Solving systems of transcendental equations involving the Heun functions
Solving systems of transcendental equations involving the Heun functions
复制标题
求解涉及 Heun 函数的超越方程组
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
D. Staicova
中科院分区：
文献类型：
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作者：
P. Fiziev;D. Staicova
Heun函数在现代物理中有着广泛的应用，有望在21世纪的物理问题中取代超几何函数。然而，这些函数的数值计算是复杂的，需要填补Heun函数理论中的空白，同时还需要创建能够有效处理它们的新算法。在Muller算法的基础上，提出了求解两个复变量非线性超越方程组的新算法。新算法在以Heun函数为特征的系统中特别有用，对于它们，新算法给出的结果明显优于牛顿和布洛登的方法。作为其在物理中的应用实例，将该算法用于求解由Regge-Wheeler方程描述的Schwarzschild黑洞的准正态模态（QNM）。用本文方法得到的数值结果与已发表的QNM频率进行了比较，发现两者在很大程度上吻合。还讨论了用Teukolsky主方程描述的Kerr黑洞的QNM。
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 Muller 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.