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cfinite_sequence.py
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cfinite_sequence.py
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# -*- coding: utf-8 -*-
r"""
C-Finite Sequences
C-finite infinite sequences satisfy homogenous linear recurrences with constant coefficients:
.. MATH::
a_{n+d} = c_0a_n + c_1a_{n+1} + \cdots + c_{d-1}a_{n+d-1}, \quad d>0.
CFiniteSequences are completely defined by their ordinary generating function (o.g.f., which
is always a :mod:`fraction <sage.rings.fraction_field_element>` of
:mod:`polynomials <sage.rings.polynomial.polynomial_element>` over `\mathbb{Z}` or `\mathbb{Q}` ).
EXAMPLES::
sage: fibo = CFiniteSequence(x/(1-x-x^2)) # the Fibonacci sequence
sage: fibo
C-finite sequence, generated by x/(-x^2 - x + 1)
sage: fibo.parent()
The ring of C-Finite sequences in x over Rational Field
sage: fibo.parent().category()
Category of commutative rings
sage: C.<x> = CFiniteSequences(QQ);
sage: fibo.parent() == C
True
sage: C
The ring of C-Finite sequences in x over Rational Field
sage: C(x/(1-x-x^2))
C-finite sequence, generated by x/(-x^2 - x + 1)
sage: C(x/(1-x-x^2)) == fibo
True
sage: var('y')
y
sage: CFiniteSequence(y/(1-y-y^2))
C-finite sequence, generated by y/(-y^2 - y + 1)
sage: CFiniteSequence(y/(1-y-y^2)) == fibo
False
Finite subsets of the sequence are accessible via python slices::
sage: fibo[137] #the 137th term of the Fibonacci sequence
19134702400093278081449423917
sage: fibo[137] == fibonacci(137)
True
sage: fibo[0:12]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
sage: fibo[14:4:-2]
[377, 144, 55, 21, 8]
They can be created also from the coefficients and start values of a recurrence::
sage: r = C.from_recurrence([1,1],[0,1])
sage: r == fibo
True
Given enough values, the o.g.f. of a C-finite sequence
can be guessed::
sage: r = C.guess([0,1,1,2,3,5,8])
sage: r == fibo
True
.. SEEALSO::
:func:`fibonacci`, :class:`BinaryRecurrenceSequence`
AUTHORS:
- Ralf Stephan (2014): initial version
REFERENCES:
.. [GK82] Greene, Daniel H.; Knuth, Donald E. (1982), "2.1.1 Constant
coefficients - A) Homogeneous equations", Mathematics for the Analysis
of Algorithms (2nd ed.), Birkhauser, p. 17.
.. [SZ94] Bruno Salvy and Paul Zimmermann. - Gfun: a Maple package for
the manipulation of generating and holonomic functions in one variable.
- Acm transactions on mathematical software, 20.2:163-177, 1994.
.. [Z11] Zeilberger, Doron. "The C-finite ansatz." The Ramanujan Journal
(2011): 1-10.
"""
#*****************************************************************************
# Copyright (C) 2014 Ralf Stephan <gtrwst9@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.categories.fields import Fields
from sage.misc.inherit_comparison import InheritComparisonClasscallMetaclass
from sage.rings.ring import CommutativeRing
from sage.rings.integer import Integer
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.arith.all import gcd
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.polynomial.polynomial_ring import PolynomialRing_general
from sage.rings.laurent_series_ring import LaurentSeriesRing
from sage.rings.power_series_ring import PowerSeriesRing
from sage.rings.fraction_field import FractionField
from sage.structure.element import FieldElement
from sage.structure.unique_representation import UniqueRepresentation
from sage.interfaces.gp import Gp
from sage.misc.all import sage_eval
_gp = None
def CFiniteSequences(base_ring, names = None, category = None):
r"""
Return the ring of C-Finite sequences.
The ring is defined over a base ring (`\mathbb{Z}` or `\mathbb{Q}` )
and each element is represented by its ordinary generating function (ogf)
which is a rational function over the base ring.
INPUT:
- ``base_ring`` -- the base ring to construct the fraction field
representing the C-Finite sequences
- ``names`` -- (optional) the list of variables.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: C
The ring of C-Finite sequences in x over Rational Field
sage: C.an_element()
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: C.category()
Category of commutative rings
sage: C.one()
Finite sequence [1], offset = 0
sage: C.zero()
Constant infinite sequence 0.
sage: C(x)
Finite sequence [1], offset = 1
sage: C(1/x)
Finite sequence [1], offset = -1
sage: C((-x + 2)/(-x^2 - x + 1))
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
TESTS::
sage: TestSuite(C).run()
"""
if isinstance(base_ring, PolynomialRing_general):
polynomial_ring = base_ring
base_ring = polynomial_ring.base_ring()
if names is None:
names = ['x']
elif len(names)>1:
raise NotImplementedError("Multidimensional o.g.f. not implemented.")
if category is None:
category = Fields()
if not(base_ring in (QQ, ZZ)):
raise ValueError("O.g.f. base not rational.")
polynomial_ring = PolynomialRing(base_ring, names)
return CFiniteSequences_generic(polynomial_ring, category)
class CFiniteSequence(FieldElement):
r"""
Create a C-finite sequence given its ordinary generating function.
INPUT:
- ``ogf`` -- a rational function, the ordinary generating function
(can be a an element from the symbolic ring, fraction field or polynomial
ring)
OUTPUT:
- A CFiniteSequence object
EXAMPLES::
sage: CFiniteSequence((2-x)/(1-x-x^2)) # the Lucas sequence
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: CFiniteSequence(x/(1-x)^3) # triangular numbers
C-finite sequence, generated by x/(-x^3 + 3*x^2 - 3*x + 1)
Polynomials are interpreted as finite sequences, or recurrences of degree 0::
sage: CFiniteSequence(x^2-4*x^5)
Finite sequence [1, 0, 0, -4], offset = 2
sage: CFiniteSequence(1)
Finite sequence [1], offset = 0
This implementation allows any polynomial fraction as o.g.f. by interpreting
any power of `x` dividing the o.g.f. numerator or denominator as a right or left shift
of the sequence offset::
sage: CFiniteSequence(x^2+3/x)
Finite sequence [3, 0, 0, 1], offset = -1
sage: CFiniteSequence(1/x+4/x^3)
Finite sequence [4, 0, 1], offset = -3
sage: P = LaurentPolynomialRing(QQ.fraction_field(), 'X')
sage: X=P.gen()
sage: CFiniteSequence(1/(1-X))
C-finite sequence, generated by 1/(-X + 1)
The o.g.f. is always normalized to get a denominator constant coefficient of `+1`::
sage: CFiniteSequence(1/(x-2))
C-finite sequence, generated by -1/2/(-1/2*x + 1)
The given ``ogf`` is used to create an appropriate parent: it can
be a symbolic expression, a polynomial , or a fraction field element
as long as it can be coerced into a proper fraction field over the
rationals::
sage: var('x')
x
sage: f1 = CFiniteSequence((2-x)/(1-x-x^2))
sage: P.<x> = QQ[]
sage: f2 = CFiniteSequence((2-x)/(1-x-x^2))
sage: f1 == f2
True
sage: f1.parent()
The ring of C-Finite sequences in x over Rational Field
sage: f1.ogf().parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: CFiniteSequence(log(x))
Traceback (most recent call last):
...
TypeError: unable to convert log(x) to a rational
TESTS::
sage: P.<x> = QQ[]
sage: CFiniteSequence(0.1/(1-x))
C-finite sequence, generated by 1/10/(-x + 1)
sage: CFiniteSequence(pi/(1-x))
Traceback (most recent call last):
...
TypeError: unable to convert -pi to a rational
sage: P.<x,y> = QQ[]
sage: CFiniteSequence(x*y)
Traceback (most recent call last):
...
NotImplementedError: Multidimensional o.g.f. not implemented.
"""
__metaclass__ = InheritComparisonClasscallMetaclass
@staticmethod
def __classcall_private__(cls, ogf):
r"""
Ensures that elements created by :class:`CFiniteSequence` have the same
parent than the ones created by the parent itself and follow the category
framework (they should be instance of :class:`CFiniteSequences` automatic
element class).
This method is called before the ``__init__`` method, it checks the
o.g.f to create the appropriate parent.
INPUT:
- ``ogf`` - a rational function
TESTS::
sage: f1 = CFiniteSequence((2-x)/(1-x-x^2))
sage: f1
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: C.<x> = CFiniteSequences(QQ);
sage: f2 = CFiniteSequence((2-x)/(1-x-x^2))
sage: f2
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: f3 = C((2-x)/(1-x-x^2))
sage: f3
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: f1 == f2 and f2 == f3
True
sage: f1.parent() == f2.parent() and f2.parent() == f3.parent()
True
sage: type(f1)
<class 'sage.rings.cfinite_sequence.CFiniteSequences_generic_with_category.element_class'>
sage: type(f1) == type(f2) and type(f2) == type(f3)
True
sage: CFiniteSequence(log(x))
Traceback (most recent call last):
...
TypeError: unable to convert log(x) to a rational
sage: CFiniteSequence(pi)
Traceback (most recent call last):
...
TypeError: Unable to coerce pi (<class 'sage.symbolic.constants.Pi'>) to Rational
sage: var('y')
y
sage: f4 = CFiniteSequence((2-y)/(1-y-y^2))
sage: f4
C-finite sequence, generated by (-y + 2)/(-y^2 - y + 1)
sage: f4 == f1
False
sage: f4.parent() == f1.parent()
False
sage: f4.parent()
The ring of C-Finite sequences in y over Rational Field
"""
br = ogf.base_ring()
if not(br in (QQ, ZZ)):
br = QQ # if the base ring of the o.g.f is not QQ, we force it to QQ and see if the o.g.f converts nicely
# trying to figure out the ogf variables
variables = []
if not ogf in br:
if hasattr(ogf, 'variables'):
variables = ogf.variables()
elif hasattr(ogf.parent(), 'gens'):
variables = ogf.parent().gens()
# for some reason, fraction field elements don't have the variables
# method, but symbolic elements don't have the gens method so we check both
if len(variables)==0:
parent = CFiniteSequences(QQ) # if we cannot find variables, we create the default parent (with x)
else:
parent = CFiniteSequences(QQ, variables)
return parent(ogf) # if ogf cannot be converted to a fraction field, this will break and raise the proper error
def __init__(self, parent, ogf):
r"""
Initialize the C-Finite sequence.
The ``__init__`` method can only be called by the :class:`CFiniteSequences`
class. By Default, a class call reaches the ``__classcall_private__``
which first creates a proper parent and then call the ``__init__``.
INPUT:
- ``ogf`` -- the ordinary generating function, a fraction of polynomials over the rationals
- ``parent`` -- the parent of the C-Finite sequence, an occurence of :class:`CFiniteSequences`
OUTPUT:
- A CFiniteSequence object
TESTS::
sage: C.<x> = CFiniteSequences(QQ);
sage: C((2-x)/(1-x-x^2)) # indirect doctest
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
"""
br = parent.base_ring()
ogf = parent.fraction_field()(ogf)
P = parent.polynomial_ring()
num = ogf.numerator()
den = ogf.denominator()
FieldElement.__init__(self, parent)
if den == 1:
self._c = []
self._off = num.valuation()
self._deg = 0
if ogf == 0:
self._a = [0]
else:
self._a = P((num / (P.gen()) ** self._off)).list()
else:
# Transform the ogf numerator and denominator to canonical form
# to get the correct offset, degree, and recurrence coeffs and
# start values.
self._off = 0
self._deg = 0
x = P.gen()
if num.constant_coefficient() == 0:
self._off = num.valuation()
num = P(num / x ** self._off)
elif den.constant_coefficient() == 0:
self._off = -den.valuation()
den = P(den * x ** self._off)
f = den.constant_coefficient()
num = P(num / f)
den = P(den / f)
f = gcd(num, den)
num = P(num / f)
den = P(den / f)
self._deg = den.degree()
self._c = [-den.list()[i] for i in range(1, self._deg + 1)]
if self._off >= 0:
num = x ** self._off * num
else:
den = x ** (-self._off) * den
# determine start values (may be different from _get_item_ values)
alen = max(self._deg, num.degree() + 1)
R = LaurentSeriesRing(br, parent.variable_name(), default_prec=alen)
rem = num % den
if den != 1:
self._a = R(num / den).list()
self._aa = R(rem / den).list()[:self._deg] # needed for _get_item_
else:
self._a = num.list()
if len(self._a) < alen:
self._a.extend([0] * (alen - len(self._a)))
ogf = num / den
self._ogf = ogf
def _repr_(self):
"""
Return textual definition of sequence.
TESTS::
sage: CFiniteSequence(1/x^5)
Finite sequence [1], offset = -5
sage: CFiniteSequence(x^3)
Finite sequence [1], offset = 3
"""
if self._deg == 0:
if self.ogf() == 0:
return 'Constant infinite sequence 0.'
else:
return 'Finite sequence ' + str(self._a) + ', offset = ' + str(self._off)
else:
return 'C-finite sequence, generated by ' + str(self.ogf())
def __hash__(self):
r"""
Hash value for C finite sequence.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: hash(C((2-x)/(1-x-x^2))) # random
42
"""
return hash(self.parent()) ^ hash(self._ogf)
def _add_(self, other):
"""
Addition of C-finite sequences.
TESTS::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C(1/(1-2*x))
sage: r[0:5] # a(n) = 2^n
[1, 2, 4, 8, 16]
sage: s = C.from_recurrence([1],[1])
sage: (r + s)[0:5] # a(n) = 2^n + 1
[2, 3, 5, 9, 17]
sage: r + 0 == r
True
sage: (r + x^2)[0:5]
[1, 2, 5, 8, 16]
sage: (r + 3/x)[-1]
3
sage: r = CFiniteSequence(x)
sage: r + 0 == r
True
sage: CFiniteSequence(0) + CFiniteSequence(0)
Constant infinite sequence 0.
"""
return CFiniteSequence(self.ogf() + other.numerator() / other.denominator())
def _sub_(self, other):
"""
Subtraction of C-finite sequences.
TESTS::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C(1/(1-2*x))
sage: r[0:5] # a(n) = 2^n
[1, 2, 4, 8, 16]
sage: s = C.from_recurrence([1],[1])
sage: (r - s)[0:5] # a(n) = 2^n + 1
[0, 1, 3, 7, 15]
"""
return CFiniteSequence(self.ogf() - other.numerator() / other.denominator())
def _mul_(self, other):
"""
Multiplication of C-finite sequences.
TESTS::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C.guess([1,2,3,4,5,6])
sage: (r*r)[0:6] # self-convolution
[1, 4, 10, 20, 35, 56]
sage: r = C(x)
sage: r*1 == r
True
sage: r*-1
Finite sequence [-1], offset = 1
sage: C(0) * C(1)
Constant infinite sequence 0.
"""
return CFiniteSequence(self.ogf() * other.numerator() / other.denominator())
def _div_(self, other):
"""
Division of C-finite sequences.
TESTS::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C.guess([1,2,3,4,5,6])
sage: (r/2)[0:6]
[1/2, 1, 3/2, 2, 5/2, 3]
sage: s = C(x)
sage: s/(s*-1 + 1)
C-finite sequence, generated by x/(-x + 1)
"""
return CFiniteSequence(self.ogf() / (other.numerator() / other.denominator()))
def coefficients(self):
"""
Return the coefficients of the recurrence representation of the
C-finite sequence.
OUTPUT:
- A list of values
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: lucas = C((2-x)/(1-x-x^2)) # the Lucas sequence
sage: lucas.coefficients()
[1, 1]
"""
return self._c
def __eq__(self, other):
"""
Compare two CFiniteSequences.
EXAMPLES::
sage: f = CFiniteSequence((2-x)/(1-x-x^2))
sage: f2 = CFiniteSequence((2-x)/(1-x-x^2))
sage: f == f2
True
sage: f == (2-x)/(1-x-x^2)
False
sage: (2-x)/(1-x-x^2) == f
False
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C.from_recurrence([1,1],[2,1])
sage: s = C.from_recurrence([-1],[1])
sage: r == s
False
sage: r = C.from_recurrence([-1],[1])
sage: s = C(1/(1+x))
sage: r == s
True
"""
if not isinstance(other, CFiniteSequence):
return False
return self.ogf() == other.ogf()
def __getitem__(self, key):
r"""
Return a slice of the sequence.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C.from_recurrence([3,3],[2,1])
sage: r[2]
9
sage: r[101]
16158686318788579168659644539538474790082623100896663971001
sage: r = C(1/(1-x))
sage: r[5]
1
sage: r = C(x)
sage: r[0]
0
sage: r[1]
1
sage: r = C(0)
sage: r[66]
0
sage: lucas = C.from_recurrence([1,1],[2,1])
sage: lucas[5:10]
[11, 18, 29, 47, 76]
sage: r = C((2-x)/x/(1-x-x*x))
sage: r[0:4]
[1, 3, 4, 7]
sage: r = C(1-2*x^2)
sage: r[0:4]
[1, 0, -2, 0]
sage: r[-1:4] # not tested, python will not allow this!
[0, 1, 0 -2, 0]
sage: r = C((-2*x^3 + x^2 + 1)/(-2*x + 1))
sage: r[0:5] # handle ogf > 1
[1, 2, 5, 8, 16]
sage: r[-2]
0
sage: r = C((-2*x^3 + x^2 - x + 1)/(2*x^2 - 3*x + 1))
sage: r[0:5]
[1, 2, 5, 9, 17]
sage: s=C((1-x)/(-x^2 - x + 1))
sage: s[0:5]
[1, 0, 1, 1, 2]
sage: s=C((1+x^20+x^40)/(1-x^12)/(1-x^30))
sage: s[0:20]
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
sage: s=C(1/((1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)))
sage: s[999998]
289362268629630
"""
if isinstance(key, slice):
m = max(key.start, key.stop)
return [self[ii] for ii in xrange(*key.indices(m + 1))]
elif isinstance(key, (int, Integer)):
from sage.matrix.constructor import Matrix
d = self._deg
if (self._off <= key and key < self._off + len(self._a)):
return self._a[key - self._off]
elif d == 0:
return 0
(quo, rem) = self.numerator().quo_rem(self.denominator())
wp = quo[key - self._off]
if key < self._off:
return wp
A = Matrix(QQ, 1, d, self._c)
B = Matrix.identity(QQ, d - 1)
C = Matrix(QQ, d - 1, 1, 0)
if quo == 0:
V = Matrix(QQ, d, 1, self._a[:d][::-1])
else:
V = Matrix(QQ, d, 1, self._aa[:d][::-1])
M = Matrix.block([[A], [B, C]], subdivide=False)
return wp + list(M ** (key - self._off) * V)[d - 1][0]
else:
raise TypeError("invalid argument type")
def ogf(self):
"""
Return the ordinary generating function associated with
the CFiniteSequence.
This is always a fraction of polynomials in the base ring.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C.from_recurrence([2],[1])
sage: r.ogf()
1/(-2*x + 1)
sage: C(0).ogf()
0
"""
return self._ogf
def numerator(self):
r"""
Return the numerator of the o.g.f of ``self``.
EXAMPLES::
sage: f = CFiniteSequence((2-x)/(1-x-x^2)); f
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: f.numerator()
-x + 2
"""
return self.ogf().numerator()
def denominator(self):
r"""
Return the numerator of the o.g.f of ``self``.
EXAMPLES::
sage: f = CFiniteSequence((2-x)/(1-x-x^2)); f
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: f.denominator()
-x^2 - x + 1
"""
return self.ogf().denominator()
def recurrence_repr(self):
"""
Return a string with the recurrence representation of
the C-finite sequence.
OUTPUT:
- A string
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: C((2-x)/(1-x-x^2)).recurrence_repr()
'Homogenous linear recurrence with constant coefficients of degree 2: a(n+2) = a(n+1) + a(n), starting a(0...) = [2, 1]'
sage: C(x/(1-x)^3).recurrence_repr()
'Homogenous linear recurrence with constant coefficients of degree 3: a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n), starting a(1...) = [1, 3, 6]'
sage: C(1).recurrence_repr()
'Finite sequence [1], offset 0'
sage: r = C((-2*x^3 + x^2 - x + 1)/(2*x^2 - 3*x + 1))
sage: r.recurrence_repr()
'Homogenous linear recurrence with constant coefficients of degree 2: a(n+2) = 3*a(n+1) - 2*a(n), starting a(0...) = [1, 2, 5, 9]'
sage: r = CFiniteSequence(x^3/(1-x-x^2))
sage: r.recurrence_repr()
'Homogenous linear recurrence with constant coefficients of degree 2: a(n+2) = a(n+1) + a(n), starting a(3...) = [1, 1, 2, 3]'
"""
if self._deg == 0:
return 'Finite sequence %s, offset %d' % (str(self._a), self._off)
else:
if self._c[0] == 1:
cstr = 'a(n+%d) = a(n+%d)' % (self._deg, self._deg - 1)
elif self._c[0] == -1:
cstr = 'a(n+%d) = -a(n+%d)' % (self._deg, self._deg - 1)
else:
cstr = 'a(n+%d) = %s*a(n+%d)' % (self._deg, str(self._c[0]), self._deg - 1)
for i in range(1, self._deg):
j = self._deg - i - 1
if self._c[i] < 0:
if self._c[i] == -1:
cstr = cstr + ' - a(n+%d)' % (j,)
else:
cstr = cstr + ' - %d*a(n+%d)' % (-(self._c[i]), j)
elif self._c[i] > 0:
if self._c[i] == 1:
cstr = cstr + ' + a(n+%d)' % (j,)
else:
cstr = cstr + ' + %d*a(n+%d)' % (self._c[i], j)
cstr = cstr.replace('+0', '')
astr = ', starting a(%s...) = [' % str(self._off)
maxwexp = self.numerator().quo_rem(self.denominator())[0].degree() + 1
for i in range(maxwexp + self._deg):
astr = astr + str(self[self._off + i]) + ', '
astr = astr[:-2] + ']'
return 'Homogenous linear recurrence with constant coefficients of degree ' + str(self._deg) + ': ' + cstr + astr
def series(self, n):
"""
Return the Laurent power series associated with the
CFiniteSequence, with precision `n`.
INPUT:
- `n` -- a nonnegative integer
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: r = C.from_recurrence([-1,2],[0,1])
sage: s = r.series(4); s
x + 2*x^2 + 3*x^3 + 4*x^4 + O(x^5)
sage: type(s)
<type 'sage.rings.laurent_series_ring_element.LaurentSeries'>
"""
R = LaurentSeriesRing(QQ, 'x', default_prec=n)
return R(self.ogf())
class CFiniteSequences_generic(CommutativeRing, UniqueRepresentation):
r"""
The class representing the ring of C-Finite Sequences
TESTS::
sage: C.<x> = CFiniteSequences(QQ)
sage: from sage.rings.cfinite_sequence import CFiniteSequences_generic
sage: isinstance(C,CFiniteSequences_generic)
True
sage: type(C)
<class 'sage.rings.cfinite_sequence.CFiniteSequences_generic_with_category'>
sage: C
The ring of C-Finite sequences in x over Rational Field
"""
Element = CFiniteSequence
def __init__(self, polynomial_ring, category):
r"""
Create the ring of CFiniteSequences over ``base_ring``
INPUT:
- ``base_ring`` -- the base ring for the o.g.f (either ``QQ`` or ``ZZ``)
- ``names`` -- an iterable of variables (shuould contain only one variable)
- ``category`` -- the category of the ring (default: ``Fields()``)
TESTS::
sage: C.<y> = CFiniteSequences(QQ); C
The ring of C-Finite sequences in y over Rational Field
sage: C.<x> = CFiniteSequences(QQ); C
The ring of C-Finite sequences in x over Rational Field
sage: C.<x> = CFiniteSequences(ZZ); C
The ring of C-Finite sequences in x over Integer Ring
sage: C.<x,y> = CFiniteSequences(ZZ)
Traceback (most recent call last):
...
NotImplementedError: Multidimensional o.g.f. not implemented.
sage: C.<x> = CFiniteSequences(CC)
Traceback (most recent call last):
...
ValueError: O.g.f. base not rational.
"""
base_ring = polynomial_ring.base_ring()
self._polynomial_ring = polynomial_ring
self._fraction_field = FractionField(self._polynomial_ring)
CommutativeRing.__init__(self,base_ring, self._polynomial_ring.gens(), category)
def _repr_(self):
r"""
Return the string representation of ``self``
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ)
sage: C
The ring of C-Finite sequences in x over Rational Field
"""
return "The ring of C-Finite sequences in {} over {}".format(self.gen(), self.base_ring())
def _element_constructor_(self, ogf):
r"""
Construct a C-Finite Sequence
INPUT:
- ``ogf`` -- the ordinary generating function, a fraction of polynomials over the rationals
TESTS::
sage: C.<x> = CFiniteSequences(QQ)
sage: C((2-x)/(1-x-x^2))
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
sage: C(x/(1-x)^3)
C-finite sequence, generated by x/(-x^3 + 3*x^2 - 3*x + 1)
sage: C(x^2-4*x^5)
Finite sequence [1, 0, 0, -4], offset = 2
sage: C(x^2+3/x)
Finite sequence [3, 0, 0, 1], offset = -1
sage: C(1/x + 4/x^3)
Finite sequence [4, 0, 1], offset = -3
sage: P = LaurentPolynomialRing(QQ.fraction_field(), 'X')
sage: X = P.gen()
sage: C(1/(1-X))
C-finite sequence, generated by 1/(-x + 1)
sage: C = CFiniteSequences(QQ)
sage: C(x)
Finite sequence [1], offset = 1
"""
ogf = self.fraction_field()(ogf)
return self.element_class(self, ogf)
def ngens(self):
r"""
Return the number of generators of ``self``
EXAMPLES::
sage: from sage.rings.cfinite_sequence import CFiniteSequences
sage: C.<x> = CFiniteSequences(QQ);
sage: C.ngens()
1
"""
return 1
def gen(self,i=0):
r"""
Return the i-th generator of ``self``.
INPUT:
- ``i`` -- an integer (default:0)
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ);
sage: C.gen()
x
sage: x == C.gen()
True
TESTS::
sage: C.gen(2)
Traceback (most recent call last):
...
ValueError: The ring of C-Finite sequences in x over Rational Field has only one generator (i=0)
"""
if i!= 0:
raise ValueError("{} has only one generator (i=0)".format(self))
return self.polynomial_ring().gen()
def an_element(self):
r"""
Return an element of C-Finite Sequences.
OUTPUT:
The Lucas sequence.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ);
sage: C.an_element()
C-finite sequence, generated by (-x + 2)/(-x^2 - x + 1)
"""
x = self.gen()
return self((2-x)/(1-x-x**2))
def __contains__(self, x):
"""
Return True if x is an element of ``CFinteSequences`` or
canonically coerces to this ring.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ);
sage: x in C
True
sage: 1/x in C
True
sage: 5 in C
True
sage: pi in C
False
sage: Cy.<y> = CFiniteSequences(QQ);
sage: y in C
False
sage: y in Cy
True
"""
if x.parent() == self:
return True
try:
self._coerce_(x)
except TypeError:
return False
return True
def fraction_field(self):
r"""
Return the faction field used to represent the elements of ``self``.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ);
sage: C.fraction_field()
Fraction Field of Univariate Polynomial Ring in x over Rational Field
"""
return self._fraction_field
def polynomial_ring(self):
r"""
Return the polynomial ring used to represent the elements of ``self``.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ);
sage: C.polynomial_ring()
Univariate Polynomial Ring in x over Rational Field
"""
return self._polynomial_ring
def _coerce_map_from_(self, S):
"""
A coercion from `S` exists, if `S` coerces into ``self``'s fraction
field.
EXAMPLES::
sage: C.<x> = CFiniteSequences(QQ);
sage: C.has_coerce_map_from(C.fraction_field())
True
sage: C.has_coerce_map_from(QQ)
True
sage: C.has_coerce_map_from(QQ[x])
True
sage: C.has_coerce_map_from(ZZ)
True
"""
if self.fraction_field().has_coerce_map_from(S):
return True
def from_recurrence(self, coefficients, values):
"""
Create a C-finite sequence given the coefficients $c$ and
starting values $a$ of a homogenous linear recurrence.
.. MATH::