A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

محورهای موضوعی : History and biography

M. Amirfakhrian ¹ , F. Mohammad ²

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran

تاریخ دریافت : 1390/10/11 تاریخ پذیرش : 1393/10/11 تاریخ انتشار : 1391/03/12

کلید واژه: convergence, Eigenvalue problem, inexact inverse iteration, subspace iteration, Approximation, inner-outer iteration,

چکیده مقاله :

In this paper, we represent an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax = Bx [Q. Ye and P. Zhang,Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra andits Application, 434 (2011) 1697-1715 ]. In particular, the linear convergence property of theinverse subspace iteration is preserved.

منابع و مأخذ:

[1] J. Berns-Muler, I. G. Graham and A. Spence, Inexact inverse iteration for symmtric matrices, Linear Algebra Appl, 416 (2006), 389-413.

[2] J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the solution of algebraic eigenvalue problems: a practical guide, Philadelphia, PA, USA, 2000.

[3] M. A. Freitag, A. Spence, Convergence rates for inexact inverse iteration with application to preconditioned iterative solves, BIT, 47 (2007), 27-44.

[4] G. H. Golub and Q. Ye, Inexact inverse iterations for the generalized eigenvalue problems, BIT, 40 (1999), 672-684.

[5] G. H. Golub and C. F. Van loan, Matrix computation, Baltimore, MD, USA, 1989.

[6] G. H. Golub, Z. Zhang and H. Zha, Large sparse symmetric eigenvalue prob- lems with homogeneous linear constraints:the lanczos process with inner-outer iteration, Linear Algebra And Its Applications, 309 (2000), 289-306.

[7] Z. Jia, On convergence of the inexact rayleigh quotient iteration without and with minres, 2009.

[8] Y. Lai, K. Lin, W. Lin . An inexact inverse iteration for large sparse eigenvalue problems, Numerical Linear Algebra With Application, (1997), 425-437.

[9] R. B. Lehoucq and Karl Meerbergen, Using generalized cayley transformations within an inexact rational krylov sequence method, SIAM J. Matrix Anal. Appl., 20, 131-148.

[10] R. B. Morgan and D. S. Scott, Preconditioning the lanczos algorithm for sparse symmetric eigenvalue problems, SIAM J. Sci. Comput., 14 (1993), no. 3, 585-593.

[11] A. Ruhe, Rational krylov: A practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comput., 19 (1998), no. 5, 1535-1551.

[12] A. Ruhe And T. Wiberg, The method of conjugate gradients used in inverse iter- ation, BIT, 12 (1972), 543-554.

[13] P. Smit and M. H. C. Paardekooper, The eects of inexact solvers in algorithms for symmetric eigenvalue problems, Linear Algebra and its Applications, 287 (1999), 337-357.

[14] V. Simoncini and L. Eldn, Inexact rayleigh quotient-type methods for eigenvalue computations, BIT, 42 (2002), 159-182.

[15] G. Sleijpen and H. Van Der Vorst, A jacobi-davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (2000), 401-425.

[16] D. C. Sorensen and C. Yang, A truncated rq iteration for large scale eigenvalue calculations, SIAM J. Matrix Anal. Appl., 19 (1998), no. 4, 1045-1073.

[17] A. Stathopoulos, Y. Saad and F. Fischer, Robust preconditioning of large sparse symmetric eigenvalue problems, Journal of Computational and Applied Mathematics, 64 (1994), 197-215.