非线性特征值问题的鲁棒有理逼近

S. Güttel, G. Porzio, F. Tisseur
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引用次数: 7

摘要

. 我们开发了在复平面的给定子集上构造矩阵值函数的鲁棒(即,对给定容限可靠且与尺度无关)有理逼近的算法。我们考虑以分裂形式(即标量函数乘以常系数矩阵的和)和黑盒形式提供的矩阵值函数。我们开发了一种新的误差分析方法,并将其用于构建停止准则,每种形式一个停止准则。我们的分割表单标准添加了相对于每个标量函数的重要性所选择的权重,从而产生加权AAA算法,这是集值AAA算法的一种变体,可以保证以用户选择的精度返回有理近似值。我们提出了黑盒矩阵值函数的两阶段方法,该方法在第一阶段构建代理AAA近似,并在第二阶段对其进行细化,从而得到具有精确搜索的代理AAA算法和具有循环Leja-Bagby细化的代理AAA算法。在阶段二的每一步更新黑盒矩阵值函数的停止准则,以包含前一步的信息。当收敛发生时,我们的两阶段方法返回具有用户选择精度的有理近似值。我们从NLEVP集合中选择问题,这些问题代表了各种不同大小和性质的矩阵值函数,并使用它们对我们的算法进行基准测试。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Robust Rational Approximations of Nonlinear Eigenvalue Problems
. We develop algorithms that construct robust (i.e., reliable for a given tolerance, and scaling independent) rational approximants of matrix-valued functions on a given subset of the complex plane. We consider matrix-valued functions provided in both split form (i.e., as a sum of scalar functions times constant coefficient matrices) and in black box form. We develop a new error analysis and use it for the construction of stopping criteria, one for each form. Our criterion for split forms adds weights chosen relative to the importance of each scalar function, leading to the weighted AAA algorithm, a variant of the set-valued AAA algorithm that can guarantee to return a rational approximant with a user-chosen accuracy. We propose two-phase approaches for black box matrix-valued functions that construct a surrogate AAA approximation in phase one and refine it in phase two, leading to the surrogate AAA algorithm with exact search and the surrogate AAA algorithm with cyclic Leja–Bagby refinement. The stopping criterion for black box matrix-valued functions is updated at each step of phase two to include information from the previous step. When convergence occurs, our two-phase approaches return rational approximants with a user-chosen accuracy. We select problems from the NLEVP collection that represent a variety of matrix-valued functions of different sizes and properties and use them to benchmark our algorithms.
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