On Regular and Logarithmic Solutions of Ordinary Linear Differential Systems

  • S. A. Abramov
  • M. Bronstein
  • D. E. Khmelnov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


We present an approach to construct all the regular solutions of systems of linear ordinary differential equations using the desingularization algorithm of Abramov & Bronstein (2001) as an auxiliary tool. A similar approach to find all the solutions with entries in C(z) [log z] is presented as well, together with a new hybrid method for constructing the denominator of rational and logarithmic solutions.


Regular Solution Initial Segment Laurent Series Polynomial Solution Recurrence System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramov, S.: EG–eliminations. Journal of Difference Equations and Applications 5, 393–433 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abramov, S., Bronstein, M.: On solutions of linear functional systems. In: Proceedings of ISSAC 2001, pp. 1–6. ACM Press, New York (2001)CrossRefGoogle Scholar
  3. 3.
    Abramov, S., Bronstein, M., Khmelnov, D.: Regularization of linear recurrence systems. Transactions of the A.M. Liapunov Institute 4, 158–171 (2003)Google Scholar
  4. 4.
    Abramov, S., Bronstein, M., Petkovšek, M.: On polynomial solutions of linear operator equations. In: Proceedings of ISSAC 1995, pp. 290–296. ACM Press, New York (1995)CrossRefGoogle Scholar
  5. 5.
    Barkatou, M.A.: On rational solutions of systems of linear differential equations. Journal of Symbolic Computation 28, 547–567 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Barkatou, M., Pflügel, E.: An algorithm computing the regular formal solutions of a system of linear differential equations. Journal of Symbolic Computation 28, 569–587 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bronstein, M., Trager, B.: A reduction for regular differential systems. In: CD-ROM, Proceedings of MEGA 2003 (2003)Google Scholar
  8. 8.
    Coddington, E., Levinson, N.: Theory of ordinary differential equations. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  9. 9.
    Frobenius, G.: Über die Integration der linearen Differentialgleichungen mit veränder Koefficienten. Journal für die reine und angewandte Mathematik 76, 214–235 (1873)CrossRefGoogle Scholar
  10. 10.
    Heffter, L.: Einleitung in die Theorie der linearen Differentialgleichungen. Teubner, Leipzig (1894)zbMATHGoogle Scholar
  11. 11.
    Hilali, A., Wazner, A.: Formes super–irréductibles des systèmes différentiels linéaires. Numerical Mathematics 50, 429–449 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    van der Hoeven, J.: Fast evaluation of holonomic functions near and in regular singularities. Journal of Symbolic Computation 31, 717–743 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kaltofen, E., Villard, G.: On the complexity of computing determinants. Computational Complexity 13, 91–130 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Poole, E.: Introduction to the Theory of Linear Differential Equations. Dover Publications Inc., New York (1960)zbMATHGoogle Scholar
  15. 15.
    Storjohann, A., Villard, G.: Computing the rank and a small nullspace basis of a polynomial matrix. In: Proceedings of ISSAC 2005. ACM Press, New York (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • S. A. Abramov
    • 1
  • M. Bronstein
    • 2
  • D. E. Khmelnov
    • 1
  1. 1.Dorodnicyn Comp. Center of the Russ. Acad. of SciencesMoscowRussia
  2. 2.INRIA – CAFÉSophia AntipolisFrance

Personalised recommendations