CHAPTER 6

 CONVERGENCE PROPERTIES



 A solution procedure is only useful if it will find the solution

to the set of equations to which it is applied. When tear variables are

present, their values must be converged. A problem frequently encoun-

tered in the implementation of solution procedures is that they will not

converge. When automatically generating a solution procedure, it is

desirable to gain some prior knowledge about its convergence properties.

In that way, solution procedures which could be expected to perform

poorly can be modified prior to their implementation. This chapter

reviews previous studies in this area and discusses new developments.


6.1 Review

 Consider the following set of linear equations:

 Ax -b 6-1

Equation 6-1 can be rewritten as

 (L+U)x=b 6-2

where

 L = The lower triangular matrix composed of diagonal and below

 diagonal elements of A

 U = The upper triangular matrix composed of the abova diagonal

 (zero diagonal) elements of A.

A solution to these equations by Gauss-Seidel can ibe represented by: