Papers by Laurent Bernardin
We p r e s e n t a new parallel algorithm for performing linear Hensel lifting of bivariate polyn... more We p r e s e n t a new parallel algorithm for performing linear Hensel lifting of bivariate polynomials over a nite eld. The sequential version of our algorithm has a running time of O(mn 4) for lifting m univariate polynomials of degree n with respect to a bivariate polynomial of degree n in both variables, assuming that we use classical polynomial multiplication. Our parallel algorithm further reduces this complexity t o O(m n s n 3) o n s processing nodes, assuming that s < n. We also present an asymptotically faster algorithm, which has a complexity of O((ln m)n 2 ln n) operations in the coe cient eld, using fast polynomial multiplication and O(n ln m) processors. Experimental results on a massively parallel, distributed memory machine con rm that our algorithm scales well on high numbers of processing nodes.
This thesis Covers algorithms for factoring multivariate polynomials with coefficients from a fin... more This thesis Covers algorithms for factoring multivariate polynomials with coefficients from a finite field. Contributionsto all stages of the factorization process lead to an efficient practical implementation, enlarging the class of polynomials which can be factored in reasonabletime on given hardware.
Synergy Awards for Innovation (2008) - Members
Maple Transactions, 2021
Maple was conceived over forty years ago as a general purpose system for mathematical calculation... more Maple was conceived over forty years ago as a general purpose system for mathematical calculations. Its strength, however, has always been its community. The work of hundreds of researchers from around the world has produced a mathematical engine unique in its depth, breath and efficiency. Forward thinking educators have used Maple to transform the way mathematics is taught, all the way supporting each other with advice, examples and myriads of Maple worksheets. Scientists and engineers have been taking advantage of the power and ease of use of the Maple system to help them in their discovery and the development of new products. Together we have tackled environmental issues, taken on disease and reached for the stars. At Maplesoft, we are firm believers that Math Matters and our mission is to provide technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. This mission, we sh...
System and method for creating and presenting mathematical documents
System and method of gesture feature recognition
High performance symbolic computing
Mathematics and Computers in Simulation
Polynomial Factorization Challenges
ABSTRACT
Theoretical Computer Science, 1997
In this paper we present a new deterministic algorithm for computing the square-free decompositio... more In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field. Our algorithm is based on Yun's square-free factorization algorithm for characteristic 0. The new algorithm is more efficient than existing, deterministic algorithms based on Musser's squarefree algorithm. We will show that the modular approach presented by Yun has no significant performance advantage over our algorithm. The new algorithm is also simpler to implement and it can rely on any existing GCD algorithm without having to worry about choosing "good" evaluation points. To demonstrate this, we present some timings using implementations in Maple (Char et al., 1991), where the new algorithm is used for Release 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system known to the author to use an implementation of Yun's modular algorithm mentioned above.
Proceedings of the 1998 international symposium on Symbolic and algebraic computation, 1998
We present a new parallel algorithm for performing linear Hensel lifting of bivariate polynomials... more We present a new parallel algorithm for performing linear Hensel lifting of bivariate polynomials over a nite eld. The sequential version of our algorithm has a running time of Omn 4 for lifting m univariate polynomials of degree n with respect to a bivariate polynomial of degree n in both variables, assuming that we use classical polynomial multiplication. Our parallel algorithm further reduces this complexity to Om n s n 3 o n s processing nodes, assuming that s n. W e also present an asymptotically faster algorithm, which has a complexity of Oln mn 2 ln n operations in the coe cient eld, using fast polynomial multiplication and On ln m processors. Experimental results on a massively parallel, distributed memory machine con rm that our algorithm scales well on high numbers of processing nodes.
Computer Algebra
Computer algebra is a branch of scientific computation. There are several characteristic features... more Computer algebra is a branch of scientific computation. There are several characteristic features that distinguish computer algebra from numeric alanalysis, the other principal branch of scientific computation. (1) Computer algebra involves computation in algebraic structures, such as finitely presented groups, polynomial rings, rational function fields, algebraic and transcendental extensions of the rational numbers, or differential and difference fields. (2) Computer algebra manipulates formulas. Whereas in numerical computation the input and output of algorithms are basically (integer or floating point) numbers, the input and output of computer algebra algorithms are generally formulas.
Lecture Notes in Computer Science, 1997
We describe the Maple [23] implementation of multivariate factorization over general finite field... more We describe the Maple [23] implementation of multivariate factorization over general finite fields. Our first implementation is available in Maple V Release 3. We give selected details of the algorithms and show several ideas that were used to improve its efficiency. Most of the improvements presented here are incorporated in Maple V Release 4. In particular, we show that we needed a general tool for implementing computations in GF(pk)[xl, x2,..., x,,]. We also needed an efficient implementation of our algorithms in Zp[y][x] because any multivariate factorization may depend on several bivariate factorizations. The efficiency of our implementation is illustrated by the ability to factor bivariate polynomials with over a million monomials over a small prime field.
Adding Comments to a Procedure Consider the following example.
IFAC Proceedings Volumes, 2008
We present a high-level modeling formulation based on a conserved quantities approach, with the g... more We present a high-level modeling formulation based on a conserved quantities approach, with the goal of making the physical modeling process reliable and repeatable. The system of equations generated as a result of this formulation will, in general, be non-linear differential algebraic equations (DAEs). We make use of symbolic reduction techniques in order to eliminate spurious, non-physical solutions as well as to reduce to a system of ordinary differential equations, if possible.
Proceedings of the second international symposium on Parallel symbolic computation - PASCO '97, 1997
We ported the computer algebra system Maple V to the Intel Paragon, a massively parallel, distrib... more We ported the computer algebra system Maple V to the Intel Paragon, a massively parallel, distributed memory machine. In order to take advantage of the parallel architecture, we extended the Maple kernel with a set of message pawing primitives baaed on the Paragon's native message passing library. Using these primitives, we implemented a parallel version of Karatsuba multiplication for univariate polynomials over 2P Our speedup timings illustrate the practicability of our approach. On top of the message p=ing primitives we have implemented a higher level model of parallel processing baaed on the manager-worker scheme; a Maple application on one node of the parallel machine submits jobs to Maple processes residing on different nodes, then asynchronously collects the results. This model proves to be convenient for interactive usage of a distributed memory machine. Apart from the message passing parallelism we also use localized multi-threading to achieve symmetric multiprocessing within each node of the Paragon. We combine both approaches and apply them to the multiplication of large bivariate polynomials over small prime fields.
ACM SIGSAM Bulletin, 1996
Solving equations and systems of equations symbolically is a key feature of every Computer Algebr... more Solving equations and systems of equations symbolically is a key feature of every Computer Algebra System. This review examines the capabilities of the six best known general purpose systems to date in the area of general algebraic and transcendental equation solving. Areas explicitly not covered by this review are differential equations and numeric or polynomial system solving as special purpose systems exist for these kinds of problems.The aim is to provide a benchmark for comparing Computer Algebra Systems in a specific domain. We do not intend to give a rating of overall capabilities as for example [8].
Mathematics and Computers in Simulation, 1999
Here we present the ISSAC 1999 poster abstracts;. The abstracts are as distributed at the confere... more Here we present the ISSAC 1999 poster abstracts;. The abstracts are as distributed at the conference. This is the first of two parts, containing abstracts of the first poster session. Eugene V. Zima, Mohamed O. Rayes.
Bioinformatics, 2000
Motivation: We announce the availability of the second release of Darwin v. 2.0, an interpreted c... more Motivation: We announce the availability of the second release of Darwin v. 2.0, an interpreted computer language especially tailored to researchers in the biosciences. The system is a general tool applicable to a wide range of problems. Results: This second release improves Darwin version 1.6 in several ways: it now contains (1) a larger set of libraries touching most of the classical problems from computational biology (pairwise alignment, all versus all alignments, tree construction, multiple sequence alignment), (2) an expanded set of general purpose algorithms (search algorithms for discrete problems, matrix decomposition routines, complex/long integer arithmetic operations), (3) an improved language with a cleaner syntax, (4) better on-line help, and (5) a number of fixes to user-reported bugs. Availability: Darwin is made available for most operating systems free of charge from the Computational Biochemistry Research Group (CBRG), reachable at http://cbrg.inf.ethz.ch. Contact...
We ported the computer algebra system Maple V to the Intel Paragon a massively parallel distribut... more We ported the computer algebra system Maple V to the Intel Paragon a massively parallel distributed memory machine In or der to take advantage of the parallel architecture we extended the Maple kernel with a set of message passing primitives based on the Paragon s native message passing library Using these primitives we implemented a parallel version of Karatsuba multiplication for univariate polynomials overZp Our speedup timings illustrate the practicability of our approach On top of the message passing primitives we have implemented a higher level model of parallel processing based on the manager worker scheme a managing Maple process on one node of the paral lel machine submits processing requests to Maple processes residing on di erent nodes then asynchronously collects the results This model proves to be convenient for interactive usage of a distributed memory machine
Symbolic computation in Java
Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99, 1999
implementation of algebraic algorithms. We suggest uses of many of Java&#x27;s innovative fea... more implementation of algebraic algorithms. We suggest uses of many of Java&#x27;s innovative features for symbolic software de- sign problems. We discuss the suitability of Java for generic programming, a methodology whose origins actually are in computer algebra. We believe the software component ap- proach is required for designing modern systems that in- clude computer algebra. We describe ways in which Java can be used to adapt legacy software into components, and we present our ideas how component interfaces can be struc- tured.
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Papers by Laurent Bernardin