Extending Newton'S Method For Finding The Roots Of An Arbitrary System Of Equations

Abstract

The author presents a new technique for determining the roots of an arbitrary system of equations iteratively, where the equations can be nonlinear algebraic or transcendental, and the number of equations of the system can be greater than, less than or equal to the number of variables. When the number of equations is equal to the number of variables, the method is similar to Newton's method. The method is a hybrid symbolic-numerical method, in that it uses symbolic techniques to transform a system of algebraic equations into a regular form whenever the system has some redundant parts or does not have a full-rank Jacobian matrix. The method has a wide range of applicability. It is especially useful for applications in which we need to find some particular roots from a nonzero dimensional system of equations, i.e., a system whose set of zeroes is nonzero dimensional. Some conditions for convergence of the method are considered, with the convergence of the method claimed to still be quadratic. Several examples are presented to show that the method works nicely.