vectorization - how to efficiently/conveniently compute A-inner product (ie. a bilinear form) of many vectors in Matlab? -

i want compute thing v'*m*v in matlab v taken columns of given matrix a, square , potentially large. ie. v=a(:,j)

what convenient , computationally efficient way this? thinking of using bsxfun , possibly reshape not sure how work.

i remember reading similar post long time ago. can't locate it.

the fastest way came loop, not elegant way. maybe else think of better.

function compare(s, v) m = rand(s); = rand(s, v);  %method 1: loop tic r1 = zeros(1, size(a,2)); = 1:size(a,2)     r1(i) = a(:,i)'*m*a(:,i); end dt = toc; disp(['for loop ', num2str(size(a,2)), ' vectors of length ', num2str(size(a,1)), ' ', num2str(dt), ' s.'])  %method 2: cell functions anonymous function tic ap = num2cell(a, 1); r2 = cell2mat(cellfun(@(x) x'*m*x, ap, 'uni', 0)); dt = toc; disp(['cell functions using anonymous function ', num2str(size(a,2)), ' vectors of length ', num2str(size(a,1)), ' ', num2str(dt), ' s.'])  %method 3: vector multiplication tic r3 = diag(a'*m*a); dt = toc; disp(['vector multiplication ', num2str(size(a,2)), ' vectors of length ', num2str(size(a,1)), ' ', num2str(dt), ' s.'])  end 

edit: outputs

>> compare(20, 200)     loop 200 vectors of length 20 0.0016883 s.     cell functions using anonymous function 200 vectors of length 20 0.0079001 s.     vector multiplication 200 vectors of length 20 0.0035036 s. >> compare(20, 400)     loop 400 vectors of length 20 0.0035246 s.     cell functions using anonymous function 400 vectors of length 20 0.010177 s.     vector multiplication 400 vectors of length 20 0.0076295 s. >> compare(20, 800)     loop 800 vectors of length 20 0.0069367 s.     cell functions using anonymous function 800 vectors of length 20 0.022697 s.     vector multiplication 800 vectors of length 20 0.0075474 s. >> compare(20, 1600)     loop 1600 vectors of length 20 0.013802 s.     cell functions using anonymous function 1600 vectors of length 20 0.037844 s.     vector multiplication 1600 vectors of length 20 0.029591 s. >> compare(20, 3200)     loop 3200 vectors of length 20 0.026893 s.     cell functions using anonymous function 3200 vectors of length 20 0.078213 s.     vector multiplication 3200 vectors of length 20 0.084117 s. >> compare(20, 6400)     loop 6400 vectors of length 20 0.053695 s.     cell functions using anonymous function 6400 vectors of length 20 0.15759 s.     vector multiplication 6400 vectors of length 20 0.3524 s. >> compare(40, 1600)     loop 1600 vectors of length 40 0.01514 s.     cell functions using anonymous function 1600 vectors of length 40 0.040556 s.     vector multiplication 1600 vectors of length 40 0.028335 s. >> compare(80, 1600)     loop 1600 vectors of length 80 0.022824 s.     cell functions using anonymous function 1600 vectors of length 80 0.053713 s.     vector multiplication 1600 vectors of length 80 0.047412 s. >> compare(160, 1600)     loop 1600 vectors of length 160 0.045606 s.     cell functions using anonymous function 1600 vectors of length 160 0.096006 s.     vector multiplication 1600 vectors of length 160 0.052472 s. >> compare(320, 1600)     loop 1600 vectors of length 320 0.074407 s.     cell functions using anonymous function 1600 vectors of length 320 0.1386 s.     vector multiplication 1600 vectors of length 320 0.19317 s. >> compare(640, 1600)     loop 1600 vectors of length 640 0.21931 s.     cell functions using anonymous function 1600 vectors of length 640 0.36021 s.     vector multiplication 1600 vectors of length 640 0.24102 s. >> compare(1280, 1600)     loop 1600 vectors of length 1280 1.6893 s.     cell functions using anonymous function 1600 vectors of length 1280 1.8245 s.     vector multiplication 1600 vectors of length 1280 0.57957 s. >> compare(2560, 1600)     loop 1600 vectors of length 2560 6.8812 s.     cell functions using anonymous function 1600 vectors of length 2560 7.0459 s.     vector multiplication 1600 vectors of length 2560 1.2919 s.