Flexible rational approximation and its application for matrix functions

TL;DR

This paper introduces a flexible optimization-based method for minimax rational approximation of matrix functions, allowing constraints to control properties like condition number, demonstrated through spectrum filtering applications.

Contribution

It presents a novel optimization approach for generalized rational approximations with constraints, enhancing control over matrix function evaluations.

Findings

Efficient approximation of matrix functions using the proposed method.

Ability to impose constraints such as condition number bounds.

Successful application to spectrum filtering tasks.

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.