Overview

The matrix of that equation system is negative definite-which is a positive definite system that has been multiplied through by ?1. For all practical geometries the common finite difference Laplacian operator gives rise to these, the best of all possible matrices. Just about any standard solution method will succeed, and many theorems are available for your pleasure.

- FORMAN S. ACTION, Numerical Methods That Work (1970)

Many years ago we made out of half a dozen transformers a simple and rather inaccurate machine for solving simultaneous equations-the solutions being represented as flux in the cores of the transformers. During the course of our experiments we set the machine to solve the equations-

X + Y + Z = 1

X + Y + Z = 2

X + Y + Z = 3

The machine reacted sharply-it blew the main fuse and put all the lights out.

- B. V. BOWDEN, The Organization of a Typical Machine (1953)

There does seem to be some misunderstanding about the purpose of an a priori backward error analysis. All too often, too much attention is paid to the precise error bound that has been established. The main purpose of such an analysis is either to establish the essential numerical stability of an algorithm or to show why it is unstable and in doing so to expose what sort of change is necessary to make it stable. The precise...