ESTIMATION OF THE SENSITIVITY OF LINEAR AND NONLINEAR ALGEBRAIC PROBLEMS

Methods are presented for performing a rigorous sensitivity analysis for general systems of linear and nonlinear equations w.r.t. weighted perturbations in the input data. The weights offer the advantage that all or part of the input data may be perturbed relatively or absolutely. System zeros may, depending on the application, stay zero or not. The main purpose of the paper is to give methods for computing rigorous bounds on the sensitivity of each individual component of the solution on the computer. The methods presented are very effective, with the additional property that, due to an automatic error-control mechanism, every computed result is guaranteed to be correct. Examples are given for linear and nonlinear systems, demonstrating that the computed bounds are in general very sharp. Interesting comparisons with traditional condition numbers are given. For linear systems the solution set for finite perturbations in the coefficients is estimated. Moreover some theoretical results for eigenvectors, eigenvalues, and singular values are given.