On the Convergence of Ritz Values, Ritz Vectors, and Refined Ritz Vectors\symbolmark
Abstract
This paper concerns the Rayleigh--Ritz method for computing an
approximation to an eigenpair $(\lambda, x)$ of a non-Hermitian matrix
$A$. Given a subspace $\clw$ that contains an approximation to $x$,
this method returns an approximation $(\mu, \tilde x)$ to $(\lambda,
x)$. We establish four convergence results that hold as the deviation
$\epsilon$ of $x$ from $\clw$ approaches zero. First, the Ritz value
$\mu$ converges to $\lambda$. Second, if the residual $A\tilde
x-\mu\tilde x$ approaches zero, then the Ritz vector $\tilde x$
converges to $x$. Third, we give a condition on the eigenvalues of
the Rayleigh quotient from which the Ritz pair is computed that
insures convergence of the Ritz vector. Finally, we show that certain
unconditionally.
(Also cross-referenced as UMIACS-TR-99-08)