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A VARIATION ON THE BLOCK ARNOLDIMETHOD FOR large unsymmetric MATRIX EIGENPROBLEMS

Acta Mathematicae Applicatae Sinica 1998年第4期14卷 425-432页

作者：贾仲孝 Department of Applied Mathematics Dalian University of Technology Dalian China

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.

关键词： large unsymmetric block Arnoldi process block Arnoldi method Ritz value,Ritz vector modified approximate eigenvector

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IMPROVING EIGENVECTORS IN ARNOLDI'S METHOD

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Journal of Computational Mathematics 2000年第3期18卷 265-276页

作者： Zhong-xiao Jia (Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China) Ludwig Elsner (Fakultat fur Mathematik, University Bielefeld, Postfach 100131, 33501 Bielefeld,Germany) 大连理工大学辽宁大连 116024 Universitt Bielefeld 德国 Bielefeld 33501

The Ritz vectors obtained by Arnoldi's method may not be good approxima- tions and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m + 1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modi- fied m-step 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, we present a modified m-step restarted Arnoldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m+ 1)-step restarted ones.

关键词： large unsymmetric The m-step Arnoldi process The m-step Arnoldi method Eigenvalue Ritz value Eigenvector Ritz vector Modified

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