function handle as array

I define a function handle as

P = @(t) mtimes(expm(t.*M),P_initial);

Where P_initial is a 4x1 array and M a 4x4 matrix. This works fine, but now I have a function with for every t a 4x1 answer, for example at t = 0 I get:

P(0)

ans =

    0.0000
    0.0000
    0.0011
    0.9988 

What I want is to get this first element as a separate function of time, so I want a P_1(t) as a function handle, but I do not know how to do so.

Thanks already! :)

Answer 1

[Torsten]

M = rand(4,4);
P_initial = rand(4,1);
P = @(t) mtimes(expm(t.*M),P_initial);
P(0)
g = @(a,x)a(x);
g(P(0),1)

[Wout Laeremans]

Hi Torsten, this comes close to what I want thank you! Do you also know how I can plot g(P(t),1) for a general t?

[Torsten]

M = rand(4,4);
P_initial = rand(4,1);
P = @(t) mtimes(expm(t.*M),P_initial);
g = @(a,x)a(x);
t = 0:0.01:1;
plot(t,arrayfun(@(t)g(P(t),1),t))

Answer 2

[Davide Masiello]

That's because mtimes(a,b) = a*b, and if a is 4x4 and b is 4x1 than the operation yields a 4x1 array.

I think what you might want to do is

P = @(t) expm(t.*M).*P_initial;

Which will yield a 4x4 matrix where each column is .

But I am not completely sure I interpreted this correctly.

[Wout Laeremans]

No no, I will give more context: P gives a probability to be in a certain state. At every moment in time t, the first entry gives the probability to be in state 1, the second to be in state 2 and so on. However, I also need the first entry as a function of time, it est the probability to be in state 1 as a function handle of variable t.

[Davide Masiello]

you can define a cell array of functions

M = rand(4,4);
P_initial =  rand(1,4);
for i = 1:length(M)
P{i} = @(t)mtimes(expm(t.*M),P_initial);
end
P

P = 1×4 cell array
    {@(t)mtimes(expm(t.*M),P_initial)}    {@(t)mtimes(expm(t.*M),P_initial)}    {@(t)mtimes(expm(t.*M),P_initial)}    {@(t)mtimes(expm(t.*M),P_initial)}

With your formulation, however, they are all the same.

Are you sure you must use expm and mtimes?

I think the problem might be ill-posed in that aspect.

[Wout Laeremans]

Yes, I want the solution of:

The general solution is given by:

Where I call P = [P_O; P_3; P_2; P_1] and M the transition matrix. I know I can also solve this using an ode solver or just define a time array with a small time increment. However I really want P_O(t) as a function handle.