# Writing a Matlab function after calling the ode45 solver

After using ode45 to solve a set of ODEs, I want to write a Matlab function to take the initial conditions x_0 as inputs and gives the final state x_1 at time T as the output, thus creating a discrete dynamical system f.

However, I'm having some trouble writing the function that would embed the initial conditions, the ode45 solver, and the final state as the output.

My attempt:

% F is the function handle to the set of ODEs to be solved

x_0 = some initial conditions here

T = 10

function x_1 = f(x_0);

[t, y] = ode45( F, [0 T], x_0 );

% store final state values in variables

x_final     = y( end, 4 );
y_final     = y( end, 5 );
theta_final   = y( end, 3 );

% now define the output of f to take the above final state values

x_1 = [ x_final y_final theta_final ];

end


This doesn't seem to work. I get the messages:

(1) The function 'f' might be unused

(2) The value assigned here to 't' appears to be unused. Consider replacing it with ~.

What am I missing?

Thanks,

• The problem is that you are not calling the function $f$ which you have defined. You call it by writing $x_1 = f(x_0)$ without the function keyword. – Christoph Aug 31 '20 at 3:08

You want to solve the initial value problem $$\begin{equation} \dot{x} = f(t,x), \quad x(0) = x_0, \end{equation}$$ and then to evaluate the function $$F : x_0 \mapsto y = x(T)$$, where $$T > 0$$ denotes the final time. In MATLAB this can be done with a nested function, for example:

function y = F(x0)
T = ...
[t,x] = ode45(@f,[0 T],x0);
y = x(end,:);

function xp = f(t,x)
xp = ...
end
end


Once defined you can call the function $$F$$ in a MATLAB script with specific initial values $$x_0$$ to obtain the corresponding final values $$y$$.

• the function $f(t,x)$ should evaluate the right-hand side of your ODE system. The function $F$ is called with an input (initial value) $x_0$ and it outputs the final value $y = x(T;x_0)$. You can use the function $F$ inside a script, for example, where you can store the pairs $(x_0,y)$. – Christoph Aug 30 '20 at 11:24
• Yes, the code fragment that I gave should be stored in a separate file F.m. Then the function F can be called inside a script (i. e. a different file, say main.m) which is stored inside the same directory as F.m. – Christoph Aug 30 '20 at 11:30
• No, @f is just a handle to a function that needs to be defined somewhere. This happens in the nested function definition. – Christoph Aug 30 '20 at 18:24
• Variables used in a MATLAB function are local, so it is normal that you cannot access it from the command window. But if you call the function F as in >> y = F([...]), then $y$ should be defined afterwards. The other errors about F and t I don't understand. We might have to continue the discussion in a chat. – Christoph Aug 30 '20 at 18:44
• Hi Christoph, I've posted a code sample of mine - thanks. – user36920 Aug 30 '20 at 22:16