A Perturbation Analysis of the Generalized Sylvester Equation $(AR - LB, DR - LE) = (C, F)$

Authors:
Bo Kagstrom
Affiliations:
-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
1994

Citing 0
Cited 8

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Abstract

Perturbation and error bounds for the generalized Sylvester equation $(AR - LB, DR - LE) = (C, F)$ are presented. An explicit expression for the normwise relative backward error associated with an approximate solution of the generalized Sylvester equation is derived and conditions when it can be much greater than the relative residual are given. This analysis is applicable to any method that solves the generalized Sylvester equation. A condition number that reflects the structure of the problem and a normwise forward error bound based on ${\rm Dif}^{-1}[(A,D),(B,E)]$ and the residual are derived. The structure-preserving condition number can be arbitrarily smaller than a ${\rm Dif}^{-1}$-based condition number. The normwise error bound can be evaluated robustly and at moderate cost by using a reliable ${\rm Dif}^{-1}$ estimator. A componentwise LAPACK-style forward error bound that can be stronger than the normwise error bound is also presented. A componentwise approximate error bound that can be evaluated to a much lower cost is also proposed. Finally, some computational experiments that validate and evaluate the perturbation and error bounds are presented.