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International Workshop on Computer Algebra in Scientific Computing

CASC 2016: Computer Algebra in Scientific Computing pp 1–14Cite as

On the Differential and Full Algebraic Complexities of Operator Matrices Transformations

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

Abstract

We consider \(n\times n\)-matrices whose entries are scalar ordinary differential operators of order \(\leqslant d\) over a constructive differential field K. We show that to choose an algorithm to solve a problem related to such matrices it is reasonable to take into account the complexity measured as the number not only of arithmetic operations in K in the worst case but of all operations including differentiation. The algorithms that have the same complexity in terms of the number of arithmetic operations can though differ in the context of the full algebraic complexity that includes the necessary differentiations. Following this, we give a complexity analysis, first, of finding a superset of the set of singular points for solutions of a system of linear ordinary differential equations, and, second, of the unimodularity testing for an operator matrix and of constructing the inverse matrix if it exists.

S.A. Abramov—Supported in part by the Russian Foundation for Basic Research, project no. 16-01-00174.

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Notes

  1. 1.

    For the difference case, this was used in first versions [1] of algorithm EG. In a discussion related to the differential case, A. Storjohann drew the author’s attention to the fact that the complexity of this approach is less than of one which uses solving of linear algebraic systems (see also [24]).

References

  1. Abramov, S.: EG-eliminations. J. Differ. Equ. Appl. 5, 393–433 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abramov, S., Barkatou, M.: On the dimension of solution spaces of full rank linear differential systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 1–9. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  3. Abramov, S., Barkatou, M.: On solution spaces of products of linear differential or difference operators. ACM Comm. Comput. Algebra 4, 155–165 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Abramov, S., Bronstein, M.: On solutions of linear functional systems. In: ISSAC 2001 Proceedings, pp. 1–6 (2001)

    Google Scholar 

  5. Abramov, S., Bronstein, M.: Linear algebra for skew-polynomial matrices. Rapport de Recherche INRIA RR-4420, March 2002. http://www.inria.fr/RRRT/RR-4420.html

  6. Abramov, S., Khmelnov, D.: On singular points of solutions of linear differential systems with polynomial coefficients. J. Math. Sci. 185(3), 347–359 (2012). (Translated from: Fundamentalnaya i prikladnaya matematika 17(1), 3–21 (2011/2012))

    Article  MathSciNet  MATH  Google Scholar 

  7. Abramov, S., Khmelnov, D., Ryabenko, A.: Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients. Program. Comput. Softw. 42(2), 55–64 (2016)

    Article  MATH  Google Scholar 

  8. Barkatou, M., Cluzeau, T., El Bacha, C.: Simple form of higher-order linear differential systems and their application in computing regular solutions. J. Symb. Comput. 46(6), 633–658 (2011)

    Article  MATH  Google Scholar 

  9. Barkatou, M., El Bacha, C., Labahn, G., Pflügel, E.: On simultaneous row and column reduction of higher-order linear differential systems. J. Symb. Comput. 49(1), 45–64 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beckermann, B., Labahn, G.: Fraction-free computation of matrix rational interpolants and matrix GCDs. SIAM J. Matrix Anal. Appl. 77(1), 114–144 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Beckermann, B., Cheng, H., Labahn, G.: Fraction-free row reduction of matrices of Ore polynomials. J. Symb. Comput. 41, 513–543 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Benoit, A., Bostan, A., van der Hoeven, J.: Quasi-optimal multiplication of linear differential operators. In: FOCS 2012 Proceedings, pp. 524–530 (2012)

    Google Scholar 

  13. Bostan, A., Chyzak, F., Le Roix, N.: Products of ordinary differential operators by evaluating and interpolation. In: ISSAC 2008 Proceedings, pp. 23–30 (2008)

    Google Scholar 

  14. Bostan, A., Schost, E.: Polynomial evaluation and interpolation on special sets of points. J. Complex. 21(4), 420–446 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften, vol. 315. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  16. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jeannerod, C.-P., Villard, G.: Essentially optimal computation of the inverse of generic polynomial matrices. J. Complex. 21(1), 72–86 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giesbrecht, M., Sub Kim, M.: Computation of the Hermite form of a matrix of Ore polynomials. J. Algebra 376, 341–362 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gustavson, R., Kondratieva, M., Ovchinnikov, A.: New effective differential Nullstellensatz. Adv. Math. 290, 1138–1158 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. van der Hoeven, J.: FFT-like multiplication of linear differential operators. J. Symb. Comput. 33, 123–127 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Knuth, D.E.: Big omicron and big omega and big theta. ACM SIGACT News 8(2), 18–23 (1976)

    Article  Google Scholar 

  22. Middeke, J.: A polynomial-time algorithm for the Jacobson form for matrices of differential operators. Technical report No. 08–13 in RISC Report Series (2008)

    Google Scholar 

  23. Miyake, M.: Remarks on the formulation of the Cauchy problem for general system of ordinary differential equations. Tôhoku Math. J. 32(2), 79–89 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mulders, T., Storjohann, A.: On lattice reduction for polynomial matrices. J. Symb. Comput. 37(4), 485–510 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. van der Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 328. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  26. Rosenlicht, M.: Integration in finite terms. Amer. Math. Mon. 79(9), 963–972 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  27. Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vassilevska Williams, V.: Multiplying matrices faster than Coppersmith-Winograd. In: STOC 2012 Proceedings, pp. 887–898 (2012)

    Google Scholar 

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Acknowledgments

The author is thankful to M. Barkatou and A. Storjohann for interesting discussions, and to anonymous referees for useful comments.

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Authors and Affiliations

  1. Dorodnitsyn Computing Centre, Federal Research Center Computer Science and Control of Russian Academy of Sciences, Vavilova, 40, Moscow, 119333, Russia

    Sergei A. Abramov

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  1. Sergei A. Abramov
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Correspondence to Sergei A. Abramov .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Abramov, S.A. (2016). On the Differential and Full Algebraic Complexities of Operator Matrices Transformations. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_1

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_1

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