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

1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
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.
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Conference papers
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https://hal.archives-ouvertes.fr/hal-03249123
Contributor : Guillaume Moroz Connect in order to contact the contributor
Submitted on : Friday, November 19, 2021 - 2:04:35 PM
Last modification on : Wednesday, May 4, 2022 - 3:48:15 AM

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### Citation

Guillaume Moroz. New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems. FOCS 2021 - 62nd Annual IEEE Symposimum on Foundations of Computer Science, Feb 2022, Denver, United States. ⟨10.1109/FOCS52979.2021.00108⟩. ⟨hal-03249123v2⟩

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