Research ArticleOpen AccessGoogle Scholar indexed
A Compact Heart Iteration for Large Eigenvalues Problems
Hydrological Service of Israel, Jerusalem, Israel
- 1 Hydrological Service of Israel, Jerusalem, Israel
Advances in Linear Algebra & Matrix Theory·Volume 12 (2022)·Pages 24–38·Published 18 January 2022·DOI10.4236/alamt.2022.121002
Copy link · social · email
Abstract
In this paper, we present a compact version of the Heart iteration. One that requires less matrix-vector products per iteration and attains faster convergence. The Heart iteration is a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process and the use of implicit restarts. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the computed Ritz values toward their limits. Numerical experiments illustrate the usefulness of the proposed approach.
KeywordsLarge Sparse MatricesRestarted Krylov MethodsExterior EigenvaluesSymmetric MatricesMonotonicityStarting Vectors
- Baglama, J., Calvetti, D. and Reichel, L. (2003) IRBL: An Implicitly Restarted Block Lanczos Method for Large-Scale Hermitian Eigen Problems. SIAM Journal on Scientific Computing, 24, 1650-1677. https://doi.org/10.1137/S1064827501397949
- Bai, A., Demmel, J., Dongarra, J., Ruhe, A. and van der Vorst, H. (1999) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9780898719581
- Calvetti, D., Reichel, L. and Sorenson, D.C. (1994) An Implicitly Restarted Lanczos Method for Large Symmetric Eigenvalue Problems. Electronic Transactions on Numerical Analysis, 2, 1-21.
- Dax, A. (2017) The Numerical Rank of Krylov Matrices. Linear Algebra and Its Applications, 528, 185-205. https://doi.org/10.1016/j.laa.2016.07.022
- Dax, A. (2017) A New Type of Restarted Krylov Methods. Advances in Linear Algebra & Matrix Theory, 7, 18-28. https://doi.org/10.4236/alamt.2017.71003
- Dax, A. (2019) A Restarted Krylov Method with Inexact Inversions. Numerical Linear Algebra with Applications, 26, Article No. e2213. https://doi.org/10.1002/nla.2213
- Dax, A. (2019) Computing the Smallest Singular Triplets of a Large Matrix. Results in Applied Mathematic, 3, Article ID: 100006. https://doi.org/10.1016/j.rinam.2019.100006
- Dax, A. (2019) A Cross-Product Approach for Low-Rank Approximations of Large Matrices. Journal of Computational and Applied Mathematics, 369, Article ID: 112576. https://doi.org/10.1016/j.cam.2019.112576
- Demmel, J.W. (1997) Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9781611971446
- Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th Edition, Johns Hopkins University Press, Baltimore.
- Horn, R.A. and Johnson, C.R. (1985) Matrix Analysis. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511810817
- Larsen, R.M. (2001) Combining Implicit Restarts and Partial Reorthogonalization in Lanczos Bidiagonalization. Technical Report, Stanford University.
- Li, R., Xi, Y., Vecharynski, E., Yang, C. and Saad, Y. (2016) A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems. SIAM Journal on Scientific Computing, 38, A2512-A2534. https://doi.org/10.1137/15M1054493