A Universal Bound for the Average Cost of Root Finding

We analyze a path-lifting algorithm for finding an approxim ate zero of a complex polynomial, and show that for any polynomial with distinct r oots in the unit disk, the average number of iterates this algorithm requires is universally bounded by a constant times the log of the condition number. In particular, this bound is independent of the degr e d of the polynomial. The average is taken over initial valueszwith |z| = 1+1/d using uniform measure. Stony Brook IMS Preprint #2009/1 March 2009 CONTENTS

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