## Taiwanese Journal of Mathematics

### A Global Arnoldi Method for Large Non-Hermitian Eigenproblems with Special Applications to Multiple Eigenproblems

#### Abstract

Global projection methods have been used for solving numerous large matrix equations, but nothing has been known on if and how this method can be proposed for solving large eigenproblems. In this paper, a global Arnold method is proposed for large eigenproblems. It computes certain F-Ritz pairs that are used to approximate some eigenpairs. The global Arnoldi method inherits convergence properties of the standard Arnoldi method applied to a larger matrix whose distinct eigenvalues are the eigenvalues of the original given matrix. As an application, assuming that A is diagonalizable, we show that the global Arnoldi method is able to solve multiple eigenvalue problems. To be practical, we develop an implicitly restarted global Arnoldi algorithm with certain F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 4 (2011), 1497-1525.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406361

Digital Object Identifier
doi:10.11650/twjm/1500406361

Mathematical Reviews number (MathSciNet)
MR2848971

Zentralblatt MATH identifier
1232.65057

#### Citation

Duan, Congying; Jia, Zhongxiao. A Global Arnoldi Method for Large Non-Hermitian Eigenproblems with Special Applications to Multiple Eigenproblems. Taiwanese J. Math. 15 (2011), no. 4, 1497--1525. doi:10.11650/twjm/1500406361. https://projecteuclid.org/euclid.twjm/1500406361

#### References

• W. E. Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), 11-29.
• Z. Bai, J. Demmel, J. Dongarra, A. Ruhe and H. A. van der Vorst (eds.), Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
• Z. Bai, R. Barrett, D. Day, J. Demmel and J. Dongarra, Test matrix collection (non-Hermitian eigenvalue problems), http://math.nist.gov/MarketMatrix.
• L. Bao, Y. Lin and Y. Wei, A new projection method for solving large Sylvester equations, Appl. Numer. Math., 57 (2007), 521-532.
• M. Bellalij, K. Jbilou and H. Sadok, New convergence results on the global GMRES method for diagonalizable matrices, Technical Report, L. M. P. A., 2007.
• F. Chatelin and D. Ho, Arnoldi-Tchebychev procedure for large scale nonsymmetric matrices, Math. Model. Numer. Anal., 24 (1990), 53-65.
• C. C. Chu, M. H. Lai and W. S. Feng, MIMO interconnects order reductions by using the multiple point adaptive-order rational global Arnoldi algorithm, IEICE Trans. Elect., E89-C (2006), 792-808.
• C. C. Chu, M. H. Lai and W. S. Feng, The multiple point global Lanczos method for mutiple-inputs multiple-outputs interconnect order reduction, IEICE Trans. Elect., E89-A (2006), 2706-2716.
• C. C. Chu, M. H. Lai and W. S. Feng, Model-order reductions for MIMO systems using global Krylov subspace methods, Math. Comput. Simulat., to appear.
• J. K. Cullum and W. E. Donath, A block Lanczos algorithm for computing the $q$ algebraically largest eigenvalues and a corresponding eigenspace for large, sparse symmetric matrices, in: Proceedings of the 1994 IEEE Conference on Decision and Control, IEEE Press, Piscataway, NJ, 1974, pp. 505-509.
• J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
• G. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, in: Mathematical Software III, J. Rice (ed.), Academic Press, New York, 1977, pp. 364-377.
• C. Gu and Z. Yang, Global SCD algorithm for real positive definite linear systems with multiple right-hand sides, Appl. Math. Comput., 189 (2007), 59-67.
• M. Heyouni, The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides, Numer. Algor., 26 (2001), 317-332.
• M. Heyouni and K. Jbilou, Matrix Krylov subspace methods for large scale model reduction problems, Appl. Math. Comput., 181 (2006), 1215-1228.
• K. Jbilou, A. Messaoudi and H. Sadok, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math., 31 (1999), 49-63.
• K. Jbilou, An Arnoldi based algorithm for large algebraic Riccati equations, Appl. Math. Lett., 19 (2006), 437-444.
• K. Jbilou, Low rank approximate solutions to large Sylvester matrix equations, Appl. Math. Comput., 177 (2006), 365-376.
• Z. Jia, The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM J. Matrix Anal. Appl., 16 (1995), 843-862.
• Z. Jia, Generalized block Lanczos methods for large unsymmetric eigenproblems, Numer. Math., 80 (1998), 239-266.
• Z. Jia, Arnoldi type algorithms for large unsymmetric multiple eigenvalue problems, J. Comput. Math., 17 (1999), 257-274.
• W. Karush, An iterative method for finding characteristics vectors of a symmetric matrix, Pacific J. Math., 1 (1951), 233-248.
• R. B. Lehoucq and K. J. Maschhoff, Implementation of an implicitly restarted block Arnoldi method, Preprint MCS-P649-0297, Argonne National Laboratory, Argonne, IL, 1997.
• Y. Lin, Implicitly restarted global FOM and GMRES for nonsymmetric matrix equations and Sylvester equations, Appl. Math. Comput., 167 (2005), 1004-1025.
• C. C. Paige, The computation of eigenvalues and eigenvectors of very large sparse matrices, Ph. D. thesis, London University, London, England, 1971.
• Y. Saad, Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl., 34 (1980), 269-295.
• Y. Saad, Projection methods for solving large sparse eigenvalue problems, in: Matrix Pencils, B. Kågström, A. Ruhe (eds.), Lecture Notes in Mathematics 973, Springer, Berlin, 1983, pp. 121-144.
• Y. Saad, Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comput., 42 (1984), 567-588.
• Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, 1992, UK.
• D. K. Salkuyeh, CG-type algorithms to solve symmetric matrix equations, Appl. Math. Comput., 172 (2006), 985-999.
• D. K. Salkuyeh and F. Toutounian, New approaches for solving large Sylvester equations, Appl. Math. Comput., 173 (2006), 9-18.
• V. Simoncini and E. Gallopoulos, Convergence properties of block GMRES and matrix polynomials, Linear Algebra Appl., 247 (1996), 97-119.
• D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), 357-385.
• G. W. Stewart, Matrix Algorithms II: Eigensystems, SIAM, Philadelphia, 2001.