Computation of the Leading Eigenvalue and the Corresponding Eigenelement of Eigenvalue Problems with Nonlinear Dependence on the Spectral Parameter

The paper studies a symmetric eigenvalue problem with nonlinear dependence on thespectral 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 normalisedpositive eigenelement is established. The approximation of the problem in a finite-dimensionalsubspace is investigated. Results on the convergence and error of approximations to the minimumeigenvalue and the corresponding positive eigenelement are obtained. Computational methods forsolving matrix eigenvalue problems with nonlinear dependence on the spectral parameter aredeveloped and justified. The results of numerical experiments illustrating theoretical conclusionsare given.