improve documentation of partial_fraction_decomposition

[zimmerma]

How can one compute a partial fraction decomposition over the complex numbers? Consider the following:

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: r = 1 /(x^4 + 1)
sage: r.partial_fraction_decomposition()
(0, [1/(x^4 + 1)])

This is ok since we explicitely work over QQ. Now compare with:

sage: P.<y> = PolynomialRing(RationalField())
sage: Qbar.<y> = QuotientRing(P, y^2+1)
sage: x = PolynomialRing(Qbar, 'x').gen()
sage: r = 1 /(x^4 + 1)
sage: r.partial_fraction_decomposition()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)

apply trac_8499.patch​

[zimmerma]

I found out the solution by myself. If one wants say a decomposition over Q[sqrt(2)] or Q[I], then simply use QQ[sqrt(2)] or QQ[sqrt(-1)]:

sage: R.<x> = QQ[sqrt(2)][]
sage: r=1/(x^4+1)
sage: r.partial_fraction_decomposition()
(0,
 [(-1/4*sqrt2*x + 1/2)/(x^2 - sqrt2*x + 1),
  (1/4*sqrt2*x + 1/2)/(x^2 + sqrt2*x + 1)])

or:

sage: R.<x> = QQ[sqrt(-1)][]            
sage: r=1/(x^4+1)                       
sage: r.partial_fraction_decomposition()
(0, [(-1/2*I)/(x^2 - I), 1/2*I/(x^2 + I)])

Now if you want Sage to automatically find the extension where the denominator fully factors, use QQbar:

sage: R.<x> = QQbar[]                   
sage: r=1/(x^4+1)                       
sage: r.partial_fraction_decomposition()
(0,
 [([-0.17677669529663690 .. -0.17677669529663686] - [0.17677669529663686 .. 0.17677669529663690]*I)/(x + [-0.70710678118654758 .. -0.70710678118654746] - [0.70710678118654746 .. 0.70710678118654758]*I),
  ([-0.17677669529663690 .. -0.17677669529663686] + [0.17677669529663686 .. 0.17677669529663690]*I)/(x + [-0.70710678118654758 .. -0.70710678118654746] + [0.70710678118654746 .. 0.70710678118654758]*I),
  ([0.17677669529663686 .. 0.17677669529663690] - [0.17677669529663686 .. 0.17677669529663690]*I)/(x + [0.70710678118654746 .. 0.70710678118654758] - [0.70710678118654746 .. 0.70710678118654758]*I),
  ([0.17677669529663686 .. 0.17677669529663690] + [0.17677669529663686 .. 0.17677669529663690]*I)/(x + [0.70710678118654746 .. 0.70710678118654758] + [0.70710678118654746 .. 0.70710678118654758]*I)])

I'll add some examples to the documentation and then we can close this ticket.

Paul

[zimmerma]

I've changed the ticket summary, and attached a patch (against Sage 5.9) which improves the documentation of partial_fraction_decomposition.

Paul

[lftabera]

Looks good, but I have a question. How is computing the rational fraction decomposition over QQbar gives you the extension where the denominator splits? It just returns the full decomposition over QQbar no more no less. The extension would be in any case the splitting field of the denominator, isn't it?

I would expect something like "Now if you want Sage to compute an extension where the denominator fully factors, use QQbar: "

[zimmerma]

thank you for your feedback. You have two questions in fact:

How is computing the rational fraction decomposition over QQbar gives you the extension where the denominator splits?

this is given in the denominators of the output.

The extension would be in any case the splitting field of the denominator, isn't it?

yes.

Feel free to propose a reviewer patch!

Paul

[lftabera]

My complain is that using QQbar does not give you the splitting field of the denominator in an obvious way. I have changed the documentation accordingly. Paul, could you take a look at my changes and check if you agree with them?

Luis

[zimmerma]

I get an error when I try this example:

sage: R.<x> = QQ[]
sage: r = 1/(x^4+2)
sage: N = r.denominator().splitting_field('a')
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-9-c9d534f911a3> in <module>()
----> 1 N = r.denominator().splitting_field('a')

/usr/local/sage-6.0-x86_64-Linux/local/lib/python2.7/site-packages/sage/structure/element.so in sage.structure.element.Element.__getattr__ (sage/structure/element.c:3873)()

/usr/local/sage-6.0-x86_64-Linux/local/lib/python2.7/site-packages/sage/structure/misc.so in sage.structure.misc.getattr_from_other_class (sage/structure/misc.c:1696)()

AttributeError: 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint' object has no attribute 'splitting_field'

[lftabera]

Sorry for not telling, you need sage 6.1 released today...

[zimmerma]

I tried the added examples with Sage 6.1 and they work, thus positive review for me.