# Kutta-Merson method

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A five-stage Runge–Kutta method with fourth-order accuracy. Applied to the Cauchy problem (1)

the method is as follows: (2)      The number serves as an estimate of the error and is used for automatic selection of the integration step. If is the prescribed accuracy of the computation, the integration step is selected as follows. First choose some initial step and start the computation by (2) to obtain the number . If , divide the integration step by 2; if , double it. If , the selected integration step is satisfactory. Now replace the initial point by and repeat the entire procedure. This yields an approximate solution ; the quantity is mainly auxiliary.

Since i.e. the formula for is as it were "nested" in the formula for , the method described here for the estimation of the error and the selection of the integration step is known as an imbedded Runge–Kutta method.

Standard programs for the Kutta–Merson method are available in Algol , .