Vectorizing MATLAB function

Question

I have double summation over m = 1:M and n = 1:N for polar point with coordinates rho, phi, z:

I have written vectorized notation of it:

N = 10;
M = 10;
n = 1:N;
m = 1:M;

rho = 1;
phi = 1;
z = 1;

summ =  cos (n*z)  * besselj(m'-1, n*rho) * cos(m*phi)';

Now I need to rewrite this function for accepting vectors (columns) of coordinates rho, phi, z. I tried arrayfun, cellfun, simple for loop - they work too slow for me. I know about "MATLAB array manipulation tips and tricks", but as MATLAB beginner I can't understand repmat and other functions.

Can anybody suggest vectorized solution?

Answer 1

I think your code is already well vectorized (for n and m). If you want this function to also accept an array of rho/phi/z values, I suggest you simply process the values in a for-loop, as I doubt any further vectorization will bring significant improvements (plus the code will be harder to read).

Having said that, in the code below, I tried to vectorize the part where you compute the various components being multiplied {row N} * { matrix N*M } * {col M} = {scalar}, by making a single call to the BESSELJ and COS functions (I place each of the row/matrix/column in the third dimension). Their multiplication is still done in a loop (ARRAYFUN to be exact):

%# parameters
N = 10; M = 10;
n = 1:N; m = 1:M;

num = 50;
rho = 1:num; phi = 1:num; z = 1:num;

%# straightforward FOR-loop
tic
result1 = zeros(1,num);
for i=1:num
    result1(i) = cos(n*z(i)) * besselj(m'-1, n*rho(i)) * cos(m*phi(i))';
end
toc

%# vectorized computation of the components
tic
a = cos( bsxfun(@times, n, permute(z(:),[3 2 1])) );
b = besselj(m'-1, reshape(bsxfun(@times,n,rho(:))',[],1)');             %'
b = permute(reshape(b',[length(m) length(n) length(rho)]), [2 1 3]);    %'
c = cos( bsxfun(@times, m, permute(phi(:),[3 2 1])) );
result2 = arrayfun(@(i) a(:,:,i)*b(:,:,i)*c(:,:,i)', 1:num);            %'
toc

%# make sure the two results are the same
assert( isequal(result1,result2) )

I did another benchmark test using the TIMEIT function (gives more fair timings). The result agrees with the previous:

0.0062407    # elapsed time (seconds) for the my solution
0.015677     # elapsed time (seconds) for the FOR-loop solution

Note that as you increase the size of the input vectors, the two methods will start to have similar timings (the FOR-loop even wins on some occasions)

Answer 2

You need to create two matrices, say m_ and n_ so that by selecting element i,j of each matrix you get the desired index for both m and n.

Most MATLAB functions accept matrices and vectors and compute their results element by element. So to produce a double sum, you compute all elements of the sum in parallel by f(m_, n_) and sum them.

In your case (note that the .* operator performs element-wise multiplication of matrices)

N = 10;
M = 10;
n = 1:N;
m = 1:M;

rho = 1;
phi = 1;
z = 1;

% N rows x M columns for each matrix
% n_ - all columns are identical
% m_ - all rows are identical
n_ = repmat(n', 1,  M);
m_ = repmat(m , N,  1);

element_nm =  cos (n_*z) .* besselj(m_-1, n_*rho) .* cos(m_*phi);
sum_all = sum( element_nm(:) );