Numerical Review of Mathieu Function Programs for Integer Orders and Real Parameters

IF 8.2 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Review Pub Date : 2025-11-06 DOI:10.1137/23m1572726

Ho-Chul Shin

引用次数: 0

Abstract

SIAM Review, Volume 67, Issue 4, Page 661-733, December 2025.
Abstract.The Mathieu function is a special function satisfying the Mathieu differential equation. Since its inception in 1868, numerous algorithms and programs have been published to calculate it, and so it is about time to review the performance of available software. First, the fundamentals of Mathieu functions are summarized such as definition, normalization, nomenclature, and methods of solution. Then, we review several programs for Mathieu functions of integer orders with real parameters and compare the results numerically by running individual software; in addition, Bessel function routines are also compared. Finally, a straightforward algorithm is recommended with codes written in MATLAB and GNU Octave.

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整数阶和实参数的Mathieu函数程序的数值回顾

SIAM评论，第67卷，第4期，661-733页，2025年12月。摘要。马蒂厄函数是满足马蒂厄微分方程的特殊函数。自1868年开始以来，已经发表了许多算法和程序来计算它，因此是时候审查可用软件的性能了。首先，总结了Mathieu函数的基本原理，如定义、归一化、命名法和求解方法。然后，我们回顾了几种具有实参数的整数阶Mathieu函数的程序，并通过运行单个软件对结果进行了数值比较；此外，还对贝塞尔函数例程进行了比较。最后，用MATLAB和GNU Octave编写了一个简单的算法。

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

SIAM Review 数学-应用数学

CiteScore

16.90

自引率

0.00%

发文量

期刊介绍： Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter. Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.