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Gill's Method

A formula for numerical solution of differential equations,

y_{n+1}=y_n+{\textstyle{1\over 6}}[k_1+(2-\sqrt{2}\,)k_2+(2+\sqrt{2}\,)k_3+k_4]+{\mathcal O}(h^5),

$\displaystyle k_1$ $\textstyle =$ $\displaystyle hf(x_n,y_n)$  
$\displaystyle k_2$ $\textstyle =$ $\displaystyle hf(x_n+{\textstyle{1\over 2}}h, y_n+{\textstyle{1\over 2}}k_1)$  
$\displaystyle k_3$ $\textstyle =$ $\displaystyle hf[x_n+{\textstyle{1\over 2}}h, y_n+{\textstyle{1\over 2}}(-1+\sqrt{2}\,)k_1+(1-{\textstyle{1\over 2}}\sqrt{2}\,)k_2]$  
$\displaystyle k_4$ $\textstyle =$ $\displaystyle hf[x_n+h, y_n-{\textstyle{1\over 2}}\sqrt{2}\,k_2+(1+{\textstyle{1\over 2}}\sqrt{2}\,)k_3].$  

See also Adams' Method, Milne's Method, Predictor-Corrector Methods, Runge-Kutta Method


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 896, 1972.

© 1996-9 Eric W. Weisstein