Real Root Isolation of Polynomial Equations Based on Hybrid Computation

Abstract

A new algorithm for real root isolation of zero-dimensional nonsingular square polynomial systems based on hybrid computation is presented in this paper. First, approximate the (complex) roots of the given polynomial equations via homotopy continuation method. Then, for each approximate root, an initial box relying on the Kantorovich theorem is constructed, which contains the corresponding accurate root. Finally, the Krawczyk interval iteration with interval arithmetic is applied to the initial boxes so as to check whether or not the corresponding approximate roots are real and to obtain the real root isolation boxes. Moreover, an empirical construction of initial box is provided for speeding-up the computation in practice. Our experiments on many benchmarks show that the new hybrid method is very efficient. The method can find all real roots of any given zero-dimensional nonsingular square polynomial systems provided that the homotopy continuation method can find all complex roots of the equations.