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.
Highlights
Many theoretical and applied problems of mathematical physics, mechanics and engineering sciences lead to eigenvalue problems
2) The knowledge of boundaries for eigenvalues makes it possible in many cases to estimate the reliability of iterative approximation, that is at every step of iterative process to obtain the comfortable a posteriori estimate of error calculations
This paper presents a new approach to construction the methods and algorithms of bilateral approximations of eigenvalues for nonlinear eigenvalue problems, wich have supra-linear speed of convergence
Summary
Many theoretical and applied problems of mathematical physics, mechanics and engineering sciences lead to eigenvalue problems. The idea of the approach proposed is that for a continuous monotone in a neighborhood of simple zero a,b function f : a,b R , which describes the nonlinear equation some auxiliary function g : a,b R that has the same zero as the function f and such necessary properties that allow one to build the iterative processes which give monotonous bilateral (alternating or including) approximations to the root of a nonlinear equation [17,18,19] is constructed and investigated Within this approach, the algorithms of bilateral analogues of Newton’s method for finding the eigenvalues of nonlinear spectral problems are constructed and justified. The conditions on the initial approximations which provide the alternate approximations to the eigenvalue from two sides and ensure the convergence of iterative process, are obtained
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