Monotone convergence of the extended Krylov subspace method for Laplace–Stieltjes functions of Hermitian positive definite matrices

The extended Krylov subspace method is known to be very efficient in many cases in which one wants to approximate the action of a matrix function f(A)f(A) on a vector b, in particular when f belongs to the class of Laplace–Stieltjes functions. We prove that the Euclidean norm of the error decreases strictly monotonically in this situation when A   is Hermitian positive definite. Similar results are known for the (polynomial) Lanczos method for f(A)bf(A)b, and we demonstrate how the techniques of proof used in the polynomial Krylov case can be transferred to the extended Krylov case.