Abstract
We establish a general convergence theory of the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple eigenvalue nearest to a given target σ and the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimics the exact SIRA well, i.e., the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with low or modest accuracy during outer iterations. Based on the theory, we design practical stopping criteria for inner solves. Our analysis is on one step expansion of subspace and the approach applies to the Jacobi-Davidson (JD) method with the fixed target σ as well, and a similar general convergence theory is obtained for it. Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.
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References
Bai Z, Barret R, Day D, et al. Test matrix collection for non-Hermitian eigenvalue problems. http://math.nist.gov/MatrixMarket/
Bai Z, Demmel J, Dongarra J, et al. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Philadelphia, PA: SIAM, 2000
Freitag M A, Spence A. Shift-and-invert Arnoldi’s method with preconditioned iterative solvers. SIAM J Matrix Anal Appl, 2009, 31: 942–969
Hochstenbach M E, Notay Y. Controlling inner iterations in the Jacobi-Davidson method. SIAM J Matrix Anal Appl, 2009, 31: 460–477
Jia Z. The convergence of generalized Lanczos methods for large unsymmetric eigenproblems. SIAM J Matrix Anal Appl, 1995, 16: 843–862
Jia Z. Refined iterative algorithms based on Arnoldi’s process for unsymmmetric eigenproblems. Linear Algebra Appl, 1997, 259: 1–23
Jia Z. Generalized block Lanczos methods for large unsymmetric eigenproblems. Numer Math, 1998, 80: 239–266
Jia Z. The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors. Math Comput, 2005, 74: 1441–1456
Jia Z, Stewart G W, An analysis of the Rayleigh-Ritz method for approximating eigenspaces. Math Comput, 2001, 70: 637–648
Lee C. Residual Arnoldi method: Theory, package and experiments. PhD thesis, TR-4515, Department of Computer Science, University of Maryland at College Park, 2007
Lee C, Stewart G W. Analysis of the residual Arnoldi method. TR-4890, Department of Computer Science, University of Maryland at College Park, 2007
Morgan R B. Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J Matrix Anal Appl, 2000, 21: 1112–1135
Notay Y. Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem. Numer Linear Algebra Appl, 2002, 9: 21–44
Parlett B N. The Symmetric Eigenvalue Problem. Philadelphia, PA: SIAM, 1998
Saad Y. Numerical Methods for Large Eigenvalue Problems. Manchester: Manchester University Press, 1992
Simoncini V. Variable accuracy of matrix-vector products in projection methods for eigencomputation. SIAM J Numer Anal, 2005, 43: 1155–1174
Simoncini V, Szyld D B. Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J Sci Comput, 2003, 25: 454–477
Sleijpen G L G, Van der Vorst H. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J Matrix Anal Appl, 1996, 17: 401–425
Stathopoulos A. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory, I: Seeking one eigenvalue. SIAM J Sci Comput, 2007, 29: 2162–2188
Stewart G W. Matrix Algorithms II: Eigensystems. Philadelphia, PA: SIAM, 2001
van der Vorst H. Computational Methods for Large Eigenvalue Problems. North Hollands: Elsevier, 2002
Voss H. A new justification of the Jacobi-Davidson method for large eigenproblems. Linear Algebra Appl, 2007, 424: 448–455
Xue F, Elman H. Fast inexact implicitly restarted Arnoldi method for generalized eigenvalue problems with spectral transformation. SIAM J Matrix Anal Appl, 2012, 33: 433–459
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Jia, Z., Li, C. Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method. Sci. China Math. 57, 1733–1752 (2014). https://doi.org/10.1007/s11425-014-4791-5
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DOI: https://doi.org/10.1007/s11425-014-4791-5
Keywords
- subspace expansion
- expansion vector
- inexact
- low or modest accuracy
- the SIRA method
- the JD method
- inner iteration
- outer iteration