replacing for loop with more efficient code

Hi, given symmetric matrices für i=1,...,m, positiv definite and symmetric and ,

I want to compute with .

I solved this with the following for loop:

for i=1:m
        M_row=X*A(:,(i-1)*n+1:i*n)*inv_S;
        for j=i:m  %since M is symmetric
            M(i,j)=trace(M_row*A(:,(j-1)*n+1:j*n));
        end
end

Is there a more efficient way to solve this in matlab? with vectorization?

[Bruno Luong]

OP wants to compute

M(i,j) = Trace(X * A(:,:,i) * inv(S) * A(:,:,j))

where A(:,:,i), X and S are n x n symmetric, matrices, S and X are definite positive. i=1,...m

His original code that needs to be optimized is

M = zeros(m,m);
inv_S = inv(S);
for i=1:m
    M_row=X*A(:,(i-1)*n+1:i*n)*inv_S;
    for j=i:m  %since M is symmetric
        M(i,j)=trace(M_row*A(:,(j-1)*n+1:j*n));
    end
end

[Anusha Sridharan]

[Answers Dev] Restored Edits

Answer 1

[Bruno Luong]

% Generate test matrices
m = 5;
n = 10;
S = rand(n,n);
S = 0.5*(S + S.');
X = rand(n,n);
X = 0.5*(X + X.');
A = zeros(n,n,m);
for k=1:m
    Ak = rand(n,n);
    Ak = 0.5*(Ak + Ak.');
    A(:,:,k) = Ak;
end


AA = reshape(A,[n n m]);
M = zeros(m,m);
for i=1:m
    Ti = S\(AA(:,:,i)*X);
    M(i,:) = Ti(:).'*reshape(AA(:,:,:),[],m);
end

[Bruno Luong]

The loop can be replaced by vectorized code if you use this mtimesxmtimesx FEX

AA = reshape(A,[n n m]);
Ti = mtimesx(mtimesx(inv(S),AA),X);
M = reshape(Ti,[],m).' *reshape(AA(:,:,:),[],m)

Answer 2

[Lukas Müller]

Thanks for you fast answer. So I guess it doesnt work without at least one for loop. Still way better then before thanks a lot :)