本文讨论了矩阵值函数有理逼近的几种算法，包括插值AAA法、基于近似最小二乘拟合的RKFIT法、向量拟合的RKFIT法和基于块Loewner矩阵低秩逼近的RKFIT法。本文提出了一种基于带矩阵权值的广义重心公式的块- AAA算法。对模型降阶问题和非线性特征值问题，包括有噪声数据的例子，从近似精度和运行时间两方面对所有算法进行了比较。研究发现，基于插值的方法通常运行成本较低，但在存在噪声的情况下会受到影响，而基于近似的方法性能更好。
原文题目：Algorithms for the rational approximation of matrix-valued functions
原文：A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory AAA method, the 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.
原文作者：Ion Victor Gosea, Stefan Güttel
原文地址：https://arxiv.org/abs/2003.06410
矩阵值函数的有理逼近算法（CS NA）.pdf