From time to time, I have to compute the the square of norms for a large number of vectors. This usually happens when I'm modeling some physic phenomenon, such as a solar system plus a few extra bodies or a EM system.
Over the time - and my knowledge of Octave growing - I've used a few different techniques for this. The first one was the obvious for loops over the all the vectors, then, using a single for loop, simply calling norm on each line and squaring.
Lately, I found two interesting techniques: if all my vectors are the lines in a matrix M, calculating the square norms is also possible by extracting the diagonal of M*M'. The second one is to take advantadge of the .* operator and use sum to get, well, the sum.
This works fine, and recently, I've just started wondering which method would be the fastest. I deliberately skipped the very first one (the two nested for loops), and timed the three others, for sets of size 1,2,5,10,20,50,100,200,500,1000,2000,5000 and 10000.
An interesting note is that diag(M*M') doesn't work well with values above 16000, with a limit on my machine of 16383. I assume the explanation lies in the fact that this generates a resulting matrix 16383x16383, or a matrix containing 268402689 elements.
Here are the results. All the times are in second.
| Size | Using norm | Using diag | Using .* + sum |
|---|
| 1 | 9.0804e-09 | 8.8476e-09 | 8.7311e-09 |
| 2 | 5.7044e-09 | 6.5193e-09 | 5.1223e-09 |
| 5 | 6.4028e-09 | 8.2655e-09 | 5.0059e-09 |
| 10 | 7.7998e-09 | 6.5193e-09 | 5.0059e-09 |
| 20 | 1.0477e-08 | 5.3551e-09 | 5.0059e-09 |
| 50 | 1.9209e-08 | 5.4715e-09 | 5.2387e-09 |
| 100 | 3.3295e-08 | 6.1700e-09 | 5.4715e-09 |
| 200 | 6.0885e-08 | 8.7311e-09 | 6.9849e-09 |
| 500 | 1.4703e-07 | 3.5390e-08 | 5.8208e-09 |
| 1000 | 2.9348e-07 | 1.3306e-07 | 5.3551e-09 |
| 2000 | 5.9220e-07 | 5.4494e-07 | 5.2387e-09 |
| 5000 | 1.5607e-06 | 4.8558e-06 | 5.7044e-09 |
| 10000 | 4.1565e-06 | 2.4022e-05 | 6.1700e-09 |
A few interesting things there.
- The time to run the computation using for/norm is almost linear in size(input)
- The time to run the computation using diag is quadratic in size(input)
- The time to run the computation using .*/sum has an initial bump then is almost linear in size(input)
I continued the comparison past 16000 for the first and last methods. For size(input)=500000, the for/norm method took 0.0054697 seconds, the .*/sum 1.6822e-07. So far, I haven't seen a profiler for Octave, so I can just either guess or imagine why such a difference.
Bottom line, this test was interesting in multiple aspects. It also helped me decide which square norm function to choose. Starting now, my function is:
function N=sqnorm(X)
N=sum(X.*X);
endfunction
