Source code for sympy.core.numbers

from __future__ import print_function, division

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

from .containers import Tuple
from .sympify import converter, sympify, _sympify, SympifyError, _convert_numpy_types
from .singleton import S, Singleton
from .expr import Expr, AtomicExpr
from .decorators import _sympifyit
from .cache import cacheit, clear_cache
from .logic import fuzzy_not
from sympy.core.compatibility import (
    as_int, integer_types, long, string_types, with_metaclass, HAS_GMPY,
    SYMPY_INTS, int_info)
import mpmath
import mpmath.libmp as mlib
from mpmath.libmp.backend import MPZ
from mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed
from mpmath.ctx_mp import mpnumeric
from mpmath.libmp.libmpf import (
    finf as _mpf_inf, fninf as _mpf_ninf,
    fnan as _mpf_nan, fzero as _mpf_zero, _normalize as mpf_normalize,
    prec_to_dps)
from sympy.utilities.misc import debug, filldedent
from .evaluate import global_evaluate

from sympy.utilities.exceptions import SymPyDeprecationWarning

rnd = mlib.round_nearest

_LOG2 = math.log(2)


def comp(z1, z2, tol=None):
    """Return a bool indicating whether the error between z1 and z2 is <= tol.

    If ``tol`` is None then True will be returned if there is a significant
    difference between the numbers: ``abs(z1 - z2)*10**p <= 1/2`` where ``p``
    is the lower of the precisions of the values. A comparison of strings will
    be made if ``z1`` is a Number and a) ``z2`` is a string or b) ``tol`` is ''
    and ``z2`` is a Number.

    When ``tol`` is a nonzero value, if z2 is non-zero and ``|z1| > 1``
    the error is normalized by ``|z1|``, so if you want to see if the
    absolute error between ``z1`` and ``z2`` is <= ``tol`` then call this
    as ``comp(z1 - z2, 0, tol)``.
    """
    if type(z2) is str:
        if not isinstance(z1, Number):
            raise ValueError('when z2 is a str z1 must be a Number')
        return str(z1) == z2
    if not z1:
        z1, z2 = z2, z1
    if not z1:
        return True
    if not tol:
        if tol is None:
            if type(z2) is str and getattr(z1, 'is_Number', False):
                return str(z1) == z2
            a, b = Float(z1), Float(z2)
            return int(abs(a - b)*10**prec_to_dps(
                min(a._prec, b._prec)))*2 <= 1
        elif all(getattr(i, 'is_Number', False) for i in (z1, z2)):
            return z1._prec == z2._prec and str(z1) == str(z2)
        raise ValueError('exact comparison requires two Numbers')
    diff = abs(z1 - z2)
    az1 = abs(z1)
    if z2 and az1 > 1:
        return diff/az1 <= tol
    else:
        return diff <= tol


def mpf_norm(mpf, prec):
    """Return the mpf tuple normalized appropriately for the indicated
    precision after doing a check to see if zero should be returned or
    not when the mantissa is 0. ``mpf_normlize`` always assumes that this
    is zero, but it may not be since the mantissa for mpf's values "+inf",
    "-inf" and "nan" have a mantissa of zero, too.

    Note: this is not intended to validate a given mpf tuple, so sending
    mpf tuples that were not created by mpmath may produce bad results. This
    is only a wrapper to ``mpf_normalize`` which provides the check for non-
    zero mpfs that have a 0 for the mantissa.
    """
    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
        # (see issue 6639)
        if not bc:
            return _mpf_zero
        else:
            # don't change anything; this should already
            # be a well formed mpf tuple
            return mpf

    # Necessary if mpmath is using the gmpy backend
    from mpmath.libmp.backend import MPZ
    rv = mpf_normalize(sign, MPZ(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 _as_integer_ratio(p): neg_pow, man, expt, bc = getattr(p, '_mpf_', mpmath.mpf(p)._mpf_) p = [1, -1][neg_pow % 2]*man if expt < 0: q = 2**-expt else: q = 1 p *= 2**expt return int(p), int(q) def _decimal_to_Rational_prec(dec): """Convert an ordinary decimal instance to a Rational.""" if not dec.is_finite(): raise TypeError("dec must be finite, got %s." % dec) s, d, e = dec.as_tuple() prec = len(d) if e >= 0: # it's an integer rv = Integer(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(*args): """Computes nonnegative integer greatest common divisor. The algorithm is based on the well known Euclid's algorithm. To improve speed, igcd() has its own caching mechanism implemented. Examples ======== >>> from sympy.core.numbers import igcd >>> igcd(2, 4) 2 >>> igcd(5, 10, 15) 5 """ if len(args) < 2: raise TypeError( 'igcd() takes at least 2 arguments (%s given)' % len(args)) if 1 in args: a = 1 k = 0 else: a = abs(as_int(args[0])) k = 1 if a != 1: while k < len(args): b = args[k] k += 1 try: a = _gcdcache[(a, b)] except KeyError: b = as_int(b) if not b: continue if b == 1: a = 1 break if b < 0: b = -b t = a, b a = igcd2(a, b) _gcdcache[t] = _gcdcache[t[1], t[0]] = a while k < len(args): ok = as_int(args[k]) k += 1 return a
try: from math import gcd as igcd2 except ImportError: def igcd2(a, b): """Compute gcd of two Python integers a and b.""" if (a.bit_length() > BIGBITS and b.bit_length() > BIGBITS): return igcd_lehmer(a, b) a, b = abs(a), abs(b) while b: a, b = b, a % b return a # Use Lehmer's algorithm only for very large numbers. # The limit could be different on Python 2.7 and 3.x. # If so, then this could be defined in compatibility.py. BIGBITS = 5000 def igcd_lehmer(a, b): """Computes greatest common divisor of two integers. Euclid's algorithm for the computation of the greatest common divisor gcd(a, b) of two (positive) integers a and b is based on the division identity a = q*b + r, where the quotient q and the remainder r are integers and 0 <= r < b. Then each common divisor of a and b divides r, and it follows that gcd(a, b) == gcd(b, r). The algorithm works by constructing the sequence r0, r1, r2, ..., where r0 = a, r1 = b, and each rn is the remainder from the division of the two preceding elements. In Python, q = a // b and r = a % b are obtained by the floor division and the remainder operations, respectively. These are the most expensive arithmetic operations, especially for large a and b. Lehmer's algorithm is based on the observation that the quotients qn = r(n-1) // rn are in general small integers even when a and b are very large. Hence the quotients can be usually determined from a relatively small number of most significant bits. The efficiency of the algorithm is further enhanced by not computing each long remainder in Euclid's sequence. The remainders are linear combinations of a and b with integer coefficients derived from the quotients. The coefficients can be computed as far as the quotients can be determined from the chosen most significant parts of a and b. Only then a new pair of consecutive remainders is computed and the algorithm starts anew with this pair. References ========== .. [1] https://en.wikipedia.org/wiki/Lehmer%27s_GCD_algorithm """ a, b = abs(as_int(a)), abs(as_int(b)) if a < b: a, b = b, a # The algorithm works by using one or two digit division # whenever possible. The outer loop will replace the # pair (a, b) with a pair of shorter consecutive elements # of the Euclidean gcd sequence until a and b # fit into two Python (long) int digits. nbits = 2*int_info.bits_per_digit while a.bit_length() > nbits and b != 0: # Quotients are mostly small integers that can # be determined from most significant bits. n = a.bit_length() - nbits x, y = int(a >> n), int(b >> n) # most significant bits # Elements of the Euclidean gcd sequence are linear # combinations of a and b with integer coefficients. # Compute the coefficients of consequtive pairs # a' = A*a + B*b, b' = C*a + D*b # using small integer arithmetic as far as possible. A, B, C, D = 1, 0, 0, 1 # initial values while True: # The coefficients alternate in sign while looping. # The inner loop combines two steps to keep track # of the signs. # At this point we have # A > 0, B <= 0, C <= 0, D > 0, # x' = x + B <= x < x" = x + A, # y' = y + C <= y < y" = y + D, # and # x'*N <= a' < x"*N, y'*N <= b' < y"*N, # where N = 2**n. # Now, if y' > 0, and x"//y' and x'//y" agree, # then their common value is equal to q = a'//b'. # In addition, # x'%y" = x' - q*y" < x" - q*y' = x"%y', # and # (x'%y")*N < a'%b' < (x"%y')*N. # On the other hand, we also have x//y == q, # and therefore # x'%y" = x + B - q*(y + D) = x%y + B', # x"%y' = x + A - q*(y + C) = x%y + A', # where # B' = B - q*D < 0, A' = A - q*C > 0. if y + C <= 0: break q = (x + A) // (y + C) # Now x'//y" <= q, and equality holds if # x' - q*y" = (x - q*y) + (B - q*D) >= 0. # This is a minor optimization to avoid division. x_qy, B_qD = x - q*y, B - q*D if x_qy + B_qD < 0: break # Next step in the Euclidean sequence. x, y = y, x_qy A, B, C, D = C, D, A - q*C, B_qD # At this point the signs of the coefficients # change and their roles are interchanged. # A <= 0, B > 0, C > 0, D < 0, # x' = x + A <= x < x" = x + B, # y' = y + D < y < y" = y + C. if y + D <= 0: break q = (x + B) // (y + D) x_qy, A_qC = x - q*y, A - q*C if x_qy + A_qC < 0: break x, y = y, x_qy A, B, C, D = C, D, A_qC, B - q*D # Now the conditions on top of the loop # are again satisfied. # A > 0, B < 0, C < 0, D > 0. if B == 0: # This can only happen when y == 0 in the beginning # and the inner loop does nothing. # Long division is forced. a, b = b, a % b continue # Compute new long arguments using the coefficients. a, b = A*a + B*b, C*a + D*b # Small divisors. Finish with the standard algorithm. while b: a, b = b, a % b return a
[docs]def ilcm(*args): """Computes integer least common multiple. Examples ======== >>> from sympy.core.numbers import ilcm >>> ilcm(5, 10) 10 >>> ilcm(7, 3) 21 >>> ilcm(5, 10, 15) 30 """ if len(args) < 2: raise TypeError( 'ilcm() takes at least 2 arguments (%s given)' % len(args)) if 0 in args: return 0 a = args[0] for b in args[1:]: a = a*b // igcd(a, b) return a
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) def mod_inverse(a, m): """ Return the number c such that, ( a * c ) % m == 1 where c has the same sign as a. If no such value exists, a ValueError is raised. Examples ======== >>> from sympy import S >>> from sympy.core.numbers import mod_inverse Suppose we wish to find multiplicative inverse x of 3 modulo 11. This is the same as finding x such that 3 * x = 1 (mod 11). One value of x that satisfies this congruence is 4. Because 3 * 4 = 12 and 12 = 1 mod(11). This is the value return by mod_inverse: >>> mod_inverse(3, 11) 4 >>> mod_inverse(-3, 11) -4 When there is a common factor between the numerators of ``a`` and ``m`` the inverse does not exist: >>> mod_inverse(2, 4) Traceback (most recent call last): ... ValueError: inverse of 2 mod 4 does not exist >>> mod_inverse(S(2)/7, S(5)/2) 7/2 References ========== - https://en.wikipedia.org/wiki/Modular_multiplicative_inverse - https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm """ c = None try: a, m = as_int(a), as_int(m) if m > 1: x, y, g = igcdex(a, m) if g == 1: c = x % m if a < 0: c -= m except ValueError: a, m = sympify(a), sympify(m) if not (a.is_number and m.is_number): raise TypeError(filldedent(''' Expected numbers for arguments; symbolic `mod_inverse` is not implemented but symbolic expressions can be handled with the similar function, sympy.polys.polytools.invert''')) big = (m > 1) if not (big is S.true or big is S.false): raise ValueError('m > 1 did not evaluate; try to simplify %s' % m) elif big: c = 1/a if c is None: raise ValueError('inverse of %s (mod %s) does not exist' % (a, m)) return c
[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_number = True is_Number = True __slots__ = [] # Used to make max(x._prec, y._prec) return x._prec when only x is a float _prec = -1 def __new__(cls, *obj): if len(obj) == 1: obj = obj[0] if isinstance(obj, Number): return obj if isinstance(obj, SYMPY_INTS): 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, string_types): val = sympify(obj) if isinstance(val, Number): return val else: raise ValueError('String "%s" does not denote a Number' % obj) msg = "expected str|int|long|float|Decimal|Number object but got %r" raise TypeError(msg % type(obj).__name__) def invert(self, other, *gens, **args): from sympy.polys.polytools import invert if getattr(other, 'is_number', True): return mod_inverse(self, other) return invert(self, other, *gens, **args) 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 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 floor(self): raise NotImplementedError('%s needs .floor() method' % (self.__class__.__name__)) def ceiling(self): raise NotImplementedError('%s needs .ceiling() method' % (self.__class__.__name__)) def _eval_conjugate(self): return self def _eval_order(self, *symbols): from sympy import Order # Order(5, x, y) -> Order(1,x,y) return Order(S.One, *symbols) def _eval_subs(self, old, new): if old == -self: return -new return self # there is no other possibility def _eval_is_finite(self): return True @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) and global_evaluate[0]: 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) and global_evaluate[0]: 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) and global_evaluate[0]: if other is S.NaN: return S.NaN elif other is S.Infinity: if self.is_zero: return S.NaN elif self.is_positive: return S.Infinity else: return S.NegativeInfinity elif other is S.NegativeInfinity: if self.is_zero: return S.NaN elif self.is_positive: return S.NegativeInfinity else: return S.Infinity elif isinstance(other, Tuple): return NotImplemented return AtomicExpr.__mul__(self, other) @_sympifyit('other', NotImplemented) def __div__(self, other): if isinstance(other, Number) and global_evaluate[0]: 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): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) raise NotImplementedError('%s needs .__lt__() method' % (self.__class__.__name__)) def __le__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) raise NotImplementedError('%s needs .__le__() method' % (self.__class__.__name__)) def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) return _sympify(other).__lt__(self) def __ge__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) return _sympify(other).__le__(self) def __hash__(self): return super(Number, self).__hash__() def is_constant(self, *wrt, **flags): return True def as_coeff_mul(self, *deps, **kwargs): # a -> c*t if self.is_Rational or not kwargs.pop('rational', True): 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 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) if self else (S.One, self)
[docs] def as_coeff_Add(self, rational=False): """Efficiently extract the coefficient of a summation. """ if not rational: return self, S.Zero return S.Zero, self
[docs] def gcd(self, other): """Compute GCD of `self` and `other`. """ from sympy.polys import gcd return gcd(self, other)
[docs] def lcm(self, other): """Compute LCM of `self` and `other`. """ from sympy.polys import lcm return lcm(self, other)
[docs] def cofactors(self, other): """Compute GCD and cofactors of `self` and `other`. """ from sympy.polys import cofactors return cofactors(self, other)
[docs]class Float(Number): """Represent a floating-point number of arbitrary precision. Examples ======== >>> from sympy import Float >>> Float(3.5) 3.50000000000000 >>> Float(3) 3.00000000000000 Creating Floats from strings (and Python ``int`` and ``long`` types) will give a minimum precision of 15 digits, but the precision will automatically increase to capture all digits entered. >>> Float(1) 1.00000000000000 >>> Float(10**20) 100000000000000000000. >>> Float('1e20') 100000000000000000000. However, *floating-point* numbers (Python ``float`` types) retain only 15 digits of precision: >>> Float(1e20) 1.00000000000000e+20 >>> Float(1.23456789123456789) 1.23456789123457 It may be preferable to enter high-precision decimal numbers as strings: Float('1.23456789123456789') 1.23456789123456789 The desired number of digits can also be specified: >>> Float('1e-3', 3) 0.00100 >>> Float(100, 4) 100.0 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 >>> Float(3, '') 3. If a number is written in scientific notation, only the digits before the exponent are considered significant if a decimal appears, otherwise the "e" signifies only how to move the decimal: >>> Float('60.e2', '') # 2 digits significant 6.0e+3 >>> Float('60e2', '') # 4 digits significant 6000. >>> Float('600e-2', '') # 3 digits significant 6.00 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(int(m), Pow(2, int(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: >>> _._mpf_ (1, 5, 0, 3) This is not needed for instantiation and is not the same thing as the precision. The mpf tuple and the precision are two separate quantities that Float tracks. """ __slots__ = ['_mpf_', '_prec'] # A Float represents many real numbers, # both rational and irrational. is_rational = None is_irrational = None is_number = True is_real = True is_Float = True def __new__(cls, num, dps=None, prec=None, precision=None): if prec is not None: SymPyDeprecationWarning( feature="Using 'prec=XX' to denote decimal precision", useinstead="'dps=XX' for decimal precision and 'precision=XX' "\ "for binary precision", issue=12820, deprecated_since_version="1.1").warn() dps = prec del prec # avoid using this deprecated kwarg if dps is not None and precision is not None: raise ValueError('Both decimal and binary precision supplied. ' 'Supply only one. ') if isinstance(num, string_types): 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 isinstance(num, float) and num == 0: num = '0' elif isinstance(num, (SYMPY_INTS, Integer)): num = str(num) # faster than mlib.from_int elif num is S.Infinity: num = '+inf' elif num is S.NegativeInfinity: num = '-inf' elif type(num).__module__ == 'numpy': # support for numpy datatypes num = _convert_numpy_types(num) elif isinstance(num, mpmath.mpf): if precision is None: if dps is None: precision = num.context.prec num = num._mpf_ if dps is None and precision is None: dps = 15 if isinstance(num, Float): return num if isinstance(num, string_types) and _literal_float(num): try: Num = decimal.Decimal(num) except decimal.InvalidOperation: pass else: isint = '.' not in num num, dps = _decimal_to_Rational_prec(Num) if num.is_Integer and isint: dps = max(dps, len(str(num).lstrip('-'))) dps = max(15, dps) precision = mlib.libmpf.dps_to_prec(dps) elif precision == '' and dps is None or precision is None and dps == '': if not isinstance(num, string_types): raise ValueError('The null string can only be used when ' 'the number to Float is passed as a string or an integer.') ok = None if _literal_float(num): try: Num = decimal.Decimal(num) except decimal.InvalidOperation: pass else: isint = '.' not in num num, dps = _decimal_to_Rational_prec(Num) if num.is_Integer and isint: dps = max(dps, len(str(num).lstrip('-'))) precision = mlib.libmpf.dps_to_prec(dps) ok = True if ok is None: raise ValueError('string-float not recognized: %s' % num) # decimal precision(dps) is set and maybe binary precision(precision) # as well.From here on binary precision is used to compute the Float. # Hence, if supplied use binary precision else translate from decimal # precision. if precision is None or precision == '': precision = mlib.libmpf.dps_to_prec(dps) precision = int(precision) if isinstance(num, float): _mpf_ = mlib.from_float(num, precision, rnd) elif isinstance(num, string_types): _mpf_ = mlib.from_str(num, precision, rnd) elif isinstance(num, decimal.Decimal): if num.is_finite(): _mpf_ = mlib.from_str(str(num), precision, rnd) elif num.is_nan(): _mpf_ = _mpf_nan elif num.is_infinite(): if num > 0: _mpf_ = _mpf_inf else: _mpf_ = _mpf_ninf else: raise ValueError("unexpected decimal value %s" % str(num)) 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) # If we're loading an object pickled in Python 2 into # Python 3, we may need to strip a tailing 'L' because # of a shim for int on Python 3, see issue #13470. if num[1].endswith('L'): num[1] = num[1][:-1] num[1] = MPZ(num[1], 16) _mpf_ = tuple(num) else: if len(num) == 4: # handle normalization hack return Float._new(num, precision) else: return (S.NegativeOne**num[0]*num[1]*S(2)**num[2]).evalf(precision) else: try: _mpf_ = num._as_mpf_val(precision) except (NotImplementedError, AttributeError): _mpf_ = mpmath.mpf(num, prec=precision)._mpf_ # special cases if _mpf_ == _mpf_zero: pass # we want a Float elif _mpf_ == _mpf_nan: return S.NaN obj = Expr.__new__(cls) obj._mpf_ = _mpf_ obj._prec = precision return obj @classmethod def _new(cls, _mpf_, _prec): # special cases if _mpf_ == _mpf_zero: return S.Zero # XXX this is different from Float which gives 0.0 elif _mpf_ == _mpf_nan: return S.NaN obj = Expr.__new__(cls) obj._mpf_ = mpf_norm(_mpf_, _prec) # XXX: Should this be obj._prec = obj._mpf_[3]? 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 Integer(int(mlib.to_int( mlib.mpf_floor(self._mpf_, self._prec)))) def ceiling(self): return 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) if rv != self._mpf_ and self._prec == prec: debug(self._mpf_, rv) return rv def _as_mpf_op(self, prec): return self._mpf_, max(prec, self._prec) def _eval_is_finite(self): if self._mpf_ in (_mpf_inf, _mpf_ninf): return False return True def _eval_is_infinite(self): if self._mpf_ in (_mpf_inf, _mpf_ninf): return True return False def _eval_is_integer(self): return self._mpf_ == _mpf_zero def _eval_is_negative(self): if self._mpf_ == _mpf_ninf: return True if self._mpf_ == _mpf_inf: return False return self.num < 0 def _eval_is_positive(self): if self._mpf_ == _mpf_inf: return True if self._mpf_ == _mpf_ninf: return False return self.num > 0 def _eval_is_zero(self): return self._mpf_ == _mpf_zero def __nonzero__(self): return self._mpf_ != _mpf_zero __bool__ = __nonzero__ def __neg__(self): return Float._new(mlib.mpf_neg(self._mpf_), self._prec) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number) and global_evaluate[0]: 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) and global_evaluate[0]: 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) and global_evaluate[0]: 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) and other != 0 and global_evaluate[0]: 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, Rational) and other.q != 1 and global_evaluate[0]: # calculate mod with Rationals, *then* round the result return Float(Rational.__mod__(Rational(self), other), precision=self._prec) if isinstance(other, Float) and global_evaluate[0]: r = self/other if r == int(r): return Float(0, precision=max(self._prec, other._prec)) if isinstance(other, Number) and global_evaluate[0]: 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, Float) and global_evaluate[0]: return other.__mod__(self) if isinstance(other, Number) and global_evaluate[0]: 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 self == 0: if expt.is_positive: return S.Zero if expt.is_negative: return Float('inf') 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) elif isinstance(expt, Rational) and \ expt.p == 1 and expt.q % 2 and self.is_negative: return Pow(S.NegativeOne, expt, evaluate=False)*( -self)._eval_power(expt) expt, prec = expt._as_mpf_op(self._prec) mpfself = self._mpf_ try: y = mpf_pow(mpfself, expt, prec, rnd) return Float._new(y, prec) except mlib.ComplexResult: re, im = mlib.mpc_pow( (mpfself, _mpf_zero), (expt, _mpf_zero), 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): if self._mpf_ == _mpf_zero: return 0 return int(mlib.to_int(self._mpf_)) # uses round_fast = round_down __long__ = __int__ 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 NotImplemented if other.is_NumberSymbol: if other.is_irrational: return False return other.__eq__(self) if other.is_Float: return bool(mlib.mpf_eq(self._mpf_, other._mpf_)) if other.is_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 == other def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) if other.is_NumberSymbol: return other.__lt__(self) if other.is_Rational and not other.is_Integer: self *= other.q other = _sympify(other.p) elif other.is_comparable: other = other.evalf() if other.is_Number and other is not S.NaN: return _sympify(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: raise TypeError("Invalid comparison %s >= %s" % (self, other)) if other.is_NumberSymbol: return other.__le__(self) if other.is_Rational and not other.is_Integer: self *= other.q other = _sympify(other.p) elif other.is_comparable: other = other.evalf() if other.is_Number and other is not S.NaN: return _sympify(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: raise TypeError("Invalid comparison %s < %s" % (self, other)) if other.is_NumberSymbol: return other.__gt__(self) if other.is_Rational and not other.is_Integer: self *= other.q other = _sympify(other.p) elif other.is_comparable: other = other.evalf() if other.is_Number and other is not S.NaN: return _sympify(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: raise TypeError("Invalid comparison %s <= %s" % (self, other)) if other.is_NumberSymbol: return other.__ge__(self) if other.is_Rational and not other.is_Integer: self *= other.q other = _sympify(other.p) elif other.is_comparable: other = other.evalf() if other.is_Number and other is not S.NaN: return _sympify(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="1e-15"): return abs(self - other) < Float(epsilon) def _sage_(self): import sage.all as sage return sage.RealNumber(str(self)) def __format__(self, format_spec): return format(decimal.Decimal(str(self)), format_spec)
# Add sympify converters converter[float] = converter[decimal.Decimal] = Float # this is here to work nicely in Sage RealNumber = Float
[docs]class Rational(Number): """Represents integers and rational numbers (p/q) of any size. Examples ======== >>> from sympy import Rational, nsimplify, S, pi >>> Rational(3) 3 >>> Rational(1, 2) 1/2 Rational is unprejudiced in accepting input. If a float is passed, the underlying value of the binary representation will be returned: >>> Rational(.5) 1/2 >>> Rational(.2) 3602879701896397/18014398509481984 If the simpler representation of the float is desired then consider limiting the denominator to the desired value or convert the float to a string (which is roughly equivalent to limiting the denominator to 10**12): >>> Rational(str(.2)) 1/5 >>> Rational(.2).limit_denominator(10**12) 1/5 An arbitrarily precise Rational is obtained when a string literal is passed: >>> Rational("1.23") 123/100 >>> Rational('1e-2') 1/100 >>> Rational(".1") 1/10 >>> Rational('1e-2/3.2') 1/320 The conversion of other types of strings can be handled by the sympify() function, and conversion of floats to expressions or simple fractions can be handled with nsimplify: >>> S('.[3]') # repeating digits in brackets 1/3 >>> S('3**2/10') # general expressions 9/10 >>> nsimplify(.3) # numbers that have a simple form 3/10 But if the input does not reduce to a literal Rational, an error will be raised: >>> Rational(pi) Traceback (most recent call last): ... TypeError: invalid input: pi 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 0.75 See Also ======== sympify, sympy.simplify.simplify.nsimplify """ is_real = True is_integer = False is_rational = True is_number = True __slots__ = ['p', 'q'] is_Rational = True @cacheit def __new__(cls, p, q=None, gcd=None): if q is None: if isinstance(p, Rational): return p if isinstance(p, string_types): if p.count('/') > 1: raise TypeError('invalid input: %s' % p) pq = p.rsplit('/', 1) if len(pq) == 2: p, q = pq fp = fractions.Fraction(p) fq = fractions.Fraction(q) f = fp/fq return Rational(f.numerator, f.denominator, 1) p = p.replace(' ', '') try: p = fractions.Fraction(p) except ValueError: pass # error will raise below if not isinstance(p, string_types): try: if isinstance(p, fractions.Fraction): return Rational(p.numerator, p.denominator, 1) except NameError: pass # error will raise below if isinstance(p, (float, Float)): return Rational(*_as_integer_ratio(p)) if not isinstance(p, SYMPY_INTS + (Rational,)): raise TypeError('invalid input: %s' % p) q = q or S.One gcd = 1 else: p = Rational(p) q = Rational(q) if isinstance(q, Rational): p *= q.q q = q.p if isinstance(p, Rational): q *= p.q p = p.p # p and q are now integers if q == 0: if p == 0: if _errdict["divide"]: raise ValueError("Indeterminate 0/0") else: return S.NaN return S.ComplexInfinity if q < 0: q = -q p = -p if not gcd: gcd = igcd(abs(p), q) if gcd > 1: p //= gcd q //= gcd 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 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 """ f = fractions.Fraction(self.p, self.q) return Rational(f.limit_denominator(fractions.Fraction(int(max_denominator))))
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 global_evaluate[0]: if isinstance(other, Integer): return Rational(self.p + self.q*other.p, self.q, 1) elif isinstance(other, Rational): #TODO: this can probably be optimized more 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) return Number.__add__(self, other) __radd__ = __add__ @_sympifyit('other', NotImplemented) def __sub__(self, other): if global_evaluate[0]: if isinstance(other, Integer): return Rational(self.p - self.q*other.p, self.q, 1) elif 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) return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __rsub__(self, other): if global_evaluate[0]: if isinstance(other, Integer): return Rational(self.q*other.p - self.p, self.q, 1) elif isinstance(other, Rational): return Rational(self.q*other.p - self.p*other.q, self.q*other.q) elif isinstance(other, Float): return -self + other else: return Number.__rsub__(self, other) return Number.__rsub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if global_evaluate[0]: if isinstance(other, Integer): return Rational(self.p*other.p, self.q, igcd(other.p, self.q)) elif isinstance(other, Rational): return Rational(self.p*other.p, self.q*other.q, igcd(self.p, other.q)*igcd(self.q, other.p)) elif isinstance(other, Float): return other*self else: return Number.__mul__(self, other) return Number.__mul__(self, other) __rmul__ = __mul__ @_sympifyit('other', NotImplemented) def __div__(self, other): if global_evaluate[0]: if isinstance(other, Integer): if self.p and other.p == S.Zero: return S.ComplexInfinity else: return Rational(self.p, self.q*other.p, igcd(self.p, other.p)) elif isinstance(other, Rational): return Rational(self.p*other.q, self.q*other.p, igcd(self.p, other.p)*igcd(self.q, other.q)) elif isinstance(other, Float): return self*(1/other) else: return Number.__div__(self, other) return Number.__div__(self, other) @_sympifyit('other', NotImplemented) def __rdiv__(self, other): if global_evaluate[0]: if isinstance(other, Integer): return Rational(other.p*self.q, self.p, igcd(self.p, other.p)) elif isinstance(other, Rational): return Rational(other.p*self.q, other.q*self.p, igcd(self.p, other.p)*igcd(self.q, other.q)) elif isinstance(other, Float): return other*(1/self) else: return Number.__rdiv__(self, other) return Number.__rdiv__(self, other) __truediv__ = __div__ @_sympifyit('other', NotImplemented) def __mod__(self, other): if global_evaluate[0]: 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): # calculate mod with Rationals, *then* round the answer return Float(self.__mod__(Rational(other)), precision=other._prec) return Number.__mod__(self, other) return Number.__mod__(self, other) @_sympifyit('other', NotImplemented) def __rmod__(self, other): if isinstance(other, Rational): return Rational.__mod__(other, self) 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.is_negative: 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, 1) 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 -int(-p//q) return int(p//q) __long__ = __int__ def floor(self): return Integer(self.p // self.q) def ceiling(self): return -Integer(-self.p // self.q) def __eq__(self, other): try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_NumberSymbol: if other.is_irrational: return False return other.__eq__(self) if other.is_Number: if other.is_Rational: # a Rational is always in reduced form so will never be 2/4 # so we can just check equivalence of args return self.p == other.p and self.q == other.q if other.is_Float: return mlib.mpf_eq(self._as_mpf_val(other._prec), other._mpf_) return False def __ne__(self, other): return not self == other def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) if other.is_NumberSymbol: return other.__lt__(self) expr = self if other.is_Number: if other.is_Rational: return _sympify(bool(self.p*other.q > self.q*other.p)) if other.is_Float: return _sympify(bool(mlib.mpf_gt( self._as_mpf_val(other._prec), other._mpf_))) elif other.is_number and other.is_real: expr, other = Integer(self.p), self.q*other return Expr.__gt__(expr, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) if other.is_NumberSymbol: return other.__le__(self) expr = self if other.is_Number: if other.is_Rational: return _sympify(bool(self.p*other.q >= self.q*other.p)) if other.is_Float: return _sympify(bool(mlib.mpf_ge( self._as_mpf_val(other._prec), other._mpf_))) elif other.is_number and other.is_real: expr, other = Integer(self.p), self.q*other return Expr.__ge__(expr, other) def __lt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) if other.is_NumberSymbol: return other.__gt__(self) expr = self if other.is_Number: if other.is_Rational: return _sympify(bool(self.p*other.q < self.q*other.p)) if other.is_Float: return _sympify(bool(mlib.mpf_lt( self._as_mpf_val(other._prec), other._mpf_))) elif other.is_number and other.is_real: expr, other = Integer(self.p), self.q*other return Expr.__lt__(expr, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) expr = self if other.is_NumberSymbol: return other.__ge__(self) elif other.is_Number: if other.is_Rational: return _sympify(bool(self.p*other.q <= self.q*other.p)) if other.is_Float: return _sympify(bool(mlib.mpf_le( self._as_mpf_val(other._prec), other._mpf_))) elif other.is_number and other.is_real: expr, other = Integer(self.p), self.q*other return Expr.__le__(expr, 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 factorrat return factorrat(self, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose).copy()
@_sympifyit('other', NotImplemented) def gcd(self, other): if isinstance(other, Rational): if other is S.Zero: return other return Rational( Integer(igcd(self.p, other.p)), Integer(ilcm(self.q, other.q))) return Number.gcd(self, other) @_sympifyit('other', NotImplemented) def lcm(self, other): if isinstance(other, Rational): return Rational( self.p*other.p//igcd(self.p, other.p), igcd(self.q, other.q)) return Number.lcm(self, other) 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, clear=True): """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
[docs] def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """ return self, S.One
[docs] def as_coeff_Add(self, rational=False): """Efficiently extract the coefficient of a summation. """ return self, S.Zero
# 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_number = True is_Integer = True __slots__ = ['p'] def _as_mpf_val(self, prec): return mlib.from_int(self.p, prec, rnd) 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, string_types): 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. try: ival = int(i) except TypeError: raise TypeError( 'Integer can only work with integer expressions.') 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 __long__ = __int__ def floor(self): return Integer(self.p) def ceiling(self): return Integer(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) and global_evaluate[0]: return Tuple(*(divmod(self.p, other.p))) else: return Number.__divmod__(self, other) def __rdivmod__(self, other): from .containers import Tuple if isinstance(other, integer_types) and global_evaluate[0]: 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 global_evaluate[0]: if isinstance(other, integer_types): return Integer(self.p + other) elif isinstance(other, Integer): return Integer(self.p + other.p) elif isinstance(other, Rational): return Rational(self.p*other.q + other.p, other.q, 1) return Rational.__add__(self, other) else: return Add(self, other) def __radd__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(other + self.p) elif isinstance(other, Rational): return Rational(other.p + self.p*other.q, other.q, 1) return Rational.__radd__(self, other) return Rational.__radd__(self, other) def __sub__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(self.p - other) elif isinstance(other, Integer): return Integer(self.p - other.p) elif isinstance(other, Rational): return Rational(self.p*other.q - other.p, other.q, 1) return Rational.__sub__(self, other) return Rational.__sub__(self, other) def __rsub__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(other - self.p) elif isinstance(other, Rational): return Rational(other.p - self.p*other.q, other.q, 1) return Rational.__rsub__(self, other) return Rational.__rsub__(self, other) def __mul__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(self.p*other) elif isinstance(other, Integer): return Integer(self.p*other.p) elif isinstance(other, Rational): return Rational(self.p*other.p, other.q, igcd(self.p, other.q)) return Rational.__mul__(self, other) return Rational.__mul__(self, other) def __rmul__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(other*self.p) elif isinstance(other, Rational): return Rational(other.p*self.p, other.q, igcd(self.p, other.q)) return Rational.__rmul__(self, other) return Rational.__rmul__(self, other) def __mod__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(self.p % other) elif isinstance(other, Integer): return Integer(self.p % other.p) return Rational.__mod__(self, other) return Rational.__mod__(self, other) def __rmod__(self, other): if global_evaluate[0]: if isinstance(other, integer_types): return Integer(other % self.p) elif isinstance(other, Integer): return Integer(other.p % self.p) return Rational.__rmod__(self, other) return Rational.__rmod__(self, other) def __eq__(self, other): if isinstance(other, integer_types): 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 == other def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) if other.is_Integer: return _sympify(self.p > other.p) return Rational.__gt__(self, other) def __lt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) if other.is_Integer: return _sympify(self.p < other.p) return Rational.__lt__(self, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) if other.is_Integer: return _sympify(self.p >= other.p) return Rational.__ge__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) if other.is_Integer: return _sympify(self.p <= other.p) return Rational.__le__(self, other) def __hash__(self): return hash(self.p) 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 isinstance(expt, Float): # Rational knows how to exponentiate by a Float return super(Integer, self)._eval_power(expt) if not isinstance(expt, Rational): return if expt is S.Half and self.is_negative: # we extract I for this special case since everyone is doing so return S.ImaginaryUnit*Pow(-self, expt) if expt.is_negative: # invert base and change sign on exponent ne = -expt if self.is_negative: 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.is_negative: result *= S.NegativeOne**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 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 _eval_is_composite(self): if self > 1: return fuzzy_not(self.is_prime) else: return False 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)
# Add sympify converters for i_type in integer_types: converter[i_type] = Integer
[docs]class AlgebraicNumber(Expr): """Class for representing algebraic numbers in SymPy. """ __slots__ = ['rep', 'root', 'alias', 'minpoly'] is_AlgebraicNumber = True is_algebraic = True is_number = True def __new__(cls, expr, coeffs=None, alias=None, **args): """Construct a new algebraic number. """ from sympy import Poly from sympy.polys.polyclasses import ANP, DMP from sympy.polys.numberfields import minimal_polynomial from sympy.core.symbol import Symbol expr = sympify(expr) if isinstance(expr, (tuple, Tuple)): minpoly, root = expr if not minpoly.is_Poly: minpoly = Poly(minpoly) elif expr.is_AlgebraicNumber: minpoly, root = expr.minpoly, expr.root else: minpoly, root = minimal_polynomial( expr, args.get('gen'), polys=True), expr dom = minpoly.get_domain() if coeffs is not None: if not isinstance(coeffs, ANP): rep = DMP.from_sympy_list(sympify(coeffs), 0, dom) scoeffs = Tuple(*coeffs) else: rep = DMP.from_list(coeffs.to_list(), 0, dom) scoeffs = Tuple(*coeffs.to_list()) if rep.degree() >= minpoly.degree(): rep = rep.rem(minpoly.rep) else: rep = DMP.from_list([1, 0], 0, dom) scoeffs = Tuple(1, 0) if root.is_negative: rep = -rep scoeffs = Tuple(-1, 0) sargs = (root, scoeffs) if alias is not None: if not isinstance(alias, Symbol): alias = Symbol(alias) sargs = sargs + (alias,) obj = Expr.__new__(cls, *sargs) obj.rep = rep obj.root = root obj.alias = alias obj.minpoly = minpoly return obj def __hash__(self): return super(AlgebraicNumber, self).__hash__() def _eval_evalf(self, prec): return self.as_expr()._evalf(prec) @property def is_aliased(self): """Returns ``True`` if ``alias`` was set. """ return self.alias is not None
[docs] def as_poly(self, x=None): """Create a Poly instance from ``self``. """ from sympy import Dummy, Poly, PurePoly if x is not None: return Poly.new(self.rep, x) else: if self.alias is not None: return Poly.new(self.rep, self.alias) else: return PurePoly.new(self.rep, Dummy('x'))
[docs] def as_expr(self, x=None): """Create a Basic expression from ``self``. """ return self.as_poly(x or self.root).as_expr().expand()
[docs] def coeffs(self): """Returns all SymPy coefficients of an algebraic number. """ return [ self.rep.dom.to_sympy(c) for c in self.rep.all_coeffs() ]
[docs] def native_coeffs(self): """Returns all native coefficients of an algebraic number. """ return self.rep.all_coeffs()
[docs] def to_algebraic_integer(self): """Convert ``self`` to an algebraic integer. """ from sympy import Poly f = self.minpoly if f.LC() == 1: return self coeff = f.LC()**(f.degree() - 1) poly = f.compose(Poly(f.gen/f.LC())) minpoly = poly*coeff root = f.LC()*self.root return AlgebraicNumber((minpoly, root), self.coeffs())
def _eval_simplify(self, ratio, measure): from sympy.polys import CRootOf, minpoly for r in [r for r in self.minpoly.all_roots() if r.func != CRootOf]: if minpoly(self.root - r).is_Symbol: # use the matching root if it's simpler if measure(r) < ratio*measure(self.root): return AlgebraicNumber(r) return self
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)
[docs]class Zero(with_metaclass(Singleton, IntegerConstant)): """The number zero. Zero is a singleton, and can be accessed by ``S.Zero`` Examples ======== >>> from sympy import S, Integer, zoo >>> Integer(0) is S.Zero True >>> 1/S.Zero zoo References ========== .. [1] http://en.wikipedia.org/wiki/Zero """ p = 0 q = 1 is_positive = False is_negative = False is_zero = True is_number = True __slots__ = [] @staticmethod def __abs__(): return S.Zero @staticmethod def __neg__(): return S.Zero def _eval_power(self, expt): if expt.is_positive: return self if expt.is_negative: return S.ComplexInfinity 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.ComplexInfinity**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 __nonzero__(self): return False __bool__ = __nonzero__
[docs] def as_coeff_Mul(self, rational=False): # XXX this routine should be deleted """Efficiently extract the coefficient of a summation. """ return S.One, self
[docs]class One(with_metaclass(Singleton, IntegerConstant)): """The number one. One is a singleton, and can be accessed by ``S.One``. Examples ======== >>> from sympy import S, Integer >>> Integer(1) is S.One True References ========== .. [1] http://en.wikipedia.org/wiki/1_%28number%29 """ is_number = True 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 else: return {}
[docs]class NegativeOne(with_metaclass(Singleton, IntegerConstant)): """The number negative one. NegativeOne is a singleton, and can be accessed by ``S.NegativeOne``. Examples ======== >>> from sympy import S, Integer >>> Integer(-1) is S.NegativeOne True See Also ======== One References ========== .. [1] http://en.wikipedia.org/wiki/%E2%88%921_%28number%29 """ is_number = True 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
[docs]class Half(with_metaclass(Singleton, RationalConstant)): """The rational number 1/2. Half is a singleton, and can be accessed by ``S.Half``. Examples ======== >>> from sympy import S, Rational >>> Rational(1, 2) is S.Half True References ========== .. [1] http://en.wikipedia.org/wiki/One_half """ is_number = True p = 1 q = 2 __slots__ = [] @staticmethod def __abs__(): return S.Half
[docs]class Infinity(with_metaclass(Singleton, Number)): r"""Positive infinite quantity. In real analysis the symbol `\infty` denotes an unbounded limit: `x\to\infty` means that `x` grows without bound. Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled `+\infty` and `-\infty` can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. Infinity is a singleton, and can be accessed by ``S.Infinity``, or can be imported as ``oo``. Examples ======== >>> from sympy import oo, exp, limit, Symbol >>> 1 + oo oo >>> 42/oo 0 >>> x = Symbol('x') >>> limit(exp(x), x, oo) oo See Also ======== NegativeInfinity, NaN References ========== .. [1] http://en.wikipedia.org/wiki/Infinity """ is_commutative = True is_positive = True is_infinite = True is_number = True is_prime = False __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"\infty" def _eval_subs(self, old, new): if self == old: return new @_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'): return S.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'): return S.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 == 0: return S.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'): return S.NaN elif other.is_nonnegative: 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`` ================ ======= ============================== See Also ======== Pow NaN NegativeInfinity """ from sympy.functions import re 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 S.ComplexInfinity: return S.NaN if expt.is_real is False and expt.is_number: expt_real = re(expt) if expt_real.is_positive: return S.ComplexInfinity if expt_real.is_negative: return S.Zero if expt_real.is_zero: return S.NaN 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): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) if other.is_real: return S.false return Expr.__lt__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) if other.is_real: if other.is_finite or other is S.NegativeInfinity: return S.false elif other.is_nonpositive: return S.false elif other.is_infinite and other.is_positive: return S.true return Expr.__le__(self, other) def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) if other.is_real: if other.is_finite or other is S.NegativeInfinity: return S.true elif other.is_nonpositive: return S.true elif other.is_infinite and other.is_positive: return S.false return Expr.__gt__(self, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) if other.is_real: return S.true return Expr.__ge__(self, other) def __mod__(self, other): return S.NaN __rmod__ = __mod__ def floor(self): return self def ceiling(self): return self
oo = S.Infinity
[docs]class NegativeInfinity(with_metaclass(Singleton, Number)): """Negative infinite quantity. NegativeInfinity is a singleton, and can be accessed by ``S.NegativeInfinity``. See Also ======== Infinity """ is_commutative = True is_negative = True is_infinite = True is_number = True __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"-\infty" def _eval_subs(self, old, new): if self == old: return new @_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'): 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'): 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 is S.NaN or other.is_zero: return S.NaN elif other.is_positive: return Float('-inf') else: return Float('inf') else: if other.is_positive: 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 is S.NaN: return S.NaN elif other.is_nonnegative: 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 ================ ======= ============================== See Also ======== Infinity Pow NaN """ if expt.is_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): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) if other.is_real: if other.is_finite or other is S.Infinity: return S.true elif other.is_nonnegative: return S.true elif other.is_infinite and other.is_negative: return S.false return Expr.__lt__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) if other.is_real: return S.true return Expr.__le__(self, other) def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) if other.is_real: return S.false return Expr.__gt__(self, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) if other.is_real: if other.is_finite or other is S.Infinity: return S.false elif other.is_nonnegative: return S.false elif other.is_infinite and other.is_negative: return S.true return Expr.__ge__(self, other) def __mod__(self, other): return S.NaN __rmod__ = __mod__ def floor(self): return self def ceiling(self): return self
[docs]class NaN(with_metaclass(Singleton, Number)): """ Not a Number. This serves as a place holder for numeric values that are indeterminate. Most operations on NaN, produce another NaN. Most indeterminate forms, such as ``0/0`` or ``oo - oo` produce NaN. Two exceptions are ``0**0`` and ``oo**0``, which all produce ``1`` (this is consistent with Python's float). NaN is loosely related to floating point nan, which is defined in the IEEE 754 floating point standard, and corresponds to the Python ``float('nan')``. Differences are noted below. NaN is mathematically not equal to anything else, even NaN itself. This explains the initially counter-intuitive results with ``Eq`` and ``==`` in the examples below. NaN is not comparable so inequalities raise a TypeError. This is in constrast with floating point nan where all inequalities are false. NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported as ``nan``. Examples ======== >>> from sympy import nan, S, oo, Eq >>> nan is S.NaN True >>> oo - oo nan >>> nan + 1 nan >>> Eq(nan, nan) # mathematical equality False >>> nan == nan # structural equality True References ========== .. [1] http://en.wikipedia.org/wiki/NaN """ is_commutative = True is_real = None is_rational = None is_algebraic = None is_transcendental = None is_integer = None is_comparable = False is_finite = None is_zero = None is_prime = None is_positive = None is_negative = None is_number = True __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"\mathrm{NaN}" @_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 floor(self): return self def ceiling(self): return self def _as_mpf_val(self, prec): return _mpf_nan def _sage_(self): import sage.all as sage return sage.NaN def __hash__(self): return super(NaN, self).__hash__() def __eq__(self, other): # NaN is structurally equal to another NaN return other is S.NaN def __ne__(self, other): return other is not S.NaN def _eval_Eq(self, other): # NaN is not mathematically equal to anything, even NaN return S.false # Expr will _sympify and raise TypeError __gt__ = Expr.__gt__ __ge__ = Expr.__ge__ __lt__ = Expr.__lt__ __le__ = Expr.__le__
nan = S.NaN
[docs]class ComplexInfinity(with_metaclass(Singleton, AtomicExpr)): r"""Complex infinity. In complex analysis the symbol `\tilde\infty`, called "complex infinity", represents a quantity with infinite magnitude, but undetermined complex phase. ComplexInfinity is a singleton, and can be accessed by ``S.ComplexInfinity``, or can be imported as ``zoo``. Examples ======== >>> from sympy import zoo, oo >>> zoo + 42 zoo >>> 42/zoo 0 >>> zoo + zoo nan >>> zoo*zoo zoo See Also ======== Infinity """ is_commutative = True is_infinite = True is_number = True is_prime = False is_complex = True is_real = False __slots__ = [] def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"\tilde{\infty}" @staticmethod def __abs__(): return S.Infinity def floor(self): return self def ceiling(self): return self @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 def _sage_(self): import sage.all as sage return sage.UnsignedInfinityRing.gen()
zoo = S.ComplexInfinity
[docs]class NumberSymbol(AtomicExpr): is_commutative = 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 NotImplemented if self is other: return True if other.is_Number and self.is_irrational: return False return False # NumberSymbol != non-(Number|self) def __ne__(self, other): return not self == other def __le__(self, other): if self is other: return S.true return Expr.__le__(self, other) def __ge__(self, other): if self is other: return S.true return Expr.__ge__(self, other) def __int__(self): # subclass with appropriate return value raise NotImplementedError def __long__(self): return self.__int__() def __hash__(self): return super(NumberSymbol, self).__hash__()
[docs]class Exp1(with_metaclass(Singleton, NumberSymbol)): r"""The `e` constant. The transcendental number `e = 2.718281828\ldots` is the base of the natural logarithm and of the exponential function, `e = \exp(1)`. Sometimes called Euler's number or Napier's constant. Exp1 is a singleton, and can be accessed by ``S.Exp1``, or can be imported as ``E``. Examples ======== >>> from sympy import exp, log, E >>> E is exp(1) True >>> log(E) 1 References ========== .. [1] http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 """ is_real = True is_positive = True is_negative = False # XXX Forces is_negative/is_nonnegative is_irrational = True is_number = True is_algebraic = False is_transcendental = True __slots__ = [] def _latex(self, printer): return r"e" @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): from sympy import exp return exp(expt) def _eval_rewrite_as_sin(self): from sympy import sin I = S.ImaginaryUnit return sin(I + S.Pi/2) - I*sin(I) def _eval_rewrite_as_cos(self): from sympy import cos I = S.ImaginaryUnit return cos(I) + I*cos(I + S.Pi/2) def _sage_(self): import sage.all as sage return sage.e
E = S.Exp1
[docs]class Pi(with_metaclass(Singleton, NumberSymbol)): r"""The `\pi` constant. The transcendental number `\pi = 3.141592654\ldots` represents the ratio of a circle's circumference to its diameter, the area of the unit circle, the half-period of trigonometric functions, and many other things in mathematics. Pi is a singleton, and can be accessed by ``S.Pi``, or can be imported as ``pi``. Examples ======== >>> from sympy import S, pi, oo, sin, exp, integrate, Symbol >>> S.Pi pi >>> pi > 3 True >>> pi.is_irrational True >>> x = Symbol('x') >>> sin(x + 2*pi) sin(x) >>> integrate(exp(-x**2), (x, -oo, oo)) sqrt(pi) References ========== .. [1] http://en.wikipedia.org/wiki/Pi """ is_real = True is_positive = True is_negative = False is_irrational = True is_number = True is_algebraic = False is_transcendental = True __slots__ = [] def _latex(self, printer): return r"\pi" @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
[docs]class GoldenRatio(with_metaclass(Singleton, NumberSymbol)): r"""The golden ratio, `\phi`. `\phi = \frac{1 + \sqrt{5}}{2}` is algebraic number. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities, i.e. their maximum. GoldenRatio is a singleton, and can be accessed by ``S.GoldenRatio``. Examples ======== >>> from sympy import S >>> S.GoldenRatio > 1 True >>> S.GoldenRatio.expand(func=True) 1/2 + sqrt(5)/2 >>> S.GoldenRatio.is_irrational True References ========== .. [1] http://en.wikipedia.org/wiki/Golden_ratio """ is_real = True is_positive = True is_negative = False is_irrational = True is_number = True is_algebraic = True is_transcendental = False __slots__ = [] def _latex(self, printer): return r"\phi" 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 _eval_rewrite_as_sqrt = _eval_expand_func
[docs]class EulerGamma(with_metaclass(Singleton, NumberSymbol)): r"""The Euler-Mascheroni constant. `\gamma = 0.5772157\ldots` (also called Euler's constant) is a mathematical constant recurring in analysis and number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm: .. math:: \gamma = \lim\limits_{n\to\infty} \left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right) EulerGamma is a singleton, and can be accessed by ``S.EulerGamma``. Examples ======== >>> from sympy import S >>> S.EulerGamma.is_irrational >>> S.EulerGamma > 0 True >>> S.EulerGamma > 1 False References ========== .. [1] http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant """ is_real = True is_positive = True is_negative = False is_irrational = None is_number = True __slots__ = [] def _latex(self, printer): return r"\gamma" 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
[docs]class Catalan(with_metaclass(Singleton, NumberSymbol)): r"""Catalan's constant. `K = 0.91596559\ldots` is given by the infinite series .. math:: K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2} Catalan is a singleton, and can be accessed by ``S.Catalan``. Examples ======== >>> from sympy import S >>> S.Catalan.is_irrational >>> S.Catalan > 0 True >>> S.Catalan > 1 False References ========== .. [1] http://en.wikipedia.org/wiki/Catalan%27s_constant """ is_real = True is_positive = True is_negative = False is_irrational = None is_number = True __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
[docs]class ImaginaryUnit(with_metaclass(Singleton, AtomicExpr)): r"""The imaginary unit, `i = \sqrt{-1}`. I is a singleton, and can be accessed by ``S.I``, or can be imported as ``I``. Examples ======== >>> from sympy import I, sqrt >>> sqrt(-1) I >>> I*I -1 >>> 1/I -I References ========== .. [1] http://en.wikipedia.org/wiki/Imaginary_unit """ is_commutative = True is_imaginary = True is_finite = True is_number = True is_algebraic = True is_transcendental = False __slots__ = [] def _latex(self, printer): return r"i" @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 @property def _mpc_(self): return (Float(0)._mpf_, Float(1)._mpf_)
I = S.ImaginaryUnit def sympify_fractions(f): return Rational(f.numerator, f.denominator) converter[fractions.Fraction] = sympify_fractions try: if HAS_GMPY == 2: import gmpy2 as gmpy elif HAS_GMPY == 1: import gmpy else: raise ImportError def sympify_mpz(x): return Integer(long(x)) def sympify_mpq(x): return Rational(long(x.numerator), long(x.denominator)) 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()