Muller's method

[fgittins]

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

[avik-pal]

Given that this method only works for univariate functions, it might be too restrictive for NonlinearSolve.jl

This would be great to have in SimpleNonlinearSolve

[fgittins]

Thanks for the feedback. I had some time so I put together an initial implementation at this algorithm with some simple tests. If it looks reasonable, I can make a pull request.

One thing to note is that differently to other non-linear solvers, Muller's method requires three initial guesses. This would need to be made clear in subsequent documentation.

[avik-pal]

yes open the pr