Implementing a polynomial factorization and GCD package

Moore, P. M. A.; Norman, A. C.

doi:10.1145/800206.806379

Cited by 22 publications

(5 citation statements)

References 3 publications

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“…This would generally be quite a substantial piece of code. It is worth noting that [MN81] described their multivariate g.c.d./factorization package as being "at least as large" as their (fairly complete) implementation of the Risch-Norman integration method.…”

Section: Multivariate Liftingmentioning

confidence: 99%

Small algorithms for small systems

Davenport

2012

ACM Commun. Comput. Algebra

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Knuth is said to have described Computer Science as "that part of mathematics in which log log n = 3". This paper only considers some parts of Computer Algebra, and the even more special case when log n ≈ 3, or even less, and where compactness of the algorithm itself, as well as the data structures, is important. We begin with a few remarks on integration, which, while in some sense the culmination of computer algebra for the "compact computer algebra" market, also inspired many of our other suggestions. Acknowledgements. This paper was prepared when the author was visiting the Symbolic Com putation Group in the David R. Cheriton School of Computer Science, University of Waterloo, and the author is grateful to Prof. Giesbrecht and colleagues for their hospitality. Its preparation was inspired by an invitation to talk at the Compact Computer Algebra workshop at CICM 2009, and the author is grateful to Dr Smirnova for the invitation. At that workshop Prof. Geddes remarked that the subresultant algorithm did not work well for multivariates, which inspired section 5.

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Section: Multivariate Liftingmentioning

confidence: 99%

Small algorithms for small systems

Davenport

2012

ACM Commun. Comput. Algebra

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“…The $ can then be used in an attempt to group extraneous factors mod pi. For example, V2={2,2,4} gives 9 ={O, 4,8) which means the factors corresponding to 1 and 3 in DI, or 2 and 2 in D2, should be multiplied together into one factor for the subsequent lifting stage.…”

Section: Parallel Degree Reconciliationmentioning

confidence: 99%

“…Among the modern algorithms for factoring over Q, the most efficient in practice are those based on (a) the Berlekamp algorithm [l], [6] for factoring a univariate polynomial over a small prime field and (b) the EEZ algorithm, a simultaneous p-adic lifting procedure of all factors suggested by Wang [14]. These algorithms are fast and effective [8], [12], [15] for practical purposes despite the fact that they are considered of exponential time complexity in the degree of the polynomial.…”

Section: Introductionmentioning

confidence: 99%

Parallel univariate polynomial factorization on shared-memory multiprocessors

Wang

1990

Proceedings of the International Symposium on Symbolic and Algebraic Computation

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Using parallelism afforded by shared-memory multiprocessors to speed up systems for polynomial factorization is discussed. The approach is to take the fastest known factoring algorithm for practical purposes and parallelize key parts of it. The univariate factoring algorithm consists of two major tasks (a) factoring modulo small integer primes and (b) EEZ lifting and recovery of true factors. A C coded system PFACTOR that implements (a) in parallel is described in detail. PFAC-TOR is a stand-alone parallel factorizer that can take input from a file, a pipe or a socket connection over a network. It can also be used interactively as a UNIX command. PFACTOR consists of parallel selection of primes, automatic balancing of work, parallel Berlekamp algorithm, and parallel reconciliation of degrees of factors modulo different primes. Actual timings on the Encore Multimax show the effectiveness of the approach. Work on part (b) is still on going.

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“…In factoring the above sample polynomial we followed the algorithm by P. Various implementation issues can be found in [30]. A good set of polynomials for benchmarking an actual implementation of the factorization algorithm can be found in [9].…”

Section: Factorization Of Multivariate Polynomialsmentioning

confidence: 99%

Factorization of Polynomials

Kaltofen

1982

Computing Supplementum

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Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squarefree decomposition algorithms and Hensel lifting techniques are analyzed. An attempt is made to establish a complete historic trace for today's methods. The exponential worst case complexity nature of these algorithms receives attention.

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Implementing a polynomial factorization and GCD package

Cited by 22 publications

References 3 publications

Small algorithms for small systems

Small algorithms for small systems

Parallel univariate polynomial factorization on shared-memory multiprocessors

Factorization of Polynomials

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