  We propose a new algorithm for solving a system of two nonlinear
transcendental equations with two complex variables based on the Muller
algorithm. The two-dimensional Muller algorithm is tested on systems of
different type and is found to work comparably to Newton's method and Broyden's
method in many cases. The new algorithm is particularly useful in systems
featuring the Heun functions whose complexity may make the already known
algorithms not efficient enough or not working at all. In those specific cases,
the new algorithm gives distinctly better results than the other two 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.
