Improving the efficiency of a polynomial system solver via a reordering technique

Methods of interval arithmetic can be used to reliably find with certainty all solutions to nonlinear systems of equations. In such methods, the system is transformed into a linear interval system and a preconditioned interval Gauss-Seidel method may then be used to compute such solution bounds. In this work, a new heuristic for solving polynomial systems is presented, called reordering technique. The proposed technique constitutes a preprocessing step to interval Gauss-Seidel method to improve the overall efficiency of an interval Newton method. The key idea is to exploit some properties of the original polynomial system, expressed by two suitable permutation matrices, by reordering the resulted linearized system. Numerical experiments have been shown that the permuted system can be solved efficiently when it is combined with an interval Newton method, like Hansen’s algorithm. We present the motivation for the reordering scheme and support these arguments by a sample of several test results.