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 ];

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?


  • $\begingroup$ 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. $\endgroup$
    – Christoph
    Aug 31, 2020 at 3:08

1 Answer 1


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 = ...

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

  • $\begingroup$ 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 $(x_0,y)$. $\endgroup$
    – Christoph
    Aug 30, 2020 at 11:24
  • $\begingroup$ 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. $\endgroup$
    – Christoph
    Aug 30, 2020 at 11:30
  • $\begingroup$ No, @f is just a handle to a function that needs to be defined somewhere. This happens in the nested function definition. $\endgroup$
    – Christoph
    Aug 30, 2020 at 18:24
  • $\begingroup$ 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. $\endgroup$
    – Christoph
    Aug 30, 2020 at 18:44
  • $\begingroup$ Hi Christoph, I've posted a code sample of mine - thanks. $\endgroup$
    – user36920
    Aug 30, 2020 at 22:16

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