The E² Bathe subspace iteration method

Abstract

Since its development in 1971, the Bathe subspace iteration method has been widely-used to solve the generalized symmetric-definite eigenvalue problem. The method is particularly useful for solving large eigenvalue problems when only a few of the least dominant eigenpairs are sought. In reference [18], an enriched subspace iteration method was proposed that accelerated the convergence of the basic method by replacing some of the iteration vectors with more effective turning vectors. In this thesis, we build upon this recent acceleration effort and further enrich the subspace of each iteration by replacing additional iteration vectors with our new turning-of-turning vectors. We begin by reviewing the underpinnings of the subspace iteration methodology. Then, we present the steps of our new algorithm, which we refer to as the Enriched- Enriched (E2 ) Bathe subspace iteration method. This is followed by a tabulation of the number of floating point operations incurred during a general iteration of the E2 algorithm. Additionally, we perform a simplified convergence analysis showing that the E2 method converges asymptotically at a faster rate than the enriched method. Finally, we examine the results from several test problems that were used to illustrate the E2 method and to assess its potential computational savings compared to the enriched method. The sample results for the E2 method are consistent with the theoretical asymptotic convergence rate that was obtained in our convergence analysis. Further, the results from the CPU time tests suggest that the E2 method can often provide a useful reduction in computational effort compared to the enriched method, particularly when relatively few iteration vectors are used in comparison with the number of eigenpairs that are sought.