Adjusting the Rayleigh Quotient in Semiorthogonal Lanczos Methods

In a semiorthogonal Lanczos algorithm, the orthogonality of theLanczos vectors is allowed to deteriorate to roughly the square root
of the rounding unit, after which the current vectors are
reorthogonalized. A theorem of Simon \cite{simo:84} shows that the
Rayleigh quotient\,---\,i.e., the tridiagonal matrix produced by the
Lanczos recursion\,---\,contains fully accurate approximations to the
Ritz values in spite of the lack of orthogonality. Unfortunately, the
same lack of orthogonality can cause the Ritz vectors to fail to
converge. It also makes the classical estimate for the residual norm
misleadingly small. In this note we show how to adjust the Rayleigh
quotient to overcome this problem.
(Cross-referenced as UMIACS-TR-2001-31)