How can I improve the efficiency of an equation with double summation

In some of my calculations, I have to perform a double-summation and my current approach seems to be very inefficient - taking several seconds to complete. The equation that I am trying to calculate is shown below (it has been simplified for clarity).

In this equation:

• f(t) and y(t) are each [1 x 10001] vectors
• M = 100
• x(n) is a 101 (M+1) vector of positive numbers (type: double)
• i is an imaginary number

I looked at a handful of loop optimization answers such as https://www.mathworks.com/matlabcentral/answers/877868-suggestions-for-vectorizing-double-triple-for-loops-in-matlab but none of the solutions that I found appeared to have more than small impact on my calculation time.

My approach

(I split it up into multiple terms within the for loops because there are a handful of scalar factors omitted here)

rng("default")
M = 100;
% Some temporary vectors so that the code can run
t = linspace(2,10,10001);
y = sqrt(t);
x = abs(randn(1,M+1));
tic
% Preallocate 3D Matrix 
fsum = zeros(length(y),M+1,M+1); 
% Loop through to calculate each M term
for m = 0:M
    for n = 1:M+1
        term1 = 1i.*y/((m+1)*x(n));
        num2 = 2*m + 5*x(n);
        den2 = 7;
        term2 = (num2/den2 + 1i.*y).^(-1);
        fsum(:,m+1,n) = term1.*term2;
    end
end
% Now sum away the 2nd and 3rd dimensions
fsum = sum(fsum,[2 3]);
% Now that we have the summation term as a vector that matches y(t),
% we can insert it into the f equation
f = 1./(1 - fsum);
toc

Elapsed time is 2.938936 seconds.

% Simple plot to make sure that f became a function of t
figure;
plot(t,abs(f),'r')
xlabel('t')
grid on

Is there anything that I can do to speed up the calculation of this vector f(t)? Currently the y(t) vector is only 10,001 points but I would like to increase that in the future to address some resolution issues that I am having with subsequent calculations.

Answer 1

[Torsten]

rng("default")
M = 100;
% Some temporary vectors so that the code can run
t = linspace(2,10,10001);
y = sqrt(t);
x = abs(randn(1,M+1)).';
tic
%mat = zeros(M+1);
f = zeros(size(t));
v = 0:M;
f = arrayfun(@(i)sum(1i*y(i)./((v + 1).*x)./((2*v + 5*x)/7 + 1i*y(i)),'All'),1:numel(y));
%for i=1:numel(y)
%    mat = 1i*y(i)./((v + 1).*x)./((2*v + 5*x)/7 + 1i*y(i));
%    f(i) = sum(mat,'All');
%end
f = 1./(1-f);
toc

Elapsed time is 0.771715 seconds.

figure;
plot(t,abs(f),'r')
xlabel('t')
grid on

[Emma Farnan]

Thank you! This is so much faster and I definitely wouldn't have thought of using arrayfun on my own