simplify_full returns odd result from symbolic series input

[rws]

SR.series will lose the order term when passed to Maxima. Thus only the coefficients may be simplified, and this must be done in all simplify* functions.

sage: x=var('x')
sage: s=(1/(1-x)).series(x,6)
sage: s.coeffs()
[[x^5 + x^4 + x^3 + x^2 + x + Order(x^6) + 1, 0]]
sage: s.simplify_full().coeffs()
[[Order(x^6) + 1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]]

See also the related #17399.

Originally found in ​http://ask.sagemath.org/question/24968/coefficients-in-polynomial-ring-over-symbolic-ring/

Also, series should simplify its terms on a per-term basis:

sage: var('x,y')
(x, y)
sage: ex=1/(1-x*y-x^2)
sage: ex.series(x,5)
1 + (y)*x + (y^2 + 1)*x^2 + ((y^2 + 1)*y + y)*x^3 + (((y^2 + 1)*y + y)*y + y^2 + 1)*x^4 + Order(x^5)

Compare with e.g. Pari:

? 1/(1-x*y-x^2) + O(x^5)
%1 = 1 + y*x + (y^2 + 1)*x^2 + (y^3 + 2*y)*x^3 + (y^4 + 3*y^2 + 1)*x^4 + O(x^5)

Both issues can be fixed by writing series simplification methods.

[rws]

Probably with symbolic series, only coefficients should be simplified in the simplify* functions.

[rws]

There are no power series objects in Maxima, just conversion to infinite sums, i.e. formal power series:

sage: maxima.powerseries(x^2+1/(1-x),x,0)
'sum(_SAGE_VAR_x^i2,i2,0,inf)+_SAGE_VAR_x^2
sage: maxima.powerseries(x^2+1/(1-x),x,0).sage()
x^2 + sum(x^i3, i3, 0, +Infinity)

The Taylor series objects have an order parameter on creation, but this does not get output or translated to Sage:

sage: maxima.taylor(1+x+x^2+x^3,x,0,3)
1+_SAGE_VAR_x+_SAGE_VAR_x^2+_SAGE_VAR_x^3
sage: maxima.taylor(1+x+x^2+x^3,x,0,3).sage()
x^3 + x^2 + x + 1

so there is no way around it that SR.series will lose the order term when passed to Maxima. Thus only the coefficients may be simplified, and this must be done in or called from all simplify* functions.

[rws]

The following functions call Maxima, have a meaning /wrt series, and therefore need to be modified: simplify, simplify_trig, simplify_rectform, simplify_rational, simplify_log, expand_log, expand_trig.

[rws]

Proof of concept.

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New commits:

​a4d2084	17659: make symbolic series subclass of Expression
​ed003f0	16203: make imports more specific to prevent import loops
​32ae67c	17399: do not let maxima handle ex.series coefficients
​99820cf	17399: roll back previous commit to allow merge of 17428
​168b659	Merge branch 'u/rws/coefficients_of_symbolic_expressions_revamp' of trac.sagemath.org:sage into t/17399/fix_coefficients_for_symbolic_series
​6cd5286	17399: handle series in ex.coefficients()
​0b9914f	17399: remove whitespace changes
​2e74911	Merge commit '0b9914f8b6cd4cb0f885f0550bfd375d80495a71' of trac.sagemath.org:sage into t/17400/simplify_full_returns_odd_result_from_symbolic_series_input
​2660235	17400: implement series.simplify_full()

[rws]

Squashed it all into one commit.

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New commits:

​2643553

17400: merge 17659, add simplify/expand methods for SymbolicSeries

[rws]

Squashed it all into one commit.

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New commits:

​73cd5bf

17400: symbolic series simplification methods

[rws]

Pending because #17659 is pending.

[vbraun]

Still pending?

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New commits:

​611f93b

Merge branch 'u/rws/17400' of trac.sagemath.org:sage into tmp05

[rws]

No, you're right. Maybe there are conflicts with #17402.

[vbraun]

Merge conflict, possibly with #20088

[nbruin]

Replying to rws:

There are no power series objects in Maxima, [...] The Taylor series objects have an order parameter on creation, but this does not get output or translated to Sage

So there are power series-type object in maxima. We could just access those then and use them to translate back-and-forth. It's a bit of a question how much functionality maxima has for it and whether maxima itself works with them faithfully.

The internal representation seems to be a bit complicated:

sage: P=maxima_calculus.taylor(exp(x),x,0,5)
sage: P
1+_SAGE_VAR_x+_SAGE_VAR_x^2/2+_SAGE_VAR_x^3/6+_SAGE_VAR_x^4/24+_SAGE_VAR_x^5/120
sage: P.ecl()
<ECL: ((MRAT SIMP (((MEXPT SIMP) $%E |$_SAGE_VAR_x|) |$_SAGE_VAR_x|)
  (#:|%e^_SAGE_VAR_x2396| #:|_SAGE_VAR_x2397|)
  ((|$_SAGE_VAR_x| ((5 . 1)) 0 NIL #:|_SAGE_VAR_x2397| . 2)) TRUNC)
 PS (#:|_SAGE_VAR_x2397| . 2) ((5 . 1)) ((0 . 1) 1 . 1) ((1 . 1) 1 . 1)
 ((2 . 1) 1 . 2) ((3 . 1) 1 . 6) ((4 . 1) 1 . 24) ((5 . 1) 1 . 120))>

but fundamentally everything is there to create/peel apart these objects.

(note that it would be a bit harder to work with these objects via the expect-interface to maxima)

A basic explanation of the MRAT format is here: ​http://def.fe.up.pt/pipermail/maxima-discuss/2005/010416.html

EDIT: never mind, this isn't suitable for power series. This whole "TRUNC" property is very fickle (and it doesn't carry proper order information). It disappears with very trivial arithmetic already.

[git]

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

​e3907ec

Merge branch 'develop' into t/17400/17400-1

[git]

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

​895d900	Merge branch 'develop' into t/17400/17400-1
​22c947a	17400: adapt doctest to changed simplify behaviour

[rws]

Two doctests failing, see patchbot.