Decay bounds for the numerical quasiseparable preservation in matrix functions

Given matrices A and B such that B=f(A), where f(z) is a holomorphic function, we analyze the relation between the singular values of the off-diagonal submatrices of A and B. We provide a family of bounds which depend on the interplay between the spectrum of the argument A and the singularities of the function. In particular, these bounds guarantee the numerical preservation of quasiseparable structures under mild hypotheses. We extend the Dunford-Cauchy integral formula to the case in which some poles are contained inside the contour of integration. We use this tool together with the technology of hierarchical matrices (H-matrices) for the effective computation of matrix functions with quasiseparable arguments.