Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials
-
Miloud Sadkane
Abstract
An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.
Acknowledgements
The author thanks Mickaël Robbé and Roger Sidje for their remarks on an earlier version of the paper. He also thanks the referees for their helpful comments and suggestions.
References
[1] J. Berns-Müller and A. Spence, Inexact inverse iteration with variable shift for nonsymmetric generalized eigenvalue problems, SIAM J. Matrix Anal. Appl. 28 (2006), no. 4, 1069–1082. 10.1137/050623255Search in Google Scholar
[2] T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder and F. Tisseur, NLEVP: A collection of nonlinear eigenvalue problems, ACM Trans. Math. Software 39 (2013), no. 2, Article ID 7. 10.1145/2427023.2427024Search in Google Scholar
[3] T. Betcke and D. Kressner, Perturbation, extraction and refinement of invariant pairs for matrix polynomials, Linear Algebra Appl. 435 (2011), no. 3, 514–536. 10.1016/j.laa.2010.06.029Search in Google Scholar
[4] M. A. Freitag, P. Kürschner and J. Pestana, GMRES convergence bounds for eigenvalue problems, Comput. Methods Appl. Math. 18 (2018), no. 2, 203–222. 10.1515/cmam-2017-0017Search in Google Scholar
[5] M. A. Freitag and A. Spence, Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem, Electron. Trans. Numer. Anal. 28 (2007/08), 40–64. Search in Google Scholar
[6] M. A. Freitag and A. Spence, A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems, IMA J. Numer. Anal. 28 (2008), no. 3, 522–551. 10.1093/imanum/drm036Search in Google Scholar
[7] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. Search in Google Scholar
[8] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University, Baltimore, 1996. Search in Google Scholar
[9] G. H. Golub and Q. Ye, Inexact inverse iteration for generalized eigenvalue problems, BIT 40 (2000), no. 4, 671–684. 10.1023/A:1022388317839Search in Google Scholar
[10] S. Hammarling, C. J. Munro and F. Tisseur, An algorithm for the complete solution of quadratic eigenvalue problems, ACM Trans. Math. Software 39 (2013), no. 3, Article ID 18. 10.1145/2450153.2450156Search in Google Scholar
[11] D. Kressner, A block Newton method for nonlinear eigenvalue problems, Numer. Math. 114 (2009), no. 2, 355–372. 10.1007/s00211-009-0259-xSearch in Google Scholar
[12] Y.-L. Lai, K.-Y. Lin and W.-W. Lin, An inexact inverse iteration for large sparse eigenvalue problems, Numer. Linear Algebra Appl. 4 (1997), no. 5, 425–437. 10.1002/(SICI)1099-1506(199709/10)4:5<425::AID-NLA117>3.0.CO;2-GSearch in Google Scholar
[13] R. Nasser and M. Sadkane, Convergence and preconditioning of inexact inverse subspace iteration for generalized eigenvalue problems, Comput. Methods Appl. Math. 20 (2020), no. 2, 343–359. 10.1515/cmam-2018-0212Search in Google Scholar
[14] M. Robbé and M. Sadkane, Exact and inexact breakdowns in the block GMRES method, Linear Algebra Appl. 419 (2006), no. 1, 265–285. 10.1016/j.laa.2006.04.018Search in Google Scholar
[15] M. Robbé, M. Sadkane and A. Spence, Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 (2009), no. 1, 92–113. 10.1137/060673795Search in Google Scholar
[16] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2011. 10.1137/1.9781611970739Search in Google Scholar
[17] G. W. Stewart and J. G. Sun, Matrix Perturbation Theory, Academic Press, Boston, 1990. Search in Google Scholar
[18] D. B. Szyld and F. Xue, Several properties of invariant pairs of nonlinear algebraic eigenvalue problems, IMA J. Numer. Anal. 34 (2014), no. 3, 921–954. 10.1093/imanum/drt026Search in Google Scholar
[19] F. Xue and H. C. Elman, Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation, Linear Algebra Appl. 435 (2011), no. 3, 601–622. 10.1016/j.laa.2010.06.021Search in Google Scholar
[20] Q. Ye and P. Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra Appl. 434 (2011), no. 7, 1697–1715. 10.1016/j.laa.2010.08.001Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Abstract
An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.
Acknowledgements
The author thanks Mickaël Robbé and Roger Sidje for their remarks on an earlier version of the paper. He also thanks the referees for their helpful comments and suggestions.
References
[1] J. Berns-Müller and A. Spence, Inexact inverse iteration with variable shift for nonsymmetric generalized eigenvalue problems, SIAM J. Matrix Anal. Appl. 28 (2006), no. 4, 1069–1082. 10.1137/050623255Search in Google Scholar
[2] T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder and F. Tisseur, NLEVP: A collection of nonlinear eigenvalue problems, ACM Trans. Math. Software 39 (2013), no. 2, Article ID 7. 10.1145/2427023.2427024Search in Google Scholar
[3] T. Betcke and D. Kressner, Perturbation, extraction and refinement of invariant pairs for matrix polynomials, Linear Algebra Appl. 435 (2011), no. 3, 514–536. 10.1016/j.laa.2010.06.029Search in Google Scholar
[4] M. A. Freitag, P. Kürschner and J. Pestana, GMRES convergence bounds for eigenvalue problems, Comput. Methods Appl. Math. 18 (2018), no. 2, 203–222. 10.1515/cmam-2017-0017Search in Google Scholar
[5] M. A. Freitag and A. Spence, Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem, Electron. Trans. Numer. Anal. 28 (2007/08), 40–64. Search in Google Scholar
[6] M. A. Freitag and A. Spence, A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems, IMA J. Numer. Anal. 28 (2008), no. 3, 522–551. 10.1093/imanum/drm036Search in Google Scholar
[7] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. Search in Google Scholar
[8] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University, Baltimore, 1996. Search in Google Scholar
[9] G. H. Golub and Q. Ye, Inexact inverse iteration for generalized eigenvalue problems, BIT 40 (2000), no. 4, 671–684. 10.1023/A:1022388317839Search in Google Scholar
[10] S. Hammarling, C. J. Munro and F. Tisseur, An algorithm for the complete solution of quadratic eigenvalue problems, ACM Trans. Math. Software 39 (2013), no. 3, Article ID 18. 10.1145/2450153.2450156Search in Google Scholar
[11] D. Kressner, A block Newton method for nonlinear eigenvalue problems, Numer. Math. 114 (2009), no. 2, 355–372. 10.1007/s00211-009-0259-xSearch in Google Scholar
[12] Y.-L. Lai, K.-Y. Lin and W.-W. Lin, An inexact inverse iteration for large sparse eigenvalue problems, Numer. Linear Algebra Appl. 4 (1997), no. 5, 425–437. 10.1002/(SICI)1099-1506(199709/10)4:5<425::AID-NLA117>3.0.CO;2-GSearch in Google Scholar
[13] R. Nasser and M. Sadkane, Convergence and preconditioning of inexact inverse subspace iteration for generalized eigenvalue problems, Comput. Methods Appl. Math. 20 (2020), no. 2, 343–359. 10.1515/cmam-2018-0212Search in Google Scholar
[14] M. Robbé and M. Sadkane, Exact and inexact breakdowns in the block GMRES method, Linear Algebra Appl. 419 (2006), no. 1, 265–285. 10.1016/j.laa.2006.04.018Search in Google Scholar
[15] M. Robbé, M. Sadkane and A. Spence, Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 (2009), no. 1, 92–113. 10.1137/060673795Search in Google Scholar
[16] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2011. 10.1137/1.9781611970739Search in Google Scholar
[17] G. W. Stewart and J. G. Sun, Matrix Perturbation Theory, Academic Press, Boston, 1990. Search in Google Scholar
[18] D. B. Szyld and F. Xue, Several properties of invariant pairs of nonlinear algebraic eigenvalue problems, IMA J. Numer. Anal. 34 (2014), no. 3, 921–954. 10.1093/imanum/drt026Search in Google Scholar
[19] F. Xue and H. C. Elman, Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation, Linear Algebra Appl. 435 (2011), no. 3, 601–622. 10.1016/j.laa.2010.06.021Search in Google Scholar
[20] Q. Ye and P. Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra Appl. 434 (2011), no. 7, 1697–1715. 10.1016/j.laa.2010.08.001Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity
- Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation
- Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers
- Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces
- A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space
- Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
- Dual System Least-Squares Finite Element Method for a Hyperbolic Problem
- Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes
- Numerical Analysis of a Finite Element Approximation to a Level Set Model for Free-Surface Flows
- Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials
- Convergence Theory for IETI-DP Solvers for Discontinuous Galerkin Isogeometric Analysis that is Explicit in ℎ and 𝑝
- Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary
Articles in the same Issue
- Frontmatter
- Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity
- Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation
- Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers
- Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces
- A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space
- Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
- Dual System Least-Squares Finite Element Method for a Hyperbolic Problem
- Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes
- Numerical Analysis of a Finite Element Approximation to a Level Set Model for Free-Surface Flows
- Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials
- Convergence Theory for IETI-DP Solvers for Discontinuous Galerkin Isogeometric Analysis that is Explicit in ℎ and 𝑝
- Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary
