Article

Fast arithmetic for triangular sets: from theory to practice

Authors:

Marc Moreno Maza,

Éric SchostAuthors Info & Claims

ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation

Pages 269 - 276

https://doi.org/10.1145/1277548.1277585

Published: 29 July 2007 Publication History

Abstract

We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implementation with languages such as AXIOM or Maple We show the potential for huge speed-ups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.

References

[1]

D. Bini. Relations between exact and approximate bilinear algorithms. Applications.Calcolo, 17(1):87--97, 1980.

[2]

D. Bini, M. Capovani, F. Romani, and G. Lotti. O(n ^2.7799) complexity for n x n approximate matrix multiplication. Inf. Proc. Lett., 8(5):234--235, 1979.

[3]

D. Bini, G. Lotti, and F. Romani. Approximate solutions for the bilinear form computational problem. SIAM J. Comput., 9(4):692--697, 1980.

Digital Library

[4]

W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235--265, 1997.

Digital Library

[5]

D.G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Informatica, 28(7):693--701, 1991.

Digital Library

[6]

S. Cook. On the minimum computation time of functions. PhD thesis, Harvard University, 1966.

[7]

T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction to Algorithms. McGraw-Hill, 2002.

Digital Library

[8]

X. Dahan, M. Moreno Maza, É. Schost, W. Wu, and Y. Xie. Lifting techniques for triangular decompositions. In ISSAC'05, pages 108--115. ACM, 2005.

Digital Library

[9]

A. Filatei, X. Li, M. Moreno Maza, and É. Schost. Implementation techniques for fast polynomial arithmetic in a high-level programming environment. In ISSAC'06, pages 93--100. ACM, 2006.

Digital Library

[10]

J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999.

Digital Library

[11]

J.R. Johnson, W. Krandick, K. Lynch, K.G. Richardson, and A.D. Ruslanov. High-performance implementations of the Descartes method. In ISSAC'06, pages 154--161. ACM, 2006.

Digital Library

[12]

J.R. Johnson, W. Krandick, and A.D. Ruslanov. Architecture-aware classical taylor shift by 1. In ISSAC'05, pages 200--207. ACM, 2005.

Digital Library

[13]

H.T. Kung. On computing reciprocals of power series. Numerische Mathematik, 22:341--348, 1974.

Digital Library

[14]

L. Langemyr. Algorithms for a multiple algebraic extension. In Effective methods in algebraic geometry), volume94 of Progr. Math., pages 235--248. Birkhäuser, 1991.

[15]

F. Lemaire, M. Moreno Maza, and Y. Xie. The RegularChains library. In Ilias S. Kotsireas, editor, Maple Conference 2005, pages 355--368, 2005.

[16]

X. Li. Low-level techniques for Montgomery multiplication, 2007. http://www.csd.uwo.ca/People/gradstudents/xli96/

[17]

X. Li and M. Moreno Maza. Efficient implementation of polynomial arithmetic in a multiple-level programming environment. In ICMS'06, pages 12--23. Springer, 2006.

Digital Library

[18]

M. van Hoeij and M. Monagan. A modular GCD algorithm over number fields presented with multiple extensions. In ISSAC'02, pages 109--116. ACM, 2002.

Digital Library

[19]

P.L. Montgomery. Modular multiplication without trial division. Mathematics of Computation, 44(170):519--521, 1985.

[20]

M. Moreno Maza. On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK, 1999. http://www.csd.uwo.ca/~moreno/

[21]

V.Y. Pan. Simple multivariate polynomial multiplication. J. Symbolic Comput., 18(3):183--186, 1994.

Digital Library

[22]

M. Püschel, J.M.F. Moura, J. Johnson, D. Padua, M. Veloso, B.W. Singer, J. Xiong, F. Franchetti, A. Gačić, Y. Voronenko, K. Chen, R.W. Johnson, and N. Rizzolo. SPIRAL: Code generation for DSP transforms. Proc 'IEEE, 93(2):232--275, 2005.

[23]

A. Schönhage. Schnelle Multiplikation von Polynomenüber Körpern der Charakteristik 2. Acta Informatica, 7:395--398, 1977.

Digital Library

[24]

A. Schönhage and V. Strassen. Schnelle Multiplikation großer Zahlen. Computing, 7:281--292, 1971.

[25]

É. Schost. Multivariate power series multiplication. In ISSAC'05, pages 293--300. ACM, 2005.

Digital Library

[26]

V. Shoup. A new polynomial factorization algorithm and its implementation. J. Symb. Comp., 20(4):363--397, 1995.

Digital Library

[27]

M. Sieveking. An algorithm for division of powerseries. Computing, 10:153--156, 1972.

Cited By

Maza MYuan H(2022)Balanced Dense Multivariate Multiplication: The General Case2022 24th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)10.1109/SYNASC57785.2022.00015(35-42)Online publication date: Sep-2022
https://doi.org/10.1109/SYNASC57785.2022.00015
Poteaux ARybowicz MRobertz DLinton SYokoyama K(2015)Improving Complexity Bounds for the Computation of Puiseux Series over Finite FieldsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756650(299-306)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756650
Schost ERobertz DLinton SYokoyama K(2015)Algorithms for Finite Field ArithmeticProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756637(7-12)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756637
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Index Terms

Fast arithmetic for triangular sets: from theory to practice
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms

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cover image ACM Conferences

ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation

July 2007

406 pages

ISBN:9781595937438

DOI:10.1145/1277548

General Chair:
Dongming Wang
Beihang University and UPMC-CNRS (China/France)

Copyright © 2007 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 29 July 2007

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ISSAC07: International Symposium on Symbolic and Algebraic Computation

July 29 - August 1, 2007

Ontario, Waterloo, Canada

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Maza MYuan H(2022)Balanced Dense Multivariate Multiplication: The General Case2022 24th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)10.1109/SYNASC57785.2022.00015(35-42)Online publication date: Sep-2022
https://doi.org/10.1109/SYNASC57785.2022.00015
Poteaux ARybowicz MRobertz DLinton SYokoyama K(2015)Improving Complexity Bounds for the Computation of Puiseux Series over Finite FieldsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756650(299-306)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756650
Schost ERobertz DLinton SYokoyama K(2015)Algorithms for Finite Field ArithmeticProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756637(7-12)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756637
Cheng JGao X(2014)Multiplicity-preserving triangular set decomposition of two polynomialsJournal of Systems Science and Complexity10.1007/s11424-014-2017-027:6(1320-1344)Online publication date: 5-Apr-2014
https://doi.org/10.1007/s11424-014-2017-0
Meng LJohnson J(2013)Automatic Parallel Library Generation for General-Size Modular FFT AlgorithmsProceedings of the 15th International Workshop on Computer Algebra in Scientific Computing - Volume 813610.1007/978-3-319-02297-0_21(243-256)Online publication date: 9-Sep-2013
https://dl.acm.org/doi/10.1007/978-3-319-02297-0_21
Chen CMoreno Maza M(2012)Algorithms for computing triangular decomposition of polynomial systemsJournal of Symbolic Computation10.1016/j.jsc.2011.12.02347:6(610-642)Online publication date: 1-Jun-2012
https://dl.acm.org/doi/10.1016/j.jsc.2011.12.023
Li XMoreno Maza MRasheed RSchost É(2011)The modpn libraryJournal of Symbolic Computation10.1016/j.jsc.2010.08.01646:7(841-858)Online publication date: 1-Jul-2011
https://dl.acm.org/doi/10.1016/j.jsc.2010.08.016
Poteaux ARybowicz M(2011)Complexity bounds for the rational Newton-Puiseux algorithm over finite fieldsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-011-0144-622:3(187-217)Online publication date: 1-May-2011
Maza MXie Y(2010)Balanced dense polynomial multiplication on multi-coresACM Communications in Computer Algebra10.1145/1823931.182394243:3/4(85-87)Online publication date: 24-Jun-2010
https://doi.org/10.1145/1823931.1823942
Li XMaza MPan WJohnson JPark HKaltofen E(2009)Computations modulo regular chainsProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576736(239-246)Online publication date: 28-Jul-2009
https://dl.acm.org/doi/10.1145/1576702.1576736
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