Computational Implementation of the Multivariate Halley Method for Solving Nonlinear Systems of Equations

摘要

Halley's method for the solution of systems of equations is an iterative procedure which converges cubically under favorable conditions. The multivariate version requires the solution of two linear systems of equations with the same coefficient matrix, following which the correction vector is computed using componentwise multiplication and division of vectors. This report describes a general-purpose computer program which implements this method. The necessary first and second derivatives are obtained by automatic differentiation, so the user need only supply code defining the functions appearing in the system of equations. The program is written in Pascal-SC, using the new data type HESSIAN to represent dependent and independent variables. Numerical examples are given for two simple systems of equations to illustrate the use of the program and the effectiveness of the method.