New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems

Abstract

We present a new data structure to approximate accurately and efficiently a polynomial f of degree d given as a list of coefficients f i . Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying notably that the standard condition number of the zeros of a polynomial is at least exponential in the number of roots of modulus less than 1/2 or greater than 2.Given a polynomial f of degree d with f 1 = |f i | ≤ 2 τ for τ ≥ 1, isolating all its complex roots or evaluating it at d points can be done with a quasi-linear number of arithmetic operations. However, considering the bit complexity, the state-of-the-art algorithms require at least d 3/2 bit operations even for well-conditioned polynomials and when the accuracy required is low. Given a positive integer m, we can compute our new data structure and evaluate f at d points in the unit disk with an absolute error less than 2 −m in O(d(τ + m)) bit operations, where O(•) means that we omit logarithmic factors. We also show that if κ is the absolute condition number of the zeros of f , then we can isolate all the roots of f in O(d(τ + log κ)) bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in Python/NumPy that is an order of magnitude faster than the state-of-the-art solver MPSolve for high degree polynomials with random coefficients.

On the Complexity of Symbolic Computation

“…The remaining case when b = o(d) was solved recently by Moroz [95]. He shows how to isolate the roots of P in time Õ(d(b + log κ)), where κ is a suitable condition number.…”