The FD-method for an eigenvalue problem in a case where the base problem has eigenvalues of arbitrary multiplicities in a Hilbert space
DOI:
https://doi.org/10.15407/dopovidi2015.05.026Keywords:
eigenvalue problem, functional-discrete method, Hilbert space, multiple eigenvalues, super-exponentially convergent algorithm×Abstract
A new algorithm for the eigenvalue problems for linear self-adjont operators in the form of sum A+B with a discrete spectrum in a Hilbert space is proposed and justified. The algorithm is based on the approximation of B by an operator B- such that the eigenvalue problem for A+B- is computationally simpler than that for A+B. The operator A+B- is allowed to have multiple eigenvalues. The algorithm for this eigenvalue problem is based on the homotopy idea. It provides the super-exponential convergence rate, i. e. the rate faster than the convergence rate of a geometrical progression with the ratio, which is inversely proportional to the index of the eigenvalue under consideration. The eigenpairs can be computed in parallel for all prescribed indices. We supply a numerical example which supports the developed theory.
Downloads
References
Akulenko L.D., Nesterov S.V. High-precision methods in eigenvalue problems and their applications, Boca
Raton: Chapman&Hall/CRC, 2005.
Pryce J. Numerical solution of Sturm–Liouville problems, Oxford: Oxford Univ. Press, 1993.
Rellich F. Math. Ann., 1937, 113, Mitt. I: 600–619 (in German).
Rellich F. Math. Ann., 1937, 113, Mitt. II: 677–685 (in German).
Rellich F. Math. Ann., 1939, 116, Mitt. III: 555–570 (in German).
Armstrong M.A. Basic topology, Berlin: Springer, 1983. https://doi.org/10.1007/978-1-4757-1793-8
Allgower E., Georg K. Introduction to numerical continuation methods, Colorado: Colorado State University, 1990. https://doi.org/10.1007/978-3-642-61257-2
Makarov V. L. Dokl. Akad. Nauk. SSSR, 1991, 320, No. 1: 34–39 (in Russian).
Bandyrskii B. I., Makarov V. L., Ukhanev O. L. J. Comp. Appl. Math., 2000, 85, No 1: 1–60 (in Ukrainian).
Gavrilyuk I.P., Makarov V. L., Popov A.M. J. Numer. Appl. Math., 2010, 100, No 1: 60–81.
Bandyrskii B. I., Gavrilyuk I.P., Lazurchak I. I., Makarov V. L. Comput. Methods Appl. Math., 2005, 5, No 4: 362–386. https://doi.org/10.2478/cmam-2005-0017
Gavrilyuk I.P., Makarov V. L. Super-exponentially convergent parallel algorithm for eigenvalue problems in Hilbert spaces: Proc. of the Intern. Conf. "DETA 2009", Kaunas: Technologija, 2009: 86–92.
Makarov V. L., Vinokur V.V. J. Math. Sci., 1995, 77, No 5: 3399–3405. https://doi.org/10.1007/BF02367984
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Reports of the National Academy of Sciences of Ukraine
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
