A PERTURBATION ANALYSIS OF THE GENERALIZED SYLVESTER EQUATION (AR-LB,DR-LE)=(C,F)

Perturbation and error bounds for the generalized Sylvester equation ( AR-LB, DR-LE) = (C, F) are presented. An explicit expression for the n ormwise relative backward error associated with an approximate solutio n of the generalized Sylvester equation is derived and conditions when it can be much greater than the relative residual are given. This ana lysis is applicable to any method that solves the generalized Sylveste r equation. A condition number that reflects the structure of the prob lem and a normwise forward error bound based on Dif-1[(A, D), (B, E)] and the residual are derived. The structure-preserving condition numbe r can be arbitrarily smaller than a Dif-1-based condition number. The normwise error bound can be evaluated robustly and at moderate cost by using a reliable Dif-1 estimator. A componentwise LAPACK-style forwar d error bound that can be stronger than the normwise error bound is al so presented. A componentwise approximate error bound that can be eval uated to a much lower cost is also proposed. Finally, some computation al experiments that validate and evaluate the perturbation and error b ounds are presented.