ABSTRACT
The Gauss elimination technique outlined in the previous chapter appears to be a
suitable method of solving general sets of equations and there is no hint that, given
some well posed equations and care with pivoting, a good solution should not be
generally available. However, there is a problem: round-off errors. These are
generated with each and every operation and can eventually spoil the solution.
Gaussian elimination is thought to be fine for solving up to about 100 equations in
100 unknowns but for larger systems round-off errors usually render the method
unsuitable and it is necessary to try different solutions. For such systems
approximate or iterative methods can often be used. The advantage is that although
the method is an approximate one, it is possible to determine how approximate (or
accurate) the solution is required to be and to continue the iterations until that
accuracy is attained. The Gauss-Seidel method is one such method. Assume one is
trying to solve:
