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The Runge-Kutta method has the desirable property of not requiring actual evaluation of the derivatives. The precise form of the function is often not known, or may be very costly to evaluate. A Runge-Kutta method is formed by approximating the second order Taylor polynomial
( is the step size) by
This is done in [pages 242-243], and the highlights follow below.
This results in the Midpoint method given by
Note, the remainder term is , making the Midpoint method a second order method. This means that the function being integrated is being approximated by a parabola at each step.
With two approximations, the quality of the approximation [page 271] is given by
This quality factor can be used to increase or decrease the value of the step size. is the tolerance desired. The new step size is given by