Algebraic complexity of computing polynomial zeros

Abstract

Using the power sum techniques of Turan, we evaluate all the complex zeros of an nth degree univariate polynomial with relative errors ⩽ ϵ using O( n 2 log n ( n log n + log(1/ ϵ))) arithmetic operations. O( n log n log(1/ ϵ)) operations suffice to approximate (with relative error ⩽ ϵ) a single complex zero using Turan's method and all the zeros if they are real using Graeffe's method. In all cases n processors suffice for cn times parallel acceleration where c is a positive constant. Incorporating Turan's techniques into another, more recent algorithm gives a single complex zero with absolute error ⩽ ϵ using O(log 2 n log log(| λ 1/ ϵ|)) arithmetic parallel steps, n 2 processors (which places the problem in NC) and also gives all the complex zeros in O( n 2 log n + log log(| λ 1/ ϵ|))) arithmetic operations, where λ 1 is the absolutely largest zero. Computations with O(log( n| λ 1/ ϵ|))) binary bits support the latter estimates for the arithmetic complexity, which leads to a simple proof of the current best estimate for the Boolean circuit complexity (bit-operation complexity) of computing all the complex zeros of a polynomial, announced by A. Schönhage in 1982 but not proven yet.