An efficient algorithm for finding all solutions of nonlinear equations using parallelogram LP test

Abstract

This paper presents an efficient algorithm for finding all solutions of nonlinear equations using linear programming. This algorithm is based on a simple test (called the LP test) for nonexistence of a solution to a system of nonlinear equations in a given region. In the conventional LP test, a system of nonlinear equations is formulated as a linear programming problem by surrounding component nonlinear functions by rectangles. In the proposed algorithm, we first use rectangles, and when the nonlinearity of functions becomes weak, we switch to parallelograms. It is shown that we can use the dual simplex method throughout the algorithm by applying the variable transformation to the oblique coordinate system, by which the LP test becomes more efficient. Moreover, since polygons with proper sizes are used, the LP test becomes more powerful. By numerical examples, it is shown that the proposed algorithm is more efficient than the conventional algorithms using rectangles only or parallelograms only. We also consider the special case where component nonlinear functions are locally convex and monotone, and propose an efficient LP test algorithm using rectangles and triangles.