How to speed up my code?

I have a code with a lot of cycles and matrix multiplication. Input data: vector M of matrices 2x2, n2 - length of vector M (number of matrices 2x2). k - current matrix. My matrix do cyclic multiplication. It goes from the k-th element to the first, after that to last and after that to the (k+1)th.

E.g. n2=6, k=2: M2*M1*M2*M3*M4*M5*M6*M5*M4*M3

I was trying to turn vector M to gpuArray, also I was trying to turn cycles, that started from "for i=1:n2" to parfor but it didn't help. So, it's the problem with number of cycles? Can you give me some advices how to solve these problems and how avoid problems like these in future?

for i=1:n2
     K(:,:,i)=M(:,:,i);
 end
 for i=1:n2 %return the original matrix for the new cycle
           J(:,:,i)=K(:,:,i);
 end
%cycle from k-th element to the beginning 
if k == 1
else
  for i=k-1:-1:1
    M(:,:,i)=M(:,:,i+1)*M(:,:,i);
    K(:,:,i)=M(:,:,i);
  end
%return the original matrix, change the first element
  for i=1:n2
    M(:,:,i)=J(:,:,i);
  end
    M(:,:,1)=K(:,:,1);
end
%loop from beginning to end
  for i=1:n2-1
    M(:,:,i+1)=M(:,:,i)*M(:,:,i+1);
    K(:,:,i+1)=M(:,:,i+1);
  end
%return the original matrix, change the "last" element
  for i=1:n2
    M(:,:,i)=J(:,:,i);
  end
M(:,:,n2)=K(:,:,n2);
if n2 == k
M(:,:,1)=K(:,:,n2-1);
else
%loop from end to (k+1)th element
  for i=n2:-1:k+2
    M(:,:,i-1)=M(:,:,i)*M(:,:,i-1);
  end
end

Answer 1

[Akira Agata]

Avoiding for-loop will enhance computational efficiency.

For example, the first for-loop:

for i=1:n2
  K(:,:,i)=M(:,:,i);
end

should be as follows:

K(:,:,1:n2)=M(:,:,1:n2);

[George Bashkatov]

That works, thanks! But can you answer one question: Can I do something similar for this cycle only with using vectors?

for i=k-1:-1:1
    M(:,:,i)=M(:,:,i+1)*M(:,:,i);
    K(:,:,i)=M(:,:,i);
  end

[Walter Roberson]

@George Bashkatov

No, for two different reasons:

1. the * operator is not vectorizable. The closest you can get to that https://www.mathworks.com/help/matlab/ref/pagemtimes.html
2. With you looping like that, you are creating a cummulative matrix product, M(:,:,end), M(:,:,end)*M(:,:,end-1), M(:,:,end)*M(:,:,end-1)*M(:,:,end-2), and so on. So the results for any given iteration depend upon the results for the previous iterations. That is not something that can be vectorized... not even with pagemtimes .

If you were using the .* operator instead of the * operator then with a permute(), reshape(), flipud(), cumprod(), flipud(), reshape(), permute() you could get the answer without a loop. But with the * operator, you need a loop.

[George Bashkatov]

Thank you a lot