New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems
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We present a new data structure to approximate accurately and efficiently a polynomial f of degree d given as a list of coefficients. 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 ≤ 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 + logκ)) bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in that is an order of magnitude faster than the state-of-the-art solver for high degree polynomials with random coefficients.
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