Weighted Harmonic Arnoldi Method for Large Interior Eigenproblems

Authors: Zhengsheng Wang, Jing Qi, Chuntao Liu, Yuanjun Li

Abstract:

The harmonic Arnoldi method can be used to find interior eigenpairs of large matrices. However, it has been shown that this method may converge erratically and even may fail to do so. In this paper, we present a new method for computing interior eigenpairs of large nonsymmetric matrices, which is called weighted harmonic Arnoldi method. The implementation of the method has been tested by numerical examples, the results show that the method converges fast and works with high accuracy.

Keywords: Harmonic Arnoldi method, weighted harmonic Arnoldi method, eigenpair, interior eigenproblem, non symmetric matrix.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1057637

References:

[1] G. Wu. A modified harmonic block Arnoldi algorithm with adaptive shifts for large interior eigenproblems. J. Comput. Appl. Math., 205(2006)343-363.
[2] Z. Jia. The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrix. Appl. Numer. Math., 42(2002)489-512.
[3] R.B. Morgan, M.Zeng. Harmonic projection methods for large non-symmetric eigenvalue problems. Numer. Linear Algebra Appl., 5(1998)33-55.
[4] R.B. Morgan. Computing interior eigenvalues of large matrices. Linear Algebra Appl., 154-156(1991)289-309.
[5] H. Saberi Najafi, H. Ghazvini. Weighted restarting method in the weighted Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix. Appl. Math. Comput., 175(2006)1276-1287.
[6] H. Saberi Najafi, H. Moosaei. A new restarting method in the harmonic projection algorithm for computing the eigenvalues of a nonsymmetric matrix. Appl. Math. Comput., 198(2008)143-149.
[7] H. Saberi Najafi, E. Khaleghi. A new restarting method in the Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix. Appl. Math. Comput. 156(2004)59-71.
[8] G. Chen, Q. Niu. A simple harmonic Arnoldi method for computing interior eigenvalues fo large matrices. J. Xiamen University, 46(2007)312- 316.
[9] G.Chen, J.Lin. A new shift scheme for the harmonic Arnoldi method. Math. Comput. Mod., 48(2008)1701-1707.
[10] Z.Bai, R.Barret, D.Day, J.Demmel, J.Dongarra. Test matrix collection for non-Hermitian eiegnvalue problems. Technical Report CS-97-355, University of Tennessee, Knoxville, 1997, LAPACK Note #123, Software and test data available at http://math.nist.gov/MatrixMarket/.
[11] Z.Bai, J.Demmel, J.Dongarra, A.Ruhe, H. van der Vorst. Templates for the solution of algebraic eigenvalue problems: A practical guide. SIAM Philadelphia, 2000.
[12] Y.Saad, Numerical methods for large eigenvalue Problems, Algorithms and Architectures for Advanced Scientific Computing. Manchester University Press, Manchester, 1992.
[13] G.H.Golub,C.F.Van Loan, Matrix Computations, third ed. The Johns Hopkins University Oress, Baltimore aand London, 1996.
[14] H.A. van Vorst. Computational methods for large eigenvalue problems. North-Holland(Elsevier), Amsterdam, 2002.