When doubling the Newton step for the computation of the largest zero of a real polynomial with all real zeros, a classical result shows that the iterates never overshoot the largest zero of the derivative of the polynomial. Here we show that when the Newton step is extended by a factor θ with 1 < θ < 2, the iterates cannot overshoot the zero of a di.erent function. When θ = 2, our result reduces to the one for the double-step case. An analogous property exists for the smallest zero.
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