I think I wouldn't use anything like @@ often enough to remember it's meaning. I'd rather see english names for anything that is not **very** common.
I find A@@-1 pretty ugly compared to inv(A) A@@(-0.5) might be nice (do we have matrix_sqrt ?) Josef On Sat, Mar 15, 2014 at 5:11 PM, Stephan Hoyer <sho...@gmail.com> wrote: > Speaking only for myself (and as someone who has regularly used matrix > powers), I would not expect matrix power as @@ to follow from matrix > multiplication as @. I do agree that matrix power is the only reasonable > use for @@ (given @), but it's still not something I would be confident > enough to know without looking up. > > We should keep in mind that each new operator imposes some (small) > cognitive burden on everyone who encounters them for the first time, and, > in this case, this will include a large fraction of all Python users, > whether they do numerical computation or not. > > Guido has given us a tremendous gift in the form of @. Let's not insist on > @@, when it is unclear if the burden of figuring out what @@ means it would > be worth using, even for heavily numeric code. I would certainly prefer to > encounter norm(A), inv(A), matrix_power(A, n), fractional_matrix_power(A, > n) and expm(A) rather than their infix equivalents. It will certainly not > be obvious which of these @@ will support for objects from any given > library. > > One useful data point might be to consider whether matrix power is > available as an infix operator in other languages commonly used for > numerical work. AFAICT from some quick searches: > MATLAB: Yes > R: No > IDL: No > > All of these languages do, of course, implement infix matrix > multiplication, but it is apparently not clear at all whether the matrix > power is useful. > > Best, > Stephan > > > > > On Sat, Mar 15, 2014 at 9:03 AM, Olivier Delalleau <sh...@keba.be> wrote: > >> 2014-03-15 11:18 GMT-04:00 Charles R Harris <charlesr.har...@gmail.com>: >> >> >>> >>> >>> On Fri, Mar 14, 2014 at 10:32 PM, Nathaniel Smith <n...@pobox.com> wrote: >>> >>>> 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. >>>> >>>> >>> I don't have a strong feeling either way on '@@' . Matrix inverses are >>> pretty common in matrix expressions, but I don't know that the new operator >>> offers much advantage over a function call. The positive integer powers >>> might be useful in some domains, as others have pointed out, but >>> computational practice one would tend to factor the evaluation. >>> >>> Chuck >>> >> >> Personally I think it should go in, because: >> - it's useful (although marginally), as in the examples previously >> mentioned >> - it's what people will expect >> - it's the only reasonable use of @@ once @ makes it in >> >> As far as the details about precedence rules and what not... Yes, someone >> should think about them and come up with rules that make sense, but since >> it will be pretty much only be used in unambiguous situations, this >> shouldn't be a blocker. >> >> -=- Olivier >> >> _______________________________________________ >> NumPy-Discussion mailing list >> NumPy-Discussion@scipy.org >> http://mail.scipy.org/mailman/listinfo/numpy-discussion >> >> > > _______________________________________________ > NumPy-Discussion mailing list > NumPy-Discussion@scipy.org > http://mail.scipy.org/mailman/listinfo/numpy-discussion > >
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