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

Question

Emma Farnan 2023-1-31

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1903520-how-can-i-improve-the-efficiency-of-an-equation-with-double-summation

评论： Emma Farnan 2023-1-31

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

% 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.