[Numpy-discussion] [RFC] should we argue for a matrix power operator, @@?

[Nathaniel Smith]

Hi all,

Here's the second thread for discussion about Guido's concerns about
PEP 465. The issue here is that PEP 465 as currently written proposes
two new operators, @ for matrix multiplication and @@ for matrix power
(analogous to * and **):
  http://legacy.python.org/dev/peps/pep-0465/

The main thing we care about of course is @; I pushed for including @@
because I thought it was nicer to have than not, and I thought the
analogy between * and ** might make the overall package more appealing
to Guido's aesthetic sense.

It turns out I was wrong :-). Guido is -0 on @@, but willing to be
swayed if we think it's worth the trouble to make a solid case.

Note that question now is *not*, how will @@ affect the reception of
@. @ itself is AFAICT a done deal, regardless of what happens with @@.
For this discussion let's assume @ can be taken for granted, and that
we can freely choose to either add @@ or not add @@ to the language.
The question is: which do we think makes Python a better language (for
us and in general)?

Some thoughts to start us off:

Here are the interesting use cases for @@ that I can think of:
- 'vector @@ 2' gives the squared Euclidean length (because it's the
same as vector @ vector). Kind of handy.
- 'matrix @@ n' of course gives the matrix power, which is of marginal
use but does come in handy sometimes, e.g., when looking at graph
connectivity.
- 'matrix @@ -1' provides a very transparent notation for translating
textbook formulas (with all their inverses) into code. It's a bit
unhelpful in practice, because (a) usually you should use solve(), and
(b) 'matrix @@ -1' is actually more characters than 'inv(matrix)'. But
sometimes transparent notation may be important. (And in some cases,
like using numba or theano or whatever, 'matrix @@ -1 @ foo' could be
compiled into a call to solve() anyway.)

(Did I miss any?)

In practice it seems to me that the last use case is the one that's
might matter a lot practice, but then again, it might not -- I'm not
sure. For example, does anyone who teaches programming with numpy have
a feeling about whether the existence of '@@ -1' would make a big
difference to you and your students? (Alan? I know you were worried
about losing the .I attribute on matrices if switching to ndarrays for
teaching -- given that ndarray will probably not get a .I attribute,
how much would the existence of @@ -1 affect you?)

On a more technical level, Guido is worried about how @@'s precedence
should work (and this is somewhat related to the other thread about
@'s precedence and associativity, because he feels that if we end up
giving @ and * different precedence, then that makes it much less
clear what to do with @@, and reduces the strength of the */**/@/@@
analogy). In particular, if we want to argue for @@ then we'll need to
figure out what expressions like
   a @@ b @@ c
and
   a ** b @@ c
and
   a @@ b ** c
should do.

A related question is what @@ should do if given an array as its right
argument. In the current PEP, only integers are accepted, which rules
out a bunch of the more complicated cases like a @@ b @@ c (at least
assuming @@ is right-associative, like **, and I can't see why you'd
want anything else). OTOH, in the brave new gufunc world, it
technically would make sense to define @@ as being a gufunc with
signature (m,m),()->(m,m), and the way gufuncs work this *would* allow
the "power" to be an array -- for example, we'd have:

   mat = randn(m, m)
   pow = range(n)
   result = gufunc_matrix_power(mat, pow)
   assert result.shape == (n, m, m)
   for i in xrange(n):
       assert np.all(result[i, :, :] == mat ** i)

In this case, a @@ b @@ c would at least be a meaningful expression to
write. OTOH it would be incredibly bizarre and useless, so probably
no-one would ever write it.

As far as these technical issues go, my guess is that the correct rule
is that @@ should just have the same precedence and the same (right)
associativity as **, and in practice no-one will ever write stuff like
a @@ b @@ c. But if we want to argue for @@ we need to come to some
consensus or another here.

It's also possible the answer is "ugh, these issues are too
complicated, we should defer this until later when we have more
experience with @ and gufuncs and stuff". After all, I doubt anyone
else will swoop in and steal @@ to mean something else! OTOH, if e.g.
there's a strong feeling that '@@ -1' will make a big difference in
pedagogical contexts, then putting that off for years might be a
mistake.

-n

-- 
Nathaniel J. Smith
Postdoctoral researcher - Informatics - University of Edinburgh
http://vorpus.org