Extend normalize() and use it instead of Maxima in simplify_rational()

[rws]

At the moment normalize() will not expand factors of fractions. If fractions are combined however, the final numerator is expanded:

sage: ((x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1)).normalize()
(x^(1/2*y) + 1)^2*(x^(1/2*y) - 1)^2/(x^y - 1)
sage: (x + y^2/(x + 2)).normalize()
(x^2 + y^2 + 2*x)/(x + 2)

The alternatives are provided atm by Maxima

sage: ((x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1)).simplify_rational(algorithm='simple')
(x^(2*y) - 2*x^y + 1)/(x^y - 1)
sage: (x + y^2/(x + 2)).simplify_rational(algorithm='noexpand')
((x + 2)*x + y^2)/(x + 2)

​https://github.com/pynac/pynac/issues/191

After Pynac will have implemented normalize() options to these effect the calls to Maxima in simplify_rational should be replaced by the respective calls to Pynac.

[rws]

This is implemented in Pynac master but one doctest depends on #21360. Also, nested application ("full") needs to be done.

[rws]

New commits:

​b996f4f	version/chksum
​91a08d2	changes affecting Sage behaviour or interface
​a4f58d7	doctest fixes
​ac7d971	revert removal of pos.char. doctests; add "known bug"
​3dd8058	address reviewer's comments
​1f29305	add doctest
​0065f62	address reviewer's comments
​129b6e6	use normal() instead of Maxima

[git]

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

​b06cfa0	Fix _eval_self(float) for "real" complex expressions
​fd9fa19	Merge branch 'u/jdemeyer/update_to_pynac_0_6_9' of trac.sagemath.org:sage into t/21335/noexpand_simple_options_for_ex_normalize__
​c69fd2e	21335: fullratsimp: replace calls to Maxima with Pynac ones

[rws]

Three doctest fails in symbolic/expression.pyx.

[git]

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

​a8eba31	Merge branch 'develop' into t/21335/noexpand_simple_options_for_ex_normalize__
​2ea5c31	21529: fix bug in factoring of general symbolic expressions
​c4932f8	Merge branch 'u/rws/bug_in_factoring_of_general_symbolic_expressions' of trac.sagemath.org:sage into t/21335/noexpand_simple_options_for_ex_normalize__

[mmezzarobba]

Minor point: "and and" should be "and".

Apart from that, I don't understand why algorithm="full" attempts to factor the simplified fraction (which is a potentially costly operation). Is there something in the way the Pynac normal() function works that makes it necessary to do so?

[rws]

Indeed the performance question was the source of some headaches to me, as well. I figured at the time consistency with Maxima behaviour was more important. Without factor the following doctests will fail:

File "src/sage/symbolic/expression.pyx", line 9427, in sage.symbolic.expression.Expression.simplify_rational
Failed example:
    f.simplify_rational()
Expected:
    -2*sqrt(x - 1)/sqrt((x + 1)*(x - 1))
Got:
    ((x - 1)^(3/2) - sqrt(x - 1)*x - sqrt(x - 1))/sqrt((x + 1)*(x - 1))
**********************************************************************
File "src/sage/symbolic/expression.pyx", line 9451, in sage.symbolic.expression.Expression.simplify_rational
Failed example:
    g.simplify_rational()
Expected:
    x^y - 1
Got:
    (x^(2*y) - 2*x^y + 1)/(x^y - 1)

[git]

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

​ab551be	Merge branch 'develop' into t/21335/noexpand_simple_options_for_ex_normalize__
​30bc79e	cosmetics

[mmezzarobba]

Replying to rws:

Indeed the performance question was the source of some headaches to me, as well. I figured at the time consistency with Maxima behaviour was more important.

I don't know exactly what either Maxima or Pynac does, but just by chance, do you think it would be okay to keep the loop but expand the denominators instead of calling factor()?

[rws]

Replying to mmezzarobba:

I don't know exactly what either Maxima or Pynac does, but just by chance, do you think it would be okay to keep the loop but expand the denominators instead of calling factor()?

This will, additionally to the two doctests above, fail this doctest:

File "src/sage/symbolic/expression.pyx", line 9458, in sage.symbolic.expression.Expression.simplify_rational
Failed example:
    f.simplify_rational()
Expected:
    (2*x^2 + 5*x + 4)/((x + 2)^2*(x + 1))
Got:
    (2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4)

[rws]

I think I see now how to resolve the three tests without factoring. Pynac's gcd needs to be a bit more aware of powers in the expression and needs to replace some of them with new symbols. This way also exponentials can be handled identically (which does not work atm).