COMPUTATION OF THE LEADING EIGENVALUE AND THE CORRESPONDING EIGENELEMENT OF EIGENVALUE PROBLEMS WITH NONLINEAR DEPENDENCE ON THE SPECTRAL PARAMETER

The paper studies the symmetric eigenvalue problem with nonlinear dependence on the spectral parameter in a Hilbert space which is a vector lattice with a cone of positive elements. The existence of a positive simple minimum eigenvalue corresponding to a single normalised positive eigenelement is established. The approximation of the problem in a finite-dimensional subspace is investigated. Results on the convergence and error of approximations to the minimum eigenvalue and the corresponding positive eigenelement are obtained. Computational methods for solving matrix eigenvalue problems with nonlinear dependence on the spectral parameter are developed and justified. The results of numerical experiments illustrating theoretical conclusions are given.