Midpoint two- steps rule for the square root method

Abstract

We present a method “Midpoint-Two-Steps rule” for the square root functional iteration for enclosing zeros of a polynomial. The method combines classical square root method in its formulation a variant of the approach adopted by (Wang and WU,1985) where a Gauss-Siedel updating formula was used to accelerate the well known Meahly –Aberth third order method for finding zeros of a polynomial in interval arithmetic. Our method has the capability of converging faster than the classical square root and Meahly-Aberth third order methods. This was possible since the
predictor and corrector iterations which the algorithm entails do not jump across paths of orientation. The convergence of the midpoints and radii of the including disks which the presented method entails for the sequence of solution are coupled via the inclusion isotonicity property of circular interval arithmetic. Interestingly, it was proved that the midpoint-two steps rule for the square root method converges if and only if the second order derivative of the polynomial P(z) is inverse forcing. Theoretical numerical example has been demonstrated with constructed algorithm with high substantial of probability 1.