vectorizing a non-explicit function in Matlab and Octave - matlab

Question

I want to vectorize a 3D function, but the function does not have an analytical expression. For instance, I can vectorize the function
F(x, y, z) = (sin(y)*x, z*y, x*y)
by doing something like
function out = Vect_fn(x, y,z)
out(1) = x.*sin(y);
out(2) = z.*y;
out(3) = x.*y;
end
And then running the script
a = linspace(0,1,10);
[xx, yy, zz] = meshgrid(a, a, a);
D = Vect_fn(xx, yy, zz)
However, suppose the function does not have an analytical expression, for example
function y = Vect_Nexplicit(y0)
%%%%%%y0 is a 3x1 vector%%%%%%%%%%%%%%
t0 = 0.0;
tf = 3.0;
[t, z] = ode45('ODE_fn', [t0,tf], y0);
sz = size(z);
n = sz(1);
y = z(n, :);
end
where ODE_fn is just some function that spits out the right-hand side of a an ODE. Thus the function simply solves an ODE and so the function is not known explicitly. Of course I can use a for loop, but those are slower (esp. in Octave, which I prefer since it has lsode for solving ODEs)
Trying something like
a = linspace(0,1,10);
[xx, yy, zz] = meshgrid(a, a, a);
D = Vect_Nexplicit(xx, yy, zz)
does not work. Also here is the code for ODF_fn:
function ydot = ODE_fn(t, yin)
A = sqrt(3.0);
B = sqrt(2.0);
C = 1.0;
x = yin(1, 1);
y = yin(2,1);
z = yin(3, 1);
M = reshape(yin(4:12), 3, 3);
ydot(1,1) = A*sin(yin(3)) + C*cos(yin(2));
ydot(2,1) = B*sin(yin(1)) + A*cos(yin(3));
ydot(3,1) = C*sin(yin(2)) + B*cos(yin(1));
DV = [0 -C*sin(y) A*cos(z); B*cos(x) 0 -A*sin(z); -B*sin(x) C*cos(y) 0];
Mdot = DV*M;
ydot(4:12,1) = reshape(Mdot, 9, 1);
end

Answer 1

You can solve systems of differential equations with ode45, so if ODE_fn is vectorised you can use your approach. y0 just needs to be a vector too.
You can create a y0 that is [x1, ..., xn, y1, ..., yn, z1, ..., zn, M1_1-9, ... ,Mn_1-9] and then use for x, y, z just the appropriate indexes i.e. 1:n, n+1:2*n, 2*n+1:3n. Then use reshape(yin(3*n+1:end),3,3,n). But I am not sure how to vectorize the matrix multiplication.