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- W4287125151 abstract "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 $|f|_1 leq 2^tau$ for $tau geq 1$, 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 $widetilde O(d(tau+m))$ bit operations, where $widetilde O(cdot)$ means that we omit logarithmic factors. We also show that if $kappa$ is the absolute condition number of the zeros of $f$, then we can isolate all the roots of $f$ in $widetilde O(d(tau + log kappa))$ bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in verb|Python/NumPy| that is an order of magnitude faster than the state-of-the-art solver verb/MPSolve/ for high degree polynomials with random coefficients." @default.
- W4287125151 created "2022-07-25" @default.
- W4287125151 creator A5024458284 @default.
- W4287125151 date "2021-06-04" @default.
- W4287125151 modified "2023-09-23" @default.
- W4287125151 title "New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems" @default.
- W4287125151 hasPublicationYear "2021" @default.
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