FINITE-RANK METHODS AND THEIR STABILITY FOR COUPLED SYSTEMS OF OPERATOR EQUATIONS

摘要

Let 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 K is reduced to a solution of a matrix Sylvester equation <(alpha)under bar>L - K<(alpha)under bar> = <(beta)under bar>. It is shown that this equation has a unique solution satisfying P-sigma<(alpha)under bar> = (0) under bar (resp., V-S<(alpha)under bar> = (0) under bar), provided P-sigma<(beta)under bar> = (0) under bar (resp., V-S<(beta)under bar> = (0) under bar) 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 K. Often K is an approximation of a bounded linear operator F on X, yielding an approximate computable solution of either a coupled system involving F or of an eigenvalue problem for 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. [References: 9]