Evaluation of matrix functions with the block Lanczos algorithm

For the approximate eigenvalues and eigenvectors obtained from a block Lanczos iteration, characterizations are given in terms of vectors of polynomials. It is proven that after r steps of block Lanczos with selfadjoint iteration matrix S, the application p(S)f of a matrix polynomial and the linear functional v, q(S)f can be evaluated exactly, if p and q are polynomials of degree r − 1 and 2r − 1, respectively. For that, the Lanczos starting block should contain the vectors f and v. Based on this property a priori error bounds for the evaluation of g(S)f and v, g(S)f for more general functions g are derived. The error bounds are applied to the case of calculating the dynamic response in forced vibration problems. A numerical example is given.