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"""Implementation of :class:`IntegerRing` class. """
from sympy.external.gmpy import MPZ, GROUND_TYPES
from sympy.core.numbers import int_valued
from sympy.polys.domains.groundtypes import (
SymPyInteger,
factorial,
gcdex, gcd, lcm, sqrt, is_square, sqrtrem,
)
from sympy.polys.domains.characteristiczero import CharacteristicZero
from sympy.polys.domains.ring import Ring
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
import math
@public
class IntegerRing(Ring, CharacteristicZero, SimpleDomain):
r"""The domain ``ZZ`` representing the integers `\mathbb{Z}`.
The :py:class:`IntegerRing` class represents the ring of integers as a
:py:class:`~.Domain` in the domain system. :py:class:`IntegerRing` is a
super class of :py:class:`PythonIntegerRing` and
:py:class:`GMPYIntegerRing` one of which will be the implementation for
:ref:`ZZ` depending on whether or not ``gmpy`` or ``gmpy2`` is installed.
See also
========
Domain
"""
rep = 'ZZ'
alias = 'ZZ'
dtype = MPZ
zero = dtype(0)
one = dtype(1)
tp = type(one)
is_IntegerRing = is_ZZ = True
is_Numerical = True
is_PID = True
has_assoc_Ring = True
has_assoc_Field = True
def __init__(self):
"""Allow instantiation of this domain. """
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
if isinstance(other, IntegerRing):
return True
else:
return NotImplemented
def __hash__(self):
"""Compute a hash value for this domain. """
return hash('ZZ')
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return SymPyInteger(int(a))
def from_sympy(self, a):
"""Convert SymPy's Integer to ``dtype``. """
if a.is_Integer:
return MPZ(a.p)
elif int_valued(a):
return MPZ(int(a))
else:
raise CoercionFailed("expected an integer, got %s" % a)
def get_field(self):
r"""Return the associated field of fractions :ref:`QQ`
Returns
=======
:ref:`QQ`:
The associated field of fractions :ref:`QQ`, a
:py:class:`~.Domain` representing the rational numbers
`\mathbb{Q}`.
Examples
========
>>> from sympy import ZZ
>>> ZZ.get_field()
QQ
"""
from sympy.polys.domains import QQ
return QQ
def algebraic_field(self, *extension, alias=None):
r"""Returns an algebraic field, i.e. `\mathbb{Q}(\alpha, \ldots)`.
Parameters
==========
*extension : One or more :py:class:`~.Expr`.
Generators of the extension. These should be expressions that are
algebraic over `\mathbb{Q}`.
alias : str, :py:class:`~.Symbol`, None, optional (default=None)
If provided, this will be used as the alias symbol for the
primitive element of the returned :py:class:`~.AlgebraicField`.
Returns
=======
:py:class:`~.AlgebraicField`
A :py:class:`~.Domain` representing the algebraic field extension.
Examples
========
>>> from sympy import ZZ, sqrt
>>> ZZ.algebraic_field(sqrt(2))
QQ<sqrt(2)>
"""
return self.get_field().algebraic_field(*extension, alias=alias)
def from_AlgebraicField(K1, a, K0):
"""Convert a :py:class:`~.ANP` object to :ref:`ZZ`.
See :py:meth:`~.Domain.convert`.
"""
if a.is_ground:
return K1.convert(a.LC(), K0.dom)
def log(self, a, b):
r"""Logarithm of *a* to the base *b*.
Parameters
==========
a: number
b: number
Returns
=======
$\\lfloor\log(a, b)\\rfloor$:
Floor of the logarithm of *a* to the base *b*
Examples
========
>>> from sympy import ZZ
>>> ZZ.log(ZZ(8), ZZ(2))
3
>>> ZZ.log(ZZ(9), ZZ(2))
3
Notes
=====
This function uses ``math.log`` which is based on ``float`` so it will
fail for large integer arguments.
"""
return self.dtype(int(math.log(int(a), b)))
def from_FF(K1, a, K0):
"""Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
return MPZ(K0.to_int(a))
def from_FF_python(K1, a, K0):
"""Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
return MPZ(K0.to_int(a))
def from_ZZ(K1, a, K0):
"""Convert Python's ``int`` to GMPY's ``mpz``. """
return MPZ(a)
def from_ZZ_python(K1, a, K0):
"""Convert Python's ``int`` to GMPY's ``mpz``. """
return MPZ(a)
def from_QQ(K1, a, K0):
"""Convert Python's ``Fraction`` to GMPY's ``mpz``. """
if a.denominator == 1:
return MPZ(a.numerator)
def from_QQ_python(K1, a, K0):
"""Convert Python's ``Fraction`` to GMPY's ``mpz``. """
if a.denominator == 1:
return MPZ(a.numerator)
def from_FF_gmpy(K1, a, K0):
"""Convert ``ModularInteger(mpz)`` to GMPY's ``mpz``. """
return MPZ(K0.to_int(a))
def from_ZZ_gmpy(K1, a, K0):
"""Convert GMPY's ``mpz`` to GMPY's ``mpz``. """
return a
def from_QQ_gmpy(K1, a, K0):
"""Convert GMPY ``mpq`` to GMPY's ``mpz``. """
if a.denominator == 1:
return a.numerator
def from_RealField(K1, a, K0):
"""Convert mpmath's ``mpf`` to GMPY's ``mpz``. """
p, q = K0.to_rational(a)
if q == 1:
# XXX: If MPZ is flint.fmpz and p is a gmpy2.mpz, then we need
# to convert via int because fmpz and mpz do not know about each
# other.
return MPZ(int(p))
def from_GaussianIntegerRing(K1, a, K0):
if a.y == 0:
return a.x
def from_EX(K1, a, K0):
"""Convert ``Expression`` to GMPY's ``mpz``. """
if a.is_Integer:
return K1.from_sympy(a)
def gcdex(self, a, b):
"""Compute extended GCD of ``a`` and ``b``. """
h, s, t = gcdex(a, b)
# XXX: This conditional logic should be handled somewhere else.
if GROUND_TYPES == 'gmpy':
return s, t, h
else:
return h, s, t
def gcd(self, a, b):
"""Compute GCD of ``a`` and ``b``. """
return gcd(a, b)
def lcm(self, a, b):
"""Compute LCM of ``a`` and ``b``. """
return lcm(a, b)
def sqrt(self, a):
"""Compute square root of ``a``. """
return sqrt(a)
def is_square(self, a):
"""Return ``True`` if ``a`` is a square.
Explanation
===========
An integer is a square if and only if there exists an integer
``b`` such that ``b * b == a``.
"""
return is_square(a)
def exsqrt(self, a):
"""Non-negative square root of ``a`` if ``a`` is a square.
See also
========
is_square
"""
if a < 0:
return None
root, rem = sqrtrem(a)
if rem != 0:
return None
return root
def factorial(self, a):
"""Compute factorial of ``a``. """
return factorial(a)
ZZ = IntegerRing()
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