Runge-Kutta formula/derivation
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## Runge-Kutta formula/derivation

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 [2][pages 242-243], and the highlights
follow below.

- To match the coefficients, and , Taylor expand
:
where the remainder term is

- Match the coefficients of similar terms between
and
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.

- The above formula is valid for a first order differential
equation. The set of equations given in becomes

- To determine the error associated with this approximation, a
higher order approximation is needed. The fourth order Runge-Kutta is
given by

With two approximations, the quality of the approximation [2][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

*mcr@ccs.carleton.ca*