标题： 求解涉及Heun函数的超越方程组 显示英文标题

标题： Solving systems of transcendental equations involving the Heun functions

摘要： Heun 函数在现代物理学中有广泛的应用，并有望在21世纪的物理问题中取代超几何函数。 然而，这些函数的数值工作很复杂，需要完善Heun函数理论中的空白，并且还需要创建能够高效处理它们的新算法。 我们提出了一种基于Müller算法的新算法，用于求解具有两个复变量的两个非线性超越方程的系统。 该新算法对于涉及Heun函数的系统特别有用，在这种情况下，与牛顿法和Broyden法相比，新算法给出了明显更好的结果。 作为其在物理学中应用的一个例子，新算法被用来寻找由Regge-Wheeler方程描述的Schwarzschild黑洞的准正模(QNM)。 我们的方法得到的数值结果与已发表的QNM频率进行了比较，发现它们在很大程度上是一致的。 还讨论了由Teukolsky主方程描述的Kerr黑洞的准正模(QNM)。 显示英文摘要

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 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.