arXiv subjects

Bernard Parisse

Publications and source records attributed to Bernard Parisse.

8 recordsLinked to original sources

arXiv2021

Certifying a probabilistic parallel modular algorithm for rational univariate representation

This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation, Berlekamp-Massey algorithm and Hankel linear system solving modulo several primes in parallel. Then it introduces a new method (theorem \ref{prop:check}) for rur certification that is effective for most polynomial systems.These algorithms are implemented in https://www-fourier.univ-grenoble-alpes.fr/~parisse/giac.html since version 1.7.0-13 or 1.7.0-17 for certification, it has (July 2021) leading performances on multiple CPU, at least for an open-source software.

cs.SC↗

arXiv2019

Computing huge Groebner basis like cyclic10 over $\Q$ with Giac

We present a short description on how to fine-tune the modular algorithm implemented in the Giac computer algebra system to reconstruct huge Groebner basis over $\Q$.The classical cyclic10 benchmark will serve as example.

cs.SC↗

arXiv2017

Compiling LATEX to computer algebra-enabled HTML5

This document explains how to create or modify an existing LATEX document with commands enabling computations in the HTML5 output: when the reader opens the HTML5 output, he can run a computation in his browser, or modify the command to be executed and run it. This is done by combining different softwares: hevea for compilation to HTML5, giac.js for the CAS computing kernel (itself compiled from the C++ Giac library with emscripten), and a modified version of itex2MML for fast and nice rendering in MathML in browsers that support MathML.

cs.SC↗

arXiv2016

Sparse multivariate factorization by mean of a few bivariate factorizations

We describe an algorithm to factor sparse multivariate polynomials using O(d) bivariate factorizations where d is the number of variables. This algorithm is implemented in the Giac/Xcas computer algebra system.

cs.SC↗

arXiv2013

A probabilistic and deterministic modular algorithm for computing Groebner basis over $\Q$

Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among CAS implementers. In this paper, I will show how to check a candidate Groebner basis (obtained by reconstruction of several Groebner basis modulo distinct prime numbers) with a given error probability, that may be 0 if a certified Groebner basis is desired. This algorithm is now the default algorithm used by the Giac/Xcas computer algebra system with competitive timings, thanks to a trick that can accelerate computing Groebner basis modulo a prime once the computation has been done modulo another prime.

cs.SC↗

arXiv2004

Jordan Normal and Rational Normal Form Algorithms

In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula: \[(λ\cdot I-A) \cdot B(λ)=P(λ) \cdot I\] where $B(λ)$ is $(λ\cdot I-A)$'s comatrix and $P(λ)$ is $A$'s characteristic polynomial. This rational Jordan normal form algorithm differs from usual algorithms since it is not based on the Frobenius/Smith normal form but rather on the idea already remarked in Gantmacher that the non-zero column vectors of $B(λ_0)$ are eigenvectors of $A$ associated to $λ_0$ for any root $λ_0$ of the characteristical polynomial. The complexity of the algorithm is $O(n^4)$ field operations if we know the factorization of the characteristic polynomial (or $O(n^5 \ln(n))$ operations for a matrix of integers of fixed size). This algorithm has been implemented using the Maple and Giac/Xcas computer algebra systems.

cs.SC↗

arXiv2002

A correct proof of the heuristic GCD algorithm

In this note, we fill a gap in the proof of the heuristic GCD in the multivariate case made by Char, Geddes and Gonnet (JSC 1989) and give some additionnal information on this method.

cs.SC↗

arXiv1998

Semi-classical study of the Quantum Hall conductivity

The semi-classical study of the integer Quantum Hall conductivity is investigated for electrons in a bi-periodic potential $V(x,y)$. The Hall conductivity is due to the tunnelling effect and we concentrate our study to potentials having three wells in a periodic cell. A non-zero topological conductivity requires special conditions for the positions, and shapes of the wells. The results are derived analytically and well confirmed by numerical calculations.

cond-mat.mes-hall↗