Efficiently ways to solve for multiple for loops, of which the indices are the parameters of a function

Hi guys, I have code as follows, which involves 3 for loops. It runs very slow. Can I get some suggestion on how to make it run faster? Thanks.

function Grid_Output = Grid_Conv_Assign(x,y,t)  % x, y, t are big vectors
Grid_Output = zeros(size(Grid_Pre));
for i = 1:length(x)
    for j = 1:length(y)
        for k = 1:length(t)
            Grid_Output(i,j,k) = Grid_Calculation(x(i),y(j),t(k));
        end
    end
end
end


function Prob = Grid_Calculation(x,y,t) 
%This function includes matrix multiplication and logrithmn

mu = rand(3,3)
sig = rand(3,3)
g = [cos(t),-sin(t),x;
     sin(t), cos(t),y;
     0     , 0     ,1];

y_m = logm(mu\g);
y_v = [y_m(1,3);y_m(2,3);y_m(2,1)];
Prob = exp(-0.5*y_v'/sig*y_v);
end

Answer 1

[Walter Roberson]

You can do at least part of it in vectorized form.

The following deliberately creates random numbers in the same order as if you had run your function over multiple iterations

N = length(x);
rands = rand(3,3,2,N);
mu_rc = reshape(rands(:,:,1,:), 3, []);     %arrange them as 3 x 3 beside each other
sigs = reshape(rands(:,:,2,:), 3, []);      %arrange them as 3 x 3 beside each other

ct = cos(t(:));
st = sin(t(:));

g_r1 = reshape([ct, -st, x(:)].', 1, []);
g_r2 = reshape([st, ct, y(:)].', 1, []);
g_r3 = repmat([0 0 1], 1, N);
g_r = [g_r1; g_r2; g_r3];

mu_c = mat2cell(mu_rc, 3, 3*ones(1,N));
g_c  = mat2cell(g_r, 3, 3*ones(1,N));

mu_blk = blkdiag(mu_c{:});
g_blk = blkdiag(g_c{:});

mug_blk = mu_blk \ g_blk;

mug_c = mat2cell(mug_blk, 3*ones(1,N), 3*ones(1,N));

y_mc = cellfun(@logm, mug_c(1:N+1:end), 'uniform', 0);

At this point, y_mc is a cell array of the y_m results. What remains is to do the vectorized calculation of the remaining two steps, namely

y_v = [y_m(1,3);y_m(2,3);y_m(2,1)];
Prob = exp(-0.5*y_v'/sig*y_v);

I am not going to do this at the moment; I am tired and need to do some other things.