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 4] 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...

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 4] 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...

A nonlinear Rayleigh-Ritz iterative (NRRIT) method for solving nonlinear eigenvalue problems is studied in this paper. It is an extension of the nonlinear Arnoldi algorithm due to Heinrich Voss. The effienicy of the NRRIT method is demonstrated by comparing with inverse iteration methods to solve a highly nonlinear eigenvalue problem arising from finite element electromagnetic simulation in acc...

Rayleigh–Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing, which has helpful implications for theory, applications, and algorithms. In contrast, few results about the Ritz values of non-Hermitian matrices are known, beyond their containment within the numerical range. To show that such Ritz values enjoy considerable structure, we establish regions within the numerical ra...

This paper concerns the Rayleigh–Ritz method for computing an approximation to an eigenspace X of a general matrix A from a subspace W that contains an approximation to X . The method produces a pair (N, X̃) that purports to approximate a pair (L,X), where X is a basis for X and AX = XL. In this paper we consider the convergence of (N, X̃) as the sine of the angle between X andW approaches zero. ...

We make use of the Pad e approximants and Krylov's sequence x; Ax; ; : : :; A m?1 x in the projection methods to compute a few Ritz values of a hermitian matrix A of order n. This process consists of approaching the poles of R x () = ((I ?A) ?1 x; x), the mean value of the resolvant of A, by those of m ? 1=m] Rx (), where m ? 1=m] Rx () is the Pad e approximant of order m of the function R x ()...

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.