"""Implementation of :class:`Domain` class. """
from __future__ import print_function, division
from sympy.polys.domains.domainelement import DomainElement
from sympy.core import Basic, sympify
from sympy.core.compatibility import HAS_GMPY, integer_types, is_sequence
from sympy.polys.polyerrors import UnificationFailed, CoercionFailed, DomainError
from sympy.polys.orderings import lex
from sympy.polys.polyutils import _unify_gens
from sympy.utilities import default_sort_key, public
@public
[docs]class Domain(object):
"""Represents an abstract domain. """
dtype = None
zero = None
one = None
has_Ring = False
has_Field = False
has_assoc_Ring = False
has_assoc_Field = False
is_FiniteField = is_FF = False
is_IntegerRing = is_ZZ = False
is_RationalField = is_QQ = False
is_RealField = is_RR = False
is_ComplexField = is_CC = False
is_AlgebraicField = is_Algebraic = False
is_PolynomialRing = is_Poly = False
is_FractionField = is_Frac = False
is_SymbolicDomain = is_EX = False
is_Exact = True
is_Numerical = False
is_Simple = False
is_Composite = False
has_CharacteristicZero = False
rep = None
alias = None
def __init__(self):
raise NotImplementedError
def __str__(self):
return self.rep
def __repr__(self):
return str(self)
def __hash__(self):
return hash((self.__class__.__name__, self.dtype))
def new(self, *args):
return self.dtype(*args)
@property
def tp(self):
return self.dtype
def __call__(self, *args):
"""Construct an element of ``self`` domain from ``args``. """
return self.new(*args)
def normal(self, *args):
return self.dtype(*args)
[docs] def convert_from(self, element, base):
"""Convert ``element`` to ``self.dtype`` given the base domain. """
if base.alias is not None:
method = "from_" + base.alias
else:
method = "from_" + base.__class__.__name__
_convert = getattr(self, method)
if _convert is not None:
result = _convert(element, base)
if result is not None:
return result
raise CoercionFailed("can't convert %s of type %s from %s to %s" % (element, type(element), base, self))
[docs] def convert(self, element, base=None):
"""Convert ``element`` to ``self.dtype``. """
if base is not None:
return self.convert_from(element, base)
if self.of_type(element):
return element
from sympy.polys.domains import PythonIntegerRing, GMPYIntegerRing, GMPYRationalField, RealField, ComplexField
if isinstance(element, integer_types):
return self.convert_from(element, PythonIntegerRing())
if HAS_GMPY:
integers = GMPYIntegerRing()
if isinstance(element, integers.tp):
return self.convert_from(element, integers)
rationals = GMPYRationalField()
if isinstance(element, rationals.tp):
return self.convert_from(element, rationals)
if isinstance(element, float):
parent = RealField(tol=False)
return self.convert_from(parent(element), parent)
if isinstance(element, complex):
parent = ComplexField(tol=False)
return self.convert_from(parent(element), parent)
if isinstance(element, DomainElement):
return self.convert_from(element, element.parent())
# TODO: implement this in from_ methods
if self.is_Numerical and getattr(element, 'is_ground', False):
return self.convert(element.LC())
if isinstance(element, Basic):
try:
return self.from_sympy(element)
except (TypeError, ValueError):
pass
else: # TODO: remove this branch
if not is_sequence(element):
try:
element = sympify(element)
if isinstance(element, Basic):
return self.from_sympy(element)
except (TypeError, ValueError):
pass
raise CoercionFailed("can't convert %s of type %s to %s" % (element, type(element), self))
[docs] def of_type(self, element):
"""Check if ``a`` is of type ``dtype``. """
return isinstance(element, self.tp) # XXX: this isn't correct, e.g. PolyElement
def __contains__(self, a):
"""Check if ``a`` belongs to this domain. """
try:
self.convert(a)
except CoercionFailed:
return False
return True
[docs] def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
raise NotImplementedError
[docs] def from_sympy(self, a):
"""Convert a SymPy object to ``dtype``. """
raise NotImplementedError
[docs] def from_FF_python(K1, a, K0):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return None
[docs] def from_ZZ_python(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return None
[docs] def from_QQ_python(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return None
[docs] def from_FF_gmpy(K1, a, K0):
"""Convert ``ModularInteger(mpz)`` to ``dtype``. """
return None
[docs] def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpz`` object to ``dtype``. """
return None
[docs] def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpq`` object to ``dtype``. """
return None
[docs] def from_RealField(K1, a, K0):
"""Convert a real element object to ``dtype``. """
return None
[docs] def from_ComplexField(K1, a, K0):
"""Convert a complex element to ``dtype``. """
return None
[docs] def from_AlgebraicField(K1, a, K0):
"""Convert an algebraic number to ``dtype``. """
return None
[docs] def from_PolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
if a.is_ground:
return K1.convert(a.LC, K0.dom)
[docs] def from_FractionField(K1, a, K0):
"""Convert a rational function to ``dtype``. """
return None
[docs] def from_ExpressionDomain(K1, a, K0):
"""Convert a ``EX`` object to ``dtype``. """
return K1.from_sympy(a.ex)
[docs] def from_GlobalPolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
if a.degree() <= 0:
return K1.convert(a.LC(), K0.dom)
def from_GeneralizedPolynomialRing(K1, a, K0):
return K1.from_FractionField(a, K0)
def unify_with_symbols(K0, K1, symbols):
if (K0.is_Composite and (set(K0.symbols) & set(symbols))) or (K1.is_Composite and (set(K1.symbols) & set(symbols))):
raise UnificationFailed("can't unify %s with %s, given %s generators" % (K0, K1, tuple(symbols)))
return K0.unify(K1)
[docs] def unify(K0, K1, symbols=None):
"""
Construct a minimal domain that contains elements of ``K0`` and ``K1``.
Known domains (from smallest to largest):
- ``GF(p)``
- ``ZZ``
- ``QQ``
- ``RR(prec, tol)``
- ``CC(prec, tol)``
- ``ALG(a, b, c)``
- ``K[x, y, z]``
- ``K(x, y, z)``
- ``EX``
"""
if symbols is not None:
return K0.unify_with_symbols(K1, symbols)
if K0 == K1:
return K0
if K0.is_EX:
return K0
if K1.is_EX:
return K1
if K0.is_Composite or K1.is_Composite:
K0_ground = K0.dom if K0.is_Composite else K0
K1_ground = K1.dom if K1.is_Composite else K1
K0_symbols = K0.symbols if K0.is_Composite else ()
K1_symbols = K1.symbols if K1.is_Composite else ()
domain = K0_ground.unify(K1_ground)
symbols = _unify_gens(K0_symbols, K1_symbols)
order = K0.order if K0.is_Composite else K1.order
if ((K0.is_FractionField and K1.is_PolynomialRing or
K1.is_FractionField and K0.is_PolynomialRing) and
(not K0_ground.has_Field or not K1_ground.has_Field) and domain.has_Field):
domain = domain.get_ring()
if K0.is_Composite and (not K1.is_Composite or K0.is_FractionField or K1.is_PolynomialRing):
cls = K0.__class__
else:
cls = K1.__class__
return cls(domain, symbols, order)
def mkinexact(cls, K0, K1):
prec = max(K0.precision, K1.precision)
tol = max(K0.tolerance, K1.tolerance)
return cls(prec=prec, tol=tol)
if K0.is_ComplexField and K1.is_ComplexField:
return mkinexact(K0.__class__, K0, K1)
if K0.is_ComplexField and K1.is_RealField:
return mkinexact(K0.__class__, K0, K1)
if K0.is_RealField and K1.is_ComplexField:
return mkinexact(K1.__class__, K1, K0)
if K0.is_RealField and K1.is_RealField:
return mkinexact(K0.__class__, K0, K1)
if K0.is_ComplexField or K0.is_RealField:
return K0
if K1.is_ComplexField or K1.is_RealField:
return K1
if K0.is_AlgebraicField and K1.is_AlgebraicField:
return K0.__class__(K0.dom.unify(K1.dom), *_unify_gens(K0.orig_ext, K1.orig_ext))
elif K0.is_AlgebraicField:
return K0
elif K1.is_AlgebraicField:
return K1
if K0.is_RationalField:
return K0
if K1.is_RationalField:
return K1
if K0.is_IntegerRing:
return K0
if K1.is_IntegerRing:
return K1
if K0.is_FiniteField and K1.is_FiniteField:
return K0.__class__(max(K0.mod, K1.mod, key=default_sort_key))
from sympy.polys.domains import EX
return EX
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, Domain) and self.dtype == other.dtype
def __ne__(self, other):
"""Returns ``False`` if two domains are equivalent. """
return not self.__eq__(other)
[docs] def map(self, seq):
"""Rersively apply ``self`` to all elements of ``seq``. """
result = []
for elt in seq:
if isinstance(elt, list):
result.append(self.map(elt))
else:
result.append(self(elt))
return result
[docs] def get_ring(self):
"""Returns a ring associated with ``self``. """
raise DomainError('there is no ring associated with %s' % self)
[docs] def get_field(self):
"""Returns a field associated with ``self``. """
raise DomainError('there is no field associated with %s' % self)
[docs] def get_exact(self):
"""Returns an exact domain associated with ``self``. """
return self
def __getitem__(self, symbols):
"""The mathematical way to make a polynomial ring. """
if hasattr(symbols, '__iter__'):
return self.poly_ring(*symbols)
else:
return self.poly_ring(symbols)
[docs] def poly_ring(self, *symbols, **kwargs):
"""Returns a polynomial ring, i.e. `K[X]`. """
from sympy.polys.domains.polynomialring import PolynomialRing
return PolynomialRing(self, symbols, kwargs.get("order", lex))
[docs] def frac_field(self, *symbols, **kwargs):
"""Returns a fraction field, i.e. `K(X)`. """
from sympy.polys.domains.fractionfield import FractionField
return FractionField(self, symbols, kwargs.get("order", lex))
[docs] def old_poly_ring(self, *symbols, **kwargs):
"""Returns a polynomial ring, i.e. `K[X]`. """
from sympy.polys.domains.old_polynomialring import PolynomialRing
return PolynomialRing(self, *symbols, **kwargs)
[docs] def old_frac_field(self, *symbols, **kwargs):
"""Returns a fraction field, i.e. `K(X)`. """
from sympy.polys.domains.old_fractionfield import FractionField
return FractionField(self, *symbols, **kwargs)
[docs] def algebraic_field(self, *extension):
"""Returns an algebraic field, i.e. `K(\\alpha, \dots)`. """
raise DomainError("can't create algebraic field over %s" % self)
[docs] def inject(self, *symbols):
"""Inject generators into this domain. """
raise NotImplementedError
[docs] def is_zero(self, a):
"""Returns True if ``a`` is zero. """
return not a
[docs] def is_one(self, a):
"""Returns True if ``a`` is one. """
return a == self.one
[docs] def is_positive(self, a):
"""Returns True if ``a`` is positive. """
return a > 0
[docs] def is_negative(self, a):
"""Returns True if ``a`` is negative. """
return a < 0
[docs] def is_nonpositive(self, a):
"""Returns True if ``a`` is non-positive. """
return a <= 0
[docs] def is_nonnegative(self, a):
"""Returns True if ``a`` is non-negative. """
return a >= 0
[docs] def abs(self, a):
"""Absolute value of ``a``, implies ``__abs__``. """
return abs(a)
[docs] def neg(self, a):
"""Returns ``a`` negated, implies ``__neg__``. """
return -a
[docs] def pos(self, a):
"""Returns ``a`` positive, implies ``__pos__``. """
return +a
[docs] def add(self, a, b):
"""Sum of ``a`` and ``b``, implies ``__add__``. """
return a + b
[docs] def sub(self, a, b):
"""Difference of ``a`` and ``b``, implies ``__sub__``. """
return a - b
[docs] def mul(self, a, b):
"""Product of ``a`` and ``b``, implies ``__mul__``. """
return a * b
[docs] def pow(self, a, b):
"""Raise ``a`` to power ``b``, implies ``__pow__``. """
return a ** b
[docs] def exquo(self, a, b):
"""Exact quotient of ``a`` and ``b``, implies something. """
raise NotImplementedError
[docs] def quo(self, a, b):
"""Quotient of ``a`` and ``b``, implies something. """
raise NotImplementedError
[docs] def rem(self, a, b):
"""Remainder of ``a`` and ``b``, implies ``__mod__``. """
raise NotImplementedError
[docs] def div(self, a, b):
"""Division of ``a`` and ``b``, implies something. """
raise NotImplementedError
[docs] def invert(self, a, b):
"""Returns inversion of ``a mod b``, implies something. """
raise NotImplementedError
[docs] def revert(self, a):
"""Returns ``a**(-1)`` if possible. """
raise NotImplementedError
[docs] def numer(self, a):
"""Returns numerator of ``a``. """
raise NotImplementedError
[docs] def denom(self, a):
"""Returns denominator of ``a``. """
raise NotImplementedError
[docs] def half_gcdex(self, a, b):
"""Half extended GCD of ``a`` and ``b``. """
s, t, h = self.gcdex(a, b)
return s, h
[docs] def gcdex(self, a, b):
"""Extended GCD of ``a`` and ``b``. """
raise NotImplementedError
[docs] def cofactors(self, a, b):
"""Returns GCD and cofactors of ``a`` and ``b``. """
gcd = self.gcd(a, b)
cfa = self.quo(a, gcd)
cfb = self.quo(b, gcd)
return gcd, cfa, cfb
[docs] def gcd(self, a, b):
"""Returns GCD of ``a`` and ``b``. """
raise NotImplementedError
[docs] def lcm(self, a, b):
"""Returns LCM of ``a`` and ``b``. """
raise NotImplementedError
[docs] def log(self, a, b):
"""Returns b-base logarithm of ``a``. """
raise NotImplementedError
[docs] def sqrt(self, a):
"""Returns square root of ``a``. """
raise NotImplementedError
[docs] def evalf(self, a, prec=None, **options):
"""Returns numerical approximation of ``a``. """
return self.to_sympy(a).evalf(prec, **options)
n = evalf
def real(self, a):
return a
def imag(self, a):
return self.zero
[docs] def almosteq(self, a, b, tolerance=None):
"""Check if ``a`` and ``b`` are almost equal. """
return a == b
[docs] def characteristic(self):
"""Return the characteristic of this domain. """
raise NotImplementedError('characteristic()')
