Skip to main content
View expanded cover

Computational Discrete Mathematics pp 119–135Cite as

Division-Free Algorithms for the Determinant and the Pfaffian: Algebraic and Combinatorial Approaches

Part of the Lecture Notes in Computer Science book series (LNCS,volume 2122)

Abstract

The most common algorithm for computing the determinant ofan n×n matrix A is Gaussian elimination and needs O(nsu3) additions, subtractions, multiplications, and divisions. On the other hand, the explicit definition of the determinant as the sum of n! products,

$$ c:\left\{ \begin{gathered} {\mathbf{ }}A^{n - 1} \to A^n \hfill \\ a_1 a_2 \ldots a_{n - 1} \mapsto a_1 a_2 \ldots a_{n - 1} a_n . \hfill \\ \end{gathered} \right. $$

shows that the determinant can be computed without divisions. The summation is taken over the set of all permutations π of n elements. Avoiding divisions seems attractive when working over a commutative ring which is not a field, for example when the entries are integers, polynomials, or rational or even more complicated expressions. Such determinants arise in combinatorial problems, see [11].

Keywords

  • Characteristic Polynomial
  • Dynamic Programming Algorithm
  • Repeated Element
  • Gaussian Elimination
  • Incremental Algorithm

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/3-540-45506-X_9
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   44.99
Price excludes VAT (USA)
  • ISBN: 978-3-540-45506-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   59.99
Price excludes VAT (USA)
Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Martin Aigner, Lattice paths and determinants. In: Computational Discrete Mathematics, ed. Helmut Alt, (this volume), Lecture Notes Comput. Sci., Vol. 2122 2001, pp. 1–12.

    Google Scholar 

  2. Erwin H. Bareiss, Sylvester’s identity and multistep integer-preserving Gaussian elimination. Math. Comput. 22 (1968), 565–578.

    MATH  CrossRef  MathSciNet  Google Scholar 

  3. Stuart J. Berkowitz, On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett. 18 (1984), 147–150.

    MATH  CrossRef  MathSciNet  Google Scholar 

  4. Allan Borodin, Joachim von zur Gathen, John Hopcroft, Fast parallel matrix and GCD computations. Inf. Control 52 (1982), 241–256

    MATH  CrossRef  Google Scholar 

  5. E. Durant, Solution numérique des équations algébriques, tome II: systémes des plusieurs équations. Masson & Cie., Paris 1961.

    Google Scholar 

  6. Jack Edmonds, Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards, Sect. B 71 (1967), 241–245.

    MATH  MathSciNet  Google Scholar 

  7. Jack Edmonds, J.-F. Maurras, Note sur les Q-matrices d’Edmonds. RAIRO, Rech. opér. 31 (1997), 203–209.

    MATH  MathSciNet  Google Scholar 

  8. D. K. Faddeyev, V. N. Faddeyeva, Vyčislitelńye metody lineynoy algebry (in Russian), Moscow, 1960. English translation: D. K. Faddeev, V. N. Faddeeva, Numerical Methods of Linear Algebra. Freeman, San Francisco 1963. German translation: D. K. Faddejew, W. N. Faddejewa, Numerische Methoden der linearen Algebra, several editions since 1964.

    Google Scholar 

  9. G. Galbiati and Franceso Maffioli, On the computation of pfaffians. Discr. Appl. Math. 51 (1994), 269–275.

    MATH  CrossRef  MathSciNet  Google Scholar 

  10. Donald E. Knuth, Overlapping Pfaffians. Electron. J. Comb. 3 (1996), No. 2, article R5, 13 pp. Printed version: J. Comb. 3 (1996), No. 2, 147-159.

    Google Scholar 

  11. Christian Krattenthaler, Advanced determinant calculus. Séminaire Lotharingien de Combinatoire B42q (1999), 67 pp.

    MathSciNet  Google Scholar 

  12. László Lovász, M. D. Plummer, Matching Theory. Ann. Discr. Math., Vol. 29. North-Holland Mathematics Studies, Vol. 121. Amsterdam 1986.

    Google Scholar 

  13. Meena Bhaskar Mahajan, P R Subramanya, V Vinay, A combinatorial algorithm for Pfaffians. In: Computing and combinatorics. Proc. 5th annual international conference. (COCOON’ 99), Tokyo, July 1999, ed. Takao Asano et al., Lecture Notes Comput. Sci. 1627, Springer-Verlag, pp. 134–143(1999). Extended version: DIMACS Technical Report 99-39, Rutgers University, July 1999.

    Google Scholar 

  14. Meena Bhaskar Mahajan, V Vinay, Determinant: Combinatorics, algorithms, and complexity. Chicago J. Theor. Comput. Sci., Vol. 1997, Article no. 1997-5, 26 pp.

    Google Scholar 

  15. Meena Mahajan, V. Vinay, Determinant: Old algorithms, new insights. SIAM J. Discrete Math. 12 (1999), 474–490.

    MATH  CrossRef  MathSciNet  Google Scholar 

  16. Thomas Muir, A Treatise on the Theory of Determinants. MacMillan and Co., London 1882; repr. Dover, New York 1960.

    Google Scholar 

  17. Günter Rote, Path problems in graphs. In: Computational graph theory, ed. Gottfried Tinhofer et al., Computing Suppl. 7, 155–189, Springer-Verlag, Wien1990.

    Google Scholar 

  18. D. E. Rutherford, The Cayley-Hamilton theorem for semi-rings. Proc. Roy. Soc. Edinburgh, Sect. A 66(1961–64), 211–215(1964).

    MathSciNet  Google Scholar 

  19. Paul A. Samuelson, A method of determining explicitly the coefficients of the characteristic equation. Ann. Math. Statist. 13 (1942), 424–429.

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. Dennis Stanton, Dennis White, Constructive combinatorics. Springer-Verlag, New York 1986.

    MATH  Google Scholar 

  21. John R. Stembridge, Nonintersecting paths, pfaffians, and plane partitions. Adv. Math. 83 (1990), 96–113.

    MATH  CrossRef  MathSciNet  Google Scholar 

  22. Volker Strassen, Vermeidung von Divisionen. J. reine angew. Math. 264 (1973), 184–202.

    MATH  MathSciNet  Google Scholar 

  23. Howard Straubing, A combinatorial proof of the Cayley-Hamilton theorem. Discrete Math. 43 (1983), 273–279.

    MATH  CrossRef  MathSciNet  Google Scholar 

  24. Leslie G. Valiant, Why is Boolean complexity theory difficult? In: Boolean Function Complexity, ed. M. S. Paterson, LMS Lecture Notes Series, Vol. 169, Cambridge Univ. Press, 1992, pp. 84–94.

    Google Scholar 

  25. Doron Zeilberger, A combinatorial approach to matrix algebra. Discrete Math. 56 (1985), 61–72.

    MATH  CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195, Berlin

    Günter Rote

Authors
  1. Günter Rote
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany

    Helmut Alt

Rights and permissions

Reprints and Permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Rote, G. (2001). Division-Free Algorithms for the Determinant and the Pfaffian: Algebraic and Combinatorial Approaches. In: Alt, H. (eds) Computational Discrete Mathematics. Lecture Notes in Computer Science, vol 2122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45506-X_9

Download citation

  • DOI: https://doi.org/10.1007/3-540-45506-X_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42775-9

  • Online ISBN: 978-3-540-45506-6

  • eBook Packages: Springer Book Archive