kepler4

Celestial Mechanics on a Graphing Calculator

## A simple example of the Runge-Kutta algorithm

 An illustration of the Runge-Kutta algorithm appliedto the differential equation y' = 2y witha step size of 1 and initial value y(0) = 1/4.The predicted value is a linear combination, withcoefficients 1/6, 2/6, 2/6, 1/6 of the values obtained by following the (blue) slope 1/2 line at (0,1/4). This gives y = 3/4. following the (green) line with the slope (1) picked up by theblue line at x = 1/2. This gives y = 5/4. following the (orange) line with the slope (3/2) picked up by thegreen line at x = 1/2. This gives y = 7/4. following the (black) line with the slope (7/2) picked up by theorange line at x = 1. This gives y = 15/4.The total is (1/24)(3+10+14+15) = 1.75.The same computational load would buy 4 iterations of Euler'smethod yielding 1.265.The exact value is y(1) = (1/4)e2 = 1.85.

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