On local quadratic convergence of inexact simplified Jacobi–Davidson method

Abstract

For the Hermitian eigenproblems, we prove local quadratic convergence of the inexact simplified Jacobi–Davidson method when the involved relaxed correction equation is solved by a standard Krylov subspace iteration. This method then shows local cubic convergence rate when the relaxed correction equation is solved to a prescribed precision proportional to the norm of the current residual. As a by-product, we obtain local cubic convergence of the simplified Jacobi–Davidson method. These results significantly improve the existing ones that show only local linear convergence for the inexact simplified Jacobi–Davidson method, which lead to local quadratic convergence for the simplified Jacobi–Davidson method when the tolerance of the inexact solve is particularly set to be zero. Numerical experiments confirm these theoretical results.