On local quadratic convergence of inexact simplified Jacobi-Davidson method

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.