  Real root finding of polynomial equations is a basic problem in computer
algebra. This task is usually divided into two parts: isolation and refinement.
In this paper, we propose two algorithms LZ1 and LZ2 to refine real roots of
univariate polynomial equations. Our algorithms combine Newton's method and the
secant method to bound the unique solution in an interval of a monotonic convex
isolation (MCI) of a polynomial, and have quadratic and cubic convergence
rates, respectively. To avoid the swell of coefficients and speed up the
computation, we implement the two algorithms by using the floating-point
interval method in Maple15 with the package intpakX. Experiments show that our
methods are effective and much faster than the function RefineBox in the
software Maple15 on benchmark polynomials.
