TABLE OF CONTENTS We seek to compute a vector x which, when multiplied by the square matrix A, yields the known vector b; that is, A x = b. In the ensuing discussion summation will not be implied over a repeated index but will be stated explicitly when required.
When the matrix A is diagonal, the unknown vector x is computed by the simple algorithm
When A is lower triangular, we use the forward substitution algorithm
When A is upper triangular, we use the backward substitution algorithm
Note that, in this case, the last unknown is computed first.
When A does not have a particular structure, the solution can be found using either a direct or an iterative method; the second class of methods are designated for systems of large size involving sparse matrices.
Gauss elimination is the simplest and most popular direct method. The associated algorithm with row pivoting proceeds according to the following steps:
Form the N ( N+1) partitioned augmented matrix C (1) ?[ A b] and introduce the matrix L whose elements are initialized to zero.
Assume that the maximum absolute value of the elements in the first column C (l) i1, i = 1, N is C (1) K1corresponding to the
