矩阵函数高阶导数的统一框架

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-02-08 DOI:10.1137/23m1580589

Emanuel H. Rubensson

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引用次数: 0

摘要

SIAM 矩阵分析与应用期刊》，第 45 卷第 1 期，第 504-528 页，2024 年 3 月。摘要。我们提出了[math]形式矩阵函数的一般偏导数理论，其中[math]是多变量矩阵路径（[math]）。基于马蒂亚斯 [SIAM J. Matrix Anal. Appl.我们利用这种分块形式推导出高阶导数的存在条件和广义 Daleckiĭ-Kreĭn 公式。我们证明，该公式的某些特殊化会导致量子扰动理论的经典公式。我们展示了我们的结果与早先关于高阶弗雷谢特导数的结果之间的关系。我们介绍了复步近似的块形式，并说明了这些块形式与通过上三角形式求导的关系。我们将用数值示例来说明这些关系。

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A Unifying Framework for Higher Order Derivatives of Matrix Functions

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 504-528, March 2024.
Abstract. We present a theory for general partial derivatives of matrix functions of the form [math], where [math] is a matrix path of several variables ([math]). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610–620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Daleckiĭ–Kreĭn formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fréchet derivatives. Block forms of complex step approximations are introduced, and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.