Norm-Preserving Dilations and Their Applications to Optimal Error Bounds
Chandler Davis, W. Kahan, Hans F. Weinberger · SIAM Journal on Numerical Analysis · 1982
The problem is, given A, B, C, to find D such that $\left\| ( {\begin{array}{*{20}c} A & C \\ B & D \\ \end{array} )} \right\| \leqq \mu $; here we deal with Hilbert-space operators, A, B, and C are given, and $\mu $ is a given positive number. We give explicit formulas for all solutions D. In case the D sought is a finite-dimensional matrix, we express the answer in a form involving explicit matrix computations. It is explained how this allows one to find best possible error bounds for certain problems of choosing algorithms for approximate solutions of linear boundary problems. The main theorem is also used (§ 5) to find exact bounds for an n-tuple of eigenvalues of a Hermitian eigenproblem in terms of the norm of the residual.