A factorization algorithm for linear ordinary differential equations

ABSTRACT

The reducibility and factorization of linear homogeneous differential equations are of great theoretical and practical importance in mathematics. Although it has been known for a long time that factorization is in principle a decision procedure, its use in an automatic differential equation solver requires a more detailed analysis of the various steps involved. Especially important are certain auxiliary equations, the so-called associated equations. An upper bound for the degree of its coefficients is derived. Another important ingredient is the computation of optimal estimates for the size of polynomial and rational solutions of certain differential equations with rational coefficients. Applying these results, the design of the factorization algorithm LODEF and its implementation in the Scratchpad II Computer Algebra System is described.

REFERENCES

• 1: E. Landau, Journal fiir die reine und angewandte Mathematik 124, 115-120(1902).
• 2: L. Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen I-III, Teubner, Leipzig, 1895-1898. These volumes have been reprinted by the Johnson Reprint Corporation in 1968.
• 3: E. Kamke, Differentialgleichungen, Ldsungsmethoden und L~sungen. 1". GewJhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, Leipzig, 1962.
• 4: A. Mambriani, Equazioni differenziali lineari aventi soluzioni polinomiali, Bolletino Unione Mat. Italiana 17, 26-32(1938).
• 5: O. Perron, Uber lineare Differenzen- und Differentialgleichungen, Mathematische Annalen 66, 446- 487(1909).
• 6: R. D. Jenks, R. S. Sutor and S. M. Watt, Scratchpad H: An Abstract Datalype System for Mathematical Computation, LNCS 296, Springer, 1987.
• 7: D. Yu. Gxigor'ev, Complezity of ~'actoring and Calculating ~he GCD of Linear Ordinary Differential Operators, Leningrad Department of Mathematics, preprint.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)

General Terms:
Algorithms, Theory