Because of their wide applications in physics, the Heun functions 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 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 tests of the new algorithm show that for systems featuring the confluent Heun functions, it gives distinctly better results than Newton's and Broyden's methods. The new algorithm is used for finding the quasinormal modes (QNMs) of nonrotating and rotating black holes.
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