The RK4 function uses the fourth-order Runge-Kutta method to advance a solution to a system of ordinary differential equations one time-step *H*, given values for the variables *Y* and their derivatives *Dydx* known at *X*.

RK4 is based on the routine rk4 described in section 16.1 of *Numerical Recipes in C: The Art of Scientific Computing* (Second Edition), published by Cambridge University Press, and is used by permission.

## Examples

To integrate the example system of differential equations for one time step, H:

; Define the step size:

H = 0.5

; Define an initial X value:

X = 0.0

; Define initial Y values:

Y = [4.0, 6.0]

; Calculate the initial derivative values:

dydx = DIFFERENTIAL(X,Y)

; Integrate over the interval (0, 0.5):

result = RK4(Y, dydx, X, H, 'differential')

; Print the result:

PRINT, result

IDL prints:

3.11523 6.85767

This is the exact solution vector to five-decimal precision.

## Syntax

*Result* = RK4( *Y*, *Dydx*, *X*, *H*, *Derivs* [, /DOUBLE] )

## Return Value

Returns the integrations of the ordinary differential equations.

## Arguments

### Y

A vector of values for *Y* at *X*

**Note: **If RK4 is complex then only the real part is used for the computation.

### Dydx

A vector of derivatives for *Y* at *X*.

### X

A scalar value for the initial condition.

### H

A scalar value giving interval length or step size.

### Derivs

A scalar string specifying the name of a user-supplied IDL function that calculates the values of the derivatives *Dydx* at *X*. This function must accept two arguments: A scalar floating value *X*, and one *n*-element vector *Y*. It must return an *n*-element vector result.

For example, suppose the values of the derivatives are defined by the following relations:

*dy*_{0} / *dx* = –0.5*y*_{0}_{, }*dy*_{1} / *dx* = 4.0 – 0.3*y*_{1} – 0.1*y*_{0}

We can write a function DIFFERENTIAL to express these relationships in the IDL language:

`FUNCTION differential, X, Y`

RETURN, [-0.5 * Y[0], 4.0 - 0.3 * Y[1] - 0.1 * Y[0]]

`END`

## Keywords

### DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

## Version History

4.0 |
Introduced |