Accuracy and Stability of Numerical Algorithms

by Nicholas J. Higham

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.

Genres: Computer Science, Mathematics

710 pages, Hardcover

First published January 1, 1996

About the author

Librarian note: There are other authors with the same name.

Nicholas John Higham FRS is a numerical analyst and Richardson Professor of Applied Mathematics at the School of Mathematics, University of Manchester.

He is a graduate of the Victoria University of Manchester gaining his BA in 1982, MSc in and 1983 and PhD 1985. His PhD thesis was entitled Nearness Problems in Numerical Linear Algebra and his supervisor was George Hall. Higham is Director of Research within the School of Mathematics, Director of the Manchester Institute for Mathematical Sciences (MIMS), and Head of the Numerical Analysis Group. He held a prestigious Royal Society Wolfson Research Merit Award (2003–2008) and as of 2006[update] is on the Institute for Scientific Information Highly Cited Researcher list.

Reviews

[Paul Floyd · Rating 5 out of 5]

Faultless. I need to read this book several times, and to work through the exercises.

You know that you can't go far wrong with a book in this category when the bibliography is almost 70 pages long.

[Leif Lui · Rating 5 out of 5]

This book discussed the accuracy and stability of various numerical schemes in great depth. Not only is this book extremely detailed, but it also covers many topics from the basics of IEEE arithmetic, error propagation, numerical differentiation and integration, matrix operations, eigenvalue problems, etc.

What I like about this book is that it provides many examples of where certain numerical algorithms go wrong and what alternatives you should turn to for these specific cases. Additionally, this book has diagrams/figures to illustrate difficult concepts. One diagram I particularly like in this book is using ladders/arrows to show how errors propagate.

I found this book particularly useful because when reading about eigenvalue algorithms in Trefethen and Bau's 'Numerical Linear Algebra', I had some trouble understanding the stability properties of QR factorization with and without shifts, and this book was a really good supplement to help aid the understanding of why householder methods and Givens rotations make the QR factorization algorithm more numerically stable.

To summarize, I would highly recommend this book to anyone working in physics, applied mathematics, or any science that involves writing numerical scheme to solve complex problems.