# Perturbation Theory and Backward Error for $AX-XB=C$

Higham, Nicholas J. (1993) Perturbation Theory and Backward Error for $AX-XB=C$. BIT Numerical Mathematics, 33. pp. 124-136. ISSN 1572-9125

Because of the special structure of the equations $AX-XB=C$ the usual relation for linear equations $\mbox{backward error} = \mbox{relative residual}$'' does not hold, and application of the standard perturbation result for $Ax=b$ yields a perturbation bound involving ${\rm sep}(A,B)^{-1}$ that is not always attainable. An expression is derived for the backward error of an approximate solution $Y$; it shows that the backward error can exceed the relative residual by an arbitrary factor. A sharp perturbation bound is derived and it is shown that the condition number it defines can be arbitrarily smaller than the ${\rm sep}(A,B)^{-1}$-based quantity that is usually used to measure sensitivity. For practical error estimation using the residual of a computed solution an LAPACK-style'' bound is shown to be efficiently computable and potentially much smaller than a sep-based bound. A Fortran~77 code has been written that solves the Sylvester equation and computes this bound, making use of LAPACK routines.