HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Conference papers

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

Guillaume Moroz 1
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.
Complete list of metadata


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

Files

preprint.pdf
Files produced by the author(s)

Identifiers

Collections

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⟩

Share

Metrics

Record views

297

Files downloads

145