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From: Juliette Salexa on 8 Apr 2010 19:34
Hello, I tried:

n=round(7^4.5);
tic;a=rand(n,1);toc;
tic;b=rand(n,n);toc;
tic;sum(b*a);toc;
tic;sum(sum(b));toc;

% Elapsed time is 0.000502 seconds.
% Elapsed time is 3.609380 seconds.
% Elapsed time is 0.161011 seconds.
% Elapsed time is 0.297674 seconds.

-----------------

Which I found strange because the last computation does the same number of summations as the second last, but doesn't have to do any multiplying!

-----------------
For sparse matrices the difference is even bigger:

n=7^6;density=0.001;
tic;a=sprand(n,1,density);toc;
tic;b=sprand(n,n,density);toc;
tic;sum(b*a);toc;
tic;sum(sum(b));toc;

% Elapsed time is 0.049862 seconds.
% Elapsed time is 70.757589 seconds.
% Elapsed time is 0.026991 seconds.
% Elapsed time is 0.206313 seconds.

-----------------

A multiplying AND adding is 10 times faster than just adding ?

Does anyone have any idea why this is true ?
From: Walter Roberson on 8 Apr 2010 19:34
Juliette Salexa wrote:
> Hello, I tried:
>
> n=round(7^4.5);
> tic;a=rand(n,1);toc;
> tic;b=rand(n,n);toc;
> tic;sum(b*a);toc;
> tic;sum(sum(b));toc;
>
> % Elapsed time is 0.000502 seconds.
> % Elapsed time is 3.609380 seconds.
> % Elapsed time is 0.161011 seconds.
> % Elapsed time is 0.297674 seconds.
>
> -----------------
>
> Which I found strange because the last computation does the same number
> of summations as the second last, but doesn't have to do any multiplying!

For non-sparse matrices, the time difference disappears if you put the
commands into a function file and allow the Just In Time (JIT) compiler to
work its magic.

If you also add

tic;sum(b(:));toc;

then that will take almost exactly twice as long as the nested sum() calls,
even though it only has to compute a single sum instead lf n+1 sums. This
suggests that the computation time is dominated by copying the matrices rather
than by the arithmetic operations.
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