Zhongxiao Jia

Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems

Abstract Arnoldis method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices. However, the approximate eigenvectors or Ritz vectors obtained by Arnoldis method cannot be guaranteed to converge in theory even if the approximate eigenvalues or Ritz values do. In order to circumvent this potential danger, a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions. It is shown that refined approximate eigenvectors converge to eigenvectors if Ritz values do. Moreover, the resulting refined algorithms converge more rapidly. We report some numerical experiments and compare the refined algorithms with their counterparts, the iterative Arnoldi and Arnoldi-Chebyshev algorithms. The results show that the refined algorithms are considerably more efficient than their counterparts.

An analysis of the Rayleigh—Ritz method for approximating eigenspaces

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,X) 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,X) as the sine e of the angle between X and W approaches zero. It is shown that under a natural hypothesis - called the uniform separation condition - the Ritz pairs (N,X) converge to the eigenpair (L,X). When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that A has distinct eigenvalues or is diagonalizable.

The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems

In this paper, we investigate the convergence theory of generalized Lanczos methods for solving the eigenproblems of large unsymmetric matrices. Bounds for the distances between normalized eigenvectors and the Krylov subspace

An Implicitly Restarted Refined Bidiagonalization Lanczos Method for Computing a Partial Singular Value Decomposition

{\cal K}_m(v_1,A)

A refined iterative algorithm based on the block arnoldi process for large unsymmetric eigenproblems

spanned by

Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm

v_1, Av_1, \ldots, A^{m-1}v_1

The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices

are established, and a priori theoretical error bounds for eigenelements are presented when matrices are defective. Using them we show that the methods will still favor the outer part eigenvalues and the associated eigenvectors of

A refined subspace iteration algorithm for large sparse eigenproblems

A

The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

usually though they may converge quite slowly in the case of

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

A