An efficient predictor-corrector method for solving nonlinear equations
Let \(F: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\). The paper presents a new algorithm for obtaining a solution of a nonlinear system of equations \(F(x)=0\). The method is based on the presentation of the system in the form \(H(x,t)=0\) where \(H:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^ n\), \(H(x,1)=F(x)\) and the system \(H(x,0)=0\) has a known solution \(x^ 0\in {\mathbb{R}}^ n\). The predictor- part of the method is an ordinary Euler method, but the corrector-part converges more rapidly than well-known types of the method and of Newton's method. Three different examples are presented which show the efficiency of the algorithm realized in FORTRAN IV. At the convergence criterion \(\epsilon =10^{-6}\) the new method gives the solution from 7 to 41 times faster (by the number of iterates) than the usual Euler method and from 15 to 82 times faster than Newton's method.
scientific article; zbMATH DE number 4028760