ABSTRACT
We introduce new ideas to improve the efficiency and rationality of a triangulation decomposition algorithm. On the one hand we identify and isolate the polynomial remainder sequences in the triangulation-decomposition algorithm. Subresultant polynomial remainder sequences are then used to compute them and their specialization properties are applied for the splittings. The gain is two fold: control of expression swell and reduction of the number of splittings. On the other hand, we remove the role that initials had in previous triangulation-decomposition algorithms. They are not needed in theoretical results and it was expected that they need not appear in the input and output of the algorithms. This is the case of the algorithm presented. New algorithms are presented to compute a subsequent characteristic decomposition from the output of the triangulation decomposition algorithm where the initials need not appear.
- P. Aubry. Ensembles triangulaires de polynômes et rsolution de systémes algèbriques. Implantation en Axiom. PhD thesis, Universit7eacute; de Paris 6, 1999.]]Google Scholar
- P. Aubry and D. Wang. Reasonning about surfaces using differential zero and ideal decomposition. In ADG 2000, number 2061 in LNAI, 2001.]] Google ScholarDigital Library
- F. Boulier and E. Hubert. textscdiffalg: description and examples of use. U. of Waterloo, 1998, \www.inria.fr/cafe/Evelyne.Hubert/diffalg.]]Google Scholar
- F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a finitely generated differential ideal. In ISSAC'95. ACM, 1995.]] Google ScholarDigital Library
- F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Computing representations for radicals of finitely generated differential ideals. Technical Report IT-306, LIFL, 1997.]]Google Scholar
- F. Boulier and F. Lemaire. Computing canonical representatives of regular differential ideals. In ISSAC 2000. ACM, 2000.]] Google ScholarDigital Library
- F. Boulier, F. Lemaire, and M. Moreno-Maza. Pardi! In ISSAC 2001. ACM, 2001.]] Google ScholarDigital Library
- D. Bouziane, A. Kandri Rody, and H. Maârouf. Unmixed-dimensional decomposition of a finitely generated perfect differential ideal. Journal of Symbolic Computation, 31(6):631--649, 2001.]] Google ScholarDigital Library
- G. Carra Ferro. Grübner bases and differential algebra. In AAECC, volume 356 of Lecture Notes in Computer Science. Springer-Verlag, 1987.]]Google Scholar
- S-C. Chou and X-S. Gao. Automated reasonning in differential geometry and mechanics using the characteristic set method. Part II. Mechanical theorem proving. Journal of Automated Reasonning, 10:173--189, 1993.]]Google ScholarCross Ref
- P. A. Clarkson and E. L. Mansfield. Symmetry reductions and exact solutions of a class of non-linear heat equations. Physica, D70:250--288, 1994.]] Google ScholarDigital Library
- G. E. Collins. Subresultants and reduced polynomial remainder sequences. J. Assoc. Comput. Mach., 14:128--142, 1967.]] Google ScholarDigital Library
- S. Dellière. Triangularisation de systèmes constructibles. Application à l'évaluation dynamique. PhD thesis, Univ. de Limoges, 1999.]]Google Scholar
- L. Ducos. Optimizations of the subresultant algorithm. Journal of Pure and Applied Algebra, 145(2):149--163, 2000.]]Google ScholarCross Ref
- M. Fliess and S.T. Glad. An algebraic approach to linear and nonlinear control. In Essays on control: Perspectives in the theory and its applications, volume 14. Birkhäuser, Boston, 1993.]]Google Scholar
- T. Gómez-Dîaz. Applications de l'évaluation dynamique. PhD thesis, Univ. de Limoges, 1994.]]Google Scholar
- E. Hubert. Factorisation free decomposition algorithms in differential algebra. Journal of Symbolic Computation, 29(4-5):641--662, 2000.]] Google ScholarDigital Library
- E. Hubert. Notes on triangular sets and triangulation-decomposition algorithms I: Polynomial systems. In Winkler and Langer.]]Google Scholar
- E. Hubert. Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems. In Winkler and Langer.]]Google Scholar
- E. Hubert and N. Le Roux. Computing power series solutions of a nonlinear pde system. In ISSAC 2003. ACM, 2003.]] Google ScholarDigital Library
- M. Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. Journal of Symbolic Computation, 15(2):143--167, 1993.]] Google ScholarDigital Library
- A. A. Kapaev and E. Hubert. A note on the Lax pairs for Painlevé equations. Journal of Physics. A. Mathematical and General, 32(46), 1999.]]Google ScholarCross Ref
- E. R. Kolchin. Differential Algebra and Algebraic Groups, volume 54 of Pure and Applied Mathematics. Academic Press, 1973.]]Google Scholar
- Francois Lemaire. Les classements les plus généraux assurant l'analycité des solutions des systèmes orthonomes pour des conditions initiales analytiques. In CASC 2002. Technische Universität München, Germany, 2002.]]Google Scholar
- H. Lombardi, M-F. Roy, and M. S. El Din. New structure theorem for subresultants. Journal of Symbolic Computation, 29(4-5):663--690, 2000.]] Google ScholarDigital Library
- E. L. Mansfield. Differential Gröbner Bases. PhD thesis, University of Sydney, 1991.]]Google Scholar
- E. L. Mansfield, G. J. Reid, and P. A. Clarkson. Nonclassical reductions of a 3+1-cubic nonlinear Schrödinger system. Computer Physics Communications, 115:460--488, 1998.]]Google ScholarCross Ref
- G. Margaria, E. Riccomagno, M. J. Chappell, and H. P. Wynn. Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. Mathematical Biosciences, 174(1):1--26, 2001.]]Google ScholarCross Ref
- B. Mishra. Algorithmic Algebra, volume XIV of Texts and Monographs in Computer Science. Springer-Verlag New York, 1993.]] Google ScholarDigital Library
- M. Moreno-Maza. Calculs de pgcd au-dessus des tours d'extensions simples et résolution des systèmes d'équations algébriques. PhD thesis, Université Paris 6, 1997.]]Google Scholar
- F. Ollivier. Standard bases of differential ideals. In AAECC'90, pages 304--321. Springer, 1991.]] Google ScholarDigital Library
- F. Ollivier and A. Sedoglavic. Algorithmes efficacxes pour tester l'identifiabilité locale. In Conférence Internationale Francophone d'Automatique. IEEE, 2002.]]Google Scholar
- C. Riquier. Les systèmes d'équations aux dérivées partielles. Gauthier-Villars, Paris, 1910.]]Google Scholar
- J. F. Ritt. Differential Algebra, volume XXXIII of Colloquium publications. American Mathematical Society, 1950. \tt http://www.ams.org/online\_bks.]]Google Scholar
- C. Rust. Rankings of derivatives for elimination algorithms and formal solvability of analytic partial differential equations. PhD thesis, University of Chicago, 1998.]] Google ScholarDigital Library
- C. J. Rust, G. J. Reid, and A. D. Wittkopf. Existence and uniqueness theorems for formal power series solutions of analytic differential systems. In ISSAC'99. ACM, 1999.]] Google ScholarDigital Library
- A. Seidenberg. An elimination theory for differential algebra. University of California Publications in Mathematics, 3(2):31--66, 1956.]]Google Scholar
- D. Wang. An elimination method for differential polynomial systems. I. Systems Science and Mathematical Sciences, 9(3):216--228, 1996.]]Google Scholar
- D. Wang. Decomposing polynomial systems into simple systems. Journal of Symbolic Computation, 25(3):295--314, 1998.]] Google ScholarDigital Library
- D. Wang. Computing triangular systems and regular systems. Journal of Symbolic Computation, 30(2):221--236, 2000.]] Google ScholarDigital Library
- F. Winkler and U. Langer, editors. Symbolic and Numerical Scientific Computing, volume 2630 of LNCS. Springer Verlag, 2003.]] Google ScholarDigital Library
- W. T. Wu. On the foundation of algebraic differential geometry. Systems Science and Mathematical Sciences, 2(4):289--312, 1989.]]Google Scholar
Index Terms
- Improvements to a triangulation-decomposition algorithm for ordinary differential systems in higher degree cases
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