Multivariable Problems

Abstract

As Thacher [1] has remarked, ‘one of the areas of numerical analysis that is most backward and, at the same time, most important from the standpoint of practical applications’ is multivariable analysis. Much of our ignorance results from a lack of adequate mathematical theory in this field. Unless a function of a large number of variables is pathologically irregular, its main features will depend only upon a few overall characteristics of its variables, such as their mean value or variance, but we have little, if any, theory to tell us when this situation appertains and, if so, what overall characteristic it is that actually describes the function. In these circumstances an empirical examination, effectively some sampling experiment on the behaviour of the function, may guide us.