quadruple summation using function
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If I use function to calculate the double summation
for
, the code is
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1491532/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1491537/image.png)
n = 100;
a = @(p) sin(p);
sum(sum(a(0:n)' * a(0:n)))
However, if I want to calculate the quadruple summation
, how to modify the code?
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1491542/image.png)
4 Commenti
Matt J
il 24 Set 2023
You should be exploiting the separability of the summation,
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1491577/image.png)
and likewise for the quadruple sum.
Risposte (2)
Steven Lord
il 24 Set 2023
Spostato: Matt J
il 24 Set 2023
You could use implicit expansion to avoid having to create quite so many large arrays. It's not as fast as the case that exploits the separability, but it is significantly faster than the original general approach.
%Random value for N for example
N = 100;
kvec = 0:1:N; %1 as increment is not necessary
tic
[p,q,r,s] = ndgrid(kvec);
arr = sin(p).*sin(q).*sin(r).*sin(s);
s1 = sum(arr,'all');
toc
tic
s2=sum(sin(kvec))^4;
toc
% Use implicit expansion
tic
n = numel(kvec);
sinK = sin(kvec);
p = reshape(sinK, n, 1, 1, 1); % unnecessary in this case, but useful for generality
q = reshape(sinK, 1, n, 1, 1);
r = reshape(sinK, 1, 1, n, 1);
s = reshape(sinK, 1, 1, 1, n);
s3 = sum(p.*q.*r.*s, 'all');
toc
The trailing 1's in the reshape calls aren't really necessary, but they do make the pattern of sizes quite easy to see.
format longg
results = [s1, s1-s2, s1-s3; s2-s1, s2, s2-s3; s3-s1, s3-s2, s3]
Those diagonal elements are in pretty good agreement, and all the off-diagonal elements (the differences between the approaches) are all quite small in magnitude,
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