Rigorous Sensitivity Analysis for Systems of Linear and Nonlinear Equations

Methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent, noncorrelated data for general systems of linear and nonlinear equations. The methods may serve for the following two purposes. First, to bound the dependency of the solution on changes in the input data. In contrast to condition numbers, a componentwise sensitivity analysis of the solution vector is performed. Second, to estimate the true solution set for problems whose input data are subject to tolerances. The methods presented are very effective and have the additional property that, owing to an automatic error control mechanism, every computed result is guaranteed to be correct. Examples are given for linear systems, demonstrating that the computed bounds are in general very sharp. Interesting comparisons to traditional condition numbers are given.