TABLE OF CONTENTS In this third part of the book we turn to a systematic discussion of two fundamental issues of numerical analysis that until now we have only skirted. Conditioning pertains to the perturbation behavior of a mathematical problem. Stability pertains to the perturbation behavior of an algorithm used to solve that problem on a computer.
In the abstract, we can view a problem as a function f : X ? Y from a normed vector space X of data to a normed vector space Y of solutions. This function f is usually nonlinear (even in linear algebra), but most of the time it is at least continuous.
Typically we shall be concerned with the behavior of a problem f at a particular data point x ? X (the behavior may vary greatly from one point to another). The combination of a problem f with prescribed data x might be called a problem instance, but it is more usual, though occasionally confusing, to use the term problem for this notion too.
A well-conditioned problem (instance) is one with the property that all small perturbations of x
