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International Journal of Mathematics and Mathematical Sciences
Volume 8, Issue 1, Pages 183-187
http://dx.doi.org/10.1155/S0161171285000187

A relationship between the modified Euler method and e

1Department of Mathematics, Colorado State University, Fort Collins 80521, Colorado, USA
2Department of Mathematics, Ashland College, Ashland 44805, Ohio, USA

Received 18 March 1984; Revised 14 November 1984

Copyright © 1985 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Approximating solutions to the differential equation dy/dx=f(x,y) where f(x,y)=y by a generalization of the modified Euler method yields a sequence of approximates that converge to e. Bounds on the rapidity of convergence are determined, with the fastest convergence occuring when the parameter value is 12, so the generalized method reduces to the standard modified Euler method. The situation is similarly examined when f is altered.