One Approach to Construction of Bilateral Approximations Methods for Solution of Nonlinear Eigenvalue Problems

Abstract

In this paper a new approach to construction of iterative methods of bilateral approximations of eigenvalue is proposed and investigated. The conditions on initial approximation, which ensure the convergence of iterative processes, are obtained.

Methods of Bilateral Approximations for Nonlinear Eigenvalue Problems

“…The proposed approach can be applied also to the linear eigenvalue problems with respect to the spectral parameter, and if its is compared with existing approaches mentioned in the introduction for obtaining lower bounds of eigenvalues of self-adjoint spectral problems, then this approach has significant advantages which are mentioned in Podlevskyi [22], namely: does not require construction an auxiliary operator with a known spectrum as in the method of intermediate operators, and also does not require knowledge of the lower bound of the next eigenvalue (assuming that the eigenvalues are arranged in ascending order) as in the algorithms, based on inclusion theorems.…”

“…Therefore, the concept and apparatus of interval analysis are used to construct methods of bilateral approximations (see, for example [18,19]). This article is a continuation of the study proposed by the author of the approach to the construction of methods and algorithms of bilateral approximations to the eigenvalues of nonlinear with respect of spectral parameters the eigenvalue problems [20,21,22,23]. This approach does not use the concepts and apparatus of interval analysis.…”

“…The idea of the proposed approach is that for a continuous monotone in the neighborhood of a simple zero that describes the nonlinear equation, is constructed and explored some auxiliary function : [ , ] g a b R  that has the same zero as the function f and the necessary properties that allows to constract the iterative processes, which give monotone bilateral (alternate or inclused) approximations to the root of nonlinear equation [24,25,26,27].…”

On a Family of Newton-Like Methods for Bilateral Approximation of the Eigenvalues of Nonlinear Spectral Problems

“…The exception is spectral problems for polynomial operator bundles of self-adjoint operators, on which can be generalized the methods based on inclusion theorems . This work is a continuation of the study proposed by the author of the approach to construct the methods and algorithms of bilateral approximations to the eigenvalues of spectral problems, nonlinear with respect to the spectral parameter [2,3,4]. Here, to calculate a simple isolated eigenvalue, a oneparameter family of bilateral methods of Newton 's type is constructed.…”