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Computer Algorithms for Solving Linear Algebraic Equations

Computer Algorithms for Solving Linear Algebraic Equations
Author: Emilio Spedicato
Publisher: Springer Science & Business Media
Total Pages: 361
Release: 2012-12-06
Genre: Computers
ISBN: 3642767176
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The NATO Advanced Study Institute on "Computer algorithms for solving linear algebraic equations: the state of the art" was held September 9-21, 1990, at II Ciocco, Barga, Italy. It was attended by 68 students (among them many well known specialists in related fields!) from the following countries: Belgium, Brazil, Canada, Czechoslovakia, Denmark, France, Germany, Greece, Holland, Hungary, Italy, Portugal, Spain, Turkey, UK, USA, USSR, Yugoslavia. Solving linear equations is a fundamental task in most of computational mathematics. Linear systems which are now encountered in practice may be of very large dimension and their solution can still be a challenge in terms of the requirements of accuracy or reasonable computational time. With the advent of supercomputers with vector and parallel features, algorithms which were previously formulated in a framework of sequential operations often need a completely new formulation, and algorithms that were not recommended in a sequential framework may become the best choice. The aim of the ASI was to present the state of the art in this field. While not all important aspects could be covered (for instance there is no presentation of methods using interval arithmetic or symbolic computation), we believe that most important topics were considered, many of them by leading specialists who have contributed substantially to the developments in these fields.

Computer Algorithms for Solving Linear Algebraic Equations

Computer Algorithms for Solving Linear Algebraic Equations
Author: Emilio Spedicato
Publisher:
Total Pages: 368
Release: 1991-08-26
Genre:
ISBN: 9783642767180
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This volume presents the lectures given by fourteen specialists in algorithms for linear algebraic systems during a NATO Advanced Study Institute held at Il Ciocco, Barga, Italy, September 1990. The lectures give an up-to-date and fairly complete coverage of this fundamental field in numerical mathematics. Topics related to sequential formulation include a review of classical methods with some new proofs, and extensive presentations of complexity results, of algorithms for linear least squares, of the recently developed ABS methods, of multigrid methods, of preconditioned conjugate gradient methods for H-matrices, of domain decomposition methods, of hierarchical basis methods, and of splitting type methods. With reference to implementations on multiprocessors, topics include algorithms for general sparse systems, factorization methods for dense matrices, Gaussian elimination on systolic arrays, and methods for linear systems arising in optimization problems. The book will be useful as an introduction to a field still in rapid growth and as a reference to the most recent results in the field.

Computer Algorithms for Solving Linear Algebraic Equations

Computer Algorithms for Solving Linear Algebraic Equations
Author: Emilio Spedicato
Publisher: Springer
Total Pages: 352
Release: 1991-08-26
Genre: Computers
ISBN: 9783540541875
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The NATO Advanced Study Institute on "Computer algorithms for solving linear algebraic equations: the state of the art" was held September 9-21, 1990, at II Ciocco, Barga, Italy. It was attended by 68 students (among them many well known specialists in related fields!) from the following countries: Belgium, Brazil, Canada, Czechoslovakia, Denmark, France, Germany, Greece, Holland, Hungary, Italy, Portugal, Spain, Turkey, UK, USA, USSR, Yugoslavia. Solving linear equations is a fundamental task in most of computational mathematics. Linear systems which are now encountered in practice may be of very large dimension and their solution can still be a challenge in terms of the requirements of accuracy or reasonable computational time. With the advent of supercomputers with vector and parallel features, algorithms which were previously formulated in a framework of sequential operations often need a completely new formulation, and algorithms that were not recommended in a sequential framework may become the best choice. The aim of the ASI was to present the state of the art in this field. While not all important aspects could be covered (for instance there is no presentation of methods using interval arithmetic or symbolic computation), we believe that most important topics were considered, many of them by leading specialists who have contributed substantially to the developments in these fields.

Computer Solution of Large Linear Systems

Computer Solution of Large Linear Systems
Author: Gerard Meurant
Publisher: Elsevier
Total Pages: 777
Release: 1999-06-16
Genre: Mathematics
ISBN: 0080529518
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This book deals with numerical methods for solving large sparse linear systems of equations, particularly those arising from the discretization of partial differential equations. It covers both direct and iterative methods. Direct methods which are considered are variants of Gaussian elimination and fast solvers for separable partial differential equations in rectangular domains. The book reviews the classical iterative methods like Jacobi, Gauss-Seidel and alternating directions algorithms. A particular emphasis is put on the conjugate gradient as well as conjugate gradient -like methods for non symmetric problems. Most efficient preconditioners used to speed up convergence are studied. A chapter is devoted to the multigrid method and the book ends with domain decomposition algorithms that are well suited for solving linear systems on parallel computers.

Algorithms for Computer Algebra

Algorithms for Computer Algebra
Author: Keith O. Geddes
Publisher: Springer Science & Business Media
Total Pages: 594
Release: 2007-06-30
Genre: Computers
ISBN: 0585332479
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Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.

Compact Numerical Methods for Computers

Compact Numerical Methods for Computers
Author: John C. Nash
Publisher: CRC Press
Total Pages: 298
Release: 1990-01-01
Genre: Mathematics
ISBN: 9780852743195
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This second edition of Compact Numerical Methods for Computers presents reliable yet compact algorithms for computational problems. As in the previous edition, the author considers specific mathematical problems of wide applicability, develops approaches to a solution and the consequent algorithm, and provides the program steps. He emphasizes useful applicable methods from various scientific research fields, ranging from mathematical physics to commodity production modeling. While the ubiquitous personal computer is the particular focus, the methods have been implemented on computers as small as a programmable pocket calculator and as large as a highly parallel supercomputer. New to the Second Edition Presents program steps as Turbo Pascal code Includes more algorithmic examples Contains an extended bibliography The accompanying software (available by coupon at no charge) includes not only the algorithm source codes, but also driver programs, example data, and several utility codes to help in the software engineering of end-user programs. The codes are designed for rapid implementation and reliable use in a wide variety of computing environments. Scientists, statisticians, engineers, and economists who prepare/modify programs for use in their work will find this resource invaluable. Moreover, since little previous training in numerical analysis is required, the book can also be used as a supplementary text for courses on numerical methods and mathematical software.

Numerical Linear Algebra on High-Performance Computers

Numerical Linear Algebra on High-Performance Computers
Author: Jack J. Dongarra
Publisher: SIAM
Total Pages: 353
Release: 1998-01-01
Genre: Computers
ISBN: 0898714281
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Provides a rapid introduction to the world of vector and parallel processing for these linear algebra applications.

Numerical Linear Algebra for Applications in Statistics

Numerical Linear Algebra for Applications in Statistics
Author: James E. Gentle
Publisher: Springer Science & Business Media
Total Pages: 229
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461206235
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Accurate and efficient computer algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Regardless of the software system used, the book describes and gives examples of the use of modern computer software for numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes the relevant properties of matrix inverses, factorisations, matrix and vector norms, and other topics in linear algebra. The book is essentially self- contained, with the topics addressed constituting the essential material for an introductory course in statistical computing. Numerous exercises allow the text to be used for a first course in statistical computing or as supplementary text for various courses that emphasise computations.

Algorithms for Sparse Linear Systems

Algorithms for Sparse Linear Systems
Author: Jennifer Scott
Publisher: Springer Nature
Total Pages: 254
Release: 2023-04-29
Genre: Mathematics
ISBN: 3031258207
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Large sparse linear systems of equations are ubiquitous in science, engineering and beyond. This open access monograph focuses on factorization algorithms for solving such systems. It presents classical techniques for complete factorizations that are used in sparse direct methods and discusses the computation of approximate direct and inverse factorizations that are key to constructing general-purpose algebraic preconditioners for iterative solvers. A unified framework is used that emphasizes the underlying sparsity structures and highlights the importance of understanding sparse direct methods when developing algebraic preconditioners. Theoretical results are complemented by sparse matrix algorithm outlines. This monograph is aimed at students of applied mathematics and scientific computing, as well as computational scientists and software developers who are interested in understanding the theory and algorithms needed to tackle sparse systems. It is assumed that the reader has completed a basic course in linear algebra and numerical mathematics.