Error Analysis of Gaussian Elimination Method for Solving System of Linear Algebraic Equations

摘要

A posteriori forward error analysis is applied to the Gaussian elimination method for solving system of linear algebraic equations of the type Az = p. By attributing the generated round-off errors properly to the matrices A and p, it is shown that the computed z satisfies a new perturbed system such that (A + delta A)z = p + delta p. For large system order n, the upper bounds for delta A and delta p in infinite norm are then shown to be proportional to n squared, instead of n cubed obtained by the usual backward error analysis where round-off errors are attributed totally to the system matrix A. This answers partially some questions raised concerning the discrepancy between the theoretical result and practical observation of the perturbations. (Author)