Adjusting the Rayleigh Quotient in Semiorthogonal Lanczos Methods

Abstract

In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos 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)