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## Embedded Runge-Kutta Methods

A Runge-Kutta method is a one-step method for approximating the solution y(x0+h) of the initial value problem with the form

y(x0,h) = y0 + c1 k1 + . . . + cs ks

where the ci are constants and k1 = h f(x0,y0), ki = h f(Xi,Yi) where Xi = x0 + ai h and
Yi = y0 + bi,1 k1 + . . . + bi,i-1 k1-1, where the ai and bi,j are constants.

An embedded Runge-Kutta method is a method in which two Runge-Kutta estimates are obtained using the same auxiliary functions ki but with a different linear combination of these functions so that one estimate has an order one greater than the other. This allows an error estimate: Suppose that Y is one estimate of order r and Z is the other estimate of order
r + 1, then the difference between the two estimates satifies

Z - Y = A h r + O(h r + 1).

If the step size is scaled by µ so that the new step size is (µ h) then Z - Y becomes

Z - Y = A (µ h) r + O(h r + 1).

If ε is the allowable error on the interval from (x0, x0 + h), µ can be chosen so that the final error is less than that ε.
 Table of Embedded Runge-Kutta Methods