Adjusting the Rayleigh Quotient in Semiorthogonal Lanczos Methods
George Walter Stewart · SIAM Journal on Scientific Computing · 2002
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 [Linear Algebra Appl., 61 (1984), pp. 101--132] 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 paper we show how to adjust the Rayleigh quotient to overcome this problem.