Fast evaluation and root finding for polynomials with floating-point coefficients
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Evaluating or finding the roots of a polynomial f(z) = f_0 + ⋯ + f_d z^d with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of f obtained with a careful use of the Newton polygon of f, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of f are given with m significant bits, we provide for the first time an algorithm that finds all the roots of f with a relative condition number lower than 2^m, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of f. Notably, our new approach handles efficiently polynomials with coefficients ranging from 2^-d to 2^d, both in theory and in practice.
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