[Numpy-discussion] vectorization vs. numpy.linalg (shape (3, 3, 777) vs shape (777, 3, 3))

[Nico Schlömer]

Hi everyone,

When trying to speed up my code, I noticed that simply by reordering my
data I could get more than twice as fast for the simplest operations:
```
import numpy
a = numpy.random.rand(50, 50, 50)

%timeit a[0] + a[1]
1000000 loops, best of 3: 1.7 µs per loop

%timeit a[:, 0] + a[:, 1]
100000 loops, best of 3: 4.42 µs per loop

%timeit a[..., 0] + a[..., 1]
100000 loops, best of 3: 5.99 µs per loop
```
This makes sense considering the fact that, by default, NumPy features
C-style memory allocation: the last index runs fastest. The blocks that are
added with `a[0] + a[1]` are contiguous in memory, so cache is nicely made
use of. As opposed to that, the blocks that are added with `a[:, 0] + a[:,
1]` are not contiguous, even more so with `a[..., 0] + a[..., 1]`; hence
the slowdown. Would that be the correct explanation?

If yes, I'm wondering why most numpy.linalg methods, when vectorized, put
the vector index up front. E.g., to mass-compute determinants, one has to do
```
a = numpy.random.rand(777, 3, 3)
numpy.linalg.det(a)
```
This way, all 3x3 matrices form a memory block, so if you do `det` block by
block, that'll be fine. However, vectorized operations (like `+` above)
will be slower than necessary.
Any background on this?

(I came across this when having to rearrange my data (swapaxes, rollaxis)
from shape (3, 3, 777) (which allows for fast vectorized operations in the
rest of the code) to (777, 3, 3) for using numpy's svd.)

Cheers,
Nico
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