Componentwise Perturbation Theory for Linear Systems With Multipte Right-Hand Sides

Existing definitions of componentwise backward error and componentwise condi tion number for linear systems are extended to systems with multiple right-hand sides and to a general class of componentwise measure of perturbations involving Holder 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. *Supported by the University of Dundee Research Initiatives Fund. E-mail: [email protected]. tE-mail: na. nhigham@na net. ornl . gov. LINEAR ALGEBRA AND ITS APPLZCATZONS 174: 11 l129 (1992) Q Elsevier Science Publishing Co., Inc., 1992 111 655 Avenue of the Americas, New York, NY 10010 0024-3795/92/$5.00 112 DESMOND J. HIGHAM AND NICHOLAS J. HIGHAM