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Comprehensive LU Factors of Polynomial Matrices

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11989)


The comprehensive LU decomposition of a parametric matrix consists of a case analysis of the LU factors for each specialization of the parameters. Special cases can be discontinuous with respect to the parameters, the discontinuities being triggered by zero pivots encountered during factorization. For polynomial matrices, we describe an implementation of comprehensive LU decomposition in Maple, using the RegularChains package.


  • Parametric linear algebra
  • LU decomposition
  • Regular chains

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  1. Corless, R.M., Jeffrey, D.J.: Well... it isn’t quite that simple. SIGSAM Bull. 26(3), 2–6 (1992)

    CrossRef  Google Scholar 

  2. Kalkbrener, M.: Three Contributions to Elimination Theory. Johannes Kepler University, Linz (1991)

    MATH  Google Scholar 

  3. Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comp. 28(1–2), 105–124 (1999)

    CrossRef  MathSciNet  Google Scholar 

  4. Yang, L., Zhang, J.: Searching dependency between algebraic equations: an algorithm applied to automated reasoning. International Atomic Energy Agency, IC/89/263, Miramare, Trieste, Italy (1991)

    Google Scholar 

  5. Corless, R.M., Jeffrey, D.J.: The Turing factorization of a rectangular matrix. SIGSAM Bull. 31(3), 20–30 (1997)

    CrossRef  Google Scholar 

  6. Weispfenning, V.: Comprehensive grobner bases. J. Symbolic Comput. 14, 1–29 (1992)

    CrossRef  MathSciNet  Google Scholar 

  7. Reid, G.: Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution. Eur. J. Appl. Math. 2, 293–318 (1991)

    CrossRef  MathSciNet  Google Scholar 

  8. Jeffrey, D.J., Corless R.M.: Linear algebra in Maple. In: Hogben, L. (ed) Chapter 89 in the CRC Handbook of Linear Algebra, 2nd ed. Chapman & Hall/CRC (2013)

    Google Scholar 

  9. Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Pan, W.: Comprehensive triangular decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 73–101. Springer, Heidelberg (2007).

    CrossRef  Google Scholar 

  10. Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)

    CrossRef  MathSciNet  Google Scholar 

  11. Boulier, F., Lemaire, F., Moreno Maza, M.: Well Known Theorems on Triangular Systems and the D5 Principle. In: Dumas, J.-G. et al. (eds.) Proceedings of Transgressive Computing 2006, Granada, Spain (2006)

    Google Scholar 

  12. Chen, C., et al.: Solving semi-algebraic systems with the RegularChains library in Maple. In: Raschau, S. (ed.) Proceedings of the Fourth International Conference on Mathematical Aspects of Computer Science and Information Sciences (MACIS 2011), pp. 38–51 (2011)

    Google Scholar 

  13. Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library in Maple 10. In: Kotsireas, I.S. (ed.) Proceedings of Maple Summer Conference 2005, Waterloo, Canada (2005)

    Google Scholar 

  14. Sit, W.Y.: An algorithm for solving parametric linear systems. J. Symb. Comp. 13, 353–394 (1992)

    CrossRef  MathSciNet  Google Scholar 

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Correspondence to David J. Jeffrey .

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Camargos Couto, A.C., Moreno Maza, M., Linder, D., Jeffrey, D.J., Corless, R.M. (2020). Comprehensive LU Factors of Polynomial Matrices. In: Slamanig, D., Tsigaridas, E., Zafeirakopoulos, Z. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2019. Lecture Notes in Computer Science(), vol 11989. Springer, Cham.

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