Strict optimal a posteriori error and residual bounds for Gaussian elimination in floating-point arithmetic

Abstract

Exact representations of errors and residuals of approximate solutions of linear algebraic systems under data perturbations and rounding errors of a floating-point arithmetic are established from which strict optimal a posteriori error and residual bounds are obtained. These bounds are formulated by means of a posteriori error and residual condition numbers. Condition numbers, error and residual bounds can be computed completely in the range of nonnegative numbers using the arithmetic operations+, x, / only. It is shown that computations in this range are numerically very stable. The general results are applied to a series of numerical examples.