A Symbolic-Numerical Method for Finding a Real Solution of an Arbitrary System of Nonlinear Algebraic Equations

Abstract

In this paper, the author presents a new method for iteratively finding a real solution of an arbitrary system of nonlinear algebraic equations, where the system can be overdetermined or underdetermined and its Jacobian matrix can be of any (positive) rank. When the number of equations is equal to the number of variables and the Jacobian matrix of the system is nonsingular, the method is similar to the well-known Newton's method.The method is a hybrid symbolic-numerical method, in that we utilize some extended procedures from classical computer algebra together with ideas and algorithmic techniques from numerical computation, namely Newton's method and pseudoinverse matrices. First the symbolic techniques are used to transform an arbitrary system of algebraic equations into a set of regular systems. By regular system we mean a system whose Jacobian matrix is of full row rank. Newton-like numerical techniques are then used to find a real solution for each regular system obtained from the symbolic part of the method.The method has a wide range of applicability. It is especially useful for applications in which we need to find some particular solutions from a nonzero-dimensional manifold of real solutions of a system of equations, i.e. the system has infinitely many solutions.We find some mild conditions for the asymptotic convergence of the numerical part of our method. We prove that the asymptotic convergence of the new method is still quadratic while the robustness of the numerical part can be enhanced by techniques like damping as in the regular case. The method has been implemented in Maple andMathematica . Several examples are presented to show that the method works nicely.