Accuracy and Stability of Numerical Algorithms: Second Edition

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SIAM, Aug 1, 2002 - Mathematics - 680 pages
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This book gives a thorough, up-to-date treatment of the behaviour of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis, all enlivened by historical perspective and informative quotations. The coverage of the first edition has been expanded and updated, involving numerous improvements. Two new chapters treat symmetric indefinite systems and skew-symmetric systems, and nonlinear systems and Newton's method. Twelve new sections include coverage of additional error bounds for Gaussian elimination, rank revealing LU factorizations, weighted and constrained least squares problems, and the fused multiply-add operation found on some modern computer architectures. This new edition is a suitable reference for an advanced course and can also be used at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises. In addition the thorough indexes and extensive, up-to-date bibliography are in a readily accessible form.
  

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Contents

Principles of Finite Precision Computation
1
Floating Point Arithmetic
35
Basics
61
Summation
79
Polynomials
93
Norms
105
Perturbation Theory for Linear Systems
119
Triangular Systems
139
Matrix Powers
339
QR Factorization
353
The Least Squares Problem
381
Underdetermined Systems
407
Vandermonde Systems
415
Fast Matrix Multiplication
433
The Fast Fourier Transform and Applications
451
Nonlinear Systems and Newtons Method
459

LU Factorization and Linear Equations
157
Cholesky Factorization
195
Symmetric Indefinite and SkewSymmetric Systems
213
Iterative Refinement
231
Block LU Factorization
245
Matrix Inversion
259
Condition Number Estimation
287
The Sylvester Equation
305
Stationary Iterative Methods
321
Automatic Error Analysis
471
Software Issues in Floating Point Arithmetic
489
A Gallery of Test Matrices
511
A Solutions to Problems
527
B Acquiring Software
573
The Matrix Computation Toolbox
583
Name Index
657
Copyright

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About the author (2002)

Nicholas J. Higham, FRS, is Richardson Professor of Applied Mathematics at the University of Manchester, UK.

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