Computational complexity of one-step methods for systems of differential equations

Author:
Arthur G. Werschulz

Journal:
Math. Comp. **34** (1980), 155-174

MSC:
Primary 65L05; Secondary 34A50, 68C25

DOI:
https://doi.org/10.1090/S0025-5718-1980-0551295-0

MathSciNet review:
551295

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Abstract: The problem is to calculate an approximate solution of an initial value problem for an autonomous system of *N* ordinary differential equations. Using fast power series techniques, we exhibit an algorithm for the *p*th-order Taylor series method requiring only arithmetic operations per step as . (Moreover, we show that any such algorithm requires at least operations per step.) We compute the order which minimizes the complexity bounds for Taylor series and linear Runge-Kutta methods and show that in all cases this optimal order increases as the error criterion decreases, tending to infinity as tends to zero. Finally, we show that if certain derivatives are easy to evaluate, then Taylor series methods are asymptotically better than linear Runge-Kutta methods for problems of small dimension *N*.

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DOI:
https://doi.org/10.1090/S0025-5718-1980-0551295-0

Article copyright:
© Copyright 1980
American Mathematical Society