The Schwarzian-Newton method for solving nonlinear equations, with applications

TL;DR

The paper introduces the Schwarzian-Newton method, a fourth-order iterative technique for solving nonlinear equations exactly for functions with constant Schwarzian derivative, with applications to distribution and elliptic integral inversions.

Contribution

It presents the Schwarzian-Newton method, a novel fourth-order solver that generalizes Halley's method and provides reliable solutions for specific mathematical problems.

Findings

The method converges under certain conditions in an interval.

It is exact for functions with constant Schwarzian derivative.

Applications include inversion of distribution functions and elliptic integrals.

Abstract

The Schwarzian-Newton method can be defined as the minimal method for solving nonlinear equations $f(x)=0$ which is exact for any function $f$ with constant Schwarzian derivative; exactness means that the method gives the exact root in one iteration for any starting value in a neighborhood of the root. This is a fourth order method which has Halley's method as limit when the Schwarzian derivative tends to zero. We obtain conditions for the convergence of the SNM in an interval and show how this method can be applied for a reliable and fast solution of some problems, like the inversion of cumulative distribution functions (gamma and beta distributions) and the inversion of elliptic integrals.