A new integration algorithm for ordinary differential equations based on continued fraction approximations

ABSTRACT

A new integration algorithm is found, and an implementation is compared with other programmed algorithms. The new algorithm is a step-by-step procedure for solving the initial value problem in ordinary differential equations. It is designed to approximate poles of small integer order in the solutions of the differential equations by continued fractions obtained by manipulating the sums of truncated Taylor series expansions. The new method is compared with the Gragg-Bulirsch-Stoer, and the Taylor series method. The Taylor series method and the new method are shown to be superior in speed and accuracy, while the new method is shown to be most superior when the solution is required near a singularity. The new method can finally be seen to pass automatically through singularities where all the other methods which are discussed will have failed.

REFERENCES

• 1: Barton, D., Willers, 1.M., and Zahar, R.V.M. Taylor series methods for ordinary differential equations--an evaluation. Proc. Math. Software Symposium, Purdue U., Lafayette, ind., 1970.
• 2: Willers, I.M. Mathematical software for the initial value problem. Ph.D. Th., U. of Cambridge, 1972.
• 3: Bulirsch, R., and Stoer, J. Numerical treatment of ordinary differential equations by extrapolation methods. Num. Math. 8 (1968), 1-13.
• 4: Gragg, W.B. On extrapolation algorithms for ordinary initial value problems. SlAM J. Nurner. Anal. 2 (1965), 384-403.
• 5: Barton, D., Willers, I.M., and Zahar, R.V.M. An implementation of the Taylor series method for ordinary differential equations. Compt. J. 14, 3 (1971), 243-248; also in The Best Computer Papers of 1971, (Ed,) Petrocelli, Auerbach Philadelphia, 1971, pp. 147-163.
• 6: Bauer, F. U The quotient difference and epsilon algorithms. On numerical approximation. U. of Wisconsin Press, Madison, Wis., 1959, pp. 361-370.
• 7: Shanks, D. Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34 (1955), 1-42.
• 8: Hall, T.E., Enright, W.H., Fellen, B.M., Sedgwick, A.E. Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9 (1972), 603-637.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Initial value problems

General Terms:
Algorithms

Keywords:
Taylor series, continued fractions, initial value problem, integration, ordinary differential equations, program comparison, singularities