Flexible rational approximation and its application for matrix functions

Abstract: This paper proposes a unique optimization approach for estimating the minimax rational approximation and its application for evaluating matrix functions. Our method enables the extension to generalized rational approximations and has the flexibility of adding constraints. In particular, the latter allows us to control specific properties preferred in matrix function evaluation. For example, in the case of a normal matrix, we can guarantee a bound over the condition number of the matrix, which one needs to invert for evaluating the rational matrix function. We demonstrate the efficiency of our approach for several applications of matrix functions based on direct spectrum filtering.