Muller's method #403
Description
What kind of problems is it mostly used for? Please describe.
Determining the roots of univariate complex functions. Muller's method is commonly used to determine the (quasi-normal) oscillation modes of neutron stars [1, 2].
Note: Given that this method only works for univariate functions, it might be too restrictive for NonlinearSolve.jl, but I thought I would suggest it anyway, since I frequently use it. Also, the algorithm is very simple so it should be easy to implement.
Describe the algorithm you’d like
The algorithm is quite simple and described well in Press et al. [3]. (See also Wikipedia [4].)
To summarise: Muller's method is a generalisation of the secant method, with the key difference being that it uses quadratic interpolation across three points (as opposed to linear interpolation among two). Solving for the roots of a quadratic is trivial and allows the method to work for complex roots.
Other implementations to know about
There is an existing Julia implementation in Roots.jl [5] and a Fortran routine in IMSL [6].
References
[1] Kokkotas & Schutz (1992)
[2] Kruger, PhD Thesis (2016)
[3] Press et al. (2007), Sec. 9.5.2, p. 466
[4] Wikipedia, Muller's method
[5] Roots.muller
[6] IMSL Math/Library Users Manual, Zanly