Verified Solutions of Systems of Nonlinear Polynomial Equations

Abstract

A common problem in scientific computing is to find global solutions of a system of nonlinear polynomial equations efficiently. In this paper, we describe a possible strategy to compute verified solutions of such systems in a given box. First we use the algorithms of Sherbrooke and Patrikalakis, extended to interval arithmetic, to find intervals, which possibly contain solutions, then we verify, if these intervals really contain a zero of the system. These tests are based on the criterion of Miranda, which we further developed for polynomial systems. We show that only checking a matrix to be strictly diagonally dominant and estimating of the remainder is required. At last we quote a further criterion that can be used alternatively.