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Guaranteed Accuracy in Numerical Linear Algebra : Mathematics and Its Applications - S. K. Godunov

Guaranteed Accuracy in Numerical Linear Algebra

Mathematics and Its Applications

Hardcover Published: 30th June 1993
ISBN: 9780792323525
Number Of Pages: 537

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There exists a vast literature on numerical methods of linear algebra. In our bibliography list, which is by far not complete, we included some monographs on the subject [46], [15], [32], [39], [11], [21]. The present book is devoted to the theory of algorithms for a single problem of linear algebra, namely, for the problem of solving systems of linear equations with non-full-rank matrix of coefficients. The solution of this problem splits into many steps, the detailed discussion of which are interest­ ing problems on their own (bidiagonalization of matrices, computation of singular values and eigenvalues, procedures of deflation of singular values, etc. ). Moreover, the theory of algorithms for solutions of the symmetric eigenvalues problem is closely related to the theory of solv­ ing linear systems (Householder's algorithms of bidiagonalization and tridiagonalization, eigenvalues and singular values, etc. ). It should be stressed that in this book we discuss algorithms which to computer programs having the virtue that the accuracy of com­ lead putations is guaranteed. As far as the final program product is con­ cerned, this means that the user always finds an unambiguous solution of his problem. This solution might be of two kinds: 1. Solution of the problem with an estimate of errors, where abso­ lutely all errors of input data and machine round-offs are taken into account. 2.

Introduction
Singular Value Decompositionp. 1
Singular Value Decomposition and Singular Values of Square Matrixp. 3
Elementary Orthogonal Transformationsp. 9
Singular Value Decomposition of Rectangular Matricesp. 24
Norm of Matrix. Singular Values and Singular Vectorsp. 34
Some Numerical Characteristics of Matricesp. 49
Some Properties of Bidiagonal Square Matrices. Singular Values and Singular Vectorsp. 58
Simplification of Matrix Form by Usage of Orthogonal Transformationsp. 71
Simplification of Matrix Form by Deflationp. 86
Extension of Results for Complex Matricesp. 98
Systems of Linear Equationsp. 109
Condition Number for Square Matrixp. 111
Systems of Linear Equations with Simplest Band Matrices of Coefficientsp. 121
Generalized Normal Solutions of Systems with Arbitrary Matrices of Coefficientsp. 136
Conditioning of Generalized Normal Solutions of Systems of Full Rankp. 153
Angles between Spaces and Their Conditioningp. 162
Conditioning of the Generalized Normal Solutions in Case of Not Full Rankp. 181
Generalized Normal r-solution of the Systems of Linear Equationsp. 193
General Scheme of Finding of r-solution of Linear Systemp. 207
Deflation Algorithms for Band Matricesp. 215
Transformations of Hessenberg Matrices by Chains of Rotationsp. 217
Deflation of Degenerate Bidiagonal Matricesp. 235
Singular Deflation of Non-Degenerate Bidiagonal Matricesp. 247
Spectral Deflation of Hessenbergian and Symmetric Tridiagonal Matricesp. 259
Theory of Perturbations of Singular Deflation of Non-Degenerate Bidiagonal Matricesp. 267
Theory of Perturbations of Singular Deflation of Degenerate Bidiagonal Matricesp. 290
Theory of Perturbations of Singular Deflation of Symmetric Tridiagonal Matricesp. 296
Sturm Sequences of Tridiagonal Matricesp. 313
Elementary Proof of Sturm Theoremp. 315
Algorithm of Computation of Eigenvalues of Symmetric Tridiagonal Matrixp. 323
Trigonometric Parametrization of Rational Relationsp. 336
Sturm Sequences of Second Kindp. 349
One-Side Sturm Sequences for Tridiagonal Matricesp. 355
Two-Side Sturm Sequences for Tridiagonal Symmetric Matricesp. 368
Examples of Calculations in Problems of Finding Eigenvalues and Sturm Sequencesp. 391
Two-Side Sturm Sequences for Bidiagonal Matricesp. 404
Examples of Computations of Singular Values and Two-Side Sturm Sequences of Bidiagonal Matricesp. 417
Peculiarities of Computer Computationsp. 425
Modelling of Errors in Computer Arithmetic Operationsp. 428
Machine Operations on Vectors and Matricesp. 443
Machine Realization of Reflectionsp. 452
Analysis of Errors in Reduction of Matrices into Bi- and Tridiagonal Formp. 462
Machine Solution of Systems of Equations with Bidiagonal Coefficient Matricesp. 471
Numerical Examplesp. 478
Machine Realization of Sturm Algorithm. Estimates of Errors in Computation of Eigenvalues and Singular Valuesp. 484
Computation of Two-Side Sturm Sequence and Components of Eigenvector of Tridiagonal Symmetric Matrixp. 493
Machine Realization of Computations of Two-Side Sturm Sequences for Bidiagonal Matricesp. 502
Machine Realization of Deflation Algorithm for Bidiagonal Matricesp. 508
Bibliographyp. 523
Indexp. 529
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792323525
ISBN-10: 0792323521
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 537
Published: 30th June 1993
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 2.1