Backward Errors And Condition Numbers Of Two Systems

Backward errors and condition numbers play an important role in modern numerical linear algebra. Backward errors reveal the stability of a numerical method. Condition numbers explain the sensitivity of the solution of a problem to perturbations in the data, and, in the case where this perturbation is a backward error, the product of condition number times backward error provides a first order error bound for the computed solution.In some mathematical problems, there are requirements in structure. Correspondingly, there are ordinary algorithm and structured algorithm. To analyze the sensitivity of the computed solution, ordinary condition numbers and structured condition numbers have been proposed. And so do backward errors.In this article, several problems are discussed: the structured backward error analysis and the condition number for the Cauchy system, the structured backward error analysis for Hermite eigenvalue problem.In the first part, the normwise structured backward error and the condition number for Cauthy equations are defined. The upper bound of the backward error is obtained by Brouwer fixed-point theorem and the lower bound of the backward error is obtained through singular-value decomposition. The results are illustrated by simple numerical examples.In the second part, the backward error analysis of the Hermite eigenprob-lems is discussed. Three kind of problems are included: generalized eigenprob-lem, matrix polynomial eigenproblem, and multiparameter eigenproblem. In the case of the approximate eigenvalue being real, these problems have already been solved by Higham and Higham(1998), Tisseur(2000), and Hachstenbach and Plestenjak(2003). The general case is considered in this paper. The results can be considered as complements of theirs.