Modifying pivot elements in Gaussian elimination

Abstract:

The rounding-error analysis of Gaussian elimination shows that the method is stable only when the elements of the matrix do not grow excessively in the course of the reduction. Usually such growth is prevented by interchanging rows and columns of the matrix so that the pivot element is acceptably large. In this paper the alternative of simply altering the pivot element is examined. The alteration, which amounts to a rank one modification of the matrix, is undone at a later stage by means of the well-known formula for the inverse of a modified matrix. The technique should prove useful in applications in which the pivoting strategy has been fixed, say to preserve sparseness in the reduction.