#6541: [with patch, needs review] functions numerator and denominator for
univariate polynomials
-------------------------+--------------------------------------------------
Reporter: lftabera | Owner: tbd
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.1.1
Component: algebra | Keywords:
Reviewer: | Author: lftabera
Merged: |
-------------------------+--------------------------------------------------
Sage univariaty polynomials do not admit numerator and denominator by
default:
{{{
sage: K.<x>=RR['x']
sage: f=x+1
sage: numerator(f)
---------------------------------------------------------------------------
AttributeError Traceback (most recent call
last)
/home/luisfe/.sage/temp/mychabol/11442/_home_luisfe__sage_init_sage_0.py
in <module>()
/opt/SAGE/sage/local/lib/python2.6/site-packages/sage/misc/functional.py
in numerator(x)
686 if isinstance(x, (int, long)):
687 return x
--> 688 return x.numerator()
689
690 def numerical_approx(x, prec=None, digits=None):
AttributeError: 'sage.rings.polynomial.polynomial_real_mpfr_dense.P'
object has no attribute 'numerator'
}}}
The same happens for multivariate polynomials, see #6496
SAGE implements denominators for some univariate polynomial ring and
numerators for
sage.rings.polynomial.polynomial_element_generic.Polynomial_rational_dense
For denominators, first, it trys to compute the lcm of the denominators of
the coefficients of the polynomial. If the coefficients do not have a
denominator method or these denominators do not have a lcm method then the
function fails with an exception.
Numerators of polynomials with rational coefficients are polynomials over
the integers.
I propose to add the following methods:
1.- Change the denominator function. First try to compute the denominator
as it is done now, if it fails, return a generic denominator
self.parent()_one_element()
2.- Add a numerator function returning self * self.denominator()
This will not change the cases where denominators are used.
A patch with the feature
{{{
diff -r ca1f31d6f6bf sage/rings/polynomial/polynomial_element.pyx
--- a/sage/rings/polynomial/polynomial_element.pyx Thu Jul 09
15:14:36 2009 -0700
+++ b/sage/rings/polynomial/polynomial_element.pyx Wed Jul 15
20:25:10 2009 -0700
@@ -1954,9 +1954,15 @@
def denominator(self):
"""
- Return the least common multiple of the denominators of the
entries
- of self, when this makes sense, i.e., when the coefficients have
a
- denominator function.
+ Return a denominator of self.
+
+ First, the lcm of the denominators of the entries of self
+ is computed and returned. If this computation fails, the
+ unit of the parent of self is returned.
+
+ Note that some subclases may implement its own denominator
+ function. For example:
+
sage.rings.polynomial.polynomial_element_generic.Polynomial_rational_dense
.. warning::
@@ -1997,19 +2003,91 @@
sage: f = x + RR('0.3'); f
1.00000000000000*x + 0.300000000000000
sage: f.denominator()
- Traceback (most recent call last):
- ...
- AttributeError: 'sage.rings.real_mpfr.RealNumber' object has
no attribute 'denominator'
+ 1.00000000000000
"""
if self.degree() == -1:
return 1
- R = self.base_ring()
+ #This code was in the original algorithm it seems useless
+ #R = self.base_ring()
x = self.list()
- d = x[0].denominator()
- for y in x:
- d = d.lcm(y.denominator())
- return d
-
+ try:
+ d = x[0].denominator()
+ for y in x:
+ d = d.lcm(y.denominator())
+ #If we return self.parent(d) instead, automatic test
+ #start to fail, ex. "sage/rings/polynomial/complex_roots.py"
+ return d
+ except(AttributeError):
+ return self.parent().one_element()
+
+ def numerator(self):
+ """
+ Return a numerator of self computed as self * self.denominator()
+
+ Note that some subclases may implement its own numerator
+ function. For example:
+
sage.rings.polynomial.polynomial_element_generic.Polynomial_rational_dense
+
+ .. warning::
+
+ This is not the numerator of the rational function
+ defined by self, which would always be self since self is a
+ polynomial.
+
+ EXAMPLES:
+
+ First we compute the numerator of a polynomial with
+ integer coefficients, which is of course self.
+
+ ::
+
+ sage: R.<x> = ZZ[]
+ sage: f = x^3 + 17*x + 1
+ sage: f.numerator()
+ x^3 + 17*x + 1
+ sage: f == f.numerator()
+ True
+
+ Next we compute the numerator of a polynomial with rational
+ coefficients.
+
+ ::
+
+ sage: R.<x> = PolynomialRing(QQ)
+ sage: f = (1/17)*x^19 - (2/3)*x + 1/3; f
+ 1/17*x^19 - 2/3*x + 1/3
+ sage: f.numerator()
+ 3*x^19 - 34*x + 17
+ sage: f == f.numerator()
+ False
+
+ We try to compute the denominator of a polynomial with
+ coefficients in the real numbers, which is a ring whose elements
do
+ not have a denominator method.
+
+ ::
+
+ sage: R.<x> = RR[]
+ sage: f = x + RR('0.3'); f
+ 1.00000000000000*x + 0.300000000000000
+ sage: f.numerator()
+ 1.00000000000000*x + 0.300000000000000
+
+ We check that the computation the numerator and denominator
+ are valid
+
+ ::
+
+ sage:
K=NumberField(symbolic_expression('x^3+2'),'a')['s,t']['x']
+ sage: f=K.random_element()
+ sage: f.numerator() / f.denominator() == f
+ True
+ sage: R=RR['x']
+ sage: f=R.random_element()
+ sage: f.numerator() / f.denominator() == f
+ True
+ """
+ return self * self.denominator()
def derivative(self, *args):
r"""
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6541>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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