The Schwarzian-Newton method for solving nonlinear equations, with applications
For researchers in numerical analysis and applied mathematics, the method offers a higher-order alternative to Newton's method for specific function classes, though its applicability is limited to functions with constant Schwarzian derivative.
The paper introduces the Schwarzian-Newton method, a fourth-order iterative method for solving nonlinear equations that is exact for functions with constant Schwarzian derivative. It demonstrates the method's convergence conditions and its application to fast and reliable inversion of cumulative distribution functions (gamma and beta) and elliptic integrals.
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.