Home > src > rk4.m

# rk4

## PURPOSE RK4 4th order Runge-Kutta integration

## SYNOPSIS function [x,Y] = rk4(f,dt,x,P1,P2,P3,Y)

## DESCRIPTION ```RK4  4th order Runge-Kutta integration

Syntax:
[x,Y] = rk4(f,dt,x,[P1,P2,P3,Y])

In:
f - Name of function in form f(x,P(:)) or
inline function taking the same parameters.
In chained case the function should be f(x,y,P(:)).
dt - Delta time as scalar.
x - Value of x from the previous time step.
P1 - Values of parameters of the function at initial time t
as a cell array (or single plain value). Defaults to empty
array (no parameters).
P2 - Values of parameters of the function at time t+dt/2 as
a cell array (or single plain value). Defaults to P1 and
each empty (or missing) value in the cell array is replaced
with the corresponding value in P1.
P3 - Values of parameters of the function at time t+dt.
Defaults to P2 similarly to above.
Y - Cell array of partial results y1,y2,y3,y4 in the RK algorithm
of the second parameter in the interated function. This can be
used for chaining the integrators. Defaults to empty.

Out:
x - Next value of X
Y - Cell array of partial results in Runge-Kutta algorithm.

Description:
Perform one fourth order Runge-Kutta iteration step
for differential equation

dx/dt = f(x(t),P{:})

or in the chained case

dx/dt = f(x(t),y(t),P{:})
dy/dt = g(y(t),P{:})

- Example 1. Simple integration of model

dx/dt = tanh(x), x(0) = 1

can be done as follows:

X = [];
x = 1;
f = inline('tanh(x)','x');
for i=1:100
x = rk4(f,0.1,x);
X = [X x];
end

- Example 2. Chaining of integrators. Consider a
model of the form

dx/dt = x+y,     x(0)=1
dy/dt = tanh(y), y(0)=2

The equations can be now integrated as follows:

XY = [];
x = 1;
y = 2;
fx = inline('x+y','x','y');
fy = inline('tanh(y)','y');
for i=1:100
[y,YY] = rk4(fy,0.1,y);
x = rk4(fx,0.1,x,{},{},{},YY);
XY = [XY [x;y]];
end

which produces exactly the same result as

XY = [];
xy = [1;2];
fxy = inline('[xy(1)+xy(2);tanh(xy(2))]','xy');
for i=1:100
xy = rk4(fxy,0.1,xy);
XY = [XY xy];
end```

## CROSS-REFERENCE INFORMATION This function calls:
This function is called by:

Generated on Fri 12-Aug-2011 15:08:47 by m2html © 2005