Convergence theory of exact interpolation scheme for computing several eigenvectors

Abstract

An asymptotic convergence analysis of a new multilevel method for numerical solution of eigenvalues and eigenvectors of symmetric and positive definite matrices is performed. The analyzed method is a generalization of the original method that has recently been proposed by R. Kužel and P. Vaněk (DOI: 10.1002/nla.1975) and uses a standard multigrid prolongator matrix enriched by one full column vector, which approximates the first eigenvector. The new generalized eigensolver is designed to compute m > 1 eigenvectors. Their asymptotic convergence in terms of the generalized residuals is proved, and its convergence factor is estimated. The theoretical analysis is illustrated by numerical examples.

Energy Error Estimates of Subspace Method and Multigrid Algorithm for Eigenvalue Problems

“…This schemes decompose a large scale eigenvalue problem into standard linear equations plus small scale eigenvalue problems such that the choices of linear solvers and eigensolvers are both free. We also find papers [20,26,35] also give the similar method for symmetric positive eigenvalue problems which is only based on the inverse iteration or Rayleigh quotient method. Actually, the method and results in this paper can also provide a reasonable analysis for their schemes.…”