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646 lines
18 KiB
646 lines
18 KiB
"""Domains of Gaussian type."""
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from sympy.core.numbers import I
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from sympy.polys.polyerrors import CoercionFailed
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from sympy.polys.domains.integerring import ZZ
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from sympy.polys.domains.rationalfield import QQ
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from sympy.polys.domains.algebraicfield import AlgebraicField
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from sympy.polys.domains.domain import Domain
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from sympy.polys.domains.domainelement import DomainElement
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from sympy.polys.domains.field import Field
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from sympy.polys.domains.ring import Ring
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class GaussianElement(DomainElement):
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"""Base class for elements of Gaussian type domains."""
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base: Domain
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_parent: Domain
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__slots__ = ('x', 'y')
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def __new__(cls, x, y=0):
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conv = cls.base.convert
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return cls.new(conv(x), conv(y))
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@classmethod
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def new(cls, x, y):
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"""Create a new GaussianElement of the same domain."""
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obj = super().__new__(cls)
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obj.x = x
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obj.y = y
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return obj
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def parent(self):
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"""The domain that this is an element of (ZZ_I or QQ_I)"""
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return self._parent
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def __hash__(self):
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return hash((self.x, self.y))
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def __eq__(self, other):
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if isinstance(other, self.__class__):
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return self.x == other.x and self.y == other.y
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else:
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return NotImplemented
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def __lt__(self, other):
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if not isinstance(other, GaussianElement):
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return NotImplemented
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return [self.y, self.x] < [other.y, other.x]
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def __pos__(self):
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return self
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def __neg__(self):
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return self.new(-self.x, -self.y)
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def __repr__(self):
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return "%s(%s, %s)" % (self._parent.rep, self.x, self.y)
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def __str__(self):
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return str(self._parent.to_sympy(self))
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@classmethod
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def _get_xy(cls, other):
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if not isinstance(other, cls):
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try:
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other = cls._parent.convert(other)
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except CoercionFailed:
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return None, None
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return other.x, other.y
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def __add__(self, other):
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x, y = self._get_xy(other)
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if x is not None:
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return self.new(self.x + x, self.y + y)
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else:
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return NotImplemented
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__radd__ = __add__
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def __sub__(self, other):
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x, y = self._get_xy(other)
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if x is not None:
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return self.new(self.x - x, self.y - y)
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else:
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return NotImplemented
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def __rsub__(self, other):
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x, y = self._get_xy(other)
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if x is not None:
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return self.new(x - self.x, y - self.y)
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else:
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return NotImplemented
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def __mul__(self, other):
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x, y = self._get_xy(other)
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if x is not None:
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return self.new(self.x*x - self.y*y, self.x*y + self.y*x)
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else:
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return NotImplemented
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__rmul__ = __mul__
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def __pow__(self, exp):
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if exp == 0:
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return self.new(1, 0)
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if exp < 0:
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self, exp = 1/self, -exp
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if exp == 1:
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return self
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pow2 = self
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prod = self if exp % 2 else self._parent.one
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exp //= 2
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while exp:
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pow2 *= pow2
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if exp % 2:
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prod *= pow2
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exp //= 2
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return prod
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def __bool__(self):
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return bool(self.x) or bool(self.y)
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def quadrant(self):
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"""Return quadrant index 0-3.
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0 is included in quadrant 0.
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"""
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if self.y > 0:
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return 0 if self.x > 0 else 1
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elif self.y < 0:
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return 2 if self.x < 0 else 3
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else:
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return 0 if self.x >= 0 else 2
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def __rdivmod__(self, other):
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try:
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other = self._parent.convert(other)
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except CoercionFailed:
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return NotImplemented
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else:
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return other.__divmod__(self)
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def __rtruediv__(self, other):
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try:
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other = QQ_I.convert(other)
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except CoercionFailed:
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return NotImplemented
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else:
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return other.__truediv__(self)
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def __floordiv__(self, other):
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qr = self.__divmod__(other)
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return qr if qr is NotImplemented else qr[0]
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def __rfloordiv__(self, other):
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qr = self.__rdivmod__(other)
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return qr if qr is NotImplemented else qr[0]
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def __mod__(self, other):
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qr = self.__divmod__(other)
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return qr if qr is NotImplemented else qr[1]
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def __rmod__(self, other):
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qr = self.__rdivmod__(other)
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return qr if qr is NotImplemented else qr[1]
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class GaussianInteger(GaussianElement):
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"""Gaussian integer: domain element for :ref:`ZZ_I`
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>>> from sympy import ZZ_I
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>>> z = ZZ_I(2, 3)
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>>> z
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(2 + 3*I)
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>>> type(z)
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<class 'sympy.polys.domains.gaussiandomains.GaussianInteger'>
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"""
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base = ZZ
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def __truediv__(self, other):
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"""Return a Gaussian rational."""
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return QQ_I.convert(self)/other
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def __divmod__(self, other):
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if not other:
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raise ZeroDivisionError('divmod({}, 0)'.format(self))
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x, y = self._get_xy(other)
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if x is None:
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return NotImplemented
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# multiply self and other by x - I*y
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# self/other == (a + I*b)/c
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a, b = self.x*x + self.y*y, -self.x*y + self.y*x
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c = x*x + y*y
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# find integers qx and qy such that
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# |a - qx*c| <= c/2 and |b - qy*c| <= c/2
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qx = (2*a + c) // (2*c) # -c <= 2*a - qx*2*c < c
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qy = (2*b + c) // (2*c)
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q = GaussianInteger(qx, qy)
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# |self/other - q| < 1 since
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# |a/c - qx|**2 + |b/c - qy|**2 <= 1/4 + 1/4 < 1
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return q, self - q*other # |r| < |other|
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class GaussianRational(GaussianElement):
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"""Gaussian rational: domain element for :ref:`QQ_I`
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>>> from sympy import QQ_I, QQ
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>>> z = QQ_I(QQ(2, 3), QQ(4, 5))
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>>> z
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(2/3 + 4/5*I)
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>>> type(z)
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<class 'sympy.polys.domains.gaussiandomains.GaussianRational'>
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"""
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base = QQ
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def __truediv__(self, other):
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"""Return a Gaussian rational."""
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if not other:
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raise ZeroDivisionError('{} / 0'.format(self))
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x, y = self._get_xy(other)
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if x is None:
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return NotImplemented
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c = x*x + y*y
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return GaussianRational((self.x*x + self.y*y)/c,
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(-self.x*y + self.y*x)/c)
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def __divmod__(self, other):
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try:
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other = self._parent.convert(other)
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except CoercionFailed:
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return NotImplemented
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if not other:
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raise ZeroDivisionError('{} % 0'.format(self))
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else:
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return self/other, QQ_I.zero
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class GaussianDomain():
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"""Base class for Gaussian domains."""
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dom = None # type: Domain
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is_Numerical = True
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is_Exact = True
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has_assoc_Ring = True
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has_assoc_Field = True
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def to_sympy(self, a):
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"""Convert ``a`` to a SymPy object. """
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conv = self.dom.to_sympy
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return conv(a.x) + I*conv(a.y)
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def from_sympy(self, a):
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"""Convert a SymPy object to ``self.dtype``."""
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r, b = a.as_coeff_Add()
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x = self.dom.from_sympy(r) # may raise CoercionFailed
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if not b:
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return self.new(x, 0)
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r, b = b.as_coeff_Mul()
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y = self.dom.from_sympy(r)
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if b is I:
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return self.new(x, y)
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else:
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raise CoercionFailed("{} is not Gaussian".format(a))
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def inject(self, *gens):
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"""Inject generators into this domain. """
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return self.poly_ring(*gens)
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def canonical_unit(self, d):
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unit = self.units[-d.quadrant()] # - for inverse power
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return unit
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def is_negative(self, element):
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"""Returns ``False`` for any ``GaussianElement``. """
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return False
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def is_positive(self, element):
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"""Returns ``False`` for any ``GaussianElement``. """
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return False
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def is_nonnegative(self, element):
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"""Returns ``False`` for any ``GaussianElement``. """
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return False
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def is_nonpositive(self, element):
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"""Returns ``False`` for any ``GaussianElement``. """
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return False
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def from_ZZ_gmpy(K1, a, K0):
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"""Convert a GMPY mpz to ``self.dtype``."""
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return K1(a)
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def from_ZZ(K1, a, K0):
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"""Convert a ZZ_python element to ``self.dtype``."""
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return K1(a)
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def from_ZZ_python(K1, a, K0):
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"""Convert a ZZ_python element to ``self.dtype``."""
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return K1(a)
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def from_QQ(K1, a, K0):
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"""Convert a GMPY mpq to ``self.dtype``."""
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return K1(a)
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def from_QQ_gmpy(K1, a, K0):
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"""Convert a GMPY mpq to ``self.dtype``."""
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return K1(a)
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def from_QQ_python(K1, a, K0):
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"""Convert a QQ_python element to ``self.dtype``."""
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return K1(a)
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def from_AlgebraicField(K1, a, K0):
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"""Convert an element from ZZ<I> or QQ<I> to ``self.dtype``."""
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if K0.ext.args[0] == I:
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return K1.from_sympy(K0.to_sympy(a))
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class GaussianIntegerRing(GaussianDomain, Ring):
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r"""Ring of Gaussian integers ``ZZ_I``
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The :ref:`ZZ_I` domain represents the `Gaussian integers`_ `\mathbb{Z}[i]`
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as a :py:class:`~.Domain` in the domain system (see
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:ref:`polys-domainsintro`).
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By default a :py:class:`~.Poly` created from an expression with
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coefficients that are combinations of integers and ``I`` (`\sqrt{-1}`)
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will have the domain :ref:`ZZ_I`.
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>>> from sympy import Poly, Symbol, I
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>>> x = Symbol('x')
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>>> p = Poly(x**2 + I)
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>>> p
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Poly(x**2 + I, x, domain='ZZ_I')
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>>> p.domain
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ZZ_I
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The :ref:`ZZ_I` domain can be used to factorise polynomials that are
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reducible over the Gaussian integers.
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>>> from sympy import factor
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>>> factor(x**2 + 1)
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x**2 + 1
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>>> factor(x**2 + 1, domain='ZZ_I')
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(x - I)*(x + I)
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The corresponding `field of fractions`_ is the domain of the Gaussian
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rationals :ref:`QQ_I`. Conversely :ref:`ZZ_I` is the `ring of integers`_
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of :ref:`QQ_I`.
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>>> from sympy import ZZ_I, QQ_I
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>>> ZZ_I.get_field()
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QQ_I
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>>> QQ_I.get_ring()
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ZZ_I
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When using the domain directly :ref:`ZZ_I` can be used as a constructor.
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>>> ZZ_I(3, 4)
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(3 + 4*I)
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>>> ZZ_I(5)
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(5 + 0*I)
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The domain elements of :ref:`ZZ_I` are instances of
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:py:class:`~.GaussianInteger` which support the rings operations
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``+,-,*,**``.
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>>> z1 = ZZ_I(5, 1)
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>>> z2 = ZZ_I(2, 3)
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>>> z1
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(5 + 1*I)
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>>> z2
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(2 + 3*I)
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>>> z1 + z2
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(7 + 4*I)
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>>> z1 * z2
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(7 + 17*I)
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>>> z1 ** 2
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(24 + 10*I)
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Both floor (``//``) and modulo (``%``) division work with
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:py:class:`~.GaussianInteger` (see the :py:meth:`~.Domain.div` method).
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>>> z3, z4 = ZZ_I(5), ZZ_I(1, 3)
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>>> z3 // z4 # floor division
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(1 + -1*I)
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>>> z3 % z4 # modulo division (remainder)
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(1 + -2*I)
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>>> (z3//z4)*z4 + z3%z4 == z3
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True
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True division (``/``) in :ref:`ZZ_I` gives an element of :ref:`QQ_I`. The
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:py:meth:`~.Domain.exquo` method can be used to divide in :ref:`ZZ_I` when
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exact division is possible.
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>>> z1 / z2
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(1 + -1*I)
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>>> ZZ_I.exquo(z1, z2)
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(1 + -1*I)
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>>> z3 / z4
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(1/2 + -3/2*I)
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>>> ZZ_I.exquo(z3, z4)
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Traceback (most recent call last):
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...
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ExactQuotientFailed: (1 + 3*I) does not divide (5 + 0*I) in ZZ_I
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The :py:meth:`~.Domain.gcd` method can be used to compute the `gcd`_ of any
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two elements.
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>>> ZZ_I.gcd(ZZ_I(10), ZZ_I(2))
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(2 + 0*I)
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>>> ZZ_I.gcd(ZZ_I(5), ZZ_I(2, 1))
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(2 + 1*I)
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.. _Gaussian integers: https://en.wikipedia.org/wiki/Gaussian_integer
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.. _gcd: https://en.wikipedia.org/wiki/Greatest_common_divisor
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"""
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dom = ZZ
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dtype = GaussianInteger
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zero = dtype(ZZ(0), ZZ(0))
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one = dtype(ZZ(1), ZZ(0))
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imag_unit = dtype(ZZ(0), ZZ(1))
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units = (one, imag_unit, -one, -imag_unit) # powers of i
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rep = 'ZZ_I'
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is_GaussianRing = True
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is_ZZ_I = True
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def __init__(self): # override Domain.__init__
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"""For constructing ZZ_I."""
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def get_ring(self):
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"""Returns a ring associated with ``self``. """
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return self
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def get_field(self):
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"""Returns a field associated with ``self``. """
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return QQ_I
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def normalize(self, d, *args):
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"""Return first quadrant element associated with ``d``.
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Also multiply the other arguments by the same power of i.
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"""
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unit = self.canonical_unit(d)
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d *= unit
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args = tuple(a*unit for a in args)
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return (d,) + args if args else d
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def gcd(self, a, b):
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"""Greatest common divisor of a and b over ZZ_I."""
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while b:
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a, b = b, a % b
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return self.normalize(a)
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def lcm(self, a, b):
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"""Least common multiple of a and b over ZZ_I."""
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return (a * b) // self.gcd(a, b)
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def from_GaussianIntegerRing(K1, a, K0):
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"""Convert a ZZ_I element to ZZ_I."""
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return a
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def from_GaussianRationalField(K1, a, K0):
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"""Convert a QQ_I element to ZZ_I."""
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return K1.new(ZZ.convert(a.x), ZZ.convert(a.y))
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ZZ_I = GaussianInteger._parent = GaussianIntegerRing()
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class GaussianRationalField(GaussianDomain, Field):
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r"""Field of Gaussian rationals ``QQ_I``
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The :ref:`QQ_I` domain represents the `Gaussian rationals`_ `\mathbb{Q}(i)`
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as a :py:class:`~.Domain` in the domain system (see
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:ref:`polys-domainsintro`).
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By default a :py:class:`~.Poly` created from an expression with
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coefficients that are combinations of rationals and ``I`` (`\sqrt{-1}`)
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will have the domain :ref:`QQ_I`.
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>>> from sympy import Poly, Symbol, I
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>>> x = Symbol('x')
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>>> p = Poly(x**2 + I/2)
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>>> p
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Poly(x**2 + I/2, x, domain='QQ_I')
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>>> p.domain
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QQ_I
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The polys option ``gaussian=True`` can be used to specify that the domain
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should be :ref:`QQ_I` even if the coefficients do not contain ``I`` or are
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all integers.
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>>> Poly(x**2)
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Poly(x**2, x, domain='ZZ')
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>>> Poly(x**2 + I)
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Poly(x**2 + I, x, domain='ZZ_I')
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>>> Poly(x**2/2)
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Poly(1/2*x**2, x, domain='QQ')
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>>> Poly(x**2, gaussian=True)
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Poly(x**2, x, domain='QQ_I')
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>>> Poly(x**2 + I, gaussian=True)
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Poly(x**2 + I, x, domain='QQ_I')
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>>> Poly(x**2/2, gaussian=True)
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Poly(1/2*x**2, x, domain='QQ_I')
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The :ref:`QQ_I` domain can be used to factorise polynomials that are
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reducible over the Gaussian rationals.
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>>> from sympy import factor, QQ_I
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>>> factor(x**2/4 + 1)
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(x**2 + 4)/4
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>>> factor(x**2/4 + 1, domain='QQ_I')
|
|
(x - 2*I)*(x + 2*I)/4
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|
>>> factor(x**2/4 + 1, domain=QQ_I)
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|
(x - 2*I)*(x + 2*I)/4
|
|
|
|
It is also possible to specify the :ref:`QQ_I` domain explicitly with
|
|
polys functions like :py:func:`~.apart`.
|
|
|
|
>>> from sympy import apart
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|
>>> apart(1/(1 + x**2))
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|
1/(x**2 + 1)
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>>> apart(1/(1 + x**2), domain=QQ_I)
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|
I/(2*(x + I)) - I/(2*(x - I))
|
|
|
|
The corresponding `ring of integers`_ is the domain of the Gaussian
|
|
integers :ref:`ZZ_I`. Conversely :ref:`QQ_I` is the `field of fractions`_
|
|
of :ref:`ZZ_I`.
|
|
|
|
>>> from sympy import ZZ_I, QQ_I, QQ
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>>> ZZ_I.get_field()
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|
QQ_I
|
|
>>> QQ_I.get_ring()
|
|
ZZ_I
|
|
|
|
When using the domain directly :ref:`QQ_I` can be used as a constructor.
|
|
|
|
>>> QQ_I(3, 4)
|
|
(3 + 4*I)
|
|
>>> QQ_I(5)
|
|
(5 + 0*I)
|
|
>>> QQ_I(QQ(2, 3), QQ(4, 5))
|
|
(2/3 + 4/5*I)
|
|
|
|
The domain elements of :ref:`QQ_I` are instances of
|
|
:py:class:`~.GaussianRational` which support the field operations
|
|
``+,-,*,**,/``.
|
|
|
|
>>> z1 = QQ_I(5, 1)
|
|
>>> z2 = QQ_I(2, QQ(1, 2))
|
|
>>> z1
|
|
(5 + 1*I)
|
|
>>> z2
|
|
(2 + 1/2*I)
|
|
>>> z1 + z2
|
|
(7 + 3/2*I)
|
|
>>> z1 * z2
|
|
(19/2 + 9/2*I)
|
|
>>> z2 ** 2
|
|
(15/4 + 2*I)
|
|
|
|
True division (``/``) in :ref:`QQ_I` gives an element of :ref:`QQ_I` and
|
|
is always exact.
|
|
|
|
>>> z1 / z2
|
|
(42/17 + -2/17*I)
|
|
>>> QQ_I.exquo(z1, z2)
|
|
(42/17 + -2/17*I)
|
|
>>> z1 == (z1/z2)*z2
|
|
True
|
|
|
|
Both floor (``//``) and modulo (``%``) division can be used with
|
|
:py:class:`~.GaussianRational` (see :py:meth:`~.Domain.div`)
|
|
but division is always exact so there is no remainder.
|
|
|
|
>>> z1 // z2
|
|
(42/17 + -2/17*I)
|
|
>>> z1 % z2
|
|
(0 + 0*I)
|
|
>>> QQ_I.div(z1, z2)
|
|
((42/17 + -2/17*I), (0 + 0*I))
|
|
>>> (z1//z2)*z2 + z1%z2 == z1
|
|
True
|
|
|
|
.. _Gaussian rationals: https://en.wikipedia.org/wiki/Gaussian_rational
|
|
"""
|
|
dom = QQ
|
|
dtype = GaussianRational
|
|
zero = dtype(QQ(0), QQ(0))
|
|
one = dtype(QQ(1), QQ(0))
|
|
imag_unit = dtype(QQ(0), QQ(1))
|
|
units = (one, imag_unit, -one, -imag_unit) # powers of i
|
|
|
|
rep = 'QQ_I'
|
|
|
|
is_GaussianField = True
|
|
is_QQ_I = True
|
|
|
|
def __init__(self): # override Domain.__init__
|
|
"""For constructing QQ_I."""
|
|
|
|
def get_ring(self):
|
|
"""Returns a ring associated with ``self``. """
|
|
return ZZ_I
|
|
|
|
def get_field(self):
|
|
"""Returns a field associated with ``self``. """
|
|
return self
|
|
|
|
def as_AlgebraicField(self):
|
|
"""Get equivalent domain as an ``AlgebraicField``. """
|
|
return AlgebraicField(self.dom, I)
|
|
|
|
def numer(self, a):
|
|
"""Get the numerator of ``a``."""
|
|
ZZ_I = self.get_ring()
|
|
return ZZ_I.convert(a * self.denom(a))
|
|
|
|
def denom(self, a):
|
|
"""Get the denominator of ``a``."""
|
|
ZZ = self.dom.get_ring()
|
|
QQ = self.dom
|
|
ZZ_I = self.get_ring()
|
|
denom_ZZ = ZZ.lcm(QQ.denom(a.x), QQ.denom(a.y))
|
|
return ZZ_I(denom_ZZ, ZZ.zero)
|
|
|
|
def from_GaussianIntegerRing(K1, a, K0):
|
|
"""Convert a ZZ_I element to QQ_I."""
|
|
return K1.new(a.x, a.y)
|
|
|
|
def from_GaussianRationalField(K1, a, K0):
|
|
"""Convert a QQ_I element to QQ_I."""
|
|
return a
|
|
|
|
QQ_I = GaussianRational._parent = GaussianRationalField()
|