Flexible rational approximation for matrix functions

A rational approximation is a powerful method for estimating functions using rational polynomial functions. Motivated by the importance of matrix function in modern applications and its wide potential, we propose a unique optimization approach to construct rational approximations for matrix function evaluation. In particular, we study the minimax rational approximation of a real function and observe that it leads to a series of quasiconvex problems. This observation opens the door for a flexible method that calculates the minimax while incorporating constraints that may enhance the quality of approximation and its properties. Furthermore, the various properties, such as denominator bounds, positivity, and more, make the output approximation more suitable for matrix function tasks. Specifically, they can guarantee the condition number of the matrix, which one needs to invert for evaluating the rational matrix function. Finally, we demonstrate the efficiency of our approach on several applications of matrix functions based on direct spectrum filtering.