On the Stability of Singular Finite-Rank Methods
Lalita N. Deshpande, Balmohan V. Limaye · SIAM Journal on Numerical Analysis · 1990
The solution of the operator equation $(\lambda - {\bf \mathcal{K}})x = y$, where ${\bf \mathcal{K}}$ is a bounded linear operator of finite rank on a Banach space X and $\lambda $ is a nonzero scalar, can be reduced to the solution of a system of linear equations $(\lambda - {\bf \tilde{\mathcal{K}}} \mathcal{K})\alpha = \beta $, where ${\bf \tilde{\mathcal{K}}}$ is a matrix. A recent result of Whitley for the case where $\lambda $ is in the resolvent set of ${\bf \mathcal{K}}$ is extended to include the case where $\lambda $ is a nonzero spectral value of ${\bf \mathcal{K}}$. If $\lambda $ is a nonzero semisimple eigenvalue of ${\bf \mathcal{K}}$, then the stability of a particular solution of the linear system is related to the stability of a particular solution of the operator equation. Bounds for the condition number of an equivalent nonsingular linear system are given. These results are applied to the implementation of some iterative refinement schemes for approximating a nonzero simple eigenvalue of a compact operator on X.