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43rd ACM International Symposium on Symbolic and Algebraic Computation (ISSAC)

作者： Aoyama, Toru Noro, Masayuki Kobe Univ Grad Sch Sci Dept Math Kobe Hyogo Japan Rikkyo Univ Dept Math Tokyo Japan

ISBN: (纸本)9781450355506

In this paper, we propose algorithms for computing minimal associated primes of ideals in polynomial rings over Q and computing radicals of ideals in polynomial rings over a field. They apply Chinese Remainder Theorem (CRT) to Laplagne's algorithm which computes minimal associated primes without producing redundant components and computes radicals. CRT reconstructs an object in a ring from its modular images in the quotient rings modulo some ideals. In Laplagne's algorithm, ideals are decomposed over rational function fields by regarding some variables as parameters. In our new algorithms, we compute the minimal associated primes and the radical of , where. is a substitution map for a parameter. Then we construct candidates of the minimal associated primes and the radical of I by applying CRT for those of modular component must coincide with that of the corresponding component of the ideal for computations of minimal associated primes, and radicals of modular images of given ideals must coincide with modular images of radicals of given ideals for radical computations. The former is realized with a high probability because a multivariate irreducible polynomial over Q remains irreducible after a substitution of integers for variables with a high probability and the latter is realized except for a finite number of moduli.

关键词： minimal associated primes radicals modular algorithms Grobner basis polynomial ring

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Fast modular algorithms for squarefree factorization and hermite integration

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2001年第3期11卷 203-226页

作者： Gerhard, J Univ Gesamthsch Paderborn Fachbereich Math Informat D-33095 Paderborn Germany

We present new modular algorithms for the squarefree factorization of a primitive polynomial in Z[x] and for computing the rational part of the integral of a rational function in Q(x). We analyze both algorithms with respect to classical and fast arithmetic and argue that the latter variants are - up to logarithmic factors - asymptotically optimal. Even for classical arithmetic, the integration algorithm is faster than previously known methods.

关键词： Chinese remainder algorithm computing time analysis fast algorithms hermite integration modular algorithms rational reconstruction squarefree factorization symbolic integration

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Efficient and modular algorithms in modeling finite inelastic deformations: Objective integration, parameter identification and sub-stepping techniques

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2007年第17-20期196卷 2269-2283页

作者： Seifert, T. Schenk, T. Schmidt, I. Fraunhofer Inst Mech MAt D-79108 Freiburg Germany Netherland Inst Met Res NL-2628 CD Delft Netherlands

This paper presents a general algorithmic formulation for constitutive equations describing inelastic material deformation. The algorithmic representation focuses on the generality and modularity which is a key aspect within the development phase of constitutive equations. It is shown explicitly how to solve the incremental constitutive equations by use of objective algorithms and an implicit integration scheme applying numerical as well as analytical derivatives. Although it is shown that using numerical gradients is significantly slower, their application is easier especially in the development phase. It is also demonstrated that having the constitutive equations modeled in the general context introduced, the issue of parameter identification can be tackled by the same implementation using a driver algorithm. Besides parameter identification, the general structure is also applied to two different FE simulations: First, the objectivity of different integration methods will be presented considering a simple shear loading superposed by a rigid rotation using a strain-rate independent material model as example. In the second example it is shown for a strain-rate-dependent material model how the simulation of the wire drawing process can be accelerated crucially by using sub-stepping techniques. (c) 2006 Elsevier B.V. All rights reserved.

关键词： material modeling modular algorithms sub-stepping

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A modular SYSTEM OF algorithms FOR UNCONSTRAINED MINIMIZATION

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 1985年第4期11卷 419-440页

作者： SCHNABEL, RB KOONTZ, JE WEISS, BE NBS DIV SCI COMP BOULDER CO 80303 USA NBS DIV STAT ENGN BOULDER CO 80303 USA

We describe a new package, UNCMIN, for finding a local minimizer of a real valued function of more than one variable. The novel feature of UNCMIN is that it is a modular system of algorithms, containing three different step selection strategies (line search, dogleg, and optimal step) that may be combined with either analytic or finite difference gradient evaluation and with either analytic, finite difference, or BFGS Hessian approximation. We present the results of a comparison of the three step selection strategies on the problems in More, Garbow, and Hillstrom in two separate cases: using finite difference gradients and Hessians, and using finite difference gradients with BFGS Hessian approximations. We also describe a second package, REVMIN, that uses optimization algorithms identical to UNCMIN but obtains values of user-supplied functions by reverse communication.

关键词： modular algorithms reverse communication

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A modular algorithm for computing the exponential solutions of a linear differential operator

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JOURNAL OF SYMBOLIC COMPUTATION 2004年第3期38卷 1043-1076页

作者： Cluzeau, T van Hoeij, M Florida State Univ Dept Math Tallahassee FL 32306 USA Univ Limoges LACO F-87060 Limoges France

We present a new algorithm for computing exponential solutions of differential operators with rational function coefficients. We use a combination of local and modular computations, which allows us to reduce the number of possibilities in the combinatorial part of the algorithm. We also show how unnecessarily large algebraic extensions of the constants can be avoided in the algorithm. (C) 2004 Elsevier Ltd. All rights reserved.

关键词： modular algorithms differential equations exponential solutions

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A modular Algorithm to Compute the Resultant of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions 26th

A Modular Algorithm to Compute the Resultant of Multivariate...

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26th International Workshop on Computer Algebra in Scientific Computing (CASC)

作者： Ansari, Mahsa Monagan, Michael Simon Fraser Univ Dept Math Burnaby BC V5A 1S6 Canada

ISBN: (纸本)9783031690693;9783031690709

Let f(1) and f(2) be two multivariate polynomials over an algebraic number field Q(alpha(1) ,..., alpha(n)). In this paper, we present MRES, a modular algorithm for computing the resultant of f(1) and f(2). To enhance the efficiency, our algorithm converts f(1) and f(2) to their corresponding polynomials over Q(gamma) where gamma is a primitive element of Q(alpha(1), ..., alpha(n)). This conversion is done modulo a prime to prevent the coefficient growth. Next, our algorithm employs evaluation and dense interpolation to reduce the problem to the computation of the resultant of two univariate polynomials where we apply the monic Euclidean algorithm. Employing the monic Euclidean algorithm, we present a new formula for computing the resultant of univariate polynomials. Finally, our modular algorithm applies the Chinese remaindering and the rational number reconstruction to recover the rational coefficients of the resultant. We have implemented our algorithm in Maple. We include the expected time complexity of the algorithm, two benchmarks, and a partial failure probability analysis.

关键词： Resultant modular algorithms Algebraic number fields Primitive elements

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modular Differential Evolution 23

Modular Differential Evolution

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Vermetten, Diederick Caraffini, Fabio Kononova, Anna V. Back, Thomas Leiden Inst Adv Comp Sci Leiden Netherlands Swansea Univ Swansea Wales

ISBN: (纸本)9798400701191

New contributions in the field of iterative optimisation heuristics are often made in an iterative manner. Novel algorithmic ideas are not proposed in isolation, but usually as extensions of a preexisting algorithm. Although these contributions are often compared to the base algorithm, it is challenging to make fair comparisons between larger sets of algorithm variants. This happens because even small changes in the experimental setup, parameter settings, or implementation details can cause results to become incomparable. modular algorithms offer a way to overcome these challenges. By implementing the algorithmic modifications into a common framework, many algorithm variants can be compared, while ensuring that implementation details match in all versions. In this work, we propose a version of a modular framework for the popular Differential Evolution (DE) algorithm. We show that this modular approach not only aids in comparison but also allows for a much more detailed exploration of the space of possible DE variants. This is illustrated by showing that tuning the settings of modular DE vastly outperforms a set of commonly used DE versions which have been recreated in our framework. We then investigate these tuned algorithms in detail, highlighting the relation between modules and performance on particular problems.

关键词： Differential Evolution Benchmarking modular algorithms Algorithm Configuration

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Probabilistic and temporal failure detectors for solving distributed problems

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JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 2021年 158卷 1-15页

作者： Guerraoui, Rachid Kozhaya, David Pignolet, Yvonne-Anne Ecole Polytech Fed Lausanne Lausanne Switzerland ABB Res Baden Switzerland DFINITY Zurich Switzerland

Failure detectors (FD)s are celebrated for their modularity in solving distributed problems. algorithms are constructed using FD building blocks. Synchrony assumptions to implement FDs are studied separately and are typically expressed as eventual guarantees that need to hold, after some point in time, foreverand deterministically. But in practice, they may hold only probabilistically and temporarily. This paper studies FDs in a realistic system N, where asynchrony is inflicted by probabilistic synchronous communication. We first address a problem with lozenge S, the weakest FD to solve consensus: an implementation of "consensus with probability 1" is possible in Nwithout randomness in the algorithm, while an implementation of "lozenge Swith probability 1" is impossible in N. We introduce lozenge S*, a new FD with probabilistic and temporal accuracy. We prove that lozenge S*(i) is implementable in Nand (ii) can replace lozenge S, in several existing deterministic consensus algorithms that use lozenge S, to yield an algorithm that solves "consensus with probability 1". We extend our results to other FD classes, e.g., lozenge P, and to a larger set of problems (beyond consensus), which we call decisive problems. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Failure detectors Probabilistic links Message loss Consensus modular algorithms

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GAPN-LA: A framework for solving graph problems using Petri nets and learning automata

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2019年 77卷 255-267页

作者： Vahidipour, S. M. Esnaashari, M. Rezvanian, A. Meybodi, M. R. Univ Kashan Dept Comp Fac Elect & Comp Engn Kashan Iran KN Toosi Univ Technol Fac Comp Engn Tehran Iran Inst Res Fundamental Sci IPM Sch Comp Sci Tehran Iran Amirkabir Univ Technol Comp Engn & Informat Technol Dept Soft Comp Lab Tehran Iran

A fusion of learning automata and Petri nets, referred to as APN-LA, has been recently introduced in the literature for achieving adaptive Petri nets. A number of extensions to this adaptive Petri net have also been introduced;together we name them the APN-LA family. Members of this family can be utilized for solving problems in the domain of graph problems;each member is suitable for a specific category within this domain. In this paper, we aim at generalizing this family into a single framework, called generalized APN-LA (GAPN-LA), which can be considered as a framework for solving graph-based problems. This framework is an adaptive Petri net, organized into a graph structure. Each place or transition in the underlying Petri net is mapped into exactly one vertex of the graph, and each vertex of the graph represents a part of the underlying Petri net. A vertex in GAPN-LA can be considered as a module, which, in cooperation with other modules in the framework, helps in solving the problem at hand. To elaborate the problem-solving capability of the GAPN-LA, several graph-based problems have been solved in this paper using the proposed framework.

关键词： Adaptive Petri net Learning automata modular algorithms Graph-based problems

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algorithms for solving linear systems over cyclotomic fields

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JOURNAL OF SYMBOLIC COMPUTATION 2010年第9期45卷 902-917页

作者： Chen, Liang Monagan, Michael Simon Fraser Univ Dept Math Burnaby BC V5A 1S6 Canada

We consider the problem of solving a linear system Ax = b over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime p for which the minimal polynomial m(z) for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear p-adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo m(z) using Chinese remaindering only. Because this representation is d = deg m(z) times more compact in general, we can compute it the fastest. We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Linear systems modular algorithms Cyclotomic fields Cyclotomic polynomials

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