A VARIATION ON THE BLOCK ARNOLDIMETHOD FOR LARGE UNSYMMETRIC MATRIX EIGENPROBLEMS

1.IntroductionLarge-scalematrixeigenproblemsariseinappliedsciencesandmanyengineeringapplications.Arnoldi'smethod1'2]anditsblockversion3--6]areverypopularforsolvingthem.Thesemethodshavebeenintensivelyinvestigatedsincethe1980s,bothintheoryandinalgorithms;wereferto7--17]fordetails.WhenmstepsoftheblockArnoldiprocessareperformed,anorthonormalbasis{K}7=1oftheblockKrylovsubspaceK.(VI,A)spannedbyVI5AVI,'IAm--1VIisgenerated,whereVIisaninitialNxporthogonalmatrix,andtherestrictionofAtoKm(V…

A variation on the block Arnoldi method for large unsymmetric matrix eigenproblems

The approximate eigenvectors or Ritz vectors obtained by the block Arnoldi method may converge very slowly and even fail to converge even if the approximate eigenvalues do. In order to improve the quality of the Ritz vectors, a modified strategy is proposed such that new approximate eigenvectors are certain combinations of the Ritz vectors and the waSted (m 1) th block basis vector and their corresponding residual norms are minimized in a certain sense. They can be cheaply computed by solving a few small "dimensional minimization problems. The resulting modified m-step block Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m 1)-step one. Based on this strategy, a modified m-step iterative block Arnoldi algorithm is presented. Numerical experiments are reported to show that the modified m-step algorithm is often considerably more efficient than the standard (m 1)-step iterative one.