A refined subspace iteration algorithm for large sparse eigenproblems

Abstract

Two innovations are presented to improve the orthogonal subspace iteration. Firstly, a refined strategy is proposed that replaces conventional Ritz vectors by refined Ritz vectors in subspace iteration. Each refined Ritz vector is chosen such that the norm of the residual formed with the Ritz value is minimized over the subspace involved, and it can be cheaply computed by solving a small problem. Secondly, a standard updating matrix, Ritz vector or Schur vector matrix, is replaced by a new one that is generated only by the refined Ritz vectors used to approximate the wanted eigenvectors. A combination of them yields a refined subspace iteration algorithm. A qualitative analysis is given for the refined algorithm, showing that it may be considerably more efficient than a conventional subspace iteration algorithm. When the subspace dimension is bigger than the number of the wanted eigenpairs, numerical examples show that the refined algorithm has a sharp superiority. Numerical comparisons are also drawn for the refined algorithm with the popular package ARPACK and an implicitly restarted version of the refined Arnoldi method. The algorithm is suited for computing a few dominant eigenpairs of large sparse matrices since the only action of a matrix A in question into the program is a subroutine to form the product AQ, where Q is a matrix having much less columns than rows.