On Some Iterative Methods for Solving Elliptic Difference Equations (Classic Reprint) - Softcover

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Excerpt from On Some Iterative Methods for Solving Elliptic Difference Equations

The theorems are used to define three subclasses, called complete image classes, of the general family. These classes contain many of the schemes in current use as well as generalizations of them. Thus it is shown that a variety of independently proposed and seemingly unrelated iterative methods are special cases of a general class of methods. These complete image classes are such that each of the eigenvalues of any scheme in the class is a given algebraic function of at least one of the eigenvalues of a particular reference scheme of the class.

Thus a knowledge of the eigenvalues of the reference schemepermits, in principle, the determination of the best scheme of the given class.

A special class of equations is considered for which all the eigenvalues of each reference scheme can be explicitly written in terms of the eigenvalues of two matrices. It is then possible to determine which of the reference schemes is best and hence it is also possible to determine that scheme which is best of all those in the complete image classes.

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