Solving systems of transcendental equations involving the Heun functions

TL;DR

This paper introduces a new algorithm based on the M"uller method for solving systems of nonlinear transcendental equations involving Heun functions, demonstrating improved performance and applications in black hole physics.

Contribution

A novel algorithm tailored for systems with Heun functions, outperforming traditional methods like Newton's and Broyden's in accuracy and efficiency.

Findings

Successfully computed black hole quasi-normal modes with high accuracy.

The new algorithm outperforms existing methods in solving Heun function systems.

Validated results against known QNM frequencies in Schwarzschild black holes.

Abstract

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 M\"uller 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.