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"""Implementation of :class:`RealField` class. """ |
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from sympy.external.gmpy import SYMPY_INTS, MPQ |
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from sympy.core.numbers import Float |
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from sympy.polys.domains.field import Field |
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from sympy.polys.domains.simpledomain import SimpleDomain |
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from sympy.polys.domains.characteristiczero import CharacteristicZero |
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from sympy.polys.polyerrors import CoercionFailed |
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from sympy.utilities import public |
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from mpmath import MPContext |
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from mpmath.libmp import to_rational as _mpmath_to_rational |
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def to_rational(s, max_denom, limit=True): |
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p, q = _mpmath_to_rational(s._mpf_) |
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p = int(p) |
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q = int(q) |
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if not limit or q <= max_denom: |
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return p, q |
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p0, q0, p1, q1 = 0, 1, 1, 0 |
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n, d = p, q |
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while True: |
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a = n//d |
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q2 = q0 + a*q1 |
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if q2 > max_denom: |
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break |
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p0, q0, p1, q1 = p1, q1, p0 + a*p1, q2 |
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n, d = d, n - a*d |
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k = (max_denom - q0)//q1 |
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number = MPQ(p, q) |
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bound1 = MPQ(p0 + k*p1, q0 + k*q1) |
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bound2 = MPQ(p1, q1) |
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if not bound2 or not bound1: |
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return p, q |
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elif abs(bound2 - number) <= abs(bound1 - number): |
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return bound2.numerator, bound2.denominator |
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else: |
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return bound1.numerator, bound1.denominator |
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@public |
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class RealField(Field, CharacteristicZero, SimpleDomain): |
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"""Real numbers up to the given precision. """ |
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rep = 'RR' |
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is_RealField = is_RR = True |
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is_Exact = False |
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is_Numerical = True |
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is_PID = False |
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has_assoc_Ring = False |
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has_assoc_Field = True |
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_default_precision = 53 |
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@property |
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def has_default_precision(self): |
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return self.precision == self._default_precision |
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@property |
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def precision(self): |
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return self._context.prec |
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@property |
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def dps(self): |
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return self._context.dps |
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@property |
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def tolerance(self): |
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return self._tolerance |
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def __init__(self, prec=None, dps=None, tol=None): |
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context = MPContext() |
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if prec is None and dps is None: |
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context.prec = self._default_precision |
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elif dps is None: |
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context.prec = prec |
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elif prec is None: |
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context.dps = dps |
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else: |
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raise TypeError("Cannot set both prec and dps") |
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self._context = context |
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self._dtype = context.mpf |
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self.zero = self.dtype(0) |
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self.one = self.dtype(1) |
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self._max_denom = max(2**context.prec // 200, 99) |
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self._tolerance = self.one / self._max_denom |
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@property |
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def tp(self): |
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return self._dtype |
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def dtype(self, arg): |
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if isinstance(arg, SYMPY_INTS): |
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arg = int(arg) |
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return self._dtype(arg) |
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def __eq__(self, other): |
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return isinstance(other, RealField) and self.precision == other.precision |
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def __hash__(self): |
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return hash((self.__class__.__name__, self._dtype, self.precision)) |
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def to_sympy(self, element): |
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"""Convert ``element`` to SymPy number. """ |
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return Float(element, self.dps) |
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def from_sympy(self, expr): |
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"""Convert SymPy's number to ``dtype``. """ |
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number = expr.evalf(n=self.dps) |
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if number.is_Number: |
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return self.dtype(number) |
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else: |
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raise CoercionFailed("expected real number, got %s" % expr) |
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def from_ZZ(self, element, base): |
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return self.dtype(element) |
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def from_ZZ_python(self, element, base): |
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return self.dtype(element) |
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def from_ZZ_gmpy(self, element, base): |
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return self.dtype(int(element)) |
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def from_QQ(self, element, base): |
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return self.dtype(element.numerator) / int(element.denominator) |
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def from_QQ_python(self, element, base): |
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return self.dtype(element.numerator) / int(element.denominator) |
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def from_QQ_gmpy(self, element, base): |
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return self.dtype(int(element.numerator)) / int(element.denominator) |
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def from_AlgebraicField(self, element, base): |
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return self.from_sympy(base.to_sympy(element).evalf(self.dps)) |
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def from_RealField(self, element, base): |
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return self.dtype(element) |
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def from_ComplexField(self, element, base): |
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if not element.imag: |
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return self.dtype(element.real) |
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def to_rational(self, element, limit=True): |
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"""Convert a real number to rational number. """ |
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return to_rational(element, self._max_denom, limit=limit) |
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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_exact(self): |
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"""Returns an exact domain associated with ``self``. """ |
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from sympy.polys.domains import QQ |
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return QQ |
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def gcd(self, a, b): |
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"""Returns GCD of ``a`` and ``b``. """ |
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return self.one |
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def lcm(self, a, b): |
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"""Returns LCM of ``a`` and ``b``. """ |
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return a*b |
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def almosteq(self, a, b, tolerance=None): |
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"""Check if ``a`` and ``b`` are almost equal. """ |
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return self._context.almosteq(a, b, tolerance) |
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def is_square(self, a): |
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"""Returns ``True`` if ``a >= 0`` and ``False`` otherwise. """ |
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return a >= 0 |
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def exsqrt(self, a): |
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"""Non-negative square root for ``a >= 0`` and ``None`` otherwise. |
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Explanation |
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=========== |
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The square root may be slightly inaccurate due to floating point |
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rounding error. |
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""" |
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return a ** 0.5 if a >= 0 else None |
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RR = RealField() |
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