A Convergence Analysis of the Inexact Simplified Jacobi–Davidson Algorithm for Polynomial Eigenvalue Problems

The simplified Jacobi–Davidson (JD) method is a variant of the JD method without subspace acceleration. If the correction equation is solved approximately, the inexact simplified JD method is obtained. In this paper, we present a new convergence analysis of the inexact simplified JD method. The analysis applies to polynomial eigenvalue problems with simple eigenpairs. We first establish a relationship between the solution of the correction equation and the residual of the approximate eigenpair. From this relationship, we find the difference of two adjacent approximate eigenvalues bounded in terms of the residual norm of the approximate eigenpair. Then we prove the convergence of the inexact simplified JD method in terms of the residual norm of the approximate eigenpair. Depending on how accurately we solve the correction equation, the convergence rate of the inexact simplified JD may take several different forms. Numerical experiments confirm the convergence analysis.