) tree reduction technique exposes more parallelism when factorizing tall and skinny matrices for solving over determined systems of linear equations or calculating the singular value decomposition. Integrated within the PLASMA library using tile algorithms, which will eventually supersede the block algorithms from LAPACK, both strategies further excel in performance in the presence of a dynamic task scheduler while targeting multicore architecture. Energy consumption measurements are reported along with parallel performance numbers on a dual-socket quad-core Intel Xeon as well as a quad-socket quad-core Intel Sandy Bridge chip, both providing component-based energy monitoring at all levels of the system, through the Power Pack framework and the Running Average Power Limit model, respectively. © 2012 IEEE.
Decomposition Theory in the Teaching of Elementary Linear Algebra.
London, R. R.; Rogosinski, H. P.
1990-01-01
Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)
Decomposition Theory in the Teaching of Elementary Linear Algebra.
London, R. R.; Rogosinski, H. P.
1990-01-01
Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)
Student learning and perceptions in a flipped linear algebra course
Love, Betty; Hodge, Angie; Grandgenett, Neal; Swift, Andrew W.
2014-04-01
The traditional lecture style of teaching has long been the norm in college science, technology, engineering, and mathematics (STEM) courses, but an innovative teaching model, facilitated by recent advances in technology, is gaining popularity across college campuses. This new model inverts or 'flips' the usual classroom paradigm, in that students learn initial course concepts outside of the classroom, while class time is reserved for more active problem-based learning and practice activities. While the flipped classroom model shows promise for improving STEM learning and increasing student interest in STEM fields, discussions to date of the model and its impact are more anecdotal than data driven - very little research has been undertaken to rigorously assess the potential effects on student learning that can result from the flipped classroom environment. This study involved 55 students in 2 sections of an applied linear algebra course, using the traditional lecture format in one section and the flipped classroom model in another. In the latter, students were expected to prepare for the class in some way, such as watching screencasts prepared by the instructor, or reading the textbook or the instructor's notes. Student content understanding and course perceptions were examined. Content understanding was measured by the performance on course exams, and students in the flipped classroom environment had a more significant increase between the sequential exams compared to the students in the traditional lecture section, while performing similarly in the final exam. Course perceptions were represented by an end-of-semester survey that indicated that the flipped classroom students were very positive about their experience in the course, and particularly appreciated the student collaboration and instructional video components.
Parallelization for MIMD multiprocessors with applications to linear algebra algorithms
Energy Technology Data Exchange (ETDEWEB)
Nelken, I.H.
1989-01-01
In this thesis, the author considers the parallelization problem. Given a sequential algorithm and a target architecture, how can the sequential algorithm be converted into a parallel algorithm suitable for the target architecture The parallel algorithm must be correct and produce the same results as the sequential one. It must also utilize the resources of the target architecture efficiently. The parallelization problem can be divided into three main stages: identification of parallelism which includes dependency analysis, partitioning the statements into atomic tasks of granularity suitable to the target architecture and scheduling these tasks into the processors. The identification of parallelism is independent of the target architecture while the partitioning and scheduling stages are very dependent on it. For example, the partitioning for a machine with many small processors is very different than the partitioning for a machine with a few large ones. It is well known that the problems arising in the partitioning and scheduling stages are NP-complete. The thesis shows that for some algorithms arising in linear algebra, simple heuristics are sufficient to produce good solutions to the partitioning and scheduling problems. He considers the Gaussian elimination and Gauss-Jordan algorithms for general dense matrices and the Cholesky decomposition algorithms for symmetric positive definite matrices. In addition he studies algorithms for the solution of simultaneous triangular systems with the same coefficient matrix and different right hand sides and for the solution of the triangular Sylvester equation. Most of the results in this thesis are related to the more difficult problems of partitioning and scheduling for message passing architectures.
Developing ontological model of computational linear algebra - preliminary considerations
Wasielewska, K.; Ganzha, M.; Paprzycki, M.; Lirkov, I.
2013-10-01
The aim of this paper is to propose a method for application of ontologically represented domain knowledge to support Grid users. The work is presented in the context provided by the Agents in Grid system, which aims at development of an agent-semantic infrastructure for efficient resource management in the Grid. Decision support within the system should provide functionality beyond the existing Grid middleware, specifically, help the user to choose optimal algorithm and/or resource to solve a problem from a given domain. The system assists the user in at least two situations. First, for users without in-depth knowledge about the domain, it should help them to select the method and the resource that (together) would best fit the problem to be solved (and match the available resources). Second, if the user explicitly indicates the method and the resource configuration, it should "verify" if her choice is consistent with the expert recommendations (encapsulated in the knowledge base). Furthermore, one of the goals is to simplify the use of the selected resource to execute the job; i.e., provide a user-friendly method of submitting jobs, without required technical knowledge about the Grid middleware. To achieve the mentioned goals, an adaptable method of expert knowledge representation for the decision support system has to be implemented. The selected approach is to utilize ontologies and semantic data processing, supported by multicriterial decision making. As a starting point, an area of computational linear algebra was selected to be modeled, however, the paper presents a general approach that shall be easily extendable to other domains.
Student learning of basis, span and linear independence in linear algebra
Stewart, Sepideh; Thomas, Michael O. J.
2010-03-01
One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have major difficulties with concepts of span and linear independence which form the requirements for a set of vectors to form a basis. In this research we applied a theoretical framework based on Tall's three worlds of mathematics learning and action-process-object-schema (APOS) theory to the learning of the concept of basis by a group of second year university students. The results suggest that an emphasis on matrix processes may not help students understand the concept, and embodied, visual ideas that could be valuable were usually lacking.
Approximate inverse preconditioning of iterative methods for nonsymmetric linear systems
Energy Technology Data Exchange (ETDEWEB)
Benzi, M. [Universita di Bologna (Italy); Tuma, M. [Inst. of Computer Sciences, Prague (Czech Republic)
1996-12-31
A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.
07071 Report on Dagstuhl Seminar -- Web Information Retrieval and Linear Algebra Algorithms
Frommer, Andreas; Mahoney, Michael W.; Szyld, Daniel B.
2007-01-01
A seminar concentrating on the intersection of the fields of information retrieval and other web-related aspects with numerical and applied linear algebra techniques was held with the attendance of scientists from industry and academia.
Proposal for an extended set of Fortran Basic Linear Algebra Subprograms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Du Croz, J.; Hammarling, S.; Hanson, R.J.
1984-12-01
This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions proposed are targeted at matrix vector operations which should provide for more efficient and portable implementations of algorithms for high performance computers.
Efficient linear algebra routines for symmetric matrices stored in packed form.
Ahlrichs, Reinhart; Tsereteli, Kakha
2002-01-30
Quantum chemistry methods require various linear algebra routines for symmetric matrices, for example, diagonalization or Cholesky decomposition for positive matrices. We present a small set of these basic routines that are efficient and minimize memory requirements.
A Simple and Practical Linear Algebra Library Interface with Static Size Checking
Directory of Open Access Journals (Sweden)
Akinori Abe
2015-12-01
Full Text Available Linear algebra is a major field of numerical computation and is widely applied. Most linear algebra libraries (in most programming languages do not statically guarantee consistency of the dimensions of vectors and matrices, causing runtime errors. While advanced type systems—specifically, dependent types on natural numbers—can ensure consistency among the sizes of collections such as lists and arrays, such type systems generally require non-trivial changes to existing languages and application programs, or tricky type-level programming. We have developed a linear algebra library interface that verifies the consistency (with respect to dimensions of matrix operations by means of generative phantom types, implemented via fairly standard ML types and module system. To evaluate its usability, we ported to it a practical machine learning library from a traditional linear algebra library. We found that most of the changes required for the porting could be made mechanically, and changes that needed human thought are minor.
Answers to selected problems in multivariable calculus with linear algebra and series
Trench, William F
1972-01-01
Answers to Selected Problems in Multivariable Calculus with Linear Algebra and Series contains the answers to selected problems in linear algebra, the calculus of several variables, and series. Topics covered range from vectors and vector spaces to linear matrices and analytic geometry, as well as differential calculus of real-valued functions. Theorems and definitions are included, most of which are followed by worked-out illustrative examples.The problems and corresponding solutions deal with linear equations and matrices, including determinants; vector spaces and linear transformations; eig
Linear algebraic theory of partial coherence: continuous fields and measures of partial coherence.
Ozaktas, Haldun M; Gulcu, Talha Cihad; Alper Kutay, M
2016-11-01
This work presents a linear algebraic theory of partial coherence for optical fields of continuous variables. This approach facilitates use of linear algebraic techniques and makes it possible to precisely define the concepts of incoherence and coherence in a mathematical way. We have proposed five scalar measures for the degree of partial coherence. These measures are zero for incoherent fields, unity for fully coherent fields, and between zero and one for partially coherent fields.
Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth-Death Processes with Immigration
Ohkubo, Jun
2014-10-01
Analytical solutions for time-inhomogeneous linear birth-death processes with immigration are derived. While time-inhomogeneous linear birth-death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth-death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.
su(1,2) Algebraic Structure of XYZ Antiferromagnetic Model in Linear Spin-Wave Frame
Institute of Scientific and Technical Information of China (English)
GUAN Yong; JIN Shuo; LIN Bing-Sheng; XIE Bing-Hao; JING Si-Cong; YU Zhao-Xian; HOU Jing-Min
2008-01-01
The XYZ antiferromagnetic model in linear spin-wave frame is shown explicitly to have an su(1,2) aigebraic structure: the Hamiltonian can be written as a linear function of the su(1,2) algebra generators. Based on it, the energy eigenvalues are obtained by making use of the similar transformations, and the algebraic diagonalization method is investigated. Some numerical solutions are given, and the results indicate that only one group solution could be accepted in physics.
Mathematical modelling in engineering: A proposal to introduce linear algebra concepts
Andrea Dorila Cárcamo; Joan Vicenç Gómez; Josep María Fortuny
2016-01-01
The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasize the development of mathematical abilities primarily associated with modelling and interpreting, which aren´t limited only to calculus abilities. Considering this, an instructional design was elaborated based on mathematic modelling and emerging heuristic models for the construction of specific linear algebra concepts: span and spanning set. This was applied to first year e...
Mathematical modelling in engineering: a proposal to introduce linear algebra concepts
Cárcamo Bahamonde, Andrea Dorila; Gómez Urgellés, Joan Vicenç; Fortuny Aymeni, José María
2016-01-01
The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasize the development of mathematical abilities primarily associated with modelling and interpreting, which aren´t limited only to calculus abilities. Considering this, an instructional design was elaborated based on mathematic modelling and emerging heuristic models for the construction of specific linear algebra concepts: span and spanning set. This was applied to first year engineeri...
Mathematical modelling in engineering: a proposal to introduce linear algebra concepts
Cárcamo Bahamonde, Andrea; Gómez Urgellés, Joan Vicenç; Fortuny Aymemi, Josep Maria
2016-01-01
The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasize the development of mathematical abilities primarily associated with modelling and interpreting, which aren´t limited only to calculus abilities. Considering this, an instructional design was elaborated based on mathematic modelling and emerging heuristic models for the construction of specific linear algebra concepts: span and spanning set. This was applied to first year engineer...
1979-09-01
without determinantal divisors, Linear and Multilinear Algebra 7(1979), 107-109. 4. The use of integral operators in number theory (with C. Ryavec and...Gersgorin revisited, to appear in Letters in Linear Algebra. 15. A surprising determinantal inequality for real matrices (with C.R. Johnson), to appear in...Analysis: An Essay Concerning the Limitations of Some Mathematical Methods in the Social , Political and Biological Sciences, David Berlinski, MIT Press
Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence.
Ozaktas, Haldun M; Yüksel, Serdar; Kutay, M Alper
2002-08-01
A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence. The mathematical definitions are related to our physical understanding of the corresponding concepts by considering them in the context of Young's experiment.
Numerical linear algebra on emerging architectures: The PLASMA and MAGMA projects
Agullo, Emmanuel; Demmel, Jim; Dongarra, Jack; Hadri, Bilel; Kurzak, Jakub; Langou, Julien; Ltaief, Hatem; Luszczek, Piotr; Tomov, Stanimire
2009-07-01
The emergence and continuing use of multi-core architectures and graphics processing units require changes in the existing software and sometimes even a redesign of the established algorithms in order to take advantage of now prevailing parallelism. Parallel Linear Algebra for Scalable Multi-core Architectures (PLASMA) and Matrix Algebra on GPU and Multics Architectures (MAGMA) are two projects that aims to achieve high performance and portability across a wide range of multi-core architectures and hybrid systems respectively. We present in this document a comparative study of PLASMA's performance against established linear algebra packages and some preliminary results of MAGMA on hybrid multi-core and GPU systems.
CONVERGENCE DOMAINS OF AOR TYPE ITERATIVE MATRICES FOR SOLVING NON-HERMITIAN LINEAR SYSTEMS
Institute of Scientific and Technical Information of China (English)
Li Wang
2004-01-01
We discuss AOR type iterative methods for solving non-Hermitian linear systems based on Hermitian splitting and skew-Hermitian splitting. Convergence domains of iterative matrices are given and optimal parameters are investigated for skew-Hermitian splitting.Numerical examples are presented to compare the effectiveness of the iterative methods in different points in the domain. In addition, a model problem of three-dimensional convection-diffusion equation is used to illustrated the application of our results.
A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
Directory of Open Access Journals (Sweden)
Zhang Cheng-yi
2016-01-01
Full Text Available It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices. However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices. This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.
Linear algebra and linear operators in engineering with applications in Mathematica
Davis, H Ted
2000-01-01
Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical ...
Weighted thinned linear array design with the iterative FFT technique
CSIR Research Space (South Africa)
Du Plessis, WP
2011-09-01
Full Text Available A version of the iterative Fourier technique (IFT) for the design of thinned antenna arrays with weighted elements is presented. The structure of the algorithm means that it is well suited to the design of weighted thinned arrays with low current...
A comparative study of iterative solutions to linear systems arising in quantum mechanics
Jing, Yan-Fei; Huang, Ting-Zhu; Duan, Yong; Carpentieri, Bruno
2010-01-01
This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods
Directory of Open Access Journals (Sweden)
Jen-Yuan Chen
2014-01-01
Full Text Available Continuing from the works of Li et al. (2014, Li (2007, and Kincaid et al. (2000, we present more generalizations and modifications of iterative methods for solving large sparse symmetric and nonsymmetric indefinite systems of linear equations. We discuss a variety of iterative methods such as GMRES, MGMRES, MINRES, LQ-MINRES, QR MINRES, MMINRES, MGRES, and others.
A comparative study of iterative solutions to linear systems arising in quantum mechanics
Jing, Yan-Fei; Huang, Ting-Zhu; Duan, Yong; Carpentieri, Bruno
2010-01-01
This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods
On the relationship between ODE solvers and iterative solvers for linear equations
Energy Technology Data Exchange (ETDEWEB)
Lorber, A.; Joubert, W.; Carey, G.F. [Univ. of Texas, Austin, TX (United States)
1994-12-31
The connection between the solution of linear systems of equations by both iterative methods and explicit time stepping techniques is investigated. Based on the similarities, a suite of Runge-Kutta time integration schemes with extended stability domains are developed using Chebyshev iteration polynomials. These Runge-Kutta schemes are applied to linear and non-linear systems arising from the numerical solution of PDE`s containing either physical or artificial transient terms. Specifically, the solutions of model linear convection and convection-diffusion equations are presented, as well as the solution of a representative non-linear Navier-Stokes fluid flow problem. Included are results of parallel computations.
Fu, Jian; Hu, Xinhua; Velroyen, Astrid; Bech, Martin; Jiang, Ming; Pfeiffer, Franz
2015-01-01
Due to the potential of compact imaging systems with magnified spatial resolution and contrast, cone-beam x-ray differential phase-contrast computed tomography (DPC-CT) has attracted significant interest. The current proposed FDK reconstruction algorithm with the Hilbert imaginary filter will induce severe cone-beam artifacts when the cone-beam angle becomes large. In this paper, we propose an algebraic iterative reconstruction (AIR) method for cone-beam DPC-CT and report its experiment results. This approach considers the reconstruction process as the optimization of a discrete representation of the object function to satisfy a system of equations that describes the cone-beam DPC-CT imaging modality. Unlike the conventional iterative algorithms for absorption-based CT, it involves the derivative operation to the forward projections of the reconstructed intermediate image to take into account the differential nature of the DPC projections. This method is based on the algebraic reconstruction technique, reconstructs the image ray by ray, and is expected to provide better derivative estimates in iterations. This work comprises a numerical study of the algorithm and its experimental verification using a dataset measured with a three-grating interferometer and a mini-focus x-ray tube source. It is shown that the proposed method can reduce the cone-beam artifacts and performs better than FDK under large cone-beam angles. This algorithm is of interest for future cone-beam DPC-CT applications.
PRECONDITIONED GAUSS-SEIDEL TYPE ITERATIVE METHOD FOR SOLVING LINEAR SYSTEMS
Institute of Scientific and Technical Information of China (English)
CHENG Guang-hui; HUANG Ting-zhu; CHENG Xiao-yu
2006-01-01
The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.
Institute of Scientific and Technical Information of China (English)
冯东太; 丁世良; 王美山
2003-01-01
The highly excited vibrational states of asymmetric linear tetratomic molecules are studied in the framework of Lie algebra. By using symmetric group U1(4) U2(4) U3(4), we construct the Hamiltonian that includes not only Casimir operators but also Majorana operators M12,M13 and M23, which are useful for getting potential energy surface and force constants in Lie algebra method. By Lie algebra treatment, we obtain the eigenvalues of the Hamiltonian, and make the concrete calculation for molecule C2HF.
Representations of general linear groups and categorical actions of Kac-Moody algebras
Losev, Ivan
2012-01-01
This is an expanded version of the lectures given by the author on the 3rd school "Lie algebras, algebraic groups and invariant theory" in Togliatti, Russia. In these notes we explain the concept of a categorical Kac-Moody action by studying an example of the category of rational representations of a general linear group in positive characteristic. We also deal with some more advanced topics: a categorical action on the polynomial representations and crystals of categorical actions.
On Representations Associated with Completely n-Positive Linear Maps on Pro-C*-Algebras
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
It is shown that an n × n matrix of continuous linear maps from a pro-C*-algebra A to L(H), which verifies the condition of complete positivity, is of the formlinear operator from H to K, and [Tij]ni,j=1 is a positive element in the C*-algebra of all Suen in Proc. Amer. Math. Soc., 112(3), 1991, 709-712. Also, a covariant version of this construction is given.
Groups, matrices, and vector spaces a group theoretic approach to linear algebra
Carrell, James B
2017-01-01
This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group. Applications involving symm etry groups, determinants, linear coding theory ...
Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji
2015-06-01
We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems—that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system—we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.
Multiprocessing linear algebra algorithms on the CRAY X-MP-2: Experiences with small granularity
Energy Technology Data Exchange (ETDEWEB)
Chen, S.S.; Dongarra, J.J.; Hsiung, C.
1984-08-01
This paper gives a brief overview of the CRAY X-MP-2 general-purpose multiprocessor system and discusses how it can be used effectively to solve problems that have small granularity. An implementation is described for linear algebra algorithms that solve systems of linear equations when the matrix is general and when the matrix is symmetric and positive definite.
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
Variational iteration method for solving non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Hemeda, A.A. [Department of Mathematics, Faculty of Science, University of Tanta, Tanta (Egypt)], E-mail: aahemeda@yahoo.com
2009-02-15
In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV-MKdV equation and Camassa-Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
Liu, Jianzhou; Zhang, Juan
2011-08-01
In this article, applying the properties of M-matrix and non-negative matrix, utilising eigenvalue inequalities of matrix's sum and product, we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic Riccati equation (DCARE). Secondly, we discuss the solution existence uniqueness condition of the DCARE using the developed upper and lower matrix bounds and a fixed point theorem. Thirdly, a new fixed iterative algorithm of the solution for the DCARE is shown. Finally, the corresponding numerical examples are given to illustrate the effectiveness of the developed results.
Aydin, Sinan
2014-08-01
Linear algebra is a basic mathematical subject taught in mathematics and science depar-tments of universities. The teaching and learning of this course has always been difficult. This study aims to contribute to the research in linear algebra education, focusing on linear dependence and independence concepts. This was done by introducing student-generated examples regarding the concepts. With the help of these examples, we have analysed students' understanding of linear dependence/independence and determined the effect of the example-generation process on student understanding of linear algebra. In addition, we identified some difficulties that were experienced by students learning the concepts of linear dependence/independence. In this study, APOS (action-process-object-schema) theory is the main tool utilized to explain students' written responses. It was also used with regard to the interview questions that were posed to students with the purpose of identifying possible difficulties with linear dependence/independence and observing the adequacy of the relations that students might form between different elements of the genetic decomposition of linear dependence/independence concepts. The findings of this study confirmed that many students do not have appropriate mental structures at object and schema levels. Moreover, in order to ensure the success of such exercises, students must be encouraged to review and validate their responses to the example requests.
On linear dependence over complete differential algebraic varieties
Freitag, James; Sanchez, Omar Leon; Simmons, William
2014-01-01
We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
Mathematical modelling in engineering: A proposal to introduce linear algebra concepts
Directory of Open Access Journals (Sweden)
Andrea Dorila Cárcamo
2016-03-01
Full Text Available The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasize the development of mathematical abilities primarily associated with modelling and interpreting, which aren´t limited only to calculus abilities. Considering this, an instructional design was elaborated based on mathematic modelling and emerging heuristic models for the construction of specific linear algebra concepts: span and spanning set. This was applied to first year engineering students. Results suggest that this type of instructional design contributes to the construction of these mathematical concepts and can also favour first year engineering students understanding of key linear algebra concepts and potentiate the development of higher order skills.
Matrix Operations for Engineers and Scientists An Essential Guide in Linear Algebra
Jeffrey, Alan
2010-01-01
Engineers and scientists need to have an introduction to the basics of linear algebra in a context they understand. Computer algebra systems make the manipulation of matrices and the determination of their properties a simple matter, and in practical applications such software is often essential. However, using this tool when learning about matrices, without first gaining a proper understanding of the underlying theory, limits the ability to use matrices and to apply them to new problems. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations. These students require a straightforward introduction to linear algebra illustrated by applications to which they can relate. It caters of the needs of undergraduate engineers in all disciplines, and provides considerable detail where it is likely to be helpful. According to the author the best way to understand the theory of matrices is by working simple exercises designe...
Mathematical modelling in engineering: an alternative way to teach Linear Algebra
Domínguez-García, S.; García-Planas, M. I.; Taberna, J.
2016-10-01
Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic classroom approach in which students modelled real-world problems and turn gain a deeper knowledge of the Linear Algebra subject. Considering that most students are digital natives, we use the e-portfolio as a tool of communication between students and teachers, besides being a good place making the work visible. In this article, we present an overview of the design and implementation of a project-based learning for a Linear Algebra course taught during the 2014-2015 at the 'ETSEIB'of Universitat Politècnica de Catalunya (UPC).
Advanced topics in linear algebra weaving matrix problems through the Weyr form
O'Meara, Kevin; Vinsonhaler, Charles
2011-01-01
The Weyr matrix canonical form is a largely unknown cousin of the Jordan canonical form. Discovered by Eduard Weyr in 1885, the Weyr form outperforms the Jordan form in a number of mathematical situations, yet it remains somewhat of a mystery, even to many who are skilled in linear algebra. Written in an engaging style, this book presents various advanced topics in linear algebra linked through the Weyr form. Kevin O'Meara, John Clark, and Charles Vinsonhaler develop the Weyr form from scratch and include an algorithm for computing it. A fascinating duality exists between the Weyr form and the
On Development of a Problem Based Learning System for Linear Algebra with Simple Input Method
Yokota, Hisashi
2011-08-01
Learning how to express a matrix using a keyboard inputs requires a lot of time for most of college students. Therefore, for a problem based learning system for linear algebra to be accessible for college students, it is inevitable to develop a simple method for expressing matrices. Studying the two most widely used input methods for expressing matrices, a simpler input method for expressing matrices is obtained. Furthermore, using this input method and educator's knowledge structure as a concept map, a problem based learning system for linear algebra which is capable of assessing students' knowledge structure and skill is developed.
Multiple Representations for Systems of Linear Equations Via the Computer Algebra System Maple
Directory of Open Access Journals (Sweden)
Dann G. Mallet
2007-02-01
Full Text Available A number of different representational methods exist for presenting the theory of linear equations and associated solution spaces. Discussed in this paper are the findings of a case study where first year undergraduate students were exposed to a new (to the department method of teaching linear systems which used visual, algebraic and data-based representations constructed using the computer algebra system Maple. Positive and negative impacts on the students are discussed as they apply to representational translation and perceived learning.
Linear addition algebra of optical nonlinearities in transparent conductive oxides
Kinsey, N; Clerici, M; Kim, J; Carnemolla, E; Shaltout, A; Kaipurath, R; Faccio, D; Shalaev, V M; Ferrera, M; Boltasseva, A
2016-01-01
The fields of nanophotonics and metamaterials have revolutionized the way we think of optical space ({\\epsilon},{\\mu}), enabling us to engineer the refractive index almost at will to confine light to the smallest of volumes as well as to manipulate optical signals with extremely small footprints and energy requirements. More recently, significant efforts have been devoted to the search for suitable materials for dynamic control, and so far, all-optical methods have primarily relied on either interband or intraband excitations. Here, we show that aluminum doped zinc oxide (AZO) supports a hybrid nonlinearity that exhibits a large and ultrafast response with controllable sign. We demonstrate that these two opposite material responses are independent and can be algebraically added together via two-color excitation, resulting in an increase in device bandwidth and unprecedented tuning capabilities. This peculiar behavior of AZO places it as a key material for next-generation ultrafast tunable nanophotonics and me...
Control of Linear Systems Over Commutative Normed Algebras with Applications.
1987-02-01
Identify by block number) System Theory, Linear Systems, Control, Systems with Time Delays, Time - Varying Systems, State- Space Models, Pole...modes for the class of linear time -varying systems. These concepts are defined in terms of a noncommutative factorization of opera- tor polynomials...classes of complex linear systems, including systems with time delays, systems with unknown parameters and time -varying systems. In the work on
LMI-based robust iterative learning controller design for discrete linear uncertain systems
Institute of Scientific and Technical Information of China (English)
Jianming XU; Mingxuan SUN; Li YU
2005-01-01
This paper addresses the design problem of robust iterative learning controllers for a class of linear discrete-time systems with norm-bounded parameter uncertainties.An iterative learning algorithm with current cycle feedback is proposed to achieve both robust convergence and robust stability.The synthesis problem of the proposed iterative learning control (ILC) system is reformulated as a γ-suboptimal H-infinity control problem via the linear fractional transformation (LFT).A sufficient condition for the convergence of the ILC algorithm is presented in terms of linear matrix inequalities (LMIs).Furthermore,the linear transfer operators of the ILC algorithm with high convergence speed are obtained by using existing convex optimization techniques.The simulation results demonstrate the effectiveness of the proposed method.
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
Iterative linear system solvers with approximate matrix-vector products
Eshof, J. van den; Sleijpen, G.L.G.; Gijzen, M.B. van
2003-01-01
There are classes of linear problems for which a matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. One important example is simulations in lattice QCD with Neuberger fermions where a matrix multiply
Directory of Open Access Journals (Sweden)
Xisheng Dai
2014-01-01
Full Text Available Iterative learning control is an intelligent control algorithm which imitates human learning process. Based on this concept, this paper discussed iterative learning control problem for a class parabolic linear distributed parameter systems with uncertainty coefficients. Iterative learning control algorithm with forgetting factor is proposed and the conditions for convergence of algorithm are established. Combining the matrix theory with the basic theory of distributed parameter systems gives rigorous convergence proof of the algorithm. Finally, by using the forward difference scheme of partial differential equation to solve the problems, the simulation results are presented to illustrate the feasibility of the algorithm.
Numerical linear algebra on emerging architectures: The PLASMA and MAGMA projects
Energy Technology Data Exchange (ETDEWEB)
Agullo, Emmanuel; Demmel, Jim; Dongarra, Jack; Hadri, Bilel; Kurzak, Jakub; Langou, Julien; Ltaief, Hatem; Luszczek, Piotr [Department of Electrical Engineering and Computer Science, University of Tennessee (United States); Tomov, Stanimire, E-mail: eagullo@eecs.utk.ed, E-mail: dongarra@eecs.utk.ed, E-mail: hadri@eecs.utk.ed, E-mail: kurzak@eecs.utk.ed, E-mail: ltaief@eecs.utk.ed, E-mail: luszczek@eecs.utk.ed, E-mail: tomov@eecs.utk.ed, E-mail: demmel@cs.berkeley.ed, E-mail: julien.langou@ucdenver.ed
2009-07-01
The emergence and continuing use of multi-core architectures and graphics processing units require changes in the existing software and sometimes even a redesign of the established algorithms in order to take advantage of now prevailing parallelism. Parallel Linear Algebra for Scalable Multi-core Architectures (PLASMA) and Matrix Algebra on GPU and Multics Architectures (MAGMA) are two projects that aims to achieve high performance and portability across a wide range of multi-core architectures and hybrid systems respectively. We present in this document a comparative study of PLASMA's performance against established linear algebra packages and some preliminary results of MAGMA on hybrid multi-core and GPU systems.
Causal structure and algebraic classification of non-dissipative linear optical media
Schuller, Frederic P.; Witte, Christof; Wohlfarth, Mattias N. R.
2010-09-01
In crystal optics and quantum electrodynamics in gravitational vacua, the propagation of light is not described by a metric, but an area metric geometry. In this article, this prompts us to study conditions for linear electrodynamics on area metric manifolds to be well-posed. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structures in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. This classification, valuable in its own right, is then employed to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes. Physically, these results classify and drastically restrict the viable constitutive tensors of non-dissipative linear optical media.
Gerbracht, Eberhard H. -A.
2007-01-01
Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.
Mathematical Modelling and the Learning Trajectory: Tools to Support the Teaching of Linear Algebra
Cárcamo Bahamonde, Andrea Dorila; Fortuny Aymemí, Josep Maria; Gómez i Urgellés, Joan Vicenç
2017-01-01
In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and…
Students’ thinking modes and the emergence of signs in learning linear algebra
Turgut, M.; Drijvers, P.H.M.
2016-01-01
To analyse the use of a dynamic geometry environment for linear algebra by two students, we combine a semiotic mediation approach with a lens of students’ thinking modes. This way of approaching the data shows promise for a detailed understanding of the observed phenomena.
Mathematical Modelling in Engineering: A Proposal to Introduce Linear Algebra Concepts
Cárcamo Bahamonde, Andrea; Gómez Urgelles, Joan; Fortuny Aymemí, Josep
2016-01-01
The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasise the development of mathematical abilities primarily associated with modelling and interpreting, which are not exclusively calculus abilities. Considering this, an instructional design was created based on mathematical modelling and…
Transforming an Introductory Linear Algebra Course with a TI-92 Hand-Held Computer.
Quesada, Antonio R.
2003-01-01
Describes how the introduction of the TI-92 transformed a traditional first semester linear algebra course into a matrix-oriented course that emphasized conceptual understanding, relevant applications, and numerical issues. Indicates an increase in students' overall performance as they found the calculator very useful, believed it helped them…
Hannah, John; Stewart, Sepideh; Thomas, Michael
2016-01-01
Linear algebra is one of the first abstract mathematics courses that students encounter at university. Research shows that many students find the dense presentation of definitions, theorems and proofs difficult to comprehend. Using a case study approach, we report on a teaching intervention based on Tall's three worlds (embodied, symbolic and…
Using Technology to Facilitate Reasoning: Lifting the Fog from Linear Algebra
Berry, John S.; Lapp, Douglas A.; Nyman, Melvin A.
2008-01-01
This article discusses student difficulties in grasping concepts from linear algebra. Using an example from an interview with a student, we propose changes that might positively impact student understanding of concepts within a problem-solving context. In particular, we illustrate barriers to student understanding and suggest technological…
A Practical Approach to Inquiry-Based Learning in Linear Algebra
Chang, J.-M.
2011-01-01
Linear algebra has become one of the most useful fields of mathematics since last decade, yet students still have trouble seeing the connection between some of the abstract concepts and real-world applications. In this article, we propose the use of thought-provoking questions in lesson designs to allow two-way communications between instructors…
Principal Component Analysis: Resources for an Essential Application of Linear Algebra
Pankavich, Stephen; Swanson, Rebecca
2015-01-01
Principal Component Analysis (PCA) is a highly useful topic within an introductory Linear Algebra course, especially since it can be used to incorporate a number of applied projects. This method represents an essential application and extension of the Spectral Theorem and is commonly used within a variety of fields, including statistics,…
A proposal for a set of Level 3 Basic Linear Algebra Subprograms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.; Du Croz, J.; Duff, I.; Hammarling, S.
1987-04-01
This paper describes a proposal for Level 3 Basic Linear Algebra Subprograms (Level 3 BLAS). The Level 3 BLAS are targeted at matrix-matrix operations with the aim of providing more efficient, but portable, implementations of algorithms on high-performance computers, especially those with hierarchical memory and parallel processing capability.
A non-linear representation of the d=2 so (4)-extended superconformal algebra
Schoutens, K.
1987-01-01
We present a non-linear representation of the so(4)-extended d=2 superconformal algebra in terms of one boson and four Majorana fermions. The matter fields and the currents can be grouped into a single N=4 superfield. Breaking the supersymmetry to N=3 or N=2 leads to new representations of the N=3,2
Mathematical Modelling in Engineering: An Alternative Way to Teach Linear Algebra
Domínguez-García, S.; García-Planas, M. I.; Taberna, J.
2016-01-01
Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic…
Student Connections of Linear Algebra Concepts: An Analysis of Concept Maps
Lapp, Douglas A.; Nyman, Melvin A.; Berry, John S.
2010-01-01
This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud's theory of conceptual fields and Pirie and Kieren's model for the growth of mathematical understanding.…
Using Technology to Facilitate Reasoning: Lifting the Fog from Linear Algebra
Berry, John S.; Lapp, Douglas A.; Nyman, Melvin A.
2008-01-01
This article discusses student difficulties in grasping concepts from linear algebra. Using an example from an interview with a student, we propose changes that might positively impact student understanding of concepts within a problem-solving context. In particular, we illustrate barriers to student understanding and suggest technological…
Mathematical Modelling and the Learning Trajectory: Tools to Support the Teaching of Linear Algebra
Cárcamo Bahamonde, Andrea Dorila; Fortuny Aymemí, Josep Maria; Gómez i Urgellés, Joan Vicenç
2017-01-01
In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and…
Principal Component Analysis: Resources for an Essential Application of Linear Algebra
Pankavich, Stephen; Swanson, Rebecca
2015-01-01
Principal Component Analysis (PCA) is a highly useful topic within an introductory Linear Algebra course, especially since it can be used to incorporate a number of applied projects. This method represents an essential application and extension of the Spectral Theorem and is commonly used within a variety of fields, including statistics,…
Gasyna, Zbigniew L.
2008-01-01
Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)
Teaching Linear Algebra: Proceeding More Efficiently by Staying Comfortably within Z
Beaver, Scott
2015-01-01
For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram-Schmidt process.
Space and frequency-multiplexed optical linear algebra processor - Fabrication and initial tests
Casasent, D.; Jackson, J.
1986-01-01
A new optical linear algebra processor architecture is described. Space and frequency-multiplexing are used to accommodate bipolar and complex-valued data. A fabricated laboratory version of this processor is described, the electronic support system used is discussed, and initial test data obtained on it are presented.
Linear Maps Preserving Idempotency of Products of Matrices on Upper Triangular Matrix Algebras
Institute of Scientific and Technical Information of China (English)
QI JING; JI GUO-XING
2009-01-01
. Let Tn be the algebra of all n × n complex upper triangular matrices. We give the concrete forms of linear injective maps on Tn which preserve the nonzero idempotency of either products of two matrices or triple Jordan products of two matrices.
A Practical Approach to Inquiry-Based Learning in Linear Algebra
Chang, J.-M.
2011-01-01
Linear algebra has become one of the most useful fields of mathematics since last decade, yet students still have trouble seeing the connection between some of the abstract concepts and real-world applications. In this article, we propose the use of thought-provoking questions in lesson designs to allow two-way communications between instructors…
A Modified Approach to Team-Based Learning in Linear Algebra Courses
Nanes, Kalman M.
2014-01-01
This paper documents the author's adaptation of team-based learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, in the linear algebra classroom. The paper discusses the standard components of TBL and the necessary changes to those components for the needs of the course in question. There is also an empirically…
Mat-Rix-Toe: Improving Writing through a Game-Based Project in Linear Algebra
Graham-Squire, Adam; Farnell, Elin; Stockton, Julianna Connelly
2014-01-01
The Mat-Rix-Toe project utilizes a matrix-based game to deepen students' understanding of linear algebra concepts and strengthen students' ability to express themselves mathematically. The project was administered in three classes using slightly different approaches, each of which included some editing component to encourage the…
An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering
Zandieh, Michelle; Wawro, Megan; Rasmussen, Chris
2017-01-01
In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor…
Transforming an Introductory Linear Algebra Course with a TI-92 Hand-Held Computer.
Quesada, Antonio R.
2003-01-01
Describes how the introduction of the TI-92 transformed a traditional first semester linear algebra course into a matrix-oriented course that emphasized conceptual understanding, relevant applications, and numerical issues. Indicates an increase in students' overall performance as they found the calculator very useful, believed it helped them…
Wawro, Megan; Sweeney, George F.; Rabin, Jeffrey M.
2011-01-01
This paper reports on a study investigating students' ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. In interviews conducted with eight undergraduates, we found students' initial descriptions of subspace often varied substantially from…
Student Reactions to Learning Theory Based Curriculum Materials in Linear Algebra--A Survey Analysis
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff
2016-01-01
In this report we examine students' perceptions of the implementation of carefully designed curriculum materials (called modules) in linear algebra courses at three different universities. The curricular materials were produced collaboratively by STEM and mathematics education faculty as members of a professional learning community (PLC) over…
A note on probabilistic models over strings: the linear algebra approach.
Bouchard-Côté, Alexandre
2013-12-01
Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the "TKF91" model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.
Mathematical Modelling in Engineering: An Alternative Way to Teach Linear Algebra
Domínguez-García, S.; García-Planas, M. I.; Taberna, J.
2016-01-01
Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic…
Correlation between Student Performance in Linear Algebra and Categories of a Taxonomy.
Wood, Leigh N.; Smith, Geoffrey H.; Petocz, Peter; Reid, Anna
This paper concerns a study of the performance of students in a linear algebra examination. Differences in performance of tasks requiring understanding of concepts with those that required only the use of routine procedures and factual recall were investigated. Central to the study was the use of a taxonomy based on Bloom's Taxonomy for…
Wawro, Megan Jean
2011-01-01
In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of…
LAPACK++: A design overview of object-oriented extensions for high performance linear algebra
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. [Oak Ridge National Lab., TN (United States). Mathematical Sciences Section]|[Univ. of Tennessee, Knoxville, TN (United States). Computer Science Dept.; Pozo, R. [Univ. of Tennessee, Knoxville, TN (United States). Dept. of Computer Science; Walker, D.W. [Oak Ridge National Lab., TN (United States). Mathematical Sciences Section
1993-12-31
LAPACK++ is an object-oriented C++ extension of the LAPACK (Linear Algebra PACKage) library for solving the common problems of numerical linear algebra: linear systems, linear least squares, and eigenvalue problems on high-performance computer architectures. The advantages of an object-oriented approach include the ability to encapsule various matrix representations, hide their implementation details, reduce the number of subroutines, simplify their calling sequences, and provide an extendible software framework that can incorporate future extensions of LAPACK such as ScaLAPACK++ for distributed memory architectures. The authors present an overview of the object-oriented design of the matrix and decomposition classes in C++ and discuss its impact on elegance, generality, and performance.
LAPACK++: A design overview of object-oriented extensions for high performance linear algebra
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. [Oak Ridge National Lab., TN (United States). Mathematical Sciences Section]|[Univ. of Tennessee, Knoxville (United States). Dept. of Computer Science; Pozo, R. [Univ. of Tennessee, Knoxville (United States). Dept. of Computer Science; Walker, D.W. [Oak Ridge National Lab., TN (United States). Mathematical Sciences Section
1993-12-31
LAPACK++ is an object-oriented C++ extension of the LAPACK (Linear Algebra PACKage) library for solving the common problems of numerical linear algebra: linear systems, linear least squares, and eigenvalue problems on high-performance computer architectures. The advantages of an object-oriented approach include the ability to encapsulate various matrix representations, hide their implementation details, reduce the number of subroutines, simplify their calling sequences, and provide an extendible software framework that can incorporate future extensions of LAPACK, such as ScaLAPACK++ for distributed memory architectures. The authors present an overview of the object-oriented design of the matrix and decomposition classes in C++ and discuss its impact on elegance, generality, and performance.
On a new iterative method for solving linear systems and comparison results
Jing, Yan-Fei; Huang, Ting-Zhu
2008-10-01
In Ujevic [A new iterative method for solving linear systems, Appl. Math. Comput. 179 (2006) 725-730], the author obtained a new iterative method for solving linear systems, which can be considered as a modification of the Gauss-Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujevic's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujevic.
Building Algebra Testlets: A Comparison of Hierarchical and Linear Structures.
Wainer, Howard; And Others
1991-01-01
Hierarchical (adaptive) and linear methods of testlet construction were compared. The performance of 2,080 ninth and tenth graders on a 4-item testlet was used to predict performance on the entire test. The adaptive test was slightly superior as a predictor, but the cost of obtaining that superiority was considerable. (SLD)
Building Algebra Testlets: A Comparison of Hierarchical and Linear Structures.
Wainer, Howard; And Others
1991-01-01
Hierarchical (adaptive) and linear methods of testlet construction were compared. The performance of 2,080 ninth and tenth graders on a 4-item testlet was used to predict performance on the entire test. The adaptive test was slightly superior as a predictor, but the cost of obtaining that superiority was considerable. (SLD)
On the linearization of the automorphism groups of algebraic domains
Zaitsev, D
1994-01-01
Let $D$ be a domain in $C^n$ and $G$ a topological group which acts effectively on $D$ by holomorphic automorphisms. In this paper we are interested in projective linearizations of the action of $G$, i.e. a linear representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$ into $\\P^N$ with respect to this representation. The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. Assume that the group $G$ acts by birational automorphisms. Our main result is the equivalence of the following conditions: 1) there exists a projective linearization, i.e. a linear representation of $G$ in some $\\C^{N+1}$ and a biregular imbedding $i\\colon \\P^n \\hookrightarrow \\P^N$ such that the restriction $i|_D$ is $G$-equivariant. 2) $G$ is a subgroup of a Lie group $\\hat G$ of birational automorphisms of $D$ which extends the action of $G$ and has finitely many connected components; 3) $G$ is a subgroup of a Nash group $\\hat G...
Nonlinearity and linearity, friends or enemies? Algebraic Analyzation of Science:)
Hanckowiak, Jerzy
2013-01-01
Certain operator-valued functions and new generating structures (instead of generating functionals) are proposed for the analysis of equations for n-point information (n-pi). Some remarks are made concerning the intertwining of linearity and nonlinearity, and functions defined on non-numerical objects.
Lectures on algebraic system theory: Linear systems over rings
Kamen, E. W.
1978-01-01
The presentation centers on four classes of systems that can be treated as linear systems over a ring. These are: (1) discrete-time systems over a ring of scalars such as the integers; (2) continuous-time systems containing time delays; (3) large-scale discrete-time systems; and (4) time-varying discrete-time systems.
Wang, Hongzhu; Yu, Tianqiu; Xiao, Jinmei
2016-08-01
From the perspective of strong transitivity, a controller design method is provided to simultaneously stabilise a collection of time-varying linear systems within the framework of nest algebras. In particular, all simultaneously stabilising controllers for a class of linear plants are characterised based on the doubly coprime factorisations. These results hold as well in the time-invariant case. An illustrative example is given to demonstrate the validity of the method.
Implementing linear algebra algorithms for dense matrices on a vector pipeline machine
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Gustavson, F.G.; Karp, A.
1984-01-01
The authors examine common implementations of linear algebra algorithms, such as matrix-vector multiplication, matrix-matrix multiplication and the solution of linear equations. The different versions are examined for efficiency on a computer architecture which uses vector processing and has pipelined instruction execution. By using the advanced architectural features of such machines, one can usually achieve maximum performance, and tremendous improvements in terms of execution speed can be seen over conventional computers. 17 references.
PRECONDITIONING HIGHER ORDER FINITE ELEMENT SYSTEMS BY ALGEBRAIC MULTIGRID METHOD OF LINEAR ELEMENTS
Institute of Scientific and Technical Information of China (English)
Yun-qing Huang; Shi Shu; Xi-jun Yu
2006-01-01
We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.
Iterated non-linear model predictive control based on tubes and contractive constraints.
Murillo, M; Sánchez, G; Giovanini, L
2016-05-01
This paper presents a predictive control algorithm for non-linear systems based on successive linearizations of the non-linear dynamic around a given trajectory. A linear time varying model is obtained and the non-convex constrained optimization problem is transformed into a sequence of locally convex ones. The robustness of the proposed algorithm is addressed adding a convex contractive constraint. To account for linearization errors and to obtain more accurate results an inner iteration loop is added to the algorithm. A simple methodology to obtain an outer bounding-tube for state trajectories is also presented. The convergence of the iterative process and the stability of the closed-loop system are analyzed. The simulation results show the effectiveness of the proposed algorithm in controlling a quadcopter type unmanned aerial vehicle.
Iotti, Robert
2015-04-01
ITER is an international experimental facility being built by seven Parties to demonstrate the long term potential of fusion energy. The ITER Joint Implementation Agreement (JIA) defines the structure and governance model of such cooperation. There are a number of necessary conditions for such international projects to be successful: a complete design, strong systems engineering working with an agreed set of requirements, an experienced organization with systems and plans in place to manage the project, a cost estimate backed by industry, and someone in charge. Unfortunately for ITER many of these conditions were not present. The paper discusses the priorities in the JIA which led to setting up the project with a Central Integrating Organization (IO) in Cadarache, France as the ITER HQ, and seven Domestic Agencies (DAs) located in the countries of the Parties, responsible for delivering 90%+ of the project hardware as Contributions-in-Kind and also financial contributions to the IO, as ``Contributions-in-Cash.'' Theoretically the Director General (DG) is responsible for everything. In practice the DG does not have the power to control the work of the DAs, and there is not an effective management structure enabling the IO and the DAs to arbitrate disputes, so the project is not really managed, but is a loose collaboration of competing interests. Any DA can effectively block a decision reached by the DG. Inefficiencies in completing design while setting up a competent organization from scratch contributed to the delays and cost increases during the initial few years. So did the fact that the original estimate was not developed from industry input. Unforeseen inflation and market demand on certain commodities/materials further exacerbated the cost increases. Since then, improvements are debatable. Does this mean that the governance model of ITER is a wrong model for international scientific cooperation? I do not believe so. Had the necessary conditions for success
An example of Learning based on competences: Use of Maxima in Linear Algebra for Engineers
2011-01-01
This paper analyzes the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is defined as an indicator of the generic ompetence: Use of Technology. Additionally, we show that using CAS could help to enhance the following generi...
Memory Hierarchy Behavior Study during the Execution of Recursive Linear Algebra Library
Directory of Open Access Journals (Sweden)
I. Šimeček
2008-01-01
Full Text Available For good performance of every computer program, good cache and TLB utilization is crucial. In numerical linear algebra libraries (such as BLAS or LAPACK, good cache utilization is achieved by explicit loop restructuring (mainly loop blocking, but this requires difficult memory pattern behavior analysis. In this paper, we represent the recursive implementation (“divide and conquer” approach of some routines from numerical algebra libraries. This implementation leads to good cache and TLB utilization with no need to analyze the memory pattern behavior due to “natural” partition of data.
A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus
Directory of Open Access Journals (Sweden)
Pablo Arrighi
2012-07-01
Full Text Available We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction.
Sepanski, Mark R
2010-01-01
Mark Sepanski's Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems
Zhang, Ruikun; Hou, Zhongsheng; Ji, Honghai; Yin, Chenkun
2016-04-01
In this paper, an adaptive iterative learning control scheme is proposed for a class of non-linearly parameterised systems with unknown time-varying parameters and input saturations. By incorporating a saturation function, a new iterative learning control mechanism is presented which includes a feedback term and a parameter updating term. Through the use of parameter separation technique, the non-linear parameters are separated from the non-linear function and then a saturated difference updating law is designed in iteration domain by combining the unknown parametric term of the local Lipschitz continuous function and the unknown time-varying gain into an unknown time-varying function. The analysis of convergence is based on a time-weighted Lyapunov-Krasovskii-like composite energy function which consists of time-weighted input, state and parameter estimation information. The proposed learning control mechanism warrants a L2[0, T] convergence of the tracking error sequence along the iteration axis. Simulation results are provided to illustrate the effectiveness of the adaptive iterative learning control scheme.
On the economical solution method for a system of linear algebraic equations
Directory of Open Access Journals (Sweden)
Jan Awrejcewicz
2004-01-01
Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22. The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.
On the economical solution method for a system of linear algebraic equations
Directory of Open Access Journals (Sweden)
Awrejcewicz Jan
2004-01-01
Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ 3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O( h x 1 2 + h x 2 2 . The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.
Linear delay-differential systems with commensurate delays an algebraic approach
Gluesing-Luerssen, Heide
2002-01-01
The book deals with linear time-invariant delay-differential equations with commensurated point delays in a control-theoretic context. The aim is to show that with a suitable algebraic setting a behavioral theory for dynamical systems described by such equations can be developed. The central object is an operator algebra which turns out to be an elementary divisor domain and thus provides the main tool for investigating the corresponding matrix equations. The book also reports the results obtained so far for delay-differential systems with noncommensurate delays. Moreover, whenever possible it points out similarities and differences to the behavioral theory of multidimensional systems, which is based on a great deal of algebraic structure itself. The presentation is introductory and self-contained. It should also be accessible to readers with no background in delay-differential equations or behavioral systems theory. The text should interest researchers and graduate students.
On the effect of linear algebra implementations in real-time multibody system dynamics
González, Manuel; González, Francisco; Dopico, Daniel; Luaces, Alberto
2008-03-01
This paper compares the efficiency of multibody system (MBS) dynamic simulation codes that rely on different implementations of linear algebra operations. The dynamics of an N-loop four-bar mechanism has been solved with an index-3 augmented Lagrangian formulation combined with the trapezoidal rule as numerical integrator. Different implementations for this method, both dense and sparse, have been developed, using a number of linear algebra software libraries (including sparse linear equation solvers) and optimized sparse matrix computation strategies. Numerical experiments have been performed in order to measure their performance, as a function of problem size and matrix filling. Results show that optimal implementations can increase the simulation efficiency in a factor of 2 3, compared with our starting classical implementations, and in some topics they disagree with widespread beliefs in MBS dynamics. Finally, advices are provided to select the implementation which delivers the best performance for a certain MBS dynamic simulation.
Ghosh, A
1988-08-01
Lanczos and conjugate gradient algorithms are important in computational linear algebra. In this paper, a parallel pipelined realization of these algorithms on a ring of optical linear algebra processors is described. The flow of data is designed to minimize the idle times of the optical multiprocessor and the redundancy of computations. The effects of optical round-off errors on the solutions obtained by the optical Lanczos and conjugate gradient algorithms are analyzed, and it is shown that optical preconditioning can improve the accuracy of these algorithms substantially. Algorithms for optical preconditioning and results of numerical experiments on solving linear systems of equations arising from partial differential equations are discussed. Since the Lanczos algorithm is used mostly with sparse matrices, a folded storage scheme to represent sparse matrices on spatial light modulators is also described.
Prospectus for the development of a linear algebra library for high-performance computers
Energy Technology Data Exchange (ETDEWEB)
Demmell, J.; Dongarra, J.J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; Sorensen, D.
1987-09-01
We propose to design and implement a transportable linear algebra library in Fortran 77 for efficient use on high-performance computers. The library is intended to provide a uniform set of subroutines to solve the most common linear algebra problems and to run efficiently on a wide range of architectures. This library, which will be freely accessible via computer network, not only will ease code development, make codes more portable among machines of different architectures, and increase efficiency, but also will provide tools for evaluating computer performance. The library will be based on the well-known and widely used LINPACK and EISPACK packages for linear equation solving, eigenvalue problems, and linear least squares. LINPACK and EISPACK have provided an important infrastructure for scientific computing on serial machines, but they were not designed to exploit the profusion of parallel and vector architectures not becoming available. We propose to restructure the algorithms in terms of calls to a small number of extended Basic Linear algebra Subroutines each of which implements a basic operation such as matrix multiplication, rank-k matrix updates, and the solution of triangular systems. These operations can be optimized for each architecture, but the underlying numerical algorithms will be the same for all machines. 11 refs.
Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry
Directory of Open Access Journals (Sweden)
K. S. Mahomed
2013-01-01
Full Text Available By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y′′′=0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.
Block quasi-minimal residual iterations for non-Hermitian linear systems
Energy Technology Data Exchange (ETDEWEB)
Freund, R.W. [AT& T Bell Labs., Murray Hill, NJ (United States)
1994-12-31
Many applications require the solution of multiple linear systems that have the same coefficient matrix, but differ only in their right-hand sides. Instead of applying an iterative method to each of these systems individually, it is usually more efficient to employ a block version of the method that generates blocks of iterates for all the systems simultaneously. An example of such an iteration is the block conjugate gradient algorithm, which was first studied by Underwood and O`Leary. On parallel architectures, block versions of conjugate gradient-type methods are attractive even for the solution of single linear systems, since they have fewer synchronization points than the standard versions of these algorithms. In this talk, the author presents a block version of Freund and Nachtigal`s quasi-minimal residual (QMR) method for the iterative solution of non-Hermitian linear systems. He describes two different implementations of the block-QMR method, one based on a block version of the three-term Lanczos algorithm and one based on coupled two-term block recurrences. In both cases, the underlying block-Lanczos process still allows arbitrary normalizations of the vectors within each block, and the author discusses different normalization strategies. To maintain linear independence within each block, it is usually necessary to reduce the block size in the course of the iteration, and the author describes a deflation technique for performing this reduction. He also present some convergence results, and reports results of numerical experiments with the block-QMR method. Finally, the author discusses possible block versions of transpose-free Lanczos-based iterations such as the TFQMR method.
EXPECTED NUMBER OF ITERATIONS OF INTERIOR-POINT ALGORITHMS FOR LINEAR PROGRAMMING
Institute of Scientific and Technical Information of China (English)
Si-ming Huang
2005-01-01
We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the expected and anticipated number of iterations of these algorithms is bounded above by O(n1,5). The random LP problem is Todd's probabilistic model with the Cauchy distribution.
Average number of iterations of some polynomial interior-point——Algorithms for linear programming
Institute of Scientific and Technical Information of China (English)
黄思明
2000-01-01
We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the average number of iterations of these algorithms, coupled with a finite termination technique, is bounded above by O( n1.5). The random LP problem is Todd’s probabilistic model with the standard Gauss distribution.
Institute of Scientific and Technical Information of China (English)
MA Qinghua; YANG Enhao
2000-01-01
An estimation method for solutions to the general linear system of Volterratype integral inequalities containing several iterated integral functionals is obtained. This method is based on a result proved by the present second author in Journ. Math. Anal. Appl.(1984). A certain two-dimensional system of nonlinear ordinary differential equations is also discussed to demonstrate the usefulness of our method.
Convergence of TTS Iterative Method for Non-Hermitian Positive Definite Linear Systems
Directory of Open Access Journals (Sweden)
Cheng-Yi Zhang
2014-01-01
Full Text Available The TTS iterative method is proposed to solve non-Hermitian positive definite linear systems and some convergence conditions are presented. Subsequently, these convergence conditions are applied to the ALUS method proposed by Xiang et al. in 2012 and comparison of some convergence theorems is made. Furthermore, an example is given to demonstrate the results obtained in this paper.
The Linear Stability Properties of Medium- to High- n TAEs in ITER
Energy Technology Data Exchange (ETDEWEB)
Gorelenkov, N N; Budny, R V; Kessel, C E; Kramer, G J; McCune, D; Manickam, J; Nazikian, R
2008-02-14
This document provides a detailed report on the successful completion of the DOE OFES Theory Milestone for FY2007: Improve the simulation resolution of linear stability properties of Toroidal Alfvén Eigenmodes (TAE) driven by energetic particles and neutral beams in ITER by increasing the numbers of toroidal modes used to 15.
Solutions to quasi-linear differential equations with iterated deviating arguments
Directory of Open Access Journals (Sweden)
Rajib Haloi
2014-12-01
Full Text Available We establish sufficient conditions for the existence and uniqueness of solutions to quasi-linear differential equations with iterated deviating arguments, complex Banach space. The results are obtained by using the semigroup theory for parabolic equations and fixed point theorems. The main results are illustrated by an example.
Average number of iterations of some polynomial interior-point--Algorithms for linear programming
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the average number of iterations of these algorithms, coupled with a finite termination technique, is bounded above by O(n1.5). The random LP problem is Todd's probabilistic model with the standard Gauss distribution.
Communication efficient basic linear algebra computations on hypercube architectures
Energy Technology Data Exchange (ETDEWEB)
Johnsson, S.L.
1987-04-01
This paper presents a few algorithms for embedding loops and multidimensional arrays in hypercubes with emphasis on proximity preserving embeddings. A proximity preserving embedding minimizes the need for communication bandwidth in computations requiring nearest neighbor communication. Two storage schemes for ''large'' problems on ''small'' machines are suggested and analyzed, and algorithms for matrix transpose, multiplying matrices, factoring matrices, and solving triangular linear systems are presented. A few complete binary tree embeddings are described and analyzed. The data movement in the matrix algorithms is analyzed and it is shown that in the majority of cases the directed routing paths intersect only at nodes of the hypercube allowing for a maximum degree of pipelining.
Directory of Open Access Journals (Sweden)
Mohammad Shahzad
2016-05-01
Full Text Available This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM by State Space Exact Linearization (SSEL technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustness of the technique implemented on the chosen chaotic system.
On the linearization of nonlinear supersymmetry based on the commutator algebra
Tsuda, Motomu
2016-01-01
We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY) based on the closure of the commutator algebra for variations of functionals of Nambu-Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov-Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless) vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.
On the linearization of nonlinear supersymmetry based on the commutator algebra
Directory of Open Access Journals (Sweden)
Motomu Tsuda
2017-01-01
Full Text Available We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY based on the closure of the commutator algebra for variations of functionals of Nambu–Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov–Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.
On the linearization of nonlinear supersymmetry based on the commutator algebra
Tsuda, Motomu
2017-01-01
We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY) based on the closure of the commutator algebra for variations of functionals of Nambu-Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov-Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless) vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.
Institute of Scientific and Technical Information of China (English)
MENG; Qingtian
2001-01-01
［1］Iachello, F, Levine, R. D., Algebraic approach to molecular rotation-vibration spectra, I. Diatomic molecules, J, Chem.Phys.. 1982, 77: 3046.［2］Iachello. F.. Oss, S., Overtone frequencies and intensities of bent XY2 molecules in the vibron model, J. Mol. Spectrosc.,1990,142: 85.［3］Van Roosmalen, O. S., Iachello, F., Levine, R. D. et al., Algebraic approach to molecular rotation-vibration spectra, II. Triatomic molecules, J. Chem. Phys., 1983, 79: 2515.［4］Iachello, F., Levine, R. D., Algebraic approach to molecular rotation-vibration spectra, Int. J. Quantum Chem., 1983, 23:1679.［5］Cooper, I. L., Levine, R. D., Computed overtone spectra of linear triatomic molecules by dynamical symmetry, J. Mol. Spectrosc., 1991, 148: 391.［6］Iachello. F., Manini. N., Oss, S., Quasi-linear four-atomic molecules in the vibron model, J. Mol. Spectrosc., 1992, 156:190.［7］Wiesenfeld, L.. The vibron model for methane: stretch-bend interactions, J. Mol. Spectrosc., 1997, 184: 277.［8］Zheng, Y.. Ding, S., Vibrational spectra of HCN and OCS from second-order expansion of the U1(4) U2(4) algebra,Phys. Lett. A. 1999. 256: 197.［9］Zheng, Y.. Ding. S., Algebraic method for determining the potential energy surface for nonlinear triatomic molecules, Chem. Phys., 1999, 247: 225.［10］Zheng, Y.. Ding, S.. Algebraic description of stretching and bending vibrational spectra of H2O and H2S, J. Mol. Spectrosc.,2000. 201: 109.［11］Meng. Q., Zheng, Y., Ding, S., Lie algebraic approach to Fermi resonance levels of CS2 and CO2, Int. J. Quantum Chem.,2001, 81: 154.［12］Ding, S., Zheng, Y., Lie algebraic approach to potential energy surface for symmetric triatomic molecules, J. Chem. Phys.,1999. 111: 4466.［13］Zheng. Y., Ding, S., Algebraic approach to the potential energy surface for the electronic ground state of ozone, Chem.Phys.. 2000. 255: 217.［14］Zheng. Y., Ding, S., Theoretical study of nonlinear triatomic molecular potential energy surfaces: Lie
Second degree generalized Jacobi iteration method for solving system of linear equations
Directory of Open Access Journals (Sweden)
Tesfaye Kebede Enyew
2016-05-01
Full Text Available In this paper, a Second degree generalized Jacobi Iteration method for solving system of linear equations, $Ax=b$ and discuss about the optimal values $a_{1}$ and $b_{1}$ in terms of spectral radius about for the convergence of SDGJ method of $x^{(n+1}=b_{1}[D_{m}^{-1}(L_{m}+U_{m}x^{(n}+k_{1m}]-a_{1}x^{(n-1}.$ Few numerical examples are considered to show that the effective of the Second degree Generalized Jacobi Iteration method (SDGJ in comparison with FDJ, FDGJ, SDJ.
Institute of Scientific and Technical Information of China (English)
Ge-mai Chen; Jin-hong You
2005-01-01
Consider a repeated measurement partially linear regression model with an unknown vector pasemiparametric generalized least squares estimator (SGLSE) ofβ, we propose an iterative weighted semiparametric least squares estimator (IWSLSE) and show that it improves upon the SGLSE in terms of asymptotic covariance matrix. An adaptive procedure is given to determine the number of iterations. We also show that when the number of replicates is less than or equal to two, the IWSLSE can not improve upon the SGLSE.These results are generalizations of those in [2] to the case of semiparametric regressions.
Achievable Rates of MIMO Systems with Linear Precoding and Iterative LMMSE Detection
Yuan, Xiaojun; Kavcic, Aleksandar
2011-01-01
We extend the area property on an additive-white-Gaussian-noise (AWGN) channel to more general linear channel models (including inter-symbol-interference (ISI) and multiple-input multiple-output (MIMO) channels) with arbitrary input constellations. We show that the theoretical limit of a generic linear channel (i.e., the input output mutual information) can be achieved using iterative minimum mean-square error (MMSE) detection under the so-called uniform and Gaussian (UG) assumption on the messages generated in iterative detection. Our major contribution is a linear precoding (LP) technique that can asymptotically ensure the UG assumption as the transmission block length tends to infinity based on the central limit theorem. We also show that superposition coded modulation (SCM) can further help to materialize the UG assumption. Numerical results are demonstrated to verify our analysis.
Winicour, Jeffrey
2017-08-01
An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to solving the constraints of Einstein’s equations which does not involve elliptic equations and has potential importance for the construction of binary black hole data. In order to shed light on the underpinnings of this approach, we consider its application to obtain solutions of the constraints for linearized perturbations of Minkowski space. In that case, we find the surprising result that there are no suitable Cauchy hypersurfaces in Minkowski space for which the linearized algebraic-hyperbolic constraint problem is well posed.
GPU TECHNOLOGIES EMBODIED IN PARALLEL SOLVERS OF LINEAR ALGEBRAIC EQUATION SYSTEMS
Directory of Open Access Journals (Sweden)
Sidorov Alexander Vladimirovich
2012-10-01
Full Text Available The author reviews existing shareware solvers that are operated by graphical computer devices. The purpose of this review is to explore the opportunities and limitations of the above parallel solvers applicable for resolution of linear algebraic problems that arise at Research and Educational Centre of Computer Modeling at MSUCE, and Research and Engineering Centre STADYO. The author has explored new applications of the GPU in the PETSc suite and compared them with the results generated absent of the GPU. The research is performed within the CUSP library developed to resolve the problems of linear algebra through the application of GPU. The author has also reviewed the new MAGMA project which is analogous to LAPACK for the GPU.
Near-infrared reflectance analysis by Gauss-Jordan linear algebra
Energy Technology Data Exchange (ETDEWEB)
Honigs, D.E.; Freelin, J.M.; Hieftje, G.M.; Hirschfeld, T.B.
1983-11-01
Near-infrared reflectance analysis is an analytical technique that uses the near-infrared diffuse reflectance of a sample at several discrete wavelengths to predict the concentration of one or more of the chemical species in that sample. However, because near-infrared bands from solid samples are both abundant and broad, the reflectance at a given wavelength usually contains contributions from several sample components, requiring extensive calculations on overlapped bands. In the present study, these calculations have been performed using an approach similar to that employed in multi-component spectrophotometry, but with Gauss-Jordan linear algebra serving as the computational vehicle. Using this approach, correlations for percent protein in wheat flour and percent benzene in hydrocarbons have been obtained and are evaluated. The advantages of a linear-algebra approach over the common one employing stepwise regression are explored.
A project for developing a linear algebra library for high-performance computers
Energy Technology Data Exchange (ETDEWEB)
Demmel, J.; Dongarra, J.; DuCroz, J.; Greenbaum, A.; Hammarling, S.; Sorensen, D.
1988-01-01
Argonne National Laboratory, the Courant Institute for Mathematical Sciences, and the Numerical Algorithms Group, Ltd., are developing a transportable linear algebra library in Fortran 77. The library is intended to provide a uniform set of subroutines to solve the most common linear algebra problems and to run efficiently on a wide range of high-performance computers. To be effective, the new library must satisfy several criteria. First, it must be highly efficient, or at least ''tunable'' to high efficiency, on each machine. Second, the user interface must be uniform across machines. Otherwise much of the convenience of portability will be lost. Third, the program must be widely available. NETLIB has demonstrated how useful and important it is for these codes to be available easily, and preferably on line. We intend to distribute the new library in a similar way, for no cost or a nominal cost only. In addition, the programs must be well documented.
Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio
2017-02-01
Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of "spacelike linearity". Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.
Batched Triangular Dense Linear Algebra Kernels for Very Small Matrix Sizes on GPUs
Charara, Ali
2017-03-06
Batched dense linear algebra kernels are becoming ubiquitous in scientific applications, ranging from tensor contractions in deep learning to data compression in hierarchical low-rank matrix approximation. Within a single API call, these kernels are capable of simultaneously launching up to thousands of similar matrix computations, removing the expensive overhead of multiple API calls while increasing the occupancy of the underlying hardware. A challenge is that for the existing hardware landscape (x86, GPUs, etc.), only a subset of the required batched operations is implemented by the vendors, with limited support for very small problem sizes. We describe the design and performance of a new class of batched triangular dense linear algebra kernels on very small data sizes using single and multiple GPUs. By deploying two-sided recursive formulations, stressing the register usage, maintaining data locality, reducing threads synchronization and fusing successive kernel calls, the new batched kernels outperform existing state-of-the-art implementations.
A correspondence between maximal abelian sub-algebras and linear logic fragments
DEFF Research Database (Denmark)
SEILLER, THOMAS
2016-01-01
We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier (Dixmier 1954. Annals of Mathematics 59 (2) 279–286) and fragments of linear logic. We expose for this purpose a modified construction of Girard's hyperfinite geometry of interact......We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier (Dixmier 1954. Annals of Mathematics 59 (2) 279–286) and fragments of linear logic. We expose for this purpose a modified construction of Girard's hyperfinite geometry...... of interaction (Girard 2011. Theoretical Computer Science 412 (20) 1860–1883). The expressivity of the logic soundly interpreted in this model is dependent on properties of a MASA which is a parameter of the interpretation. We also unveil the essential role played by MASAs in previous geometry of interaction...
Meng, Qingtian; Guan, Daren; Ding, Shiliang
2001-04-01
An algebraic construction of a Hamiltonian is used to study the rotational spectra of linear triatomic molecules on the basis of the subgroup chain of symmetry U1(4)⊗ U2(4). After considering the rotation-vibration interaction which gives the l splittings, the eigenvalue expression of the Hamiltonian has a form of the term value equation commonly used in the calculation of molecular spectra. The method is applied to calculate the rotational energy levels of vibrational transitions (0 1 10-0 0 00) for C 34S 2, (1 1 13-0 1 10) and (1 0 03-0 0 00) for C 32S 2. The obtained rotational constants can represent the rotational spectra of the three bands with small root-mean-square frequency errors. The results show that the algebraic Hamiltonian can provide an alternative description of rovibrational spectra for linear triatomic molecules.
Adaptive Algebraic Multigrid Methods
Energy Technology Data Exchange (ETDEWEB)
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
Ertekin, E.; Solak, S.; Yazici, E.
2010-01-01
The aim of this study is to identify the effects of formalism in teaching on primary and secondary school mathematics teacher trainees' algebraic and geometric interpretations of the notions of linear dependency/independency. Quantitative research methods are drawn in order to determine differences in success levels between algebraic and geometric…
A modified approach to team-based learning in linear algebra courses
Nanes, Kalman M.
2014-11-01
This paper documents the author's adaptation of team-based learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, in the linear algebra classroom. The paper discusses the standard components of TBL and the necessary changes to those components for the needs of the course in question. There is also an empirically controlled analysis of the effects of TBL on the student learning experience in the first year of TBL use.
Counterexample Construction in Linear Algebra%线性代数中的反例构造
Institute of Scientific and Technical Information of China (English)
王琮涵; 张新华
2014-01-01
Reduce the learning curve for linear algebra problems raised in the learning process of flexible linear algebra con-structed counter-examples to solve those problems. Through literature research methods and lessons learned, discuss appro-priate use of reductio ad absurdum linear algebra study issues such as definitions and theorems, with examples indicate how reasonable construct counter-examples in the learning process of linear algebra theory to understand and grasp the essence. Analysis, counter-examples can be constructed for the understanding of the definition, theorems and propositions are false master judge offers simple ideas to improve learning efficiency.%针对降低线性代数学习难度的问题，文章提出在学习线性代数过程中灵活构造反例以解决问题的思路。通过文献研究法和经验总结法，讨论线性代数中适合运用反证法研究的定义和定理等问题，结合实例指出如何在学习线性代数过程中合理构造反例以理解和掌握理论本质。分析发现，构造反例可为定义的理解、定理的掌握和命题的正误判断提供简捷的思路，提高学习效率。
ScaLAPACK: A linear algebra library for message-passing computers
Energy Technology Data Exchange (ETDEWEB)
Blackford, L.S., Cleary, A., Petitet, A., Whaley, R.C., Dongarra, J. [Dept. of Computer Science, Tennessee Univ., Knoxville, TN (United States); Choi, J., [Soongsil University (Korea); D`Azevedo, E. [Mathematical Science Section, Oak Ridge National Lab., TN (United States); Demmel, J., Dhillon, I., Stanley, K. [California Univ., Berkeley, CA (United States). Computer Science Div.; Hammarling, S. [Nag Ltd., (England); Henry, G., Walker, D. [Itel SSPD, Beaverton, OR (United States)
1997-01-06
This article outlines the content and performance of some of the ScaLAPACK software. ScaLAPACK is a collection of mathematical software for linear algebra computations on distributed-memory computers. The importance of developing standards for computational and message-passing interfaces is discussed. We present the different components and building blocks of ScaLAPACK and provide initial performance results for selected PBLAS routines and a subset of ScaLAPACK driver routines.
Many-core graph analytics using accelerated sparse linear algebra routines
Kozacik, Stephen; Paolini, Aaron L.; Fox, Paul; Kelmelis, Eric
2016-05-01
Graph analytics is a key component in identifying emerging trends and threats in many real-world applications. Largescale graph analytics frameworks provide a convenient and highly-scalable platform for developing algorithms to analyze large datasets. Although conceptually scalable, these techniques exhibit poor performance on modern computational hardware. Another model of graph computation has emerged that promises improved performance and scalability by using abstract linear algebra operations as the basis for graph analysis as laid out by the GraphBLAS standard. By using sparse linear algebra as the basis, existing highly efficient algorithms can be adapted to perform computations on the graph. This approach, however, is often less intuitive to graph analytics experts, who are accustomed to vertex-centric APIs such as Giraph, GraphX, and Tinkerpop. We are developing an implementation of the high-level operations supported by these APIs in terms of linear algebra operations. This implementation is be backed by many-core implementations of the fundamental GraphBLAS operations required, and offers the advantages of both the intuitive programming model of a vertex-centric API and the performance of a sparse linear algebra implementation. This technology can reduce the number of nodes required, as well as the run-time for a graph analysis problem, enabling customers to perform more complex analysis with less hardware at lower cost. All of this can be accomplished without the requirement for the customer to make any changes to their analytics code, thanks to the compatibility with existing graph APIs.
Adaptive Digital Predistortion with Iterative Noise Cancelation for Power Amplifier Linearization
Jeon, Sungho; Kim, Junghyun; Lee, Jaekwon; Suh, Young-Woo; Seo, Jong-Soo
In this paper, we propose a power amplifier linearization technique combined with iterative noise cancelation. This method alleviates the effect of added noises which prevents the predistorter (PD) from estimating the exact characteristics of the power amplifier (PA). To iteratively cancel the noise added in the feedback signal, the output signal of the power amplifier without noise is reconstructed by applying the inverse characteristics of the PD to the predistorted signals. The noise can be revealed by subtracting the reconstructed signals from the feedback signals. Simulation results based on the mean-square error (MSE) and power spectral density (PSD) criteria are presented to evaluate PD performance. The results show that the iterative noise cancelation significantly enhances the MSE performance, which leads to an improvement of the out-of-band power suppression. The performance of the proposed technique is verified by computer simulation and hardware test results.
The M4RIE library for dense linear algebra over small fields with even characteristic
Albrecht, Martin R
2011-01-01
In this work, we present the M4RIE library which implements efficient algorithms for linear algebra with dense matrices over GF(2^e) for 2 <= 2 <= 10. As the name of the library indicates, it makes heavy use of the M4RI library both directly (i.e., by calling it) and indirectly (i.e., by using its concepts). We provide an open-source GPLv2+ C library for efficient linear algebra over GF(2^e) for e small. In this library we implemented an idea due to Bradshaw and Boothby which reduces matrix multiplication over GF(p^k) to a series of matrix multiplications over GF(p). Furthermore, we propose a caching technique - Newton-John tables - to avoid finite field multiplications which is inspired by Kronrod's method ("M4RM") for matrix multiplication over GF(2). Using these two techniques we provide asymptotically fast triangular solving with matrices (TRSM) and PLE-based Gaussian elimination. As a result, we are able to significantly improve upon the state of the art in dense linear algebra over GF(2^e) with 2 ...
Programming methodology and performance issues for advanced computer architectures. [Linear algebra
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Sorensen, D.C.; Connolly, K.; Patterson, J.
1987-01-01
This paper will describe some recent attempts to construct transportable numerical software for high performance computers. Restructuring algorithms in terms of simple linear algebra modules is reviewed. This technique has proved very successful in obtaining a high level of transportability without severe loss of performance on a wide variety of both vector and parallel computers. The use of modules to encapsulate parallelism and reduce the ratio of data movement to floating point operations has been demonstrably effective for regular problems such as those found in dense linear algebra. In other situations it may be necessary to express explicitly parallel algorithms. We also present a programming methodology that is useful for constructing new parallel algorithms which require sophisticated synchronization at a large grain level. We describe the SCHEDULE package which provides an environment for developing and analyzing explicitly parallel programs in Fortran which aare portable. This package now includes a preprocessor to achieve complete portability of user level code and also a graphics post processor for performance analysis and debugging. We discuss details of porting both the SCHEDULE package and user code. Examples from linear algebra, and partial differential equations are used to illustrate the utility of this approach.
Maia, Julio Daniel Carvalho; Urquiza Carvalho, Gabriel Aires; Mangueira, Carlos Peixoto; Santana, Sidney Ramos; Cabral, Lucidio Anjos Formiga; Rocha, Gerd B
2012-09-11
In this study, we present some modifications in the semiempirical quantum chemistry MOPAC2009 code that accelerate single-point energy calculations (1SCF) of medium-size (up to 2500 atoms) molecular systems using GPU coprocessors and multithreaded shared-memory CPUs. Our modifications consisted of using a combination of highly optimized linear algebra libraries for both CPU (LAPACK and BLAS from Intel MKL) and GPU (MAGMA and CUBLAS) to hasten time-consuming parts of MOPAC such as the pseudodiagonalization, full diagonalization, and density matrix assembling. We have shown that it is possible to obtain large speedups just by using CPU serial linear algebra libraries in the MOPAC code. As a special case, we show a speedup of up to 14 times for a methanol simulation box containing 2400 atoms and 4800 basis functions, with even greater gains in performance when using multithreaded CPUs (2.1 times in relation to the single-threaded CPU code using linear algebra libraries) and GPUs (3.8 times). This degree of acceleration opens new perspectives for modeling larger structures which appear in inorganic chemistry (such as zeolites and MOFs), biochemistry (such as polysaccharides, small proteins, and DNA fragments), and materials science (such as nanotubes and fullerenes). In addition, we believe that this parallel (GPU-GPU) MOPAC code will make it feasible to use semiempirical methods in lengthy molecular simulations using both hybrid QM/MM and QM/QM potentials.
Iterative solution of dense linear systems arising from the electrostatic integral equation in MEG
Energy Technology Data Exchange (ETDEWEB)
Rahola, Jussi [Simulintu Oy, Espoo (Finland); Tissari, Satu [CSC - Scientific Computing Ltd, Espoo (Finland)]. E-mail: satu.tissari@csc.fi
2002-03-21
We study the iterative solution of dense linear systems that arise from boundary element discretizations of the electrostatic integral equation in magnetoencephalography (MEG). We show that modern iterative methods can be used to decrease the total computation time by avoiding the time-consuming computation of the LU decomposition of the coefficient matrix. More importantly, the modern iterative methods make it possible to avoid the explicit formation of the coefficient matrix which is needed when a large number of unknowns are used. To study the convergence of iterative solvers we examine the eigenvalue distributions of the coefficient matrices. For the sphere we show how the eigenvalues of the integral operator are approximated by the eigenvalues of the coefficient matrix when the collocation and Galerkin methods are used as discretization methods. The collocation method approximates the eigenvalues of the integral operator directly. The Galerkin method produces a coefficient matrix that needs to be preconditioned in order to maintain optimal convergence speed. With the ILU(0) preconditioner iterative methods converge fast and independent of the number of discretization points for both the collocation and Galerkin approaches. The preconditioner has no significant effect on the total computational time. (author)
Convergence of Galerkin Solutions for Linear Differential Algebraic Equations in Hilbert Spaces
Matthes, Michael; Tischendorf, Caren
2010-09-01
The simulation of complex systems describing different physical effects becomes more and more of interest in various applications. Examples are couplings describing interactions between circuits and semiconductor devices, circuits and electromagnetic fields, fluids and structures. The modeling of such complex processes [1, 2, 3, 4, 7, 8] often leads to coupled systems that are composed of ordinary differential equations (ODEs), differential-algebraic equations (DAEs) and partial differential equations (PDEs). Such coupled systems can be regarded in the general framework of abstract differential-algebraic equations. Here, we discuss a Galerkin approach for handling linear abstract differential-algebraic equations with monotone operators. It is shown to provide solutions that converge to the unique solution of the abstract differential-algebraic system. Furthermore, the solution is proved to depend continuously on the data. The most interesting point of the Galerkin approach is the choice of basis functions. They have to be chosen in proper subspaces in order to guarantee that the solution satisfies the non-dynamic constraints. In contrast to other approaches as e.g. [5, 6], this approach allows time dependent operators but needs monotonicity.
Energy Technology Data Exchange (ETDEWEB)
Lorber, A.A.; Carey, G.F.; Bova, S.W.; Harle, C.H. [Univ. of Texas, Austin, TX (United States)
1996-12-31
The connection between the solution of linear systems of equations by iterative methods and explicit time stepping techniques is used to accelerate to steady state the solution of ODE systems arising from discretized PDEs which may involve either physical or artificial transient terms. Specifically, a class of Runge-Kutta (RK) time integration schemes with extended stability domains has been used to develop recursion formulas which lead to accelerated iterative performance. The coefficients for the RK schemes are chosen based on the theory of Chebyshev iteration polynomials in conjunction with a local linear stability analysis. We refer to these schemes as Chebyshev Parameterized Runge Kutta (CPRK) methods. CPRK methods of one to four stages are derived as functions of the parameters which describe an ellipse {Epsilon} which the stability domain of the methods is known to contain. Of particular interest are two-stage, first-order CPRK and four-stage, first-order methods. It is found that the former method can be identified with any two-stage RK method through the correct choice of parameters. The latter method is found to have a wide range of stability domains, with a maximum extension of 32 along the real axis. Recursion performance results are presented below for a model linear convection-diffusion problem as well as non-linear fluid flow problems discretized by both finite-difference and finite-element methods.
Evaluation of global synchronization for iterative algebra algorithms on many-core
ul Hasan Khan, Ayaz
2015-06-01
© 2015 IEEE. Massively parallel computing is applied extensively in various scientific and engineering domains. With the growing interest in many-core architectures and due to the lack of explicit support for inter-block synchronization specifically in GPUs, synchronization becomes necessary to minimize inter-block communication time. In this paper, we have proposed two new inter-block synchronization techniques: 1) Relaxed Synchronization, and 2) Block-Query Synchronization. These schemes are used in implementing numerical iterative solvers where computation/communication overlapping is one used optimization to enhance application performance. We have evaluated and analyzed the performance of the proposed synchronization techniques using Jacobi Iterative Solver in comparison to the state of the art inter-block lock-free synchronization techniques. We have achieved about 1-8% performance improvement in terms of execution time over lock-free synchronization depending on the problem size and the number of thread blocks. We have also evaluated the proposed algorithm on GPU and MIC architectures and obtained about 8-26% performance improvement over the barrier synchronization available in OpenMP programming environment depending on the problem size and number of cores used.
River flow forecasting. Part 2. Algebraic development of linear modelling techniques
Kachroo, R. K.; Liang, G. C.
1992-04-01
The role of linear input-output models in hydrological forecasting is discussed. The algebraic analysis of linear systems with single or multiple input and single output is presented in outline. The least squares method of system identification is discussed in the context of recursive and off-line estimation, with and without volumetric and shape constraints. An alternative means of imposing shape constraints, via parametric modelling, is also discussed. A procedure for 'updating' is presented for models used in real-time forecasting.
Linear algebra programs for use on a vector computer with a secondary solid state storage device
Energy Technology Data Exchange (ETDEWEB)
Bucher, I.Y.; Jordan, T.L.
1984-01-01
A portable set of linear algebra subprograms for use on a vector computer with an attached fast secondary storage device has been developed. The set currently contains routines for matrix multiplication and for the solution of block tridiagonal, symmetric and positive definite, and general systems of linear equations. Main matrices are stored on the external device in blocked form, and block matrix techniques are used throughout. Performance data are presented which demonstrate that the speed of the routines approaches that of routines with all data in main memory and is close to the maximum speed of the processor.
Rómoli, Santiago; Serrano, Mario Emanuel; Ortiz, Oscar Alberto; Vega, Jorge Rubén; Eduardo Scaglia, Gustavo Juan
2015-07-01
Based on a linear algebra approach, this paper aims at developing a novel control law able to track reference profiles that were previously-determined in the literature. A main advantage of the proposed strategy is that the control actions are obtained by solving a system of linear equations. The optimal controller parameters are selected through Monte Carlo Randomized Algorithm in order to minimize a proposed cost index. The controller performance is evaluated through several tests, and compared with other controller reported in the literature. Finally, a Monte Carlo Randomized Algorithm is conducted to assess the performance of the proposed controller. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Beilinson, Alexander
2004-01-01
Chiral algebras form the primary algebraic structure of modern conformal field theory. Each chiral algebra lives on an algebraic curve, and in the special case where this curve is the affine line, chiral algebras invariant under translations are the same as well-known and widely used vertex algebras. The exposition of this book covers the following topics: the "classical" counterpart of the theory, which is an algebraic theory of non-linear differential equations and their symmetries; the local aspects of the theory of chiral algebras, including the study of some basic examples, such as the ch
Directory of Open Access Journals (Sweden)
Chengcai Leng
2015-01-01
Full Text Available Optical molecular imaging is a promising technique and has been widely used in physiology, and pathology at cellular and molecular levels, which includes different modalities such as bioluminescence tomography, fluorescence molecular tomography and Cerenkov luminescence tomography. The inverse problem is ill-posed for the above modalities, which cause a nonunique solution. In this paper, we propose an effective reconstruction method based on the linearized Bregman iterative algorithm with sparse regularization (LBSR for reconstruction. Considering the sparsity characteristics of the reconstructed sources, the sparsity can be regarded as a kind of a priori information and sparse regularization is incorporated, which can accurately locate the position of the source. The linearized Bregman iteration method is exploited to minimize the sparse regularization problem so as to further achieve fast and accurate reconstruction results. Experimental results in a numerical simulation and in vivo mouse demonstrate the effectiveness and potential of the proposed method.
Iterative learning based fault diagnosis for discrete linear uncer tain systems
Institute of Scientific and Technical Information of China (English)
Wei Cao; Ming Sun
2014-01-01
In order to detect and estimate faults in discrete lin-ear time-varying uncertain systems, the discrete iterative learning strategy is applied in fault diagnosis, and a novel fault detection and estimation algorithm is proposed. And the threshold limited tech-nology is adopted in the proposed algorithm. Within the chosen optimal time region, residual signals are used in the proposed algo-rithm to correct the introduced virtual faults with iterative learning rules, making the virtual faults close to these occurred in practical systems. And the same method is repeated in the rest optimal time regions, thereby reaching the aim of fault diagnosis. The proposed algorithm not only completes fault detection and estimation for dis-crete linear time-varying uncertain systems, but also improves the reliability of fault detection and decreases the false alarm rate. The final simulation results verify the validity of the proposed algorithm.
Linear-algebraic list decoding of folded Reed-Solomon codes
Guruswami, Venkatesan
2011-01-01
Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any $\\eps > 0$, the author and Rudra (2006,08) presented an $n^{O(1/\\eps)}$ time algorithm to list decode appropriate folded RS codes of rate $R$ from a fraction $1-R-\\eps$ of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices if one settles for a smaller decoding radius (but still enough for a statement of the above form). Here we give a simple linear-algebra based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for ...
Recent advances in Lanczos-based iterative methods for nonsymmetric linear systems
Freund, Roland W.; Golub, Gene H.; Nachtigal, Noel M.
1992-01-01
In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and so-called look-ahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczos-based iterative schemes for solving nonsymmetric linear systems have been proposed. This paper gives a survey of some of these recent developments.
构造法在高等代数中的应用%Method of Construction in Linear Algebra
Institute of Scientific and Technical Information of China (English)
杨培亮
2012-01-01
This paper attempts to show applications of structural method in linear algebra from seven aspects, which provide effective auxiliary approaches for teaching linear algebra.%从函数、多项式、行列式、分块矩阵、二次型、线性方程组、反例七个方面,结合具体实例介绍构造法在高等代数中的应用.
Institute of Scientific and Technical Information of China (English)
Zhong-Zhi; Yu-Mei; K.
2010-01-01
Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, I.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iterate, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix.The igenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient,and the effect of image restoration is r0easonably well.
MODELING IN MAPLE AS THE RESEARCHING MEANS OF FUNDAMENTAL CONCEPTS AND PROCEDURES IN LINEAR ALGEBRA
Directory of Open Access Journals (Sweden)
Vasil Kushnir
2016-05-01
Full Text Available The article is devoted to binary technology and "fundamental training technology." Binary training refers to the simultaneous teaching of mathematics and computer science, for example differential equations and Maple, linear algebra and Maple. Moreover the system of traditional course of Maple is not performed. The use of the opportunities of Maple-technology in teaching mathematics is based on the following fundamental concepts of computer science as an algorithm, program, a linear program, cycle, branching, relative operators, etc. That’s why only a certain system of command operators in Maple is considered. They are necessary for fundamental concepts of linear algebra and differential equations studying in Maple-environment. Relative name - "the technology of fundamental training" reflects the study of fundamental mathematical concepts and procedures that express the properties of these concepts in Maple-environment. This article deals with the study of complex fundamental concepts of linear algebra (determinant of the matrix and algorithm of its calculation, the characteristic polynomial of the matrix and the eigenvalues of matrix, canonical form of characteristic matrix, eigenvectors of matrix, elementary divisors of the characteristic matrix, etc., which are discussed in the appropriate courses briefly enough, and sometimes are not considered at all, but they are important in linear systems of differential equations, asymptotic methods for solving differential equations, systems of linear equations. Herewith complex and voluminous procedures of finding of these linear algebra concepts embedded in Maple can be performed as a result of a simple command-operator. Especially important issue is building matrix to canonical form. In fact matrix functions are effectively reduced to the functions of the diagonal matrix or matrix in Jordan canonical form. These matrices are used to rise a square matrix to a power, to extract the roots of the n
Goodman, Roe W
2016-01-01
This textbook for undergraduate mathematics, science, and engineering students introduces the theory and applications of discrete Fourier and wavelet transforms using elementary linear algebra, without assuming prior knowledge of signal processing or advanced analysis.It explains how to use the Fourier matrix to extract frequency information from a digital signal and how to use circulant matrices to emphasize selected frequency ranges. It introduces discrete wavelet transforms for digital signals through the lifting method and illustrates through examples and computer explorations how these transforms are used in signal and image processing. Then the general theory of discrete wavelet transforms is developed via the matrix algebra of two-channel filter banks. Finally, wavelet transforms for analog signals are constructed based on filter bank results already presented, and the mathematical framework of multiresolution analysis is examined.
Liu, Da-Yan
2015-04-30
This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.
Global identifiability of linear compartmental models--a computer algebra algorithm.
Audoly, S; D'Angiò, L; Saccomani, M P; Cobelli, C
1998-01-01
A priori global identifiability deals with the uniqueness of the solution for the unknown parameters of a model and is, thus, a prerequisite for parameter estimation of biological dynamic models. Global identifiability is however difficult to test, since it requires solving a system of algebraic nonlinear equations which increases both in nonlinearity degree and number of terms and unknowns with increasing model order. In this paper, a computer algebra tool, GLOBI (GLOBal Identifiability) is presented, which combines the topological transfer function method with the Buchberger algorithm, to test global identifiability of linear compartmental models. GLOBI allows for the automatic testing of a priori global identifiability of general structure compartmental models from general multi input-multi output experiments. Examples of usage of GLOBI to analyze a priori global identifiability of some complex biological compartmental models are provided.
Linear-algebraic Lambda-calculus: higher-order, encodings and confluence
Arrighi, P; Arrighi, Pablo; Dowek, Gilles
2006-01-01
We introduce a minimal language combining both higher-order computation and linear algebra. Roughly, this is nothing else than the Lambda-calculus together with the possibility to make linear combinations of terms a.t+b.u. We describe how to "execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in field of quantum computation, which we show can be easily encoded in the calculus, as well as in other domains such as the interpretation of linear logic. Finally we prove the confluence of the calculus, this is our main result. Keywords: quantum programming language, quantum control, quantum logic, probabilistic and quantitative analysis, rewriting techniques.
A Posteriori Error Estimation for Finite Element Methods and Iterative Linear Solvers
Energy Technology Data Exchange (ETDEWEB)
Melboe, Hallgeir
2001-10-01
This thesis addresses a posteriori error estimation for finite element methods and iterative linear solvers. Adaptive finite element methods have gained a lot of popularity over the last decades due to their ability to produce accurate results with limited computer power. In these methods a posteriori error estimates play an essential role. Not only do they give information about how large the total error is, they also indicate which parts of the computational domain should be given a more sophisticated treatment in order to reduce the error. A posteriori error estimates are traditionally aimed at estimating the global error, but more recently so called goal oriented error estimators have been shown a lot of interest. The name reflects the fact that they estimate the error in user-defined local quantities. In this thesis the main focus is on global error estimators for highly stretched grids and goal oriented error estimators for flow problems on regular grids. Numerical methods for partial differential equations, such as finite element methods and other similar techniques, typically result in a linear system of equations that needs to be solved. Usually such systems are solved using some iterative procedure which due to a finite number of iterations introduces an additional error. Most such algorithms apply the residual in the stopping criterion, whereas the control of the actual error may be rather poor. A secondary focus in this thesis is on estimating the errors that are introduced during this last part of the solution procedure. The thesis contains new theoretical results regarding the behaviour of some well known, and a few new, a posteriori error estimators for finite element methods on anisotropic grids. Further, a goal oriented strategy for the computation of forces in flow problems is devised and investigated. Finally, an approach for estimating the actual errors associated with the iterative solution of linear systems of equations is suggested. (author)
PB-BLAS: A set of parallel block basic linear algebra subprograms
Energy Technology Data Exchange (ETDEWEB)
Choi, Jaeyoung [Tennessee Univ., Knoxville, TN (United States). Dept. of Computer Science; Dongarra, J. [Oak Ridge National Lab., TN (United States)]|[Tennessee Univ., Knoxville, TN (United States). Dept. of Computer Science; Walker, D.W. [Oak Ridge National Lab., TN (United States)
1994-12-31
We propose a new library of routines for performing dense linear algebra computations on block-partitioned matrices. The routines are referred to as the Block Basic Linear Algebra Subprograms, and their use is restricted to computations in which one or more of the matrices involved consists of a single row or column of blocks, and in which no more than one of the matrices consists of an unrestricted two-dimensional array of blocks. The functionality of the block BLAS routines can also be provided by Level 2 and 3 BLAS routines. However, for Non-Uniform Memory Access machines the use of the block BLAS permit certain optimizations in memory access to be taken advantage of. This is particularly true for distributed memory machines, for which the block BLAS are referred to as the Parallel Block Basic Linear Algebra Subprograms (PB-BLAS). The PB-BLAS are the main focus of this paper, and for a block-cyclic data distribution, a single row or column of blocks lies in a single row or column of the processor template. The PB-BLAS consist of calls to the sequential BLAS for local computations, and calls to the BLACS for communication. The PB-BLAS are the building blocks for implementing ScaLAPACK, the distributed-memory version of LAPACK, and provide the same ease-of-use and portability for ScaLAPACK that the BLAS provide for LAPACK. The PB-BLAS consists of all nine Level 3 BLAS routines, four of the Level-2 BLAS routines, and 2 auxiliary transpose routines. The PB-BLAS are currently available for all numeric data types, i.e., single and double precision real and complex.
Hakvoort, W.B.J.; Aarts, R.G.K.M.; Dijk, van J.; Jonker, J.B.
2009-01-01
Iterative Learning Control (ILC) improves the tracking accuracy of systems that repetitively perform the same task. This paper considers model-based ILC for linear time-varying (LTV) systems. The applied feedforward iteratively minimises a quadratic norm of the feedforward update and the error in th
Multiple Measurement Vectors ISAR Imaging Algorithm Based on a Class of Linearized Bregman Iteration
Directory of Open Access Journals (Sweden)
Chen Wenfeng
2016-08-01
Full Text Available This study aims to enable steady and speedy acquisition of Inverse Synthetic Aperture Radar (ISAR images using sparse echo data. To this end, a Multiple Measurement Vectors (MMV ISAR echo model is studied. This model is then combined with the Compressive Sensing (CS theory to realize a class of MMV fast ISAR imaging algorithms based on the Linearized Bregman Iteration (LBI. The algorithms involve four methods, and the iterative framework, application conditions, and relationship between the four methods are given. The reconstructed performance of the methods, convergence, anti-noise, and selection of regularization parameters are then compared and analyzed comprehensively. Finally, the experimental results are compared with the traditional Single Measurement Vector (SMV ISAR imaging algorithm; this comparison shows that the proposed algorithm delivers an improved imaging quality with a low Signal-to-Noise Ratio (SNR.
Quasi-Newton-type optimized iterative learning control for discrete linear time invariant systems
Institute of Scientific and Technical Information of China (English)
Yan GENG; Xiaoe RUAN
2015-01-01
In this paper, a quasi-Newton-type optimized iterative learning control (ILC) algorithm is investigated for a class of discrete linear time-invariant systems. The proposed learning algorithm is to update the learning gain matrix by a quasi-Newton-type matrix instead of the inversion of the plant. By means of the mathematical inductive method, the monotone convergence of the proposed algorithm is analyzed, which shows that the tracking error monotonously converges to zero after a finite number of iterations. Compared with the existing optimized ILC algorithms, due to the superlinear convergence of quasi-Newton method, the proposed learning law operates with a faster convergent rate and is robust to the ill-condition of the system model, and thus owns a wide range of applications. Numerical simulations demonstrate the validity and effectiveness.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The Hamiltonian describing rotational spectra of linear triatomic molecules has been derived by using the dynamical Lie algebra of symmetry group U1(4)U,(4). After rovibrational interactions being considered, the eigenvalue expression of the Hamiltonian has the form of term value equation commonly used in spectrum analysis. The molecular rotational constants can be obtained by using the expression and fitting it to the observed lines. As an example, the rotational levels of v2 band for transition (0200-0110) of molecules N2O and HCN have been fitted and the fitting root-mean-square errors (RMS) are 0.00001 and 0.0014 cm-1, respectively.
An Ada Linear-Algebra Software Package Modeled After HAL/S
Klumpp, Allan R.; Lawson, Charles L.
1990-01-01
New avionics software written more easily. Software package extends Ada programming language to include linear-algebra capabilities similar to those of HAL/S programming language. Designed for such avionics applications as Space Station flight software. In addition to built-in functions of HAL/S, package incorporates quaternion functions used in Space Shuttle and Galileo projects and routines from LINPAK solving systems of equations involving general square matrices. Contains two generic programs: one for floating-point computations and one for integer computations. Written on IBM/AT personal computer running under PC DOS, v.3.1.
Kurnyavko, O. L.; Shirokov, I. V.
2016-07-01
We offer a method for constructing invariants of the coadjoint representation of Lie groups that reduces this problem to known problems of linear algebra. This method is based on passing to symplectic coordinates on the coadjoint representation orbits, which play the role of local coordinates on those orbits. The corresponding transition functions are their parametric equations. Eliminating the symplectic coordinates from the transition functions, we can obtain the complete set of invariants. The proposed method allows solving the problem of constructing invariants of the coadjoint representation for Lie groups with an arbitrary dimension and structure.
An extended set of Fortran Basic Linear Algebra Subprograms: model implementation and test programs
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Du Croz, J.; Hammarling, S.; Hanson, R.J.
1987-01-01
This paper describes a model implementation and test software for the Level 2 Basic Linear Algebra Subprograms (Level 2 BLAS). The Level 2 BLAS are targeted at matrix-vector operations with the aim of providing more efficient, but portable, implementations of algorithms on high-performance computers. The model implementation provides a portable set of Fortran 77 Level 2 BLAS for machines where specialized implementations do not exist or are not required. The test software aims to verify that specialized implementations meet the specification of the Level 2 BLAS and that implementations are correctly installed.
Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package
Directory of Open Access Journals (Sweden)
Douglas Bates
2013-01-01
Full Text Available The RcppEigen package provides access from R (R Core Team 2012a to the Eigen (Guennebaud, Jacob, and others 2012 C++ template library for numerical linear algebra. Rcpp (Eddelbuettel and François 2011, 2012 classes and specializations of the C++ templated functions as and wrap from Rcpp provide the "glue" for passing objects from R to C++ and back. Several introductory examples are presented. This is followed by an in-depth discussion of various available approaches for solving least-squares problems, including rank-revealing methods, concluding with an empirical run-time comparison. Last but not least, sparse matrix methods are discussed.
Generalized model of double random phase encoding based on linear algebra
Nakano, Kazuya; Takeda, Masafumi; Suzuki, Hiroyuki; Yamaguchi, Masahiro
2013-01-01
We propose a generalized model for double random phase encoding (DRPE) based on linear algebra. We defined the DRPE procedure in six steps. The first three steps form an encryption procedure, while the later three steps make up a decryption procedure. We noted that the first (mapping) and second (transform) steps can be generalized. As an example of this generalization, we used 3D mapping and a transform matrix, which is a combination of a discrete cosine transform and two permutation matrices. Finally, we investigated the sensitivity of the proposed model to errors in the decryption key.
Novel parallel architectures and algorithms for linear algebra processing. Semiannual report
Energy Technology Data Exchange (ETDEWEB)
Casasent, D.
1986-10-01
Advanced problems in computational fluid dynamics, finite element structural analysis, and related areas require the solution of large partial differential equations and matrices of large size and dynamic range. This project considers an advanced parallel linear algebra processor and associated novel parallel algorithms for such applications. Research on system fabrication, quantitative performance evaluation and new parallel algorithms are described and considered. Case studies in structural mechanics, dynamics and nonlinear systems, finite element methods, computational fluid dynamics, and partial differential equations are included. The novel utilization of an optical processor for the processing of such problems is the major research given attention.
An object oriented design for high performance linear algebra on distributed memory architectures
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J. [Oak Ridge National Lab., TN (United States)]|[Univ. of Tennessee, Knoxville, TN (United States). Dept. of Computer Science; Walker, D.W. [Oak Ridge National Lab., TN (United States); Pozo, R. [Univ. of Tennessee, Knoxville, TN (United States). Dept. of Computer Science
1993-12-31
We describe the design of ScaLAPACK++, an object oriented C++ library for implementing linear algebra computations on distributed memory multicomputers. This package, when complete, will support distributed dense, banded, sparse matrix operations for symmetric, positive-definite, and non-symmetric cases. In ScaLAPACK++ we have employed object oriented design methods to enchance scalability, portability, flexibility, and ease-of-use. We illustrate some of these points by describing the implementation of a right-looking LU factorization for dense systems in ScaLAPACK++.
An Ada Linear-Algebra Software Package Modeled After HAL/S
Klumpp, Allan R.; Lawson, Charles L.
1990-01-01
New avionics software written more easily. Software package extends Ada programming language to include linear-algebra capabilities similar to those of HAL/S programming language. Designed for such avionics applications as Space Station flight software. In addition to built-in functions of HAL/S, package incorporates quaternion functions used in Space Shuttle and Galileo projects and routines from LINPAK solving systems of equations involving general square matrices. Contains two generic programs: one for floating-point computations and one for integer computations. Written on IBM/AT personal computer running under PC DOS, v.3.1.
The development of an algebraic multigrid algorithm for symmetric positive definite linear systems
Energy Technology Data Exchange (ETDEWEB)
Vanek, P.; Mandel, J.; Brezina, M. [Univ. of Colorado, Denver, CO (United States)
1996-12-31
An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate blending, and shells.
JTpack90: A parallel, object-based, Fortran 90 linear algebra package
Energy Technology Data Exchange (ETDEWEB)
Turner, J.A.; Kothe, D.B. [Los Alamos National Lab., NM (United States); Ferrell, R.C. [Cambridge Power Computing Associates, Ltd., Brookline, MA (United States)
1997-03-01
The authors have developed an object-based linear algebra package, currently with emphasis on sparse Krylov methods, driven primarily by needs of the Los Alamos National Laboratory parallel unstructured-mesh casting simulation tool Telluride. Support for a number of sparse storage formats, methods, and preconditioners have been implemented, driven primarily by application needs. They describe the object-based Fortran 90 approach, which enhances maintainability, performance, and extensibility, the parallelization approach using a new portable gather/scatter library (PGSLib), current capabilities and future plans, and present preliminary performance results on a variety of platforms.
Warner, Seth
1990-01-01
Standard text provides an exceptionally comprehensive treatment of every aspect of modern algebra. Explores algebraic structures, rings and fields, vector spaces, polynomials, linear operators, much more. Over 1,300 exercises. 1965 edition.
Design and Implementation of Numerical Linear Algebra Algorithms on Fixed Point DSPs
Nikolić, Zoran; Nguyen, Ha Thai; Frantz, Gene
2007-12-01
Numerical linear algebra algorithms use the inherent elegance of matrix formulations and are usually implemented using C/C++ floating point representation. The system implementation is faced with practical constraints because these algorithms usually need to run in real time on fixed point digital signal processors (DSPs) to reduce total hardware costs. Converting the simulation model to fixed point arithmetic and then porting it to a target DSP device is a difficult and time-consuming process. In this paper, we analyze the conversion process. We transformed selected linear algebra algorithms from floating point to fixed point arithmetic, and compared real-time requirements and performance between the fixed point DSP and floating point DSP algorithm implementations. We also introduce an advanced code optimization and an implementation by DSP-specific, fixed point C code generation. By using the techniques described in the paper, speed can be increased by a factor of up to 10 compared to floating point emulation on fixed point hardware.
Design and Implementation of Numerical Linear Algebra Algorithms on Fixed Point DSPs
Directory of Open Access Journals (Sweden)
Gene Frantz
2007-01-01
Full Text Available Numerical linear algebra algorithms use the inherent elegance of matrix formulations and are usually implemented using C/C++ floating point representation. The system implementation is faced with practical constraints because these algorithms usually need to run in real time on fixed point digital signal processors (DSPs to reduce total hardware costs. Converting the simulation model to fixed point arithmetic and then porting it to a target DSP device is a difficult and time-consuming process. In this paper, we analyze the conversion process. We transformed selected linear algebra algorithms from floating point to fixed point arithmetic, and compared real-time requirements and performance between the fixed point DSP and floating point DSP algorithm implementations. We also introduce an advanced code optimization and an implementation by DSP-specific, fixed point C code generation. By using the techniques described in the paper, speed can be increased by a factor of up to 10 compared to floating point emulation on fixed point hardware.
Directory of Open Access Journals (Sweden)
Maria Joita
2007-12-01
Full Text Available In this paper we characterize the order relation on the set of all nondegenerate completely n-positive linear maps between C*-algebras in terms of a self-dual Hilbert module induced by each completely n-positive linear map.
Tissue characterization using electrical impedance spectroscopy data: a linear algebra approach.
Laufer, Shlomi; Solomon, Stephen B; Rubinsky, Boris
2012-06-01
In this study, we use a new linear algebra manipulation on electrical impedance spectroscopy measurements to provide real-time information regarding the nature of the tissue surrounding the needle in minimal invasive procedures. Using a Comsol Multiphysics three-dimensional model, a phantom based on ex vivo animal tissue and in vivo animal data, we demonstrate how tissue inhomogeneity can be characterized without any previous knowledge of the electrical properties of the different tissues, except that they should not be linearly dependent on a certain frequency range. This method may have applications in needle biopsies, radiation seeds, or minimally invasive surgery and can reduce the number of computer tomography or magnetic resonance imaging images. We conclude by demonstrating how this mathematical approach can be useful in other applications.
A linearly approximated iterative Gaussian decomposition method for waveform LiDAR processing
Mountrakis, Giorgos; Li, Yuguang
2017-07-01
Full-waveform LiDAR (FWL) decomposition results often act as the basis for key LiDAR-derived products, for example canopy height, biomass and carbon pool estimation, leaf area index calculation and under canopy detection. To date, the prevailing method for FWL product creation is the Gaussian Decomposition (GD) based on a non-linear Levenberg-Marquardt (LM) optimization for Gaussian node parameter estimation. GD follows a ;greedy; approach that may leave weak nodes undetected, merge multiple nodes into one or separate a noisy single node into multiple ones. In this manuscript, we propose an alternative decomposition method called Linearly Approximated Iterative Gaussian Decomposition (LAIGD method). The novelty of the LAIGD method is that it follows a multi-step ;slow-and-steady; iterative structure, where new Gaussian nodes are quickly discovered and adjusted using a linear fitting technique before they are forwarded for a non-linear optimization. Two experiments were conducted, one using real full-waveform data from NASA's land, vegetation, and ice sensor (LVIS) and another using synthetic data containing different number of nodes and degrees of overlap to assess performance in variable signal complexity. LVIS data revealed considerable improvements in RMSE (44.8% lower), RSE (56.3% lower) and rRMSE (74.3% lower) values compared to the benchmark GD method. These results were further confirmed with the synthetic data. Furthermore, the proposed multi-step method reduces execution times in half, an important consideration as there are plans for global coverage with the upcoming Global Ecosystem Dynamics Investigation LiDAR sensor on the International Space Station.
线性代数主线式教学探究%The Mainline Teaching Inquiry of Linear Algebra
Institute of Scientific and Technical Information of China (English)
马朝忠; 邓西云
2012-01-01
本文从大学生的认知特征和线性代数自身的特点出发，结合教学实践探讨了启发式教学，介绍了针对线性代数的主线式教学思路，同时论述了在线性代数教学过程中培养学生的科学计算能力和实际应用能力的重要性．%The article starts from the unh, ersity student's cognitive characristics and linear algebra own chaxacteristics, combined with the teaching practice of heuristic teaching, introduce the nudnline teaching ideas for linear algebra, and discusses the science students in linear algebra leaching process computing power and the practical application of the importance of the abiltiy.
线性代数在密码学中的应用%The Applications of Linear Algebra in Cryptology
Institute of Scientific and Technical Information of China (English)
李家; 李援南
2013-01-01
This paper introduces the applications of linear algebra in several aspects of cryptology , i.e. encrypt algorithm , error correcting code and spectral method etc .Based on those applications , we can see the importance of linear algebra in cryptology and interest students in learning linear algebra .%本文介绍了线性代数中线性表示、正交基、线性变换和矩阵等在密码学中加密算法、纠错码、频谱方法等方面的应用以及线性代数在密码学中的重要性。它以简洁的表达方式和方便的计算工具给出各学科中的分析结论。
Acceleration of multiple solution of a boundary value problem involving a linear algebraic system
Gazizov, Talgat R.; Kuksenko, Sergey P.; Surovtsev, Roman S.
2016-06-01
Multiple solution of a boundary value problem that involves a linear algebraic system is considered. New approach to acceleration of the solution is proposed. The approach uses the structure of the linear system matrix. Particularly, location of entries in the right columns and low rows of the matrix, which undergo variation due to the computing in the range of parameters, is used to apply block LU decomposition. Application of the approach is considered on the example of multiple computing of the capacitance matrix by method of moments used in numerical electromagnetics. Expressions for analytic estimation of the acceleration are presented. Results of the numerical experiments for solution of 100 linear systems with matrix orders of 1000, 2000, 3000 and different relations of variated and constant entries of the matrix show that block LU decomposition can be effective for multiple solution of linear systems. The speed up compared to pointwise LU factorization increases (up to 15) for larger number and order of considered systems with lower number of variated entries.
Dorier, Jean-Luc; Robert, Aline; Rogalski, Marc
2002-01-01
Frank D. Uhlig published in a recent issue of this journal (ESM 50.3) a very interesting article about the question of proof in linear algebra. We have been doing research in the field of mathematics education about the teaching of linear algebra since the 1980s. In this paper, we want to underline the common points in Uhlig’s approach and some of our work. We also want to bring a new light of some of his ideas and give perspective for a further didactical development of Uhlig’s first experim...
On New Teaching Methods of "Linear Algebra"%“线性代数”课程教学新法
Institute of Scientific and Technical Information of China (English)
张红梅
2015-01-01
本文主要针对高校线性代数课程的教学内容、教学方法等方面存在的问题进行分析，并提出将MATLAB数学软件引入到线性代数教学内容中以提高教学效率。%This paper mainly analyzed the problems existing in the teaching contents and teaching methods of advanced linear algebra, and proposed the introduction of the mathematical soft-ware MATAB into linear algebra teaching, in order to improve teaching efficiency.
Dorier, Jean-Luc; Robert, Aline; Rogalski, Marc
2002-01-01
Frank D. Uhlig published in a recent issue of this journal (ESM 50.3) a very interesting article about the question of proof in linear algebra. We have been doing research in the field of mathematics education about the teaching of linear algebra since the 1980s. In this paper, we want to underline the common points in Uhlig’s approach and some of our work. We also want to bring a new light of some of his ideas and give perspective for a further didactical development of Uhlig’s first experim...
Ertekin, E.; Solak, S.; Yazici, E.
2010-12-01
The aim of this study is to identify the effects of formalism in teaching on primary and secondary school mathematics teacher trainees' algebraic and geometric interpretations of the notions of linear dependency/independency. Quantitative research methods are drawn in order to determine differences in success levels between algebraic and geometric interpretations of the linear dependency/independency of vectors presented in two- and three-dimensional space. On the other hand, qualitative research methods were utilized in order to investigate thinking modes involved in the geometric interpretation of the same notion. The participants were a total of 144 teacher trainees registered at the Selçuk University, Education Faculty in 2007-2008 academic year. 33 participants were first year students at Secondary Mathematics Education Department, while 111 were second year students at Primary Mathematics Education Department. The results indicated that correlations between the formal definition of the notions of linear dependency/independency and the items of the test which required algebraic and geometric interpretation were both low. Yet, the correlation for the algebraic dimension of the test was higher than the geometric dimension. Likewise, algebraic mean success score was significantly higher than the geometric mean score.
A Linear Algebra Framework for Static High Performance Fortran Code Distribution
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Corinne Ancourt
1997-01-01
Full Text Available High Performance Fortran (HPF was developed to support data parallel programming for single-instruction multiple-data (SIMD and multiple-instruction multiple-data (MIMD machines with distributed memory. The programmer is provided a familiar uniform logical address space and specifies the data distribution by directives. The compiler then exploits these directives to allocate arrays in the local memories, to assign computations to elementary processors, and to migrate data between processors when required. We show here that linear algebra is a powerful framework to encode HPF directives and to synthesize distributed code with space-efficient array allocation, tight loop bounds, and vectorized communications for INDEPENDENT loops. The generated code includes traditional optimizations such as guard elimination, message vectorization and aggregation, and overlap analysis. The systematic use of an affine framework makes it possible to prove the compilation scheme correct.
FAST SOLUTION FOR LARGE SCALE LINEAR ALGEBRAIC EQUATIONS IN FINITE ELEMENT ANALYSIS
Institute of Scientific and Technical Information of China (English)
Qi Zhaohui; Liu Yuqi; Hu Ping
2001-01-01
The computational efficiency of numerical solution of linear algebraic equations in finite elements can be improved in tow wqys. One is to decrease the fill-in numbers, which are new non-ze-ro numbers in the matrix of global stiffness generated during the process of elimination.The other is to reduce the computational operation of multiplying a real number by zero.Based on the fact that the order of elimination can determine how many fill-in numbers should be generated, we present a new method for optimization of numbering nodes. This method is quite different from bandwidth optimization. Fill-in numbers can be decreased in a large scale by the use of this method. The bi-factorization method is adoted to avoid multiplying real numbers by zero.For large scale finite element analysis, the method presented in this paper is more efficient than the traditional LDLT method.
Ltaief, Hatem
2011-08-31
This paper presents the power profile of two high performance dense linear algebra libraries i.e., LAPACK and PLASMA. The former is based on block algorithms that use the fork-join paradigm to achieve parallel performance. The latter uses fine-grained task parallelism that recasts the computation to operate on submatrices called tiles. In this way tile algorithms are formed. We show results from the power profiling of the most common routines, which permits us to clearly identify the different phases of the computations. This allows us to isolate the bottlenecks in terms of energy efficiency. Our results show that PLASMA surpasses LAPACK not only in terms of performance but also in terms of energy efficiency. © 2011 Springer-Verlag.
Linear-algebraic bath transformation for simulating complex open quantum systems
Huh, Joonsuk; Fujita, Takatoshi; Yung, Man-Hong; Aspuru-Guzik, Alán
2014-01-01
In studying open quantum systems, the environment is often approximated as a collection of non-interacting harmonic oscillators, a configuration also known as the star-bath model. It is also well known that the star-bath can be transformed into a nearest-neighbor interacting chain of oscillators. The chain-bath model has been widely used in renormalization group approaches. The transformation can be obtained by recursion relations or orthogonal polynomials. Based on a simple linear algebraic approach, we propose a bath partition strategy to reduce the system-bath coupling strength. As a result, the non-interacting star-bath is transformed into a set of weakly-coupled multiple parallel chains. The transformed bath model allows complex problems to be practically implemented on quantum simulators, and it can also be employed in various numerical simulations of open quantum dynamics.
Non-Linear Integral Equations for complex Affine Toda associated to simply laced Lie algebras
Zinn-Justin, P
1998-01-01
A set of coupled non-linear integral equations is derived for a class of models connected with the quantum group $U_q(\\hat g)$ ($q=e^{i\\gamma}$ and $g$ simply laced Lie algebra), which are solvable using the Bethe Ansatz; these equations describe arbitrary excited states of a system with finite spatial length $L$. They generalize the Destri-De Vega equation for the Sine-Gordon/massive Thirring model to affine Toda field theory with imaginary coupling constant. As an application, the central charge and all the conformal weights of the UV conformal field theory are extracted in a straightforward manner. The quantum group truncation for rational values of $\\gamma/\\pi$ is discussed in detail; in the UV limit we recover through this procedure the RCFTs with extended $W(g)$ conformal symmetry.
A look at transport theory from the point of view of linear algebra
Energy Technology Data Exchange (ETDEWEB)
Faber, V.; Manteuffel, T.A.
1988-01-01
We show that the notion of ''preconditioning'' from linear algebra can provide a framework for the discussion of algorithms for the numerical solution of the transport equation. In this context we show that the conjugate gradient method yields substantial savings over the Neumann series solution of the standard integral formulation of the transport equation for optically thick regimes. Further, we show that the diffusion synthetic acceleration (DSA) algorithm is the Neumann series solution of the standard integral formulation of the transport equation preconditioning by the Green's function of a diffusion operator. The preconditioned conjugate gradient method, using DSA as a preconditioner, again yields substantial savings. 12 refs., 3 figs., 2 tabs.
A review of vector convergence acceleration methods, with applications to linear algebra problems
Brezinski, C.; Redivo-Zaglia, M.
In this article, in a few pages, we will try to give an idea of convergence acceleration methods and extrapolation procedures for vector sequences, and to present some applications to linear algebra problems and to the treatment of the Gibbs phenomenon for Fourier series in order to show their effectiveness. The interested reader is referred to the literature for more details. In the bibliography, due to space limitation, we will only give the more recent items, and, for older ones, we refer to Brezinski and Redivo-Zaglia, Extrapolation methods. (Extrapolation Methods. Theory and Practice, North-Holland, 1991). This book also contains, on a magnetic support, a library (in Fortran 77 language) for convergence acceleration algorithms and extrapolation methods.
Directory of Open Access Journals (Sweden)
Cheng Wang
2014-01-01
Full Text Available The identification of a class of linear-in-parameters multiple-input single-output systems is considered. By using the iterative search, a least-squares based iterative algorithm and a gradient based iterative algorithm are proposed. A nonlinear example is used to verify the effectiveness of the algorithms, and the simulation results show that the least-squares based iterative algorithm can produce more accurate parameter estimates than the gradient based iterative algorithm.
Li, Zhifu; Hu, Yueming; Li, Di
2016-08-01
For a class of linear discrete-time uncertain systems, a feedback feed-forward iterative learning control (ILC) scheme is proposed, which is comprised of an iterative learning controller and two current iteration feedback controllers. The iterative learning controller is used to improve the performance along the iteration direction and the feedback controllers are used to improve the performance along the time direction. First of all, the uncertain feedback feed-forward ILC system is presented by an uncertain two-dimensional Roesser model system. Then, two robust control schemes are proposed. One can ensure that the feedback feed-forward ILC system is bounded-input bounded-output stable along time direction, and the other can ensure that the feedback feed-forward ILC system is asymptotically stable along time direction. Both schemes can guarantee the system is robust monotonically convergent along the iteration direction. Third, the robust convergent sufficient conditions are given, which contains a linear matrix inequality (LMI). Moreover, the LMI can be used to determine the gain matrix of the feedback feed-forward iterative learning controller. Finally, the simulation results are presented to demonstrate the effectiveness of the proposed schemes.
Debin Fang; Qian Yu
2011-01-01
This paper proposes an improved predictor-corrector interior-point algorithm for the linear complementarity problem (LCP) based on the Mizuno-Todd-Ye algorithm. The modified corrector steps in our algorithm cannot only draw the iteration point back to a narrower neighborhood of the center path but also reduce the duality gap. It implies that the improved algorithm can converge faster than the MTY algorithm. The iteration complexity of the improved algorithm is proved to obtain √ ( ) whi...
A high-accuracy optical linear algebra processor for finite element applications
Casasent, D.; Taylor, B. K.
1984-01-01
Optical linear processors are computationally efficient computers for solving matrix-matrix and matrix-vector oriented problems. Optical system errors limit their dynamic range to 30-40 dB, which limits their accuray to 9-12 bits. Large problems, such as the finite element problem in structural mechanics (with tens or hundreds of thousands of variables) which can exploit the speed of optical processors, require the 32 bit accuracy obtainable from digital machines. To obtain this required 32 bit accuracy with an optical processor, the data can be digitally encoded, thereby reducing the dynamic range requirements of the optical system (i.e., decreasing the effect of optical errors on the data) while providing increased accuracy. This report describes a new digitally encoded optical linear algebra processor architecture for solving finite element and banded matrix-vector problems. A linear static plate bending case study is described which quantities the processor requirements. Multiplication by digital convolution is explained, and the digitally encoded optical processor architecture is advanced.
Agoras, M.; Ponte Castañeda, P.
2013-03-01
Analytical estimates are obtained for the effective constitutive response of porous viscoplastic materials consisting of aligned ellipsoidal voids that are distributed randomly with "ellipsoidal" symmetry in the matrix material. These estimates are obtained by means of a novel iterative homogenization strategy recently proposed by Ponte Castañeda (2012), and can be shown to be bounds for certain classes of multi-scale microstructures. By design, the resulting constitutive model agrees exactly with the earlier "variational linear comparison" model at the first iteration (N=1), and provides estimates that get progressively more accurate as the number of iterations increases (N→∞), especially for high-triaxiality loading conditions, and low porosity and strain-rate sensitivity. However, in practice, a small number of iterations (N≈10) is sufficient to get very accurate results. It is important to emphasize that, unlike other models that have been proposed in the literature, the new model requires no fitting parameters, solely depending on the properties of the matrix phase and microstructural information, such as the porosity, the average void shape and orientation, as well as the generally different shape and orientation of their distribution. Results are given for the yield and gauge surfaces of ideally plastic and power-law viscoplastic porous materials for the special cases of aligned spheroidal and ellipsoidal voids, and the results are compared with available numerical results and with the results of other models. Compared to available numerical results, the new estimates are found to be quite accurate, while they also provide more flexibility than competing models in terms of the characterization of the microstructure. In particular, it was found that the effect of different shapes for the average pore shape and distribution on the yield surfaces of the porous materials can be significant at high triaxialities, even for very small porosities. In addition
A non-linear iterative method for multi-layer DOT sub-surface imaging system.
Hou, Hsiang-Wen; Wu, Shih-Yang; Sun, Hao-Jan; Fang, Wai-Chi
2014-01-01
Diffuse Optical Tomography (DOT) has become an emerging non-invasive technology, and has been widely used in clinical diagnosis. Functional near-infrared (FNIR) is one of the important applications of DOT. However, FNIR is used to reconstruct two-dimensional (2D) images for the sake of good spatial and temporal resolution. In this paper we propose a multiple-input and multiple-output (MIMO) based data extraction algorithm method in order to increase the spatial and temporal resolution. The non-linear iterative method is used to reconstruct better resolution images layer by layer. In terms of theory, the simulation results and original images are nearly identical. The proposed reconstruction method performs good spatial resolution, and has a depth resolutions capacity of three layers.
Performance evaluation of simple linear iterative clustering algorithm on medical image processing.
Cong, Jinyu; Wei, Benzheng; Yin, Yilong; Xi, Xiaoming; Zheng, Yuanjie
2014-01-01
Simple Linear Iterative Clustering (SLIC) algorithm is increasingly applied to different kinds of image processing because of its excellent perceptually meaningful characteristics. In order to better meet the needs of medical image processing and provide technical reference for SLIC on the application of medical image segmentation, two indicators of boundary accuracy and superpixel uniformity are introduced with other indicators to systematically analyze the performance of SLIC algorithm, compared with Normalized cuts and Turbopixels algorithm. The extensive experimental results show that SLIC is faster and less sensitive to the image type and the setting superpixel number than other similar algorithms such as Turbopixels and Normalized cuts algorithms. And it also has a great benefit to the boundary recall, the robustness of fuzzy boundary, the setting superpixel size and the segmentation performance on medical image segmentation.
Iterative solution of general sparse linear systems on clusters of workstations
Energy Technology Data Exchange (ETDEWEB)
Lo, Gen-Ching; Saad, Y. [Univ. of Minnesota, Minneapolis, MN (United States)
1996-12-31
Solving sparse irregularly structured linear systems on parallel platforms poses several challenges. First, sparsity makes it difficult to exploit data locality, whether in a distributed or shared memory environment. A second, perhaps more serious challenge, is to find efficient ways to precondition the system. Preconditioning techniques which have a large degree of parallelism, such as multicolor SSOR, often have a slower rate of convergence than their sequential counterparts. Finally, a number of other computational kernels such as inner products could ruin any gains gained from parallel speed-ups, and this is especially true on workstation clusters where start-up times may be high. In this paper we discuss these issues and report on our experience with PSPARSLIB, an on-going project for building a library of parallel iterative sparse matrix solvers.
Review of Singular Cooling Inlet and Linear Pressure Drop for ITER Coils Cable in Conduit Conductor
Nicollet, S.; Bessette, D.; Cloez, H.; Decool, P.; Lacroix, B.; Lebailly, C. A.; Serries, J. P.
2006-04-01
New tests and measurements performed (Othello Facility, EFDA Task) on TF mock up cooling inlet and different central spirals (characteristics: hydraulic outer diameter and perforation ratio) are presented, as well as the new model of singular and linear friction factor. The ITER Coils CICC hydraulic length pressure drop is determined in operating conditions (m=8 g/s, P=0.6 MPa and T=5 K): the important result is an increase in linear pressure drop for the TF (290 Pa/m) and CS (430 Pa/m), in comparison with prototype model coils TFMC (100 Pa/m) and CSMC (180 Pa/m). The main reason is the reduction of the central spiral diameter and associated increase of friction factor and bundle to total mass flow ratio α (from 1/3 up to 2/3 typically). The ratio of singular cooling inlet to CICC linear pressure drop is estimated: TF mock up ratio (3 m) is lower than previous CS mock up tested (12 m), due to design changes. The cryogenic power necessary to compensate the CICC pressure drop is calculated for the 4 primary loop circuits: typically 2.3 kW at 5 K for TF winding system represents 40% of the whole average TF winding magnet heat loads during operation.
An iterative statistical tolerance analysis procedure to deal with linearized behavior models
Institute of Scientific and Technical Information of China (English)
Antoine DUMAS; Jean-Yves DANTAN; Nicolas GAYTON; Thomas BLES; Robin LOEBL
2015-01-01
Tolerance analysis consists of analyzing the impact of variations on the mechanism behavior due to the manufacturing process. The goal is to predict its quality level at the design stage. The technique involves computing probabilities of failure of the mechanism in a mass production process. The various analysis methods have to consider the component’s variations as random variables and the worst configuration of gaps for over-constrained systems. This consideration varies in function by the type of mechanism behavior and is realized by an optimization scheme combined with a Monte Carlo simulation. To simplify the optimization step, it is necessary to linearize the mechanism behavior into several parts. This study aims at analyzing the impact of the linearization strategy on the probability of failure estimation; a highly over-constrained mechanism with two pins and five cotters is used as an illustration for this study. The purpose is to strike a balance among model error caused by the linearization, computing time, and result accuracy. In addition, an iterative procedure is proposed for the assembly requirement to provide accurate results without using the entire Monte Carlo simulation.
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.; Hewitt, T.
1985-08-01
This note describes some experiments on simple, dense linear algebra algorithms. These experiments show that the CRAY X-MP is capable of small-grain multitasking arising from standard implementations of LU and Cholesky decomposition. The implementation described here provides the ''fastest'' execution rate for LU decomposition, 718 MFLOPS for a matrix of order 1000.
Some Problems in Linear Algebra Course%线性代数教学中的几个问题
Institute of Scientific and Technical Information of China (English)
陈维新
2011-01-01
从线性代数教学实践中,选取3个学生提出的问题,进行分析,予以解答.%We discuss three typical problems raised by students in the linear algebra course and provide detail solutions to them.