An Estimate of Approximation of an Analytic Function of Two Matrices by a Polynomial

IF 0.8 Q2 MATHEMATICS

Lobachevskii Journal of Mathematics Pub Date : 2024-08-22 DOI:10.1134/s1995080224601206

V. G. Kurbatov, I. V. Kurbatova

引用次数: 0

Abstract

Let $U,V\subseteq\mathbb{C}$ be open convex sets, and $z_{1}$, $z_{2}$, $\dots,z_{N}\in U$ and $w_{1}$, $w_{2}$, $\dots,w_{M}\in V$ be (maybe repetitive) points. Let $f:\,U\times V\to\mathbb{C}$ be an analytic function. Let the interpolating polynomial $p$ be determined by the values of $f$ on the rectangular grid $(z_{i},w_{j})$, $i=1,2,\dots,N$, $j=1,2,\dots,M$. Let $A$ and $B$ be matrices of the sizes $n\times n$ and $m\times m$, respectively. The function $f$ of $A$ and $B$ can be defined by the formula

$$f(A,B)=\frac{1}{(2\pi i)^{2}}\int\limits_{\Gamma_{1}}\int\limits_{\Gamma_{2}}f(\lambda,\mu)(\lambda\mathbf{1}-A)^{-1}\otimes(\mu\mathbf{1}-B)^{-1}\,d\mu\,d\lambda,$$

where $\Gamma_{1}$ and $\Gamma_{2}$ surround the spectra $\sigma(A)$ and $\sigma(B)$, respectively; $p(A,B)$ is defined in the same way. An estimate of $||f(A,B)-p(A,B)||$ is given.

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用多项式逼近两个矩阵的解析函数的估计值

AbstractLet$U,Vsubseteq\mathbb{C}$ be open convex sets, and $z_{1}$,$z_{2}$,$\dots,z_{N}\in U$ and $w_{1}$,$w_{2}$,$\dots,w_{M}\in V$ be (maybe repetitive) points.让（f:\,U\times Vto\mathbb{C}\) 是一个解析函数。让内插多项式（p）由矩形网格（(z_{i},w_{j})）上的（f）值决定，（i=1,2,（dots,N）,（j=1,2,（dots,M））。让（A）和（B）分别是大小为（n乘以n）和（m乘以m）的矩阵。公式$f(A,B)=frac{1}{(2\pi i)^{2}}\intlimits_{\Gamma_{1}}\intlimits_{\Gamma_{2}}f(\lambda、\其中 $\Gamma_{1}$ 和 $\Gamma_{2}$ 分別圍繞著 $\sigma(A)$ 和 $\sigma(B)$ 的光譜；\p(A,B)的定义是一样的。给出了 $||f(A,B)-p(A,B)||$ 的估计值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Lobachevskii Journal of Mathematics MATHEMATICS-

CiteScore

1.50

自引率

42.90%

发文量

127

期刊介绍： Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.