Overview

Numerical precision is the very soul of science.

-SIR D'ARCY WENTWORTH THOMPSON, On Growth and Form (1942)

There will always be a small but steady demand for error-analysts to ... expose bad algorithms' big errors and, more important, supplant bad algorithms with provably good ones.

- WILLIAM M. KAHAN, Interval Arithmetic Options in the Proposed IEEE Floating Point Arithmetic Standard (1980)

Since none of the numbers which we take out from logarithmic and trigonometric tables admit of absolute precision, but are all to a certain extent approximate only, the results of all calculations performed by the aid of these numbers can only be approximately true ... It may happen, that in special cases the effect of the errors of the tables is so augmented that we may be obliged to reject a method, otherwise the best, and substitute another in its place.

- CARL FRIEDRICH GAUSS ^[1], Theoria Motus (1809)

Backward error analysis is no panacea; it may explain errors but not excuse them.

- HEWLETT-PACKARD, HP-15 C Advanced Functions Handbook (1982)

Overview

This book is concerned with the effects of finite precision arithmetic on numerical algorithms ^[2], particularly those in numerical linear algebra. Central to any understanding of high-level algorithms is an appreciation of the basic concepts of finite precision arithmetic. This opening chapter briskly imparts the necessary background material. Various examples are used for illustration, some of them familiar (such as the...