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quadruple summation using function

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CC SS
CC SS 2023-9-24
编辑： Matt J 2023-9-24
If I use function to calculate the double summation for , the code is
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?
  4 个评论
显示 2更早的评论
Matt J
Matt J 2023-9-24
You should be exploiting the separability of the summation,
and likewise for the quadruple sum.

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回答（2 个）

Matt J
Matt J 2023-9-24
编辑：Matt J 2023-9-24
Ran in:
n = 100;
tic;
sum(sin(0:n))^4;
toc
Elapsed time is 0.002355 seconds.
Steven Lord
Steven Lord 2023-9-24
移动：Matt J 2023-9-24
Ran in:
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
Elapsed time is 3.802691 seconds.
tic
s2=sum(sin(kvec))^4;
toc
Elapsed time is 0.002488 seconds.
% 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
Elapsed time is 0.332431 seconds.
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]
results = 3×3
1.0e+00 * 0.000261548679431158 -2.30172814402047e-13 0 2.30172814402047e-13 0.00026154867966133 2.30172814402047e-13 0 -2.30172814402047e-13 0.000261548679431158
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,
  1 个评论
Matt J
Matt J 2023-9-24
编辑：Matt J 2023-9-24
Ran in:
You can simplify this by downloading ndgridVecs from the File Exchange,
https://www.mathworks.com/matlabcentral/fileexchange/74956-ndgridvecs
tic
[p,q,r,s] = ndgridVecs(sin(0:100));
result = sum( p.*q.*r.*s , 'all');
toc
Elapsed time is 0.268219 seconds.

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