Optimal and Nearly Optimal Algorithms for Approximating Polynomial Zeros

K e y w o r d s C o m p l e x polynomial zeros, Approximation, Polynomial factorization, Parallel algorithms, Computational complexity, Sequential algorithms. 1. I N T R O D U C T I O N 1.1. T h e Subject , Some History, and a S u m m a r y of Our Resul ts The problem of solving a polynomial equation p ( x ) = 0 substantially motivated the development of mathematics throughout the centuries. As particular examples of this influence, one may recall the origin of complex numbers from the solution formulae for quadratic equations (these formulae have been known already in ancient Greece), the fundamental theorem of algebra, which states the existence of a complex solution to p ( x ) = 0 (the first celebrated proof of this theorem, given by Gauss in 1799, contained a substantial flaw, corrected by Ostrowski in 1920), and the Galois theory of 1832, which extended the earlier theorem of Ruffini 1813 and Abel 1826 on nonexistence of solution formulae in radicals for a polynomial equation of a degree n if n > 5 (such formulae have been known, since the 16 th century, for n -3 [del Ferro, Tartaglia, Cardano] and n = 4 [Ferrari]). In the absence of explicit solution formulae, numerous algorithms for approximating polynomial zeros have been proposed, and they are still appearing in great number, The author is grateful to D. Bini, P. Kirrinnis, and A. Neff, for (p)reprints of [3-7], and to A. Sadikou, for helpful comments. The results of this paper are to be presented at the 27 th Annual ACM Symposium on the Theory of Computing, 1995 (see [8]); the author is grateful to the ACM for the permission to reuse them. Supported by NSF Grant CCR. 9020690 and PSC CUNY Awards Nos. 664334 and 665301.