An estimate for the spectral norm of the inverse of a matrix with the Gauss—Jordan algorithm

摘要

In this paper an algorithm is presented for calculating an estimate for the spectral norm of the inverse of a matrix. This algorithm is to be used in combination with solving a linear system by means of the Gauss—Jordan algorithm. The norm of the inverse is needed for the condition number of that matrix. The algorithm exploits the effect the Gauss—Jordan elimination is equivalent with writing the matrix as a product of n elementary matrices. These elementary matrices are sequentially used to maximize (locally) the norm of a solution vector that matches a right-hand side vector under construction. In n steps this produces a satisfactory estimate. Our algorithm uses 5n2+O(n) extra floating-point multiplications for the calculation of the required estimate and is tested for a multitude of matrices on the CYBER 205 vector computer of the Academic Computer Centre, SARA, in Amsterdam.