ALGORITHMS FOR THE RATIONAL APPROXIMATION OF MATRIX-VALUED FUNCTIONS\ast

A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory adaptive Antoulas-Anderson (AAA) method, the rational Krylov fitting (RKFIT) method based on approximate least squares fitting, vector fitting, and a method based on low-rank approximation of a block Loewner matrix. A new method, called the block-AAA algorithm, based on a generalized barycentric formula with matrix-valued weights, is proposed. All algorithms are compared in terms of obtained approximation accuracy and runtime on a set of problems from model order reduction and nonlinear eigenvalue problems, including examples with noisy data. It is found that interpolation-based methods are typically cheaper to run, but they may suffer in the presence of noise for which approximation-based methods perform better.