Abstract: In a recent work Shub (Found. Comput. Math. 9:171?178, 2009), Shub obtained a new upper bound for the number of steps needed to continue a known zero ?0 of a system f0, to a zero ?T of an input system fT , following the path of pairs ( ft, ?t), where ft, t ? [0, T ] is a polynomial system and ft(?t) = 0. He proved that if one can choose the step-size in an optimal way, then the number of steps is essentially bounded by the length of the path of ( ft, ?t) in the so-called condition metric. However, the proof of that result in Shub (Found. Comput. Math. 9:171?178, 2009) is not constructive. We give an explicit description of an algorithm which attains that complexity bound, including the choice of step-size.

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