On the Automatic Calculation of Solutions - Different from Those Previously Obtained - of Nonlinear Systems of Equations

Abstract

Given a system of N nonlinear (algebraic or transcendental) real equations in N real unknowns, there exist a variety of numerical methods which obtain solutions of those equations. This paper presents two methods which are used to find further simple solutions - in addition to those already known a priori or from an earlier calculation. These methods have the advantage of keeping away from solutions previously calculated, saving the computer user the wasted effort entailed in converging to already known, perhaps uninteresting solutions points. The technique can also be used in avoiding previously found extreme points in function minimization. Many problems have "magnetic zeros", zeros which are converged to almost regardless of the starting guesses used. These magnetic zeros often mask out the zeros of real interest. The methods discussed are particularly effective in avoiding convergence to such magnetic zeros. Results of computer experiments ar presented.