Weighted Harmonic Arnoldi Method for Large Interior Eigenproblems
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51781", author = "Zhengsheng Wang and Jing Qi and Chuntao Liu and Yuanjun Li", title = "Weighted Harmonic Arnoldi Method for Large Interior Eigenproblems", 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.", volume = "5", number = "8", pages = "1161-4", }
