Solving Systems of Transcendental Equations Involving the Heun Functions

• 作者: Denitsa R. Staicova Plamen P. Fiziev
• 来源期刊: 美国计算数学期刊(英文)
• 学科: 数学
• 关键词: ROOT-FINDING ALGORITHM Müller ALGORITHM Two-Dimensional Müller ALGORITHM Regge-Wheeler EQUATION QUASINORMAL MODES Teukolsky Master EQUATION
• 年，卷（期）: 2012,（2）
• 页码范围: 95-105
• 页数: 11页
• 分类号: O1

摘要

The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton’s and Broyden’s methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.