The FD-method for an eigenvalue problem in a case where the base problem has eigenvalues of arbitrary multiplicities in a Hilbert space

TitleThe FD-method for an eigenvalue problem in a case where the base problem has eigenvalues of arbitrary multiplicities in a Hilbert space
Publication TypeJournal Article
Year of Publication2015
AuthorsMakarov, VL, Romaniuk, NM
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2015.05.026
Issue5
SectionMathematics
Pagination26-34
Date Published5/2015
LanguageUkrainian
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 $\overline{B}$ such that the eigenvalue problem for $A+\overline{B}$ is computationally simpler than that for $A+B$. The operator $A+\overline{B}$ 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.
Keywordseigenvalue problem, functional-discrete method, Hilbert space, multiple eigenvalues, super-exponentially convergent algorithm
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