Componentwise perturbation theory for linear systems with multiple right-hand sides

Abstract

Existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right-hand sides and to a general class of componentwise measure of perturbations involving Hölder p-norms. It is shown that for a system of order n with r right-hand sides, the componentwise backward error can be computed by finding the minimum p-norm solutions to n underdetermined linear systems, and an explicit expression is obtained in the case r = 1. A perturbation bound is derived, and from this the componentwise condition number is obtained to within a multiplicative constant. Applications of the results are discussed to invariant subspace computations, quasi-Newton methods based on multiple secant equations, and an inverse ODE problem.