A New Integration Algorithm for Ordinary Differential
Equations Based on Continued Fraction Approximations

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
Gragg-Bulirsh-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.

CACM September, 1974

Willers, I. M.

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

5.17

CA740902 JB January 17, 1978  9:06 AM

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