MATLAB: Construct product of matrices without for loop

MATLABproduct of matrices

Hi, everyone:
Suppose A is a 2 by 2 by N matrix, is it possible to construct a matrix B, such that:
B=[Prod(A)_1^{N-n},   Prod(A)_2^{N-n},   ... , Prod(A)_{N-n}^{N-n};  
   Prod(A)_1^{N-n+1}, Prod(A)_2^{N-n+1}, ... , Prod(A)_{N-n}^{N-n+1};
           .                       .          .              .        
           .                       .          .              .         
   Prod(A)_1^N,       Prod(A)_2^N,       ... , Prod(A)_{N-n}^N]
where Prod(A)_m^n = A(:,:,m)*A(:,:,m+1)*A(:,:,m+2)* … * A(:,:,n); (1 << n << N)
without any loop?? (maybe use arrayfun? cellfun?)
because for loop is too slow for me.
Thanks

Best Answer

I assume a FOR loop is the best way to go. Perhaps the slow speed is caused by a forgotten pre-allocation or if you calculate the power operation by the expensive ^ operator instead of a cumulative multiplication.
C = zeros(2, 2, n+1, N-n);
for i2 = 1:N-n
  AA = A(:, :, i2);
  PA = AA ^ (N-n-1);
  for i1 = 1:n+1
    PA = PA * AA;
    C(:, :, i1, i2) = PA;
  end
end
But if N is large and n is small, the power operation will be the bottleneck again. Then try these functions written by James: FEX: MTimesX or FEX: MPower2.
Another approach could be to avoid calling the powerful BLAS function for the matrix multiplication for a tiny 2x2 matrix:
function MatrixMultTest
x = rand(2, 2);
y = rand(2, 2)
tic; for k = 1:1e6; v = mult1(x,y); end, toc
tic; for k = 1:1e6; v = mult1(x,y); end, toc
function v = mult1(x, y)
v = x * y
function v = mult2(x, y)
v(2,2) = x(2,1)*y(1,2) + x(2,2)*y(2,2);  % Implicit pre-allocation
v(1,1) = x(1,1)*y(1,1) + x(1,2)*y(2,1);
v(1,2) = x(1,1)*y(1,2) + x(1,2)*y(2,2);
v(2,1) = x(2,1)*y(1,1) + x(2,2)*y(2,1);
Well, it seems like the manual product is 10% slower under R2009a. But it was worth to try.
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