Implement a "distribute" method

[Emmanuel Charpentier]

Motivation : some "obvious" transformations, sometimes helpful in routine calculations, cannot be obtained via the current methods. Some (elementary,

1. e. high-school- or undergrad-level) exemples : given the declarations :

   var("a,b")
   var("j,p", domain="integer")
   f,g,X=function("f,g,X")
   

the following equalities, true under no or mild conditions, cannot be deduced currently :

sum(X(j)+Y(j),j,1,p)==sum(X(j),j,1,p)+sum(Y(j),j,1,p) ## (1)
prod(X(j)*Y(j),j,1,p)==prod(X(j),j,1,p)*prod(Y(j),j,1,p) ## (2)
integrate(f(x)+g(x),x,a,b)==integrate(f(x),x,a,b)+integrate(g(x),x,a,b) ## (3)

Similarly, if A, B and C are matrices over a suitable ring and of suitable dimensions, the following equalities cannot be straightforwardly obtained.

(A+B).trace()==A.trace()+B.trace() ## (4)
(A*B).det()==A.det()*B.det() ## (5)

(Curiously, (f(x)+g(x)).diff(x) does return diff(f(x),x)+diff(g(x),x) (and a way to return to the original expression does not seem to exist currently)).

The ability to generate the right-hand form from the left-hand form of these various examples is sometimes useful in (low-level) examples. The examples (1) and/or (2) arise naturally when deriving the maximum-likelihood estimators of the mean and variance of a given distribution. The third one is often encountered in elementary calculus exercises.

In all cases, an operator is "distributed" over another one : (1) : symbolic_sum is distributed over addition. (2) : symbolic product is distributed over multiplication. (3) : integration is ditributed over addition. (4) : trace is distributed over (matrix) addition. (5) : determinant is distributed over (matrix) multiplication.

Implementing (1) as an extension of the expand() method of Expression is tempting (see #22915). However, there are situations where this expansion is not useful. So, creating a method for such situations seems useful.

One should note that such "distributions" are not systematically valid : we have a collection of special cases rather than a general mechanism. So this method shoud either limit itself to a few recognized cases or take keyword argument(s) specifying what should be distributed over what.

Another design choice is to know if these distributions should be recursive or run only on the top level of a given expression (possibly controlled by another keyword argument).

The present ticket aims at implementing this method for sage.symbolic.expression.Expression, at least in cases (1) to (3) (case (2) needs an implementation of the symbolic products, see #17505). Further suggestions for other classes are welcome...

[Emmanuel Charpentier]

Discussion opened on​sage-devel

[Emmanuel Charpentier]

First implementation :

• Symbolic sum of a sum ==> sum of symbolic sums
• Integral (definite or not) of a sum ==> sum of integrals.

Passes ptestlong with the same errors as those reported for 8.0.beta4 and 8.0.beta5

==>neeeds_review

---

New commits:

​c3bcef0

First implementation of distribute() : sums.

[Emmanuel Charpentier]

Further verifications show that this does not work in all cases.

Back to the old drawing board...

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7f9666a

Fixed recursion when non-match at toplevel.

[Emmanuel Charpentier]

Passes ptestlong with the same (unrelated) failures as before +one transient timeout :

sage -t --long src/sage/modular/modform/find_generators.py 

which passes when run standalone.

==>needs_review again.

[Emmanuel Charpentier]

Planning to implement the multiplicative case allowed by #17505.

[Travis Scrimshaw]

Minor comment: Instead of using map, you can do, e.g.,

map(lambda t:treat_term(op, t, la), aa)
treat_term(op, t, la) for t in aa

(with wrapping it as a list comprehension [treat_term(op, t, la) for t in aa] when you need a list).

Addendum - I feel this is cleaner and IIRC, it is faster.

[Emmanuel Charpentier]

Replying to tscrim:

Minor comment: Instead of using map, you can do, e.g.,

map(lambda t:treat_term(op, t, la), aa)
treat_term(op, t, la) for t in aa

(with wrapping it as a list comprehension [treat_term(op, t, la) for t in aa] when you need a list).

Indeed (my Lisp roots (and my age...) are showing).

Addendum - I feel this is cleaner and IIRC, it is faster.

I'll check that. But I'll submit first the "map" version (which works...), and re-submit the "comprehension" version afterwards if it makes a serious difference.

BTW : other enhancements are possible. For example create a local (static) dictionary of distributable operators, whose values are functions processing the corresponding case of distribution (with possible generalizations/factorizations). I didn't feel the four special cases we treat in SR so fare were worth it, but if you have other ideas, you're welcome...

Commit forthcoming. Review request after ptestlong (sually a couple of hours...).

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​647ff39	17505: unevaluated symbolic product
​e4769b5	17505: symbolic product
​58119b0	17505: fix doctests
​7a56004	17505: address reviewer's suggestions
​a018030	Merge branch 'u/rws/implement_symbolic_product' of trac.sagemath.org:sage into distribute
​c67f957	Implement distribute() in the multiplicative case.

[Travis Scrimshaw]

It's about 3x slower to use map:

sage: %timeit [i for i in range(10000)]
1000 loops, best of 3: 265 µs per loop
sage: %timeit map(lambda i: i, range(10000))
1000 loops, best of 3: 897 µs per loop

Also in Python2, map returns a list, which takes extra time and memory to create. Whereas for the sum, you only need a generator object.

I cannot comment so much about the operators as it is a little beyond my expertise.

[Emmanuel Charpentier]

This version passes ptestlong with the same three failure (deemed unrelated).

I'll try Travis' version before marking the better version for review.

---

New commits:

​647ff39	17505: unevaluated symbolic product
​e4769b5	17505: symbolic product
​58119b0	17505: fix doctests
​7a56004	17505: address reviewer's suggestions
​a018030	Merge branch 'u/rws/implement_symbolic_product' of trac.sagemath.org:sage into distribute
​c67f957	Implement distribute() in the multiplicative case.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​687cc8e

Distribute : implement Travis Scrimshaw's suggestion for iterations.

[Emmanuel Charpentier]

With Travis' suggestion, passes ptestlong with the same result as before ==> needs_review

Do you think that this enhancement should be passed to Maxima ? The algorithm seems simple to port in Maxima or Lisp.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d420ec4	17505: fix latex, cosmetics
​b784a2c	Merge branch 'u/rws/implement_symbolic_product' of trac.sagemath.org:sage into distribute
​5779423	17505: fix doctests
​4b6e71b	Merge branch 'u/rws/implement_symbolic_product' of trac.sagemath.org:sage into distribute

[Emmanuel Charpentier]

Merged #17505 again. Should be reviewable.

[Travis Scrimshaw]

In general, it is always better to pass things to upstream. Although for now, we should do things on our side.

Some other small comments:

• There is a typo:

  _ Symbolic product of a product ==> product of symbolic products.
  

• self is not considered as an input.
• Small change to your INPUT::

        - ``recursive`` -- (default: ``True``) the distribution proceeds
          along the subtrees of the expression
  

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2412f7a

Distribute : cosmetics on documentation.

[Emmanuel Charpentier]

Cosmetic (but important) changes to the documentation. Thank you, Travis.

sage -t --long src/sage/symbolic/expression.pyx passes without failure. Would you mind reviewing it (I probably can't ptestlong right now...).

[Emmanuel Charpentier]

Replying to tscrim:

You are still listed as an author in the docstring (credit where credit is due...).

[Travis Scrimshaw]

I shouldn't be because I just did some simple review changes (so please remove me). But thank you. :)

Everything else looks good.

[Marcelo Forets]

is the word "indiced" correct or it is "indexed"? (i'm not native english speaker though..)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c28097a

Distribute : at his request, Travis Crimshaw removed from Author's list.

[Travis Scrimshaw]

indexed is correct adjective.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7aee739

Distribute : one last typo (I hope...).

[Emmanuel Charpentier]

RReplying to tscrim:

I shouldn't be because I just did some simple review changes

Your suggestion to do without map() was a good one (and teached me something I tend to oversee : Python is not Lisp...).

(so please remove me).

If you insist...

But thank you. :)

Everything else looks good.

So this last update (including indiced --> indexed) should be reviewed...

---

New commits:

​7aee739

Distribute : one last typo (I hope...).

[Travis Scrimshaw]

Replying to charpent:

RReplying to tscrim:

I shouldn't be because I just did some simple review changes

Your suggestion to do without map() was a good one (and teached me something I tend to oversee : Python is not Lisp...).

I am always learning things from my reviewers too.

(so please remove me).

If you insist...

Thank you. I didn't really author code other than give suggestions, so I don't think I should be listed as an author. :)

You will need to add a doctest in the dependency, but you don't have to merge it into this ticket explicitly (because it is a dependency and will be done by Volker in the develop branch).

[Emmanuel Charpentier]

Replying to tscrim:

[ Snip ... ]

You will need to add a doctest in the dependency, but you don't have to merge it into this ticket explicitly (because it is a dependency and will be done by Volker in the develop branch).

This should wait Ralf's implementation of the externally-visible product method and function (currently, we have to either import an internal function or cast to Sage a Maxima contraption...) : that will be cleaner an probably more useful to users.

[Ralf Stephan]

Interesting. Both branches apply but mine has a hunk that fuses two hunks into one; yours has the first of the two so my hunk cannot be applied. I'll upload a branch where the two are separated.

[Ralf Stephan]

Replying to rws:

Interesting. Both branches apply but mine has a hunk that fuses two hunks into one; yours has the first of the two so my hunk cannot be applied. I'll upload a branch where the two are separated.

Ah that should have gone to #22989.

[Emmanuel Charpentier]

Replying to rws:

Replying to rws:

Interesting. Both branches apply but mine has a hunk that fuses two hunks into one; yours has the first of the two so my hunk cannot be applied. I'll upload a branch where the two are separated.

Ah that should have gone to #22989.

Go ahead. Do you need me to do anything ?

[Frédéric Chapoton]

WARNING

You have used here "apply()" which is not python3-compatible. Please correct this ASAP in another ticket.

Reference: www.diveintopython3.net/porting-code-to-python-3-with-2to3.html

The Python 2 builtin apply() has been removed in Python 3

[Emmanuel Charpentier]

Replying to chapoton:

WARNING

You have used here "apply()" which is not python3-compatible. Please correct this ASAP in another ticket.

Will do that, maybe over the coming week-end, but have to think about it I'm not much of a Python coder...).

Reference: www.diveintopython3.net/porting-code-to-python-3-with-2to3.html

Thanks for the suggestion.

The Python 2 builtin apply() has been removed in Python 3

H... right.

[Frédéric Chapoton]

Do you want me to take care of that, and then be a reviewer ? Maybe this will be faster.

[Frédéric Chapoton]

I created #23030