[Numpy-discussion] Question answered incorrectly at NumPy Tutorial

[Travis O. <oli...@ie...>]

During my NumPy Tutorial at the SciPy conference last month, somebody 
asked the question about the memory requirements of index arrays that I 
gave the wrong impression about.  Here is the context and the correct 
response that should alleviate concerns about large cross-product index 
arrays.

I was noting how copy-based (advanced) indexing using index arrays works 
in multiple-dimensions by creating an array of the same-shape of the 
input index arrays constructed by selecting the elements indicated by 
respective elements of the index arrays.

If a is 2-d, then

a[[10,12,14],[13, 15, 17]]

returns a 1-d array with elements

[a[10,13], a[12,15], a[14,17]].

This is *not* the cross-product that some would expect.  The 
cross-product can be generated using the ix_ function

a[ix_([10,12,14], [13,15,17])]

is equivalent to

a[[[10,10,10],[12,12,12],[14,14,14]], [[13,15,17],[13,15,17],[13,15,17]]]

which will return

[[a[10,13] a[10,15], a[10,17]],
 [a[12,13] a[12,15], a[12,17]],
 [a[14,13] a[14,15], a[14,17]]]

The concern mentioned at the conference was that the cross-product would 
generate large intermediate index arrays for large input arrays to ix_.  
At the time, I think I validated the concern.  However, the concern is 
unfounded.  This is because the cross product function does not actually 
create a large intermediate array, but uses the broad-casting 
implementation of indexing to generate the 2-d indexing array 
"on-the-fly" (much like ogrid and other tools in NumPy).

Notice:

ix_([10,12,14], [13,15,17])

(array([[10],
       [12],
       [14]]), array([[13, 15, 17]]))

The first indexing array is 3x1, while the second is 1x3.  The result 
array will be 3x3, but the 2-d indexing array is never actually stored.

Just to set my mind at ease about possible mis-information I spread 
during the tutorial, and give a little tutorial on advanced indexing.

Best,

-Travis