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Accuracy and Stability of Numerical Algorithms, Second EditionAccuracy and Stability of Numerical Algorithms, Second Edition

Nicholas J. Higham



The ASNA Web Site

Accuracy and Stability of Numerical Algorithms gives a thorough, up-to-date treatment of the behavior 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.

This second edition expands and updates the coverage of the first edition (1996) and includes numerous improvements to the original material. 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.

An expanded treatment of Gaussian elimination incorporates rook pivoting, along with a thorough discussion of the choice of pivoting strategy and the effects of scaling. The book's detailed descriptions of floating point arithmetic and of software issues reflect the fact that IEEE arithmetic is now ubiquitous.

Although not designed specifically as a textbook, this new edition is a suitable reference for an advanced course. It can also be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises. With its thorough indexes and extensive, up-to-date bibliography, the book provides a mine of information in a readily accessible form.

From reviews of the first edition:

This book is a monumental work on the analysis of rounding error and will serve as a new standard textbook on this subject, especially for linear computation.
S. Hitotumatu, Mathematical Reviews, Issue 97a.


…This definitive source on the accuracy and stability of numerical algorithms is quite a bargain and a worthwhile addition to the library of any statistician heavily involved in computing.
Robert L. Strawderman, Journal of the American Statistical Association, March 1999.


…A monumental book that should be on the bookshelf of anyone engaged in numerics, be it as a specialist or as a user.
A. van der Sluis, ITW Nieuws.

This text may become the new 'Bible' about accuracy and stability for the solution of systems of linear equations. It covers 688 pages carefully collected, investigated, and written ...One will find that this book is a very suitable and comprehensive reference for research in numerical linear algebra, software usage and development, and for numerical linear algebra courses.
N. Köckler, Zentrallblatt für Mathematik, Band 847/96.


…Nick Higham has assembled an enormous amount of important and useful material in a coherent, readable form. His book belongs on the shelf of anyone who has more than a casual interest in rounding error and matrix computations. I hope the author will give us the 600-odd hundred page sequel. But if not, he has more than earned his respite--and our gratitude.
G. W. Stewart, SIAM Review, March 1997.


Contents
List of Figures; List of Tables; Preface to Second Edition; Preface to First Edition; About the Dedication; Chapter 1: Principles of Finite Precision Computation; Chapter 2: Floating Point Arithmetic; Chapter 3: Basics; Chapter 4: Summation; Chapter 5: Polynomials; Chapter 6: Norms; Chapter 7: Perturbation Theory for Linear Systems; Chapter 8: Triangular Systems; Chapter 9: LU Factorization and Linear Equations; Chapter 10: Cholesky Factorization; Chapter 11: Symmetric Indefinite and Skew-Symmetric Systems; Chapter 12: Iterative Refinement; Chapter 13: Block LU Factorization; Chapter 14: Matrix Inversion; Chapter 15: Condition Number Estimation; Chapter 16: The Sylvester Equation; Chapter 17: Stationary Iterative Methods; Chapter 18: Matrix Powers; Chapter 19: QR Factorization; Chapter 20: The Least Squares Problem; Chapter 21: Underdetermined Systems; Chapter 22: Vandermonde Systems; Chapter 23: Fast Matrix Multiplication; Chapter 24: The Fast Fourier Transform and Applications; Chapter 25: Nonlinear Systems and Newton’s Method; Chapter 26: Automatic Error Analysis; Chapter 27: Software Issues in Floating Point Arithmetic; Chapter 28: A Gallery of Test Matrices; Appendix A: Solutions to Problems; Appendix B: Acquiring Software; Appendix C: Program Libraries; Appendix D: The Matrix Computation Toolbox; Bibliography; Name Index; Subject Index.


2002 / xxx + 680 pages / Hardcover / ISBN-13: 978-0-898715-21-7 / ISBN-10: 0-89871-521-0 /
List Price $82.50 / SIAM Member Price $57.75 / Order Code OT80

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