On Sat, Mar 15, 2014 at 8:38 PM, <josef.p...@gmail.com> wrote: > 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 ?) >
scipy.linalg.sqrtm: http://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.sqrtm.html Warren > 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 >> >> > > _______________________________________________ > NumPy-Discussion mailing list > NumPy-Discussion@scipy.org > http://mail.scipy.org/mailman/listinfo/numpy-discussion > >
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