Background
My question is motivated by simple observations, which somewhat undermine the beliefs/assumptions often held/made by experienced MATLAB users:
- MATLAB is very well optimized when it comes to the built-in functions and the fundamental language features, such as indexing vectors and matrices.
- Loops in MATLAB are slow (despite the JIT) and should generally be avoided if the algorithm can be expressed in a native, 'vectorized' manner.
The bottom line: core MATLAB functionality is efficient and trying to outperform it using MATLAB code is hard, if not impossible.
Investigating performance of vector indexing
The example codes shown below are as fundamental as it gets: I assign a scalar value to all vector entries. First, I allocate an empty vector x:
tic; x = zeros(1e8,1); toc
Elapsed time is 0.260525 seconds.
Having x I would like to set all its entries to the same value. In practice you would do it differently, e.g., x = value*ones(1e8,1), but the point here is to investigate the performance of vector indexing. The simplest way is to write:
tic; x(:) = 1; toc
Elapsed time is 0.094316 seconds.
Let's call it method 1 (from the value assigned to x). It seems to be very fast (faster at least than memory allocation). Because the only thing I do here is operate on memory, I can estimate the efficiency of this code by calculating the obtained effective memory bandwidth and comparing it to the hardware memory bandwidth of my computer:
eff_bandwidth = numel(x) * 8 bytes per double * 2 / time
In the above, I multiply by 2 because unless SSE streaming is used, setting values in memory requires that the vector is both read from and written to the memory. In the above example:
eff_bandwidth(1) = 1e8*8*2/0.094316 = 17 Gb/s
STREAM-benchmarked memory bandwidth of my computer is around 17.9 Gb/s, so indeed - MATLAB delivers close to peak performance in this case! So far, so good.
Method 1 is suitable if you want to set all vector elements to some value. But if you want to access elements every step entries, you need to substitute the : with e.g., 1:step:end. Below is a direct speed comparison with method 1:
tic; x(1:end) = 2; toc
Elapsed time is 0.496476 seconds.
While you would not expect it to perform any different, method 2 is clearly big trouble: factor 5 slowdown for no reason. My suspicion is that in this case MATLAB explicitly allocates the index vector (1:end). This is somewhat confirmed by using explicit vector size instead of end:
tic; x(1:1e8) = 3; toc
Elapsed time is 0.482083 seconds.
Methods 2 and 3 perform equally bad.
Another possibility is to explicitly create an index vector id and use it to index x. This gives you the most flexible indexing capabilities. In our case:
tic;
id = 1:1e8; % colon(1,1e8);
x(id) = 4;
toc
Elapsed time is 1.208419 seconds.
Now that is really something - 12 times slowdown compared to method 1! I understand it should perform worse than method 1 because of the additional memory used for id, but why is it so much worse than methods 2 and 3?
Let's try to give the loops a try - as hopeless as it may sound.
tic;
for i=1:numel(x)
x(i) = 5;
end
toc
Elapsed time is 0.788944 seconds.
A big surprise - a loop beats a vectorized method 4, but is still slower than methods 1, 2 and 3. It turns out that in this particular case you can do it better:
tic;
for i=1:1e8
x(i) = 6;
end
toc
Elapsed time is 0.321246 seconds.
And that is the probably the most bizarre outcome of this study - a MATLAB-written loop significantly outperforms native vector indexing. That should certainly not be so. Note that the JIT'ed loop is still 3 times slower than the theoretical peak almost obtained by method 1. So there is still plenty of room for improvement. It is just surprising (a stronger word would be more suitable) that usual 'vectorized' indexing (1:end) is even slower.
Questions
- is simple indexing in MATLAB very inefficient (methods 2, 3, and 4 are slower than method 1), or did I miss something?
- why is method 4 (so much) slower than methods 2 and 3?
- why does using
1e8instead ofnumel(x)as a loop bound speed up the code by factor 2?
Edit After reading Jonas's comment, here is another way to do that using logical indices:
tic;
id = logical(ones(1, 1e8));
x(id) = 7;
toc
Elapsed time is 0.613363 seconds.
Much better than method 4.
For convenience:
function test
tic; x = zeros(1,1e8); toc
tic; x(:) = 1; toc
tic; x(1:end) = 2; toc
tic; x(1:1e8) = 3; toc
tic;
id = 1:1e8; % colon(1,1e8);
x(id) = 4;
toc
tic;
for i=1:numel(x)
x(i) = 5;
end
toc
tic;
for i=1:1e8
x(i) = 6;
end
toc
end
