Moulay A. Barkatou
ABSTRACT
The notion of irreducible forms of systems of linear differential equations as defined by Moser [14 ] and its generalisation, the super-irreducible forms introduced by Hilali/Wazner in [9 ] are important concepts in the context of the symbolic resolution of systems of linear differential equations [3,15,16 ]. In this paper, we give a new algorithm for computing, given an arbitrary linear differential system with formal power series coefficients as input, an equivalent system which is super-irreducible. Our algorithm is optimal in the sense that it computes transformation matrices which obtain a maximal reduction of rank in each step of the algorithm. This distinguishes it from the algorithms in [9,14,2] and generalises [7].
- M. Barkatou. An algorithm for computing a companion block diagonal form for a system of linear differential equations. Journal of App. Alg. in Eng. Comm. and Comp., 4, 1993.Google Scholar
- M. Barkatou. A rational version of Moser's Algorithm. In A. Levelt, editor, Proceedings of ISSAC'95, pages 297-?302, Montreal, Canada, 1995. ACM Press. Google ScholarDigital Library
- M. Barkatou. An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system. Journal of App. Alg. in Eng. Comm. and Comp., 8(1):1--23,1997.Google ScholarDigital Library
- M. Barkatou. On super-irreducible forms of linear differential systems with rational function coefficients. Journal of Computational and Applied Mathematics, (162):1--15,2004. Google ScholarDigital Library
- M. Barkatou and E. Pügel. The ISOLDE package. A SourceForge Open Source project, http://isolde.sourceforge.net 2006.Google Scholar
- G. Chen. An algorithm for computing the formal solutions of differential systems in the neighbourghood of an irregular singular point. In Proceedings of ISSAC '90, pages 231--235, Tokyo, Japan, 1990. ACM Press. Google ScholarDigital Library
- V. Dietrich. Zur Reduktion von linearen Differentialgleichungssystemen. Math. Ann., 237:79--95, 1978.Google ScholarCross Ref
- M. Giesbrecht and A. Storjohann. Computing rational forms of integer matrices. J. Symb. Comput., 34(3):157--172, 2002. Google ScholarDigital Library
- A. Hilali and A. Wazner. Formes super-irr éductibles des systèmes diérentiels linéaires. Numer. Math., 50:429--449, 1987.Google ScholarDigital Library
- C.-P. Jeannerod. An algorithm for the eigenvalue perturbation problem: Reduction of a kappa-matrix to a Lidskii matrix. In Proceedings of ISSAC 2000, pages 184--191, St Andrews, Scotland, 2000. ACM Press. Google ScholarDigital Library
- C.-P.Jeannerod. Formes normales de perturbations de matrices: étude et calcul exact. PhD thesis, Institut National Polytechnique de Grenoble, 2000.Google Scholar
- A. Levelt. Stabilizing Differential Operators: a method for Computing Invariants at Irregular Singularities. Differential Equations and Computer Algebra, M. Singer (ed.), pages 181--228, 1991.Google Scholar
- V. Lidskii. Perturbation theory of non-conjugate operators. U.S.S.R. Comput. Math. and Math. Phys., 1:73--85, 1965.Google Scholar
- J. Moser. The order of a singularity in Fuchs' theory. Math. Z., pages 379--398, 1960.Google Scholar
- E. Pflügel. Réesolution symbolique des systèmes differentiels linéaires. PhD thesis, LMC-IMAG, 1998.Google Scholar
- E. Pflügel. Effective formal reduction of linear differential systems. Appl.Alg.Eng. Comm. Comp., 10(2):153--187, 2000.Google ScholarCross Ref
- H. Turritin. Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point. Acta Math., 93:27--66, 1955.Google ScholarCross Ref
- W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Robert E. Krieger Publishing, 1967.Google Scholar
Index Terms
- Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach
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