STOPPING CRITERIA FOR RATIONAL MATRIX FUNCTIONS OFHERMITIAN AND SYMMETRIC MATRICES

摘要

Building upon earlier work by Golub, Meurant, Strakos, and Tichy, we derive new aposteriori error bounds for Krylov subspace approximations to f (A)b, the action of a function f ofa real symmetric or complex Hermitian matrix A on a vector b. To this purpose we assume that arational function in partial fraction expansion form is used to approximate f, and the Krylov subspaceapproximations are obtained as linear combinations of Galerkin approximations to the individualterms in the partial fraction expansion. Our error estimates come at very low computational cost. Incertain important special situations they can be shown to actually be lower bounds of the error. Ournumerical results include experiments with the matrix exponential, as used in exponential integrators,and with the matrix sign function, as used in lattice quantum chromodynamics simulations, anddemonstrate the accuracy of the estimates. The use of our error estimates within accelerationprocedures is also discussed.