Accuracy and Stability of Numerical Algorithms

Overview

What is the most accurate way to sum floating point numbers? What are the advantages of IEEE arithmetic? How accurate is Gaussian elimination and what were the key breakthroughs in the development of error analysis for the method? The answers to these and many related questions are included here.

This book gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and ...

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Overview

What is the most accurate way to sum floating point numbers? What are the advantages of IEEE arithmetic? How accurate is Gaussian elimination and what were the key breakthroughs in the development of error analysis for the method? The answers to these and many related questions are included here.

This book 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. Software practicalities are emphasized throughout, with particular reference to LAPACK and MATLAB. The best available error bounds, some of them new, are presented in a unified format with a minimum of jargon. Because of its central role in revealing problem sensitivity and providing error bounds, perturbation theory is treated in detail.

Historical perspective and insight are given, with particular reference to the fundamental work of Wilkinson and Turing, and the many quotations provide further information in an accessible format.

The book is unique in that algorithmic developments and motivations are given succinctly and implementation details minimized, so that attention can be concentrated on accuracy and stability results. Here, in one place and in a unified notation, is error analysis for most of the standard algorithms in matrix computations. Not since Wilkinson's Rounding Errors in Algebraic Processes (1963) and The Algebraic Eigenvalue Problem (1965) has any volume treated this subject in such depth. A number of topics are treated that are not usually covered in numerical analysis textbooks, including floating point summation, block LU factorization, condition number estimation, the Sylvester equation, powers of matrices, finite precision behavior of stationary iterative methods, Vandermonde systems, and fast matrix multiplication.

Although not designed specifically as a textbook, this volume is a suitable reference for an advanced course, and could be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises (many of which have never before appeared in textbooks). The book is designed to be a comprehensive reference and its bibliography contains more than 1100 references from the research literature.

Audience

Specialists in numerical analysis as well as computational scientists and engineers concerned about the accuracy of their results will benefit from this book. Much of the book can be understood with only a basic grounding in numerical analysis and linear algebra.

About the Author

Nicholas J. Higham is a Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 40 publications and is a member of the editorial boards of the SIAM Journal on Matrix Analysis and Applications and the IMA Journal of Numerical Analysis. His book Handbook of Writing for the Mathematical Sciences was published by SIAM in 1993.

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Editorial Reviews

Booknews
Treats the behavior of numerical algorithms in finite precision arithmetic, combining algorithmic derivations, perturbation theory, and rounding error analysis and emphasizing software practicalities, with particular reference to LAPACK and MATLAB. Includes historical perspectives, especially on the work of Wilkinson and Turing, with quotations introducing chapters on subjects such as floating point summation, condition number estimation, and the Sylvester equation. Although designed as a reference rather than a text, it includes problems and solutions. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From The Critics
This graduate textbook deals with the effects of finite precision arithmetic on numerical algorithms, particularly the rounding errors that arise in numerical linear algebra. Higham (applied mathematics, University of Manchester) examines Gaussian elimination, LU and QR factorization, and the least squares problem, among others. The second edition adds chapters on symmetric indefinite and skew-symmetric systems, and nonlinear systems and Newton's method. Annotation c. Book News, Inc., Portland, OR
Jaroslav Stark
An attempt successful in my opinion to produce a successor to Wilkinson's text and give both a modern treatment of the material presented there, and to give a comprehensive account of the many developments in the subject since Wilkinson's time. .I thoroughly recommend the volume to anyone who uses computers in their work.
Mathematics Today
S. Siltanen
A comprehensive book concerning linear algebraic calculations with floating-point arithmetic. In its successful effort to give deep understanding of finite precision computations spiced with historical aspects, the book honors its classical predecessors, namely Wilkinson's books. .The nearly 700 pages.are written in a lively and down-to-earth manner, keeping several aspects in mind all the time: algorithmic derivations, perturbation theory, and rounding error analysis. Well organized material combined with up-to-date examples, a bibliography with more than 1000 entries, and a collection of good exercises constitutes a convincing piece of scientific literature.
Inverse Problems
S. Hitotumatu
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.
Mathematical Reviews
From the Publisher
'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

'…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

'…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

'… 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 page sequel. But if not, he has more than earned his respite - and our gratitude.' G. W. Stewart, SIAM Review

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Product Details

  • ISBN-13: 9780898713558
  • Publisher: SIAM
  • Publication date: 1/28/1996
  • Series: Miscellaneous Bks.
  • Edition description: Older Edition
  • Pages: 688

Table of Contents

Preface
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: Iterative Refinement
Chapter 12: Block LU Factorization
Chapter 13: Matrix Inversion
Chapter 14: Condition Number Estimation
Chapter 15: The Sylvester Equation
Chapter 16: Stationary Iterative Methods
Chapter 17: Matrix Powers
Chapter 18: QR Factorization
Chapter 19: The Least Squares Problem
Chapter 20: Underdetermined Systems
Chapter 21: Vandermonde Systems
Chapter 22: Fast Matrix Multiplication
Chapter 23: The Fast Fourier Transform and Applications
Chapter 24: Automatic Error Analysis
Chapter 25: Software Issues in Floating Point Arithmetic
Chapter 26: A Gallery of Test Matrices

Appendix A: Solutions to Problems
Appendix B: Singular Value Decomposition, M-Matrices
Appendix C: Acquiring Software Appendix D: Program Libraries
Appendix E: The Test Matrix Toolbox
Bibliography
Name Index
Subject Index


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