Newton Raphson method in two dimensions of any order of convergence

Question

Are there any generalisations to the Newton Raphson method of solving nonlinear systems of equations in multiple dimensions of any order of convergence? Is that problem solved(just like in one dimension, Schroder's method, Householder's method, etc.) Does it have a proof? I am especially interested in a method that works for any preferred order of convergence. Thank you, in advance.

Answer 1

There are higher order methods, the multi-dimensional extension of Halley's method is even named as Euler-Chebyshev method. However you will need the second order derivative which is a 3rd stage tensor and do linear algebra operations with it.

A fair measure for the efficiency of a method is the Ostrowski index which measures the convergence rate in terms of function evaluations or their equivalents. Using automatic differentiation the first derivative of a general function is equivalent to 2 additional function evaluations, thus the Ostrowski index of Newtons method distributes the convergence exponent of $2$ per Newton step to the 3 function evaluations and is thus $\sqrt[3]2=1.2599…$. Higher order methods have smaller indices, $\sqrt[6]3=1.2009…$ for a 3rd order method using second derivaties etc.

Thus variations of Newtons method or, if applicable, the secant method, will in general converge fastest with the least logistic effort in computing derivatives.