Some polynomial and Toeplitz matrix computations

Abstract

Part 1. Approximate Evaluation of Polynomial Zeros O(n2(1og2n+log b)) arithmetic operations or O( n(log2n+log b) parallel steps, n processors suffice in order to approximate with absolute error ~ 2mb to all the complex zeros of an n-th degree polynomial p(x) whose coefficients have moduli < 2m• If we only need such an approximation to a single zero of p(x), then O(n log n(n+log b)) arithmetic operations or O(log n(log2n+log b)) steps and n+n/(loin+log b) processors suffice (which places the latter problem in NC); furthermore if all the zeros are real, then we arrive at the bounds O(n log n(log3n+log b)), O(log n(log3+log b)), and n, respectively. Those estimates are reached in computations with O(nb) binary bits where the polynomial· has integer coefficients. This also implies a simple proof of the Boolean circuit complexity estimates for the approximation of all the complex zeros of p(x), announced in 1982 and partly proven by Schonhage. The computations rely on recursive application of Turan's proximity test of 1968, on its more recent extensions to root radii computations, and on contour integration via FFT within our modifications of the known geometric constructions for search and exclusion.