Science China Information Sciences

, Volume 56, Issue 11, pp 1–16 | Cite as

Parallel computation of determinants of matrices with multivariate polynomial entries

Research Paper


In this paper we present an extension to the work of Björck et al. for computing the determinants of matrices with univariate or bivariate polynomials as entries to multivariate case. The algorithm supports parallel computation and has been implemented on a multi-core cluster computer system. We show how to use our approach to calculate two unsolved problems, which arise from computational geometry optimization and electric power engineering, and analyze the time complexity as well as bits complexity.


determinant interpolation parallel algorithm 


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  1. 1.
    Björck A, Pereyra V. Solution of Vandermonde systems of equations. Math Comput, 1970, 24: 893–903CrossRefGoogle Scholar
  2. 2.
    Higham N J. Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J Numer Anal, 1988, 8: 473–486MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Calvetti D, Reichel L. Fast inversion of Vandermonde-like matrices involving orthogonal polynomials. BIT Numer Math, 1993, 33: 473–484MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Zhang J, Yang L, Deng M. The parallel numerical method of mechanical theorem proving. Theor Comput Sci, 1990, 74: 253–271MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Zippel R. Interpolating polynomials from their values. J Symb Comput, 1990, 9: 375–403MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Marco A, Martínez J J. Parallel computation of determinants of matrices with polynomial entries. J Symb Comput, 2004, 37: 749–760CrossRefMATHGoogle Scholar
  7. 7.
    Chen L. Study on Several Parallel Algorithms of Symbolic Computation. Dissertation for the Doctoral Degree. Shanghai: East China Normal University. 2008Google Scholar
  8. 8.
    Li Y. An effective algorithm of computing symbolics determinants with multivariate polynomial entries. Appl Math Comput, 2007, 192: 382–388MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Li Y. An effective hybrid algorithm for computing symbolic determinants. Appl Math Comput, 2009, 215: 2495–2501MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kumara K, Anai H. Parallel computation of determinants of matrices with polynomial entries of robust control design. In: International Workshop on Parallel and Symbolic Computation (PASCO). Grenoble: ACM Press, 2010. 173–174CrossRefGoogle Scholar
  11. 11.
    Pereyra V, Scherer G. Efficient computer manipulation of tensor products with applications to multidimensional approximation. Math Comput, 1973, 27: 595–605MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Buis P E, Dyksen W R. Efficient vector and parallel manipulation of tensor products. ACM Trans Math Softw, 1996, 22: 18–23MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dean J, Ghemawat S. MapReduce: Simplified data processing on large clusters. Commun ACM, 2008, 51: 107–113CrossRefGoogle Scholar
  14. 14.
    Chiasson J N, Tolbert L M, McKenzie K J, et al. Elimination of harmonics in a multilevel converter using the theory of symmetric polynomials and resultants. IEEE Trans Control Syst Technol, 2005, 13: 216–223CrossRefGoogle Scholar
  15. 15.
    Yang K, Fu S, Hu H, et al. Real solution number of the nonlinear equations in the SHEPWM technology. In: Proceedings of International Conference on Intelligent Control and Information Processing (ICICIP). Dalian: IEEE Press, 2010. 446–450Google Scholar
  16. 16.
    Ben-Or M, Tiwari P. A deterministic algorithm for sparse multivariate polynomial interpolation. In: Proceedings of the 20th Annual ACM Symposium of Theory of Computing. New York: ACM Press, 1988. 301–309Google Scholar
  17. 17.
    Kaltofen E, Lee W H. Early termination in sparse interpolation algorithms. J Symb Comput, 2003, 36: 365–400MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Javadi S, Monagan M. Parallel sparse polynomial interpolation over finite fields. In: International Workshop on Parallel and Symbolic Computation (PASCO). Grenoble: ACM Press, 2010. 160–168CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina

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