论文标题
矩阵值函数合理近似的算法
Algorithms for the rational approximation of matrix-valued functions
论文作者
论文摘要
讨论了用于矩阵值函数的合理近似算法的选择，包括插入式AAA方法的变体，基于近似最小二乘拟合的RKFIT方法，矢量拟合，矢量拟合以及基于块loewner Matrix的低级别近似值的方法。提出了一种新方法，称为块-AAA算法，基于具有基质值重量的广义Barycentric公式。所有算法均以获得的近似准确性和运行时间的方式进行比较，从模型顺序降低和非线性特征值问题（包括带有嘈杂数据的示例）中的一组问题。发现基于插值的方法通常更便宜，但是在存在基于近似方法更好的噪声的情况下，它们可能会遭受痛苦。
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.
