Fractional double Newton step properties for polynomials with all real zeros

Abstract

When doubling the Newton step for the computation of the largest zero of a real polynomial withall real zeros, a classical result shows that the iterates never overshoot the largest zero of the derivativeof the polynomial.Here we show that when the Newton step is extended by a factor $\theta$ with $1<\theta <2$, the iterates cannotovershoot the zero of a different function. When $\theta =2$, our result reduces to the one for the double-step case.An analogous property exists for the smallest zero.