Source code for sympy.core.numbers

from .core import C
from .sympify import converter, sympify, _sympify, SympifyError
from .basic import Basic
from .singleton import S, Singleton
from .expr import Expr, AtomicExpr
from .decorators import _sympifyit, deprecated
from .cache import cacheit, clear_cache
from sympy.core.compatibility import as_int
import sympy.mpmath as mpmath
import sympy.mpmath.libmp as mlib
from sympy.mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed
from sympy.mpmath.ctx_mp import mpnumeric
from sympy.mpmath.libmp.libmpf import fnan, _normalize as mpf_normalize

import decimal
import math
import re as regex
from collections import defaultdict

rnd = mlib.round_nearest

_LOG2 = math.log(2)

def mpf_norm(mpf, prec):
    """Return the mpf tuple normalized appropriately for the indicated
    precision.

    This also contains a portion of code to not return zero if
    the mantissa is 0 since it is zero for mpf's +inf, -inf and
    nan, too.
    """
    sign, man, expt, bc = mpf
    if not man:
        # hack for mpf_normalize which does not do this;
        # it assumes that if man is zero the result is 0
        if not bc:
            return mlib.fzero
        else:
            # don't change anything; this should already
            # be a well formed mpf tuple
            return mpf
    rv = mpf_normalize(sign, man, expt, bc, prec, rnd)
    return rv

# TODO: we should use the warnings module
_errdict = {"divide": False}
[docs]def seterr(divide=False): """ Should sympy raise an exception on 0/0 or return a nan? divide == True .... raise an exception divide == False ... return nan """ if _errdict["divide"] != divide: clear_cache() _errdict["divide"] = divide
def _decimal_to_Rational_prec(dec): """Convert an ordinary decimal instance to a Rational.""" # _is_special is needed for Python 2.5 support; is_finite for Python 3.3 # support nonfinite = getattr(dec, '_is_special', None) if nonfinite is None: nonfinite = not dec.is_finite() if nonfinite: raise TypeError("dec must be finite, got %s." % dec) s, d, e = dec.as_tuple() prec = len(d) if int(dec) == dec: rv = Rational(int(dec)) else: s = (-1)**s d = sum([di*10**i for i, di in enumerate(reversed(d))]) rv = Rational(s*d, 10**-e) return rv, prec def _literal_float(f): """Return True if n can be interpreted as a floating point number.""" pat = r"[-+]?((\d*\.\d+)|(\d+\.?))(eE[-+]?\d+)?" return bool(regex.match(pat, f)) # (a,b) -> gcd(a,b) _gcdcache = {} # TODO caching with decorator, but not to degrade performance
[docs]def igcd(a, b): """Computes positive, integer greatest common divisor of two numbers. The algorithm is based on the well known Euclid's algorithm. To improve speed, igcd() has its own caching mechanism implemented. """ try: return _gcdcache[(a,b)] except KeyError: a, b = as_int(a), as_int(b) if a and b: if b < 0: b = -b while b: a, b = b, a % b else: a = abs(a or b) _gcdcache[(a,b)] = a return a
[docs]def ilcm(a, b): """Computes integer least common multiple of two numbers. """ if a == 0 and b == 0: return 0 else: return a*b // igcd(a, b)
def igcdex(a, b): """Returns x, y, g such that g = x*a + y*b = gcd(a, b). >>> from sympy.core.numbers import igcdex >>> igcdex(2, 3) (-1, 1, 1) >>> igcdex(10, 12) (-1, 1, 2) >>> x, y, g = igcdex(100, 2004) >>> x, y, g (-20, 1, 4) >>> x*100 + y*2004 4 """ if (not a) and (not b): return (0, 1, 0) if not a: return (0, b//abs(b), abs(b)) if not b: return (a//abs(a), 0, abs(a)) if a < 0: a, x_sign = -a, -1 else: x_sign = 1 if b < 0: b, y_sign = -b, -1 else: y_sign = 1 x, y, r, s = 1, 0, 0, 1 while b: (c, q) = (a % b, a // b) (a, b, r, s, x, y) = (b, c, x-q*r, y-q*s, r, s) return (x*x_sign, y*y_sign, a)
[docs]class Number(AtomicExpr): """ Represents any kind of number in sympy. Floating point numbers are represented by the Float class. Integer numbers (of any size), together with rational numbers (again, there is no limit on their size) are represented by the Rational class. If you want to represent, for example, ``1+sqrt(2)``, then you need to do:: Rational(1) + sqrt(Rational(2)) """ is_commutative = True is_bounded = True is_finite = True is_number = True __slots__ = [] # Used to make max(x._prec, y._prec) return x._prec when only x is a float _prec = -1 is_Number = True def __new__(cls, *obj): if len(obj) == 1: obj=obj[0] if isinstance(obj, Number): return obj if isinstance(obj, int): return Integer(obj) if isinstance(obj, tuple) and len(obj) == 2: return Rational(*obj) if isinstance(obj, (float, mpmath.mpf, decimal.Decimal)): return Float(obj) if isinstance(obj, str): val = sympify(obj) if isinstance(val, Number): return val else: raise ValueError('String "%s" does not denote a Number'%obj) if isinstance(obj, Number): return obj msg = "expected str|int|long|float|Decimal|Number object but got %r" raise TypeError(msg % type(obj).__name__) def __divmod__(self, other): from .containers import Tuple from sympy.functions.elementary.complexes import sign try: other = Number(other) except TypeError: msg = "unsupported operand type(s) for divmod(): '%s' and '%s'" raise TypeError(msg % (type(self).__name__, type(other).__name__)) if not other: raise ZeroDivisionError('modulo by zero') if self.is_Integer and other.is_Integer: return Tuple(*divmod(self.p, other.p)) else: rat = self/other w = sign(rat)*int(abs(rat)) # = rat.floor() r = self - other*w #w*other + r == self return Tuple(w, r) def __rdivmod__(self, other): try: other = Number(other) except TypeError: msg = "unsupported operand type(s) for divmod(): '%s' and '%s'" raise TypeError(msg % (type(other).__name__, type(self).__name__)) return divmod(other, self) def __round__(self, *args): return round(float(self), *args) def _as_mpf_val(self, prec): """Evaluation of mpf tuple accurate to at least prec bits.""" raise NotImplementedError('%s needs ._as_mpf_val() method' % \ (self.__class__.__name__)) def _eval_evalf(self, prec): return Float._new(self._as_mpf_val(prec), prec) def _as_mpf_op(self, prec): prec = max(prec, self._prec) return self._as_mpf_val(prec), prec def __float__(self): return mlib.to_float(self._as_mpf_val(53)) def _eval_conjugate(self): return self def _eval_order(self, *symbols): # Order(5, x, y) -> Order(1,x,y) return C.Order(S.One, *symbols) def _eval_subs(self, old, new): if old == -self: return -new return self # there is no other possibility @classmethod def class_key(cls): return 1, 0, 'Number' @cacheit def sort_key(self, order=None): return self.class_key(), (0, ()), (), self @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number): if other is S.NaN: return S.NaN elif other is S.Infinity: return S.Infinity elif other is S.NegativeInfinity: return S.NegativeInfinity return AtomicExpr.__add__(self, other) @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number): if other is S.NaN: return S.NaN elif other is S.Infinity: return S.NegativeInfinity elif other is S.NegativeInfinity: return S.Infinity return AtomicExpr.__sub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number): if other is S.NaN: return S.NaN elif other is S.Infinity: if self == 0: return S.NaN elif self > 0: return S.Infinity else: return S.NegativeInfinity elif other is S.NegativeInfinity: if self == 0: return S.NaN elif self > 0: return S.NegativeInfinity else: return S.Infinity return AtomicExpr.__mul__(self, other) @_sympifyit('other', NotImplemented) def __div__(self, other): if isinstance(other, Number): if other is S.NaN: return S.NaN elif other is S.Infinity or other is S.NegativeInfinity: return S.Zero return AtomicExpr.__div__(self, other) __truediv__ = __div__ def __eq__(self, other): raise NotImplementedError('%s needs .__eq__() method' % (self.__class__.__name__)) def __ne__(self, other): raise NotImplementedError('%s needs .__ne__() method' % (self.__class__.__name__)) def __lt__(self, other): raise NotImplementedError('%s needs .__lt__() method' % (self.__class__.__name__)) def __le__(self, other): raise NotImplementedError('%s needs .__le__() method' % (self.__class__.__name__)) def __gt__(self, other): return _sympify(other).__lt__(self) def __ge__(self, other): return _sympify(other).__le__(self) def __hash__(self): return super(Number, self).__hash__() def is_constant(self, *wrt, **flags): return True @property def is_number(self): return True def as_coeff_mul(self, *deps): # a -> c*t if self.is_Rational: return self, tuple() elif self.is_negative: return S.NegativeOne, (-self,) return S.One, (self,) def as_coeff_add(self, *deps): # a -> c + t if self.is_Rational: return self, tuple() return S.Zero, (self,)
[docs] def gcd(self, other): """Compute greatest common divisor of input arguments. """ return S.One
[docs] def lcm(self, other): """Compute least common multiple of input arguments. """ other = _sympify(other) return self*other
[docs] def cofactors(self, other): """Compute GCD and cofactors of input arguments. """ other = _sympify(other) return S.One, self, other
[docs] def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """ if rational and not self.is_Rational: return S.One, self return self, S.One
[docs] def as_coeff_Add(self): """Efficiently extract the coefficient of a summation. """ return self, S.Zero
[docs]class Float(Number): """ Represents a floating point number. It is capable of representing arbitrary-precision floating-point numbers. Examples ======== >>> from sympy import Float >>> Float(3.5) # convert from Python float or int 3.50000000000000 >>> Float(3) # reverts to Integer 3 >>> Float(3, '') # forced to Float 3. Floats can be created from a string representations of Python floats to force ints to Float or to enter high-precision (> 15 significant digits) values: >>> Float('.0010') 0.00100000000000000 >>> Float('1e-3') 0.00100000000000000 >>> Float('1e-3', 3) 0.00100 Float can automatically count significant figures if a null string is sent for the precision; space are also allowed in the string. (Auto- counting is only allowed for strings, ints and longs). >>> Float('123 456 789 . 123 456', '') 123456789.123456 >>> Float('12e-3', '') 0.012 Notes ===== Floats are inexact by their nature unless their value is a binary-exact value. >>> approx, exact = Float(.1, 1), Float(.125, 1) For calculation purposes, evalf needs to be able to change the precision but this will not increase the accuracy of the inexact value. The following is the most accurate 5-digit approximation of a value of 0.1 that had only 1 digit of precision: >>> approx.evalf(5) 0.099609 By contrast, 0.125 is exact in binary (as it is in base 10) and so it can be passed to Float or evalf to obtain an arbitrary precision with matching accuracy: >>> Float(exact, 5) 0.12500 >>> exact.evalf(20) 0.12500000000000000000 Trying to make a high-precision Float from a float is not disallowed, but one must keep in mind that the *underlying float* (not the apparent decimal value) is being obtained with high precision. For example, 0.3 does not have a finite binary representation. The closest rational is the fraction 5404319552844595/2**54. So if you try to obtain a Float of 0.3 to 20 digits of precision you will not see the same thing as 0.3 followed by 19 zeros: >>> Float(0.3, 20) 0.29999999999999998890 If you want a 20-digit value of the decimal 0.3 (not the floating point approximation of 0.3) you should send the 0.3 as a string. The underlying representation is still binary but a higher precision than Python's float is used: >>> Float('0.3', 20) 0.30000000000000000000 Although you can increase the precision of an existing Float using Float it will not increase the accuracy -- the underlying value is not changed: >>> def show(f): # binary rep of Float ... from sympy import Mul, Pow ... s, m, e, b = f._mpf_ ... v = Mul(m, Pow(2, e, evaluate=False), evaluate=False) ... print('%s at prec=%s' % (v, f._prec)) ... >>> t = Float('0.3', 3) >>> show(t) 4915/2**14 at prec=13 >>> show(Float(t, 20)) # higher prec, not higher accuracy 4915/2**14 at prec=70 >>> show(Float(t, 2)) # lower prec 307/2**10 at prec=10 The same thing happens when evalf is used on a Float: >>> show(t.evalf(20)) 4915/2**14 at prec=70 >>> show(t.evalf(2)) 307/2**10 at prec=10 Finally, Floats can be instantiated with an mpf tuple (n, c, p) to produce the number (-1)**n*c*2**p: >>> n, c, p = 1, 5, 0 >>> (-1)**n*c*2**p -5 >>> Float((1, 5, 0)) -5.00000000000000 An actual mpf tuple also contains the number of bits in c as the last element of the tuple, but this is not needed for instantiation: >>> _._mpf_ (1, 5, 0, 3) """ __slots__ = ['_mpf_', '_prec'] is_real = True is_irrational = False is_integer = False is_Float = True def __new__(cls, num, prec=15): if isinstance(num, str): num = num.replace(' ', '') if num.startswith('.') and len(num) > 1: num = '0' + num elif num.startswith('-.') and len(num) > 2: num = '-0.' + num[2:] elif not num: return C.Zero() if prec == '': if isinstance(num, (int, Integer)): # an int is unambiguous, but if someone enters # .99999999999999999, Python automatically converts # this to 1.0 and although 1.0 == 1, this is not # really what the user typed, so we avoid guessing -- # even if num == int(num) -- because we don't know how # it became that exact float. num = str(num) elif not isinstance(num, str): raise ValueError('The null string can only be used when ' 'the number to Float is passed as a string.') ok = None if _literal_float(num): try: Num = decimal.Decimal(num) except decimal.InvalidOperation: pass else: num, dps = _decimal_to_Rational_prec(Num) ok = True if num.is_Integer: dps = len(str(num)) if ok is None: raise ValueError('string-float not recognized: %s' % num) else: dps = prec if prec != '' and isinstance(num, (int, Integer)): # if this is changed here it has to be changed in _new, too return Integer(num) prec = mlib.libmpf.dps_to_prec(dps) if isinstance(num, float): _mpf_ = mlib.from_float(num, prec, rnd) elif isinstance(num, (str, decimal.Decimal, Integer)): _mpf_ = mlib.from_str(str(num), prec, rnd) elif isinstance(num, Rational): _mpf_ = mlib.from_rational(num.p, num.q, prec, rnd) elif isinstance(num, tuple) and len(num) in (3, 4): if type(num[1]) is str: # it's a hexadecimal (coming from a pickled object) # assume that it is in standard form num = list(num) num[1] = int(num[1], 16) _mpf_ = tuple(num) else: if not num[1] and len(num) == 4: # handle normalization hack return Float._new(num, prec) else: _mpf_ = mpmath.mpf( S.NegativeOne**num[0]*num[1]*2**num[2])._mpf_ elif isinstance(num, Float): _mpf_ = num._mpf_ if prec < num._prec: _mpf_ = mpf_norm(_mpf_, prec) else: _mpf_ = mpmath.mpf(num)._mpf_ if not num: return C.Zero() obj = Expr.__new__(cls) obj._mpf_ = _mpf_ obj._prec = prec return obj @classmethod def _new(cls, _mpf_, _prec): if _mpf_ == mlib.fzero: return S.Zero # the new Float should be normalized unless it is # an integer because Float doesn't return Floats # for Integers. If Integers can become Floats then # all the following (up to the first 'obj =' line # can be replaced with ok = mpf_norm(_mpf_, _prec) sign, man, expt, bc = _mpf_ if not man: # hack for mpf_normalize which does not do this if not bc: ok = mlib.fzero else: ok = (sign % 2, int(man), expt, bc) elif expt < 0: # this is the non-hack normalization ok = mpf_normalize(sign, man, expt, bc, _prec, rnd) else: ok = _mpf_ obj = Expr.__new__(cls) obj._mpf_ = ok obj._prec = _prec return obj # mpz can't be pickled def __getnewargs__(self): return (mlib.to_pickable(self._mpf_),) def __getstate__(self): return {'_prec': self._prec} def _hashable_content(self): return (self._mpf_, self._prec) def floor(self): return C.Integer(int(mlib.to_int( mlib.mpf_floor(self._mpf_, self._prec)))) def ceiling(self): return C.Integer(int(mlib.to_int( mlib.mpf_ceil(self._mpf_, self._prec)))) @property def num(self): return mpmath.mpf(self._mpf_) def _as_mpf_val(self, prec): rv = mpf_norm(self._mpf_, prec) # uncomment to see failures #if rv != was._mpf_ and self._prec == prec: # print was._mpf_, rv return rv def _as_mpf_op(self, prec): return self._mpf_, max(prec, self._prec) def _eval_is_positive(self): return self.num > 0 def _eval_is_negative(self): return self.num < 0 def __neg__(self): return Float._new(mlib.mpf_neg(self._mpf_), self._prec) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number): rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec) return Number.__add__(self, other) @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number): rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_sub(self._mpf_, rhs, prec, rnd), prec) return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number): rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec) return Number.__mul__(self, other) @_sympifyit('other', NotImplemented) def __div__(self, other): if isinstance(other, Number): rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_div(self._mpf_, rhs, prec, rnd), prec) return Number.__div__(self, other) __truediv__ = __div__ @_sympifyit('other', NotImplemented) def __mod__(self, other): if isinstance(other, Number): rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_mod(self._mpf_, rhs, prec, rnd), prec) return Number.__mod__(self, other) @_sympifyit('other', NotImplemented) def __rmod__(self, other): if isinstance(other, Number): rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_mod(rhs, self._mpf_, prec, rnd), prec) return Number.__rmod__(self, other) def _eval_power(self, expt): """ expt is symbolic object but not equal to 0, 1 (-p)**r -> exp(r*log(-p)) -> exp(r*(log(p) + I*Pi)) -> -> p**r*(sin(Pi*r) + cos(Pi*r)*I) """ if isinstance(expt, Number): if isinstance(expt, Integer): prec = self._prec return Float._new( mlib.mpf_pow_int(self._mpf_, expt.p, prec, rnd), prec) expt, prec = expt._as_mpf_op(self._prec) self = self._mpf_ try: y = mpf_pow(self, expt, prec, rnd) return Float._new(y, prec) except mlib.ComplexResult: re, im = mlib.mpc_pow( (self, mlib.fzero), (expt, mlib.fzero), prec, rnd) return Float._new(re, prec) + \ Float._new(im, prec)*S.ImaginaryUnit def __abs__(self): return Float._new(mlib.mpf_abs(self._mpf_), self._prec) def __int__(self): return int(mlib.to_int(self._mpf_)) # uses round_fast = round_down def __eq__(self, other): if isinstance(other, float): # coerce to Float at same precision o = Float(other) try: ompf = o._as_mpf_val(self._prec) except ValueError: return False return bool(mlib.mpf_eq(self._mpf_, ompf)) try: other = _sympify(other) except SympifyError: return False # sympy != other --> not == if isinstance(other, NumberSymbol): if other.is_irrational: return False return other.__eq__(self) if isinstance(other, Float): # hack for the nan == nan case which, to mpf_eq is not equal # but to SymPy should be equal if other._mpf_ == self._mpf_ == fnan: return True return bool(mlib.mpf_eq(self._mpf_, other._mpf_)) if isinstance(other, Number): # numbers should compare at the same precision; # all _as_mpf_val routines should be sure to abide # by the request to change the prec if necessary; if # they don't, the equality test will fail since it compares # the mpf tuples ompf = other._as_mpf_val(self._prec) return bool(mlib.mpf_eq(self._mpf_, ompf)) return False # Float != non-Number def __ne__(self, other): return not self.__eq__(other) def __gt__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other if isinstance(other, NumberSymbol): return other.__le__(self) if other.is_comparable: other = other.evalf() if isinstance(other, Number): return bool(mlib.mpf_gt(self._mpf_, other._as_mpf_val(self._prec))) return Expr.__gt__(self, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> ! <= if isinstance(other, NumberSymbol): return other.__lt__(self) if other.is_comparable: other = other.evalf() if isinstance(other, Number): return bool(mlib.mpf_ge(self._mpf_, other._as_mpf_val(self._prec))) return Expr.__ge__(self, other) def __lt__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other if isinstance(other, NumberSymbol): return other.__ge__(self) if other.is_real and other.is_number: other = other.evalf() if isinstance(other, Number): return bool(mlib.mpf_lt(self._mpf_, other._as_mpf_val(self._prec))) return Expr.__lt__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> ! <= if isinstance(other, NumberSymbol): return other.__gt__(self) if other.is_real and other.is_number: other = other.evalf() if isinstance(other, Number): return bool(mlib.mpf_le(self._mpf_, other._as_mpf_val(self._prec))) return Expr.__le__(self, other) def __hash__(self): return super(Float, self).__hash__() def epsilon_eq(self, other, epsilon="10e-16"): return abs(self - other) < Float(epsilon) def _sage_(self): import sage.all as sage return sage.RealNumber(str(self)) # Add sympify converters
converter[float] = converter[decimal.Decimal] = Float # this is here to work nicely in Sage RealNumber = Float @deprecated(useinstead="Float", issue=1721, deprecated_since_version="0.7.0")
[docs]def Real(*args, **kwargs): # pragma: no cover """Deprecated alias for the Float constructor.""" return Float(*args, **kwargs)
[docs]class Rational(Number): """Represents integers and rational numbers (p/q) of any size. Examples ======== >>> from sympy import Rational >>> from sympy.abc import x, y >>> Rational(3) 3 >>> Rational(1,2) 1/2 >>> Rational(1.5) 1 Rational can also accept strings that are valid literals for reals: >>> Rational("1.23") 123/100 >>> Rational('1e-2') 1/100 >>> Rational(".1") 1/10 Parsing needs for any other type of string for which a Rational is desired can be handled with the rational=True option in sympify() which produces rationals from strings like '.[3]' (=1/3) and '3/10' (=3/10). **Low-level** Access numerator and denominator as .p and .q: >>> r = Rational(3,4) >>> r 3/4 >>> r.p 3 >>> r.q 4 Note that p and q return integers (not sympy Integers) so some care is needed when using them in expressions: >>> r.p//r.q 0 """ is_real = True is_integer = False is_rational = True __slots__ = ['p', 'q'] is_Rational = True @cacheit def __new__(cls, p, q=None): if q is None: if isinstance(p, Rational): return p if isinstance(p, str): p = p.replace(' ', '') try: # we might have a Float neg_pow, digits, expt = decimal.Decimal(p).as_tuple() p = [1, -1][neg_pow]*int("".join(str(x) for x in digits)) if expt > 0: # TODO: this branch needs a test return Rational(p*Pow(10, expt), 1) return Rational(p, Pow(10, -expt)) except decimal.InvalidOperation: f = regex.match('^([-+]?[0-9]+)/([0-9]+)$', p) if f: n, d = f.groups() return Rational(int(n), int(d)) raise ValueError('invalid literal: %s' % p) elif not isinstance(p, Basic): return Rational(S(p)) q = S.One if isinstance(q, Rational): p *= q.q q = q.p if isinstance(p, Rational): q *= p.q p = p.p p = int(p) q = int(q) if q == 0: if p == 0: if _errdict["divide"]: raise ValueError("Indeterminate 0/0") else: return S.NaN if p < 0: return S.NegativeInfinity return S.Infinity if q < 0: q = -q p = -p n = igcd(abs(p), q) if n > 1: p //= n q //= n if q == 1: return Integer(p) if p == 1 and q == 2: return S.Half obj = Expr.__new__(cls) obj.p = p obj.q = q #obj._args = (p, q) return obj
[docs] def limit_denominator(self, max_denominator=1000000): """Closest Rational to self with denominator at most max_denominator. >>> from sympy import Rational >>> Rational('3.141592653589793').limit_denominator(10) 22/7 >>> Rational('3.141592653589793').limit_denominator(100) 311/99 """ # Algorithm notes: For any real number x, define a *best upper # approximation* to x to be a rational number p/q such that: # # (1) p/q >= x, and # (2) if p/q > r/s >= x then s > q, for any rational r/s. # # Define *best lower approximation* similarly. Then it can be # proved that a rational number is a best upper or lower # approximation to x if, and only if, it is a convergent or # semiconvergent of the (unique shortest) continued fraction # associated to x. # # To find a best rational approximation with denominator <= M, # we find the best upper and lower approximations with # denominator <= M and take whichever of these is closer to x. # In the event of a tie, the bound with smaller denominator is # chosen. If both denominators are equal (which can happen # only when max_denominator == 1 and self is midway between # two integers) the lower bound---i.e., the floor of self, is # taken. if max_denominator < 1: raise ValueError("max_denominator should be at least 1") if self.q <= max_denominator: return self p0, q0, p1, q1 = 0, 1, 1, 0 n, d = self.p, self.q while True: a = n//d q2 = q0+a*q1 if q2 > max_denominator: break p0, q0, p1, q1 = p1, q1, p0+a*p1, q2 n, d = d, n-a*d k = (max_denominator-q0)//q1 bound1 = Rational(p0+k*p1, q0+k*q1) bound2 = Rational(p1, q1) if abs(bound2 - self) <= abs(bound1-self): return bound2 else: return bound1
def __getnewargs__(self): return (self.p, self.q) def _hashable_content(self): return (self.p, self.q) def _eval_is_positive(self): return self.p > 0 def _eval_is_zero(self): return self.p == 0 def __neg__(self): return Rational(-self.p, self.q) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Rational): return Rational(self.p*other.q + self.q*other.p, self.q*other.q) elif isinstance(other, Float): return other + self else: return Number.__add__(self, other) @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Rational): return Rational(self.p*other.q - self.q*other.p, self.q*other.q) elif isinstance(other, Float): return -other + self else: return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Rational): return Rational(self.p*other.p, self.q*other.q) elif isinstance(other, Float): return other*self else: return Number.__mul__(self, other) @_sympifyit('other', NotImplemented) def __div__(self, other): if isinstance(other, Rational): return Rational(self.p*other.q, self.q*other.p) elif isinstance(other, Float): return self*(1/other) else: return Number.__div__(self, other) __truediv__ = __div__ @_sympifyit('other', NotImplemented) def __mod__(self, other): if isinstance(other, Rational): n = (self.p*other.q) // (other.p*self.q) return Rational(self.p*other.q - n*other.p*self.q, self.q*other.q) if isinstance(other, Float): return self.evalf() % other return Number.__mod__(self, other) @_sympifyit('other', NotImplemented) def __rmod__(self, other): if isinstance(other, Rational): return Rational.__mod__(other, self) if isinstance(other, Float): return other % self.evalf() return Number.__rmod__(self, other) def _eval_power(self, expt): if isinstance(expt, Number): if isinstance(expt, Float): return self._eval_evalf(expt._prec)**expt if expt.is_negative: # (3/4)**-2 -> (4/3)**2 ne = -expt if (ne is S.One): return Rational(self.q, self.p) if self < 0: if expt.q != 1: return -(S.NegativeOne)**((expt.p % expt.q) / \ S(expt.q))*Rational(self.q, -self.p)**ne else: return S.NegativeOne**ne*Rational(self.q, -self.p)**ne else: return Rational(self.q, self.p)**ne if expt is S.Infinity: # -oo already caught by test for negative if self.p > self.q: # (3/2)**oo -> oo return S.Infinity if self.p < -self.q: # (-3/2)**oo -> oo + I*oo return S.Infinity + S.Infinity*S.ImaginaryUnit return S.Zero if isinstance(expt, Integer): # (4/3)**2 -> 4**2 / 3**2 return Rational(self.p**expt.p, self.q**expt.p) if isinstance(expt, Rational): if self.p != 1: # (4/3)**(5/6) -> 4**(5/6)*3**(-5/6) return Integer(self.p)**expt*Integer(self.q)**(-expt) # as the above caught negative self.p, now self is positive return Integer(self.q)**Rational(expt.p*(expt.q-1), expt.q) / \ Integer(self.q)**Integer(expt.p) if self.is_negative and expt.is_even: return (-self)**expt return def _as_mpf_val(self, prec): return mlib.from_rational(self.p, self.q, prec, rnd) def _mpmath_(self, prec, rnd): return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd)) def __abs__(self): return Rational(abs(self.p), self.q) def __int__(self): p, q = self.p, self.q if p < 0: return -(-p//q) return p//q def __eq__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy != other --> not == if isinstance(other, NumberSymbol): if other.is_irrational: return False return other.__eq__(self) if isinstance(other, Number): if isinstance(other, Float): return mlib.mpf_eq(self._as_mpf_val(other._prec), other._mpf_) elif isinstance(other, Rational): return self.p == other.p and self.q == other.q return False def __ne__(self, other): return not self.__eq__(other) def __gt__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> not < if isinstance(other, NumberSymbol): return other.__le__(self) if other.is_real and other.is_number and not isinstance(other, Rational): other = other.evalf() if isinstance(other, Number): if isinstance(other, Float): return bool(mlib.mpf_gt( self._as_mpf_val(other._prec), other._mpf_)) return bool(self.p*other.q > self.q*other.p) return Expr.__gt__(self, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> not <= if isinstance(other, NumberSymbol): return other.__lt__(self) if other.is_real and other.is_number and not isinstance(other, Rational): other = other.evalf() if isinstance(other, Number): if isinstance(other, Float): return bool(mlib.mpf_ge( self._as_mpf_val(other._prec), other._mpf_)) return bool(self.p*other.q >= self.q*other.p) return Expr.__ge__(self, other) def __lt__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> not < if isinstance(other, NumberSymbol): return other.__ge__(self) if other.is_real and other.is_number and not isinstance(other, Rational): other = other.evalf() if isinstance(other, Number): if isinstance(other, Float): return bool(mlib.mpf_lt( self._as_mpf_val(other._prec), other._mpf_)) return bool(self.p*other.q < self.q*other.p) return Expr.__lt__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> not <= if isinstance(other, NumberSymbol): return other.__gt__(self) if other.is_real and other.is_number and not isinstance(other, Rational): other = other.evalf() if isinstance(other, Number): if isinstance(other, Float): return bool(mlib.mpf_le( self._as_mpf_val(other._prec), other._mpf_)) return bool(self.p*other.q <= self.q*other.p) return Expr.__le__(self, other) def __hash__(self): return super(Rational, self).__hash__()
[docs] def factors(self, limit=None, use_trial=True, use_rho=False, use_pm1=False, verbose=False, visual=False): """A wrapper to factorint which return factors of self that are smaller than limit (or cheap to compute). Special methods of factoring are disabled by default so that only trial division is used. """ from sympy.ntheory import factorint f = factorint(self.p, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose).copy() f = defaultdict(int, f) for p, e in list(factorint(self.q, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose).items()): f[p] += -e if len(f) > 1 and 1 in f: del f[1] if not f: f = {1: 1} if not visual: return dict(f) else: if -1 in f: f.pop(-1) args = [S.NegativeOne] else: args = [] args.extend([Pow(*i, **{'evaluate':False}) for i in sorted(f.items())]) return Mul(*args, **{'evaluate':False})
[docs] def gcd(self, other): """Compute greatest common divisor of input arguments. """ if type(other) in (int, int): p = igcd(self.p, other) if self.is_Integer: return Integer(p) else: return Rational(p, self.q) else: other = _sympify(other) if other.is_Rational: p = igcd(self.p, other.p) if other.is_Integer: if self.is_Integer: return Integer(p) else: return Rational(p, self.q) else: if self.is_Integer: return Rational(p, other.q) else: return Rational(p, ilcm(self.q, other.q)) elif other.is_Number: return S.One else: raise TypeError("expected integer or rational, got %s" % other)
[docs] def lcm(self, other): """Compute least common multiple of input arguments. """ if type(other) in (int, int): return Integer(ilcm(self.p, other)) else: other = _sympify(other) if other.is_Rational: p = ilcm(self.p, other.p) if self.is_Integer or other.is_Integer: return Integer(p) else: return Rational(p, igcd(self.q, other.q)) elif other.is_Number: return self*other else: raise TypeError("expected integer or rational, got %s" % other)
[docs] def cofactors(self, other): """Compute GCD and cofactors of input arguments. """ other = _sympify(other) gcd = self.gcd(other) if gcd is S.One: return gcd, self, other else: return gcd, self/gcd, other/gcd
def as_numer_denom(self): return Integer(self.p), Integer(self.q) def _sage_(self): import sage.all as sage return sage.Integer(self.p)/sage.Integer(self.q)
[docs] def as_content_primitive(self, radical=False): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import S >>> (S(-3)/2).as_content_primitive() (3/2, -1) See docstring of Expr.as_content_primitive for more examples. """ if self: if self.is_positive: return self, S.One return -self, S.NegativeOne return S.One, self # int -> Integer
_intcache = {} # TODO move this tracing facility to sympy/core/trace.py ? def _intcache_printinfo(): ints = sorted(_intcache.keys()) nhit = _intcache_hits nmiss= _intcache_misses if nhit == 0 and nmiss == 0: print() print('Integer cache statistic was not collected') return miss_ratio = float(nmiss) / (nhit+nmiss) print() print('Integer cache statistic') print('-----------------------') print() print('#items: %i' % len(ints)) print() print(' #hit #miss #total') print() print('%5i %5i (%7.5f %%) %5i' % (nhit, nmiss, miss_ratio*100, nhit + nmiss)) print() print(ints) _intcache_hits = 0 _intcache_misses = 0 def int_trace(f): import os if os.getenv('SYMPY_TRACE_INT', 'no').lower() != 'yes': return f def Integer_tracer(cls, i): global _intcache_hits, _intcache_misses try: _intcache_hits += 1 return _intcache[i] except KeyError: _intcache_hits -= 1 _intcache_misses += 1 return f(cls, i) # also we want to hook our _intcache_printinfo into sys.atexit import atexit atexit.register(_intcache_printinfo) return Integer_tracer
[docs]class Integer(Rational): q = 1 is_integer = True is_Integer = True __slots__ = ['p'] def _as_mpf_val(self, prec): return mlib.from_int(self.p, prec) def _mpmath_(self, prec, rnd): return mpmath.make_mpf(self._as_mpf_val(prec)) # TODO caching with decorator, but not to degrade performance @int_trace def __new__(cls, i): if isinstance(i, str): i = i.replace(' ', '') # whereas we cannot, in general, make a Rational from an # arbitrary expression, we can make an Integer unambiguously # (except when a non-integer expression happens to round to # an integer). So we proceed by taking int() of the input and # let the int routines determine whether the expression can # be made into an int or whether an error should be raised. ival = int(i) try: return _intcache[ival] except KeyError: # We only work with well-behaved integer types. This converts, for # example, numpy.int32 instances. obj = Expr.__new__(cls) obj.p = ival _intcache[ival] = obj return obj def __getnewargs__(self): return (self.p,) # Arithmetic operations are here for efficiency def __int__(self): return self.p def __neg__(self): return Integer(-self.p) def __abs__(self): if self.p >= 0: return self else: return Integer(-self.p) def __divmod__(self, other): from .containers import Tuple if isinstance(other, Integer): return Tuple(*(divmod(self.p, other.p))) else: return Number.__divmod__(self, other) def __rdivmod__(self, other): from .containers import Tuple if isinstance(other, int): return Tuple(*(divmod(other, self.p))) else: try: other = Number(other) except TypeError: msg = "unsupported operand type(s) for divmod(): '%s' and '%s'" oname = type(other).__name__ sname = type(self).__name__ raise TypeError(msg % (oname, sname)) return Number.__divmod__(other, self) # TODO make it decorator + bytecodehacks? def __add__(self, other): if isinstance(other, int): return Integer(self.p + other) elif isinstance(other, Integer): return Integer(self.p + other.p) return Rational.__add__(self, other) def __radd__(self, other): if isinstance(other, int): return Integer(other + self.p) return Rational.__add__(self, other) def __sub__(self, other): if isinstance(other, int): return Integer(self.p - other) elif isinstance(other, Integer): return Integer(self.p - other.p) return Rational.__sub__(self, other) def __rsub__(self, other): if isinstance(other, int): return Integer(other - self.p) return Rational.__rsub__(self, other) def __mul__(self, other): if isinstance(other, int): return Integer(self.p*other) elif isinstance(other, Integer): return Integer(self.p*other.p) return Rational.__mul__(self, other) def __rmul__(self, other): if isinstance(other, int): return Integer(other*self.p) return Rational.__mul__(self, other) def __mod__(self, other): if isinstance(other, int): return Integer(self.p % other) elif isinstance(other, Integer): return Integer(self.p % other.p) return Rational.__mod__(self, other) def __rmod__(self, other): if isinstance(other, int): return Integer(other % self.p) elif isinstance(other, Integer): return Integer(other.p % self.p) return Rational.__rmod__(self, other) def __eq__(self, other): if isinstance(other, int): return (self.p == other) elif isinstance(other, Integer): return (self.p == other.p) return Rational.__eq__(self, other) def __ne__(self, other): return not self.__eq__(other) def __gt__(self, other): if isinstance(other, int): return (self.p > other) elif isinstance(other, Integer): return (self.p > other.p) return Rational.__gt__(self, other) def __lt__(self, other): if isinstance(other, int): return (self.p < other) elif isinstance(other, Integer): return (self.p < other.p) return Rational.__lt__(self, other) def __ge__(self, other): if isinstance(other, int): return (self.p >= other) elif isinstance(other, Integer): return (self.p >= other.p) return Rational.__ge__(self, other) def __le__(self, other): if isinstance(other, int): return (self.p <= other) elif isinstance(other, Integer): return (self.p <= other.p) return Rational.__le__(self, other) def __hash__(self): return super(Integer, self).__hash__() def __index__(self): return self.p ######################################## def _eval_is_odd(self): return bool(self.p % 2) def _eval_power(self, expt): """ Tries to do some simplifications on self**expt Returns None if no further simplifications can be done When exponent is a fraction (so we have for example a square root), we try to find a simpler representation by factoring the argument up to factors of 2**15, e.g. - sqrt(4) becomes 2 - sqrt(-4) becomes 2*I - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7) Further simplification would require a special call to factorint on the argument which is not done here for sake of speed. """ from sympy import perfect_power if expt is S.Infinity: if self.p > S.One: return S.Infinity # cases -1, 0, 1 are done in their respective classes return S.Infinity + S.ImaginaryUnit*S.Infinity if expt is S.NegativeInfinity: return Rational(1, self)**S.Infinity if not isinstance(expt, Number): # simplify when expt is even # (-2)**k --> 2**k if self.is_negative and expt.is_even: return (-self)**expt if not isinstance(expt, Rational): return if expt is S.Half and self < 0: # we extract I for this special case since everyone is doing so return S.ImaginaryUnit*Pow(-self, expt) if expt < 0: # invert base and change sign on exponent ne = -expt if self < 0: if expt.q != 1: return -(S.NegativeOne)**((expt.p % expt.q) / \ S(expt.q))*Rational(1, -self)**ne else: return (S.NegativeOne)**ne*Rational(1, -self)**ne else: return Rational(1, self.p)**ne # see if base is a perfect root, sqrt(4) --> 2 x, xexact = integer_nthroot(abs(self.p), expt.q) if xexact: # if it's a perfect root we've finished result = Integer(x**abs(expt.p)) if self < 0: result*= (-1)**expt return result # The following is an algorithm where we collect perfect roots # from the factors of base. # if it's not an nth root, it still might be a perfect power b_pos = int(abs(self.p)) p = perfect_power(b_pos) if p is not False: dict = {p[0]: p[1]} else: dict = Integer(self).factors(limit=2**15) # now process the dict of factors if self.is_negative: dict[-1] = 1 out_int = 1 # integer part out_rad = 1 # extracted radicals sqr_int = 1 sqr_gcd = 0 sqr_dict = {} for prime, exponent in list(dict.items()): exponent *= expt.p # remove multiples of expt.q: (2**12)**(1/10) -> 2*(2**2)**(1/10) div_e, div_m = divmod(exponent, expt.q) if div_e > 0: out_int *= prime**div_e if div_m > 0: # see if the reduced exponent shares a gcd with e.q # (2**2)**(1/10) -> 2**(1/5) g = igcd(div_m, expt.q) if g != 1: out_rad *= Pow(prime, Rational(div_m//g, expt.q//g)) else: sqr_dict[prime] = div_m # identify gcd of remaining powers for p, ex in sqr_dict.items(): if sqr_gcd == 0: sqr_gcd = ex else: sqr_gcd = igcd(sqr_gcd, ex) if sqr_gcd == 1: break for k, v in sqr_dict.items(): sqr_int *= k**(v//sqr_gcd) if sqr_int == self and out_int == 1 and out_rad == 1: result = None else: result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q)) return result def _eval_is_prime(self): from sympy.ntheory import isprime return isprime(self) def as_numer_denom(self): return self, S.One def __floordiv__(self, other): return Integer(self.p // Integer(other).p) def __rfloordiv__(self, other): return Integer(Integer(other).p // self.p)
[docs] def factorial(self): """Compute factorial of `self`. """ from sympy.functions.combinatorial.factorials import factorial return Integer(factorial(int(self)))
[docs] def isqrt(self): """Compute integer square root of `self`. """ return Integer(mlib.isqrt(int(self)))
[docs] def half_gcdex(self, other): """Half Extended Euclidean Algorithm. """ s, _, h = self.gcdex(other) return s, h
[docs] def gcdex(self, other): """Extended Euclidean Algorithm. """ if isinstance(other, int): return tuple(map(Integer, igcdex(int(self), other))) b = _sympify(other) if b.is_Integer: return tuple(map(Integer, igcdex(int(self), int(b)))) else: raise ValueError("expected an integer, got %s" % b)
[docs] def invert(self, other): """Invert `self` modulo `other`, if possible. """ if isinstance(other, int): a, b = int(self), other else: b = _sympify(other) if b.is_Integer: a, b = int(self), int(b) else: raise ValueError("expected an integer, got %s" % b) s, _, h = igcdex(a, b) if h == 1: return Integer(s % b) else: raise ZeroDivisionError("zero divisor") # Add sympify converters
converter[int] = converter[int] = Integer class RationalConstant(Rational): """ Abstract base class for rationals with specific behaviors Derived classes must define class attributes p and q and should probably all be singletons. """ __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) class IntegerConstant(Integer): __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) class Zero(IntegerConstant, metaclass=Singleton): p = 0 q = 1 is_positive = False is_negative = False is_finite = False is_zero = True is_composite = False __slots__ = [] @staticmethod def __abs__(): return S.Zero @staticmethod def __neg__(): return S.Zero @_sympifyit('other', NotImplemented) def __mul__(self, other): if other is S.NaN or \ other is S.NegativeInfinity or \ other is S.Infinity or \ other is S.ComplexInfinity: return S.NaN return S.Zero def _eval_power(self, expt): if expt.is_positive: return self if expt.is_negative: return S.Infinity if expt.is_real is False: return S.NaN # infinities are already handled with pos and neg # tests above; now throw away leading numbers on Mul # exponent coeff, terms = expt.as_coeff_Mul() if coeff.is_negative: return S.Infinity**terms if coeff is not S.One: # there is a Number to discard return self**terms def _eval_order(self, *symbols): # Order(0,x) -> 0 return self def __bool__(self): return False class One(IntegerConstant, metaclass=Singleton): p = 1 q = 1 __slots__ = [] @staticmethod def __abs__(): return S.One @staticmethod def __neg__(): return S.NegativeOne def _eval_power(self, expt): return self def _eval_order(self, *symbols): return @staticmethod def factors(limit=None, use_trial=True, use_rho=False, use_pm1=False, verbose=False, visual=False): if visual: return S.One return {1: 1} class NegativeOne(IntegerConstant, metaclass=Singleton): p = -1 q = 1 __slots__ = [] @staticmethod def __abs__(): return S.One @staticmethod def __neg__(): return S.One def _eval_power(self, expt): if expt.is_odd: return S.NegativeOne if expt.is_even: return S.One if isinstance(expt, Number): if isinstance(expt, Float): return Float(-1.0)**expt if expt is S.NaN: return S.NaN if expt is S.Infinity or expt is S.NegativeInfinity: return S.NaN if expt is S.Half: return S.ImaginaryUnit if isinstance(expt, Rational): if expt.q == 2: return S.ImaginaryUnit**Integer(expt.p) i, r = divmod(expt.p, expt.q) if i: return self**i*self**Rational(r, expt.q) return class Half(RationalConstant, metaclass=Singleton): p = 1 q = 2 __slots__ = [] @staticmethod def __abs__(): return S.Half class Infinity(Number, metaclass=Singleton): is_commutative = True is_positive = True is_bounded = False is_finite = False is_infinitesimal = False is_integer = None is_rational = None is_odd = None __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number): if other is S.NegativeInfinity or other is S.NaN: return S.NaN elif other.is_Float: if other == Float('-inf') or other._mpf_ == fnan: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') else: return Float('inf') else: return S.Infinity return NotImplemented __radd__ = __add__ @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number): if other is S.Infinity or other is S.NaN: return S.NaN elif other.is_Float: if other == Float('inf') or other._mpf_ == fnan: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') else: return Float('inf') else: return S.Infinity return NotImplemented @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number): if other is S.Zero or other is S.NaN: return S.NaN elif other.is_Float: if other._mpf_ == fnan or other == 0: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') if other > 0: return Float('inf') else: return Float('-inf') else: if other > 0: return S.Infinity else: return S.NegativeInfinity return NotImplemented __rmul__ = __mul__ @_sympifyit('other', NotImplemented) def __div__(self, other): if isinstance(other, Number): if other is S.Infinity or \ other is S.NegativeInfinity or \ other is S.NaN: return S.NaN elif other.is_Float: if other == Float('-inf') or \ other == Float('inf') or \ other._mpf_ == fnan: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') elif other >= 0: return Float('inf') else: return Float('-inf') else: if other >= 0: return S.Infinity else: return S.NegativeInfinity return NotImplemented __truediv__ = __div__ def __abs__(self): return S.Infinity def __neg__(self): return S.NegativeInfinity def _eval_power(self, expt): """ ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``oo ** nan`` ``nan`` ``oo ** -p`` ``0`` ``p`` is number, ``oo`` ================ ======= ============================== """ if expt.is_positive: return S.Infinity if expt.is_negative: return S.Zero if expt is S.NaN: return S.NaN if expt.is_number: return self**expt.evalf() def _as_mpf_val(self, prec): return mlib.finf def _sage_(self): import sage.all as sage return sage.oo def __hash__(self): return super(Infinity, self).__hash__() def __eq__(self, other): return other is S.Infinity def __ne__(self, other): return other is not S.Infinity def __lt__(self, other): return False def __le__(self, other): return other is S.Infinity def __gt__(self, other): return other is not S.Infinity def __ge__(self, other): return True def __mod__(self, other): return S.NaN __rmod__ = __mod__ oo = S.Infinity class NegativeInfinity(Number, metaclass=Singleton): is_commutative = True is_real = True is_positive = False is_bounded = False is_finite = False is_infinitesimal = False is_integer = None is_rational = None __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number): if other is S.Infinity or other is S.NaN: return S.NaN elif other.is_Float: if other == Float('inf') or other._mpf_ == fnan: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') else: return Float('-inf') else: return S.NegativeInfinity return NotImplemented __radd__ = __add__ @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number): if other is S.NegativeInfinity or other is S.NaN: return S.NaN elif other.is_Float: if other == Float('-inf') or other._mpf_ == fnan: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') else: return Float('-inf') else: return S.NegativeInfinity return NotImplemented @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number): if other is S.Zero or other is S.NaN: return S.NaN elif other.is_Float: if other._mpf_ == fnan or other == 0: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') elif other > 0: return Float('-inf') else: return Float('inf') else: if other > 0: return S.NegativeInfinity else: return S.Infinity return NotImplemented __rmul__ = __mul__ @_sympifyit('other', NotImplemented) def __div__(self, other): if isinstance(other, Number): if other is S.Infinity or \ other is S.NegativeInfinity or \ other is S.NaN: return S.NaN elif other.is_Float: if other == Float('-inf') or \ other == Float('inf') or \ other == Float('nan') or \ other._mpf_ == fnan: #Used workaround because Float('nan') == Float('nan') return False return Float('nan') elif other >= 0: return Float('-inf') else: return Float('inf') else: if other >= 0: return S.NegativeInfinity else: return S.Infinity return NotImplemented __truediv__ = __div__ def __abs__(self): return S.Infinity def __neg__(self): return S.Infinity def _eval_power(self, expt): """ ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``(-oo) ** nan`` ``nan`` ``(-oo) ** oo`` ``nan`` ``(-oo) ** -oo`` ``nan`` ``(-oo) ** e`` ``oo`` ``e`` is positive even integer ``(-oo) ** o`` ``-oo`` ``o`` is positive odd integer ================ ======= ============================== """ if isinstance(expt, Number): if expt is S.NaN or \ expt is S.Infinity or \ expt is S.NegativeInfinity: return S.NaN if isinstance(expt, Integer) and expt.is_positive: if expt.is_odd: return S.NegativeInfinity else: return S.Infinity return S.NegativeOne**expt*S.Infinity**expt def _as_mpf_val(self, prec): return mlib.fninf def _sage_(self): import sage.all as sage return -(sage.oo) def __hash__(self): return super(NegativeInfinity, self).__hash__() def __eq__(self, other): return other is S.NegativeInfinity def __ne__(self, other): return other is not S.NegativeInfinity def __lt__(self, other): return other is not S.NegativeInfinity def __le__(self, other): return True def __gt__(self, other): return False def __ge__(self, other): return other is S.NegativeInfinity class NaN(Number, metaclass=Singleton): """ Not a Number. This represents the corresponding data type to floating point nan, which is defined in the IEEE 754 floating point standard, and corresponds to the Python ``float('nan')``. NaN serves as a place holder for numeric values that are indeterminate, but not infinite. Most operations on nan, produce another nan. Most indeterminate forms, such as ``0/0`` or ``oo - oo` produce nan. Three exceptions are ``0**0``, ``1**oo``, and ``oo**0``, which all produce ``1`` (this is consistent with Python's float). NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported as ``nan``. Examples ======== >>> from sympy import nan, S, oo >>> nan is S.NaN True >>> oo - oo nan >>> nan + 1 nan References ========== - http://en.wikipedia.org/wiki/NaN """ is_commutative = True is_real = None is_rational = None is_integer = None is_comparable = False is_finite = None is_bounded = None is_zero = None is_prime = None is_positive = None is_negative = None __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) @_sympifyit('other', NotImplemented) def __add__(self, other): return self @_sympifyit('other', NotImplemented) def __sub__(self, other): return self @_sympifyit('other', NotImplemented) def __mul__(self, other): return self @_sympifyit('other', NotImplemented) def __div__(self, other): return self __truediv__ = __div__ def _as_mpf_val(self, prec): return mlib.fnan def _sage_(self): import sage.all as sage return sage.NaN def __hash__(self): return super(NaN, self).__hash__() def __eq__(self, other): return other is S.NaN def __ne__(self, other): return other is not S.NaN def __gt__(self, other): return False def __ge__(self, other): return False def __lt__(self, other): return False def __le__(self, other): return False nan = S.NaN class ComplexInfinity(AtomicExpr, metaclass=Singleton): is_commutative = True is_bounded = False is_real = None is_number = False __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) @staticmethod def __abs__(): return S.Infinity @staticmethod def __neg__(): return S.ComplexInfinity def _eval_power(self, expt): if expt is S.ComplexInfinity: return S.NaN if isinstance(expt, Number): if expt is S.Zero: return S.NaN else: if expt.is_positive: return S.ComplexInfinity else: return S.Zero zoo = S.ComplexInfinity
[docs]class NumberSymbol(AtomicExpr, metaclass=Singleton): is_commutative = True is_bounded = True is_finite = True is_number = True __slots__ = [] is_NumberSymbol = True def __new__(cls): return AtomicExpr.__new__(cls)
[docs] def approximation(self, number_cls): """ Return an interval with number_cls endpoints that contains the value of NumberSymbol. If not implemented, then return None. """
def _eval_evalf(self, prec): return Float._new(self._as_mpf_val(prec), prec) def __eq__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy != other --> not == if self is other: return True if isinstance(other, Number) and self.is_irrational: return False return False # NumberSymbol != non-(Number|self) def __ne__(self, other): return not self.__eq__(other) def __lt__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> not < if self is other: return False if isinstance(other, Number): approx = self.approximation_interval(other.__class__) if approx is not None: l,u = approx if other < l: return False if other > u: return True return self.evalf() < other if other.is_real and other.is_number: other = other.evalf() return self.evalf() < other return Expr.__lt__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: return False # sympy > other --> not <= if self is other: return True if other.is_real and other.is_number: other = other.evalf() if isinstance(other, Number): return self.evalf() <= other return Expr.__le__(self, other) def __gt__(self, other): return (-self) < (-other) def __ge__(self, other): return (-self) <= (-other) def __int__(self): # subclass with appropriate return value raise NotImplementedError def __hash__(self): return super(NumberSymbol, self).__hash__()
class Exp1(NumberSymbol, metaclass=Singleton): is_real = True is_positive = True is_negative = False # XXX Forces is_negative/is_nonnegative is_irrational = True __slots__ = [] @staticmethod def __abs__(): return S.Exp1 def __int__(self): return 2 def _as_mpf_val(self, prec): return mpf_e(prec) def approximation_interval(self, number_cls): if issubclass(number_cls,Integer): return (Integer(2),Integer(3)) elif issubclass(number_cls,Rational): pass def _eval_power(self, expt): return C.exp(expt) def _sage_(self): import sage.all as sage return sage.e E = S.Exp1 class Pi(NumberSymbol, metaclass=Singleton): is_real = True is_positive = True is_negative = False is_irrational = True __slots__ = [] @staticmethod def __abs__(): return S.Pi def __int__(self): return 3 def _as_mpf_val(self, prec): return mpf_pi(prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (Integer(3), Integer(4)) elif issubclass(number_cls, Rational): return (Rational(223,71), Rational(22,7)) def _sage_(self): import sage.all as sage return sage.pi pi = S.Pi class GoldenRatio(NumberSymbol, metaclass=Singleton): is_real = True is_positive = True is_negative = False is_irrational = True __slots__ = [] def __int__(self): return 1 def _as_mpf_val(self, prec): # XXX track down why this has to be increased rv = mlib.from_man_exp(phi_fixed(prec+10), -prec-10) return mpf_norm(rv, prec) def _eval_expand_func(self, **hints): from sympy import sqrt return S.Half + S.Half*sqrt(5) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.One, Rational(2)) elif issubclass(number_cls, Rational): pass def _sage_(self): import sage.all as sage return sage.golden_ratio class EulerGamma(NumberSymbol, metaclass=Singleton): is_real = True is_positive = True is_negative = False is_irrational = None __slots__ = [] def __int__(self): return 0 def _as_mpf_val(self, prec): # XXX track down why this has to be increased v = mlib.libhyper.euler_fixed(prec+10) rv = mlib.from_man_exp(v, -prec-10) return mpf_norm(rv, prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.Zero, S.One) elif issubclass(number_cls, Rational): return (S.Half, Rational(3, 5)) def _sage_(self): import sage.all as sage return sage.euler_gamma class Catalan(NumberSymbol, metaclass=Singleton): is_real = True is_positive = True is_negative = False is_irrational = None __slots__ = [] def __int__(self): return 0 def _as_mpf_val(self, prec): # XXX track down why this has to be increased v = mlib.catalan_fixed(prec+10) rv = mlib.from_man_exp(v, -prec-10) return mpf_norm(rv, prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.Zero, S.One) elif issubclass(number_cls, Rational): return (Rational(9, 10), S.One) def _sage_(self): import sage.all as sage return sage.catalan class ImaginaryUnit(AtomicExpr, metaclass=Singleton): is_commutative = True is_imaginary = True is_bounded = True is_finite = True is_number = True __slots__ = [] @staticmethod def __abs__(): return S.One def _eval_evalf(self, prec): return self def _eval_conjugate(self): return -S.ImaginaryUnit def _eval_power(self, expt): """ b is I = sqrt(-1) e is symbolic object but not equal to 0, 1 I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal I**0 mod 4 -> 1 I**1 mod 4 -> I I**2 mod 4 -> -1 I**3 mod 4 -> -I """ if isinstance(expt, Number): if isinstance(expt, Integer): expt = expt.p % 4 if expt == 0: return S.One if expt == 1: return S.ImaginaryUnit if expt == 2: return -S.One return -S.ImaginaryUnit return (S.NegativeOne)**(expt*S.Half) return def as_base_exp(self): return S.NegativeOne, S.Half def _sage_(self): import sage.all as sage return sage.I I = S.ImaginaryUnit try: # fractions is only available for python 2.6+ import fractions def sympify_fractions(f): return Rational(f.numerator, f.denominator) converter[fractions.Fraction] = sympify_fractions except ImportError: pass try: import gmpy def sympify_mpz(x): return Integer(int(x)) def sympify_mpq(x): return Rational(int(x.numer()), int(x.denom())) converter[type(gmpy.mpz(1))] = sympify_mpz converter[type(gmpy.mpq(1, 2))] = sympify_mpq except ImportError: pass def sympify_mpmath(x): return Expr._from_mpmath(x, x.context.prec) converter[mpnumeric] = sympify_mpmath def sympify_complex(a): real, imag = list(map(sympify, (a.real, a.imag))) return real + S.ImaginaryUnit*imag converter[complex] = sympify_complex _intcache[0] = S.Zero _intcache[1] = S.One _intcache[-1]= S.NegativeOne from .power import Pow, integer_nthroot from .mul import Mul Mul.identity = One() from .add import Add Add.identity = Zero()