Finite-Rank Methods and Their Stability for Coupled Systems of Operator Equations
Alain Largillier, Balmohan V. Limaye · SIAM Journal on Numerical Analysis · 1996
Let $\mathcal{K}$ be a bounded linear operator of finite rank on a normed linear space X. The solution of a coupled system of linear equations involving $\mathcal{K}$ is reduced to a solution of a matrix Sylvester equation $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } L - K\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = \beta $. It is shown that this equation has a unique solution satisfying $P_\sigma \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} $ (resp., $V_\sigma \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} $), provided $P_\sigma \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\beta } $ (resp., $V_\sigma \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\beta } $), where $P_\sigma $ and $V_s $ are certain projections related to the spectra $\sigma (K)$ and $\sigma (L)$ of K and L, resp. The stability of such a solution of a matrix Sylvester equation is considered and is related to the stability of a similar solution of the coupled system involving the operator $\mathcal{K}$. Often $\mathcal{K}$ is an approximation of a bounded linear operator $\mathcal{F}$ on X, yielding an approximate computable solution of either a coupled system involving $\mathcal{F}$ or of an eigenvalue problem for $\mathcal{F}$ Iterative refinement of such a computed solution can be accomplished by solving suitable matrix Sylvester equations. Numerical examples are given to illustrate this procedure.