Abstract
The problem is to calculate an approximate solution of an initial value problem for a scalar autonomous differential equation. A generalized notion of a nonlinear Runge-Kutta (NRK) method is defined. We show that the order of anys-stage NRK method cannot exceed 2s−1; hence, the family of NRK methods due to Brent has the maximal order possible. Using this result, we derive complexity bounds on the problem of finding an approximate solution with error not exceeding ε. We also compute the order which minimizes these bounds, and show that this optimal order increases as ε decreases, tending to infinity as ε tends to zero.
Zusammenfassung
Es wird das Problem behandelt, die Näherungslösung für ein Anfangswertproblem für eine skalare autonome Differentialgleichung abzuschätzen. Eine Verallgemeinerung eines nichtlinearen Runge-Kutta-(NRK-) Verfahrens wird definiert. Es wird gezeigt, daß die Ordnung irgendeiners-wertigen NRK-Methode nicht größer als 2s−1 sein kann; deshalb hat die Familie der NRK-Verfahren von Brent die maximale mögliche Ordnung. Unter Benutzung dieser Resultate werden Schranken angegeben zum Problem des Auffindens von Näherungslösungen, deren Fehler nicht größer als ε ist. Es wird ebenfalls die Ordnung berechnet, welche die Schranken minimiert, und es wird gezeigt, daß die optimale Ordnung wächst, falls ε abnimmt; sie geht nach unendlich, wenn ε gegen Null geht.
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This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370 NR 044-422
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Werschulz, A.G. Computational complexity of one-step methods for a scalar autonomous differential equation. Computing 23, 345–355 (1979). https://doi.org/10.1007/BF02254863
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DOI: https://doi.org/10.1007/BF02254863