On Convergence Rates of Root-Seeking Methods

[ttw]

I'm surprised that they didn't recognize Halley's method; both Halley's method and Chebychev's method are well known. I didn't recognize it as I was careless and I was more interested in the extended method. There is a method (from the 1960s) which is reminiscent of correlated sampling in Monte Carlo. One choses another easy function g(x) computes roots of f(x)+c*g(x) for values of c approaching zero. Sometimes this will make thing easier.

In "real life" (equations I was paid for solving), none of the higher order methods usually worked. An exception was in finding roots of orthogonal polynomials for getting Gaussian Integration nodes. Usually, everything was at best linearly convergent.

I always seem to end up using some version of the Levenberg-Marquardt method.

This is also of interest: https://en.wikipedia.org/wiki/Jenkins%E2..._algorithm
https://ac.els-cdn.com/S0898122107006694...c37e83c178