Finding multiple roots of nonlinear algebraic equations using s-system methodology

Abstract

The solution of nonlinear algebraic equations is a well-known problem in many fields of science and engineering. Several types of numerical methods exist, each with their advantages and disadvantages. S-system methodology provides a novel approach to this problem. The result, as judged by extensive numerical tests, is an effective method for finding the positive real roots of nonlinear algebraic equations. The motivation for this method begins with the observation that any system of nonlinear equations composed of elementary functions can be recast in a canonical nonlinear form known as an S-system. When the derivatives of these ordinary first-order nonlinear differential equations are zero, the resulting nonlinear algebraic equations can be represented as an underdetermined set of linear algebraic equations in the logarithms of the variables. These linear equations, together with a set of simple nonlinear constraints, determine the multiple steady-state solutions of the original system that are positive real. The numerical solution is obtained by approximating the constraints as S-system equations in steady state to yield a determined set of linear algebraic equations, which then can be solved iteratively. This method has been implemented and the experience to date suggests that the method is robust and efficient. The rate of convergence to the solutions is quadratic. A combinatorial method for selecting initial conditions has led to the identification of several, and in many cases all, of the positive real solutions for the set of problems tested. It does so without resorting to analysis in the complex domain, and without having to make random or problem-specific provisions for initializing the procedure. There is an obvious parallel implementation of the algorithm, and there are additional generalizations and efficiencies yet to be realized.