Sequential and parallel complexity of approximate evaluation of polynomial zeros

Abstract

Our new sequential and parallel algorithms establish new record upper bounds on both arithmetic and Boolean complexity of approximating to complex polynomial zeros. O( n 2 log b log n) arithmetic operations or O( n log n log ( bn)) parallel steps and n log b/log ( bn) processors suffice in order to approximate with absolute errors ⩽ 2 m−b to all the complex zeros of an nth degree polynomial p( x) whose coefficients have mod ⩽ 2 m. If we only need such an approximation to a single zero of p( x), then O( n log b log n) arithmetic operations or O(log 2 n log ( bn)) steps and ( n/log n)log b/log ( bn) processors suffice (which places the latter problem in NC, that is, in the class of problems that can be solved using polylogarithmic parallel time and a polynomial number of processors). Those estimates are reached in computations with O( bn) binary bits where the polynomial has integer coefficients. We also reach the sequential Boolean time bounds O( bn 3 log ( bn)log log( bn)) for approximating to all the zeros (very minor improvement of the bound announced in 1982 by Schönhage) and O( bn 2log log n log( bn)log log( bn)) for approximating to a single zero. Among further implications are the improvements of the known algorithms and complexity estimates for computing matrix eigenvalues, for polynomial factorization over the field of complex numbers and for solving systems of polynomial equations. The computations rely on recursive application of Turan's proximity test of 1968, on its more recent extensions to root radii computations, on contour integration via Fast Fourier transform (FFT) within geometric constructions for search and exclusion, and (for the final minor improvements of the complexity bounds) on the recursive factorization of p( x) over discs on the complex plane via numerical integration and Newton's iterations.'