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ai-content-maker/.venv/Lib/site-packages/sympy/core/numbers.py

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first commit
2024-05-03 04:18:51 +03:00
from __future__ import annotations
import numbers
import decimal
import fractions
import math
import re as regex
import sys
from functools import lru_cache
from .containers import Tuple
from .sympify import (SympifyError, _sympy_converter, sympify, _convert_numpy_types,
_sympify, _is_numpy_instance)
from .singleton import S, Singleton
from .basic import Basic
from .expr import Expr, AtomicExpr
from .evalf import pure_complex
from .cache import cacheit, clear_cache
from .decorators import _sympifyit
from .logic import fuzzy_not
from .kind import NumberKind
from sympy.external.gmpy import SYMPY_INTS, HAS_GMPY, gmpy
from sympy.multipledispatch import dispatch
import mpmath
import mpmath.libmp as mlib
from mpmath.libmp import bitcount, round_nearest as rnd
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, _normalize as mpf_normalize,
prec_to_dps, dps_to_prec)
from sympy.utilities.misc import as_int, debug, filldedent
from .parameters import global_parameters
_LOG2 = math.log(2)
def comp(z1, z2, tol=None):
r"""Return a bool indicating whether the error between z1 and z2
is $\le$ ``tol``.
Examples
========
If ``tol`` is ``None`` then ``True`` will be returned if
:math:`|z1 - z2|\times 10^p \le 5` where $p$ is minimum value of the
decimal precision of each value.
>>> from sympy import comp, pi
>>> pi4 = pi.n(4); pi4
3.142
>>> comp(_, 3.142)
True
>>> comp(pi4, 3.141)
False
>>> comp(pi4, 3.143)
False
A comparison of strings will be made
if ``z1`` is a Number and ``z2`` is a string or ``tol`` is ''.
>>> comp(pi4, 3.1415)
True
>>> comp(pi4, 3.1415, '')
False
When ``tol`` is provided and $z2$ is non-zero and
:math:`|z1| > 1` the error is normalized by :math:`|z1|`:
>>> abs(pi4 - 3.14)/pi4
0.000509791731426756
>>> comp(pi4, 3.14, .001) # difference less than 0.1%
True
>>> comp(pi4, 3.14, .0005) # difference less than 0.1%
False
When :math:`|z1| \le 1` the absolute error is used:
>>> 1/pi4
0.3183
>>> abs(1/pi4 - 0.3183)/(1/pi4)
3.07371499106316e-5
>>> abs(1/pi4 - 0.3183)
9.78393554684764e-6
>>> comp(1/pi4, 0.3183, 1e-5)
True
To see if the absolute error between ``z1`` and ``z2`` is less
than or equal to ``tol``, call this as ``comp(z1 - z2, 0, tol)``
or ``comp(z1 - z2, tol=tol)``:
>>> abs(pi4 - 3.14)
0.00160156249999988
>>> comp(pi4 - 3.14, 0, .002)
True
>>> comp(pi4 - 3.14, 0, .001)
False
"""
if isinstance(z2, str):
if not pure_complex(z1, or_real=True):
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:
a, b = z1, z2
if tol == '':
return str(a) == str(b)
if tol is None:
a, b = sympify(a), sympify(b)
if not all(i.is_number for i in (a, b)):
raise ValueError('expecting 2 numbers')
fa = a.atoms(Float)
fb = b.atoms(Float)
if not fa and not fb:
# no floats -- compare exactly
return a == b
# get a to be pure_complex
for _ in range(2):
ca = pure_complex(a, or_real=True)
if not ca:
if fa:
a = a.n(prec_to_dps(min([i._prec for i in fa])))
ca = pure_complex(a, or_real=True)
break
else:
fa, fb = fb, fa
a, b = b, a
cb = pure_complex(b)
if not cb and fb:
b = b.n(prec_to_dps(min([i._prec for i in fb])))
cb = pure_complex(b, or_real=True)
if ca and cb and (ca[1] or cb[1]):
return all(comp(i, j) for i, j in zip(ca, cb))
tol = 10**prec_to_dps(min(a._prec, getattr(b, '_prec', a._prec)))
return int(abs(a - b)*tol) <= 5
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 fzero
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}
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, _ = 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
_floatpat = regex.compile(r"[-+]?((\d*\.\d+)|(\d+\.?))")
def _literal_float(f):
"""Return True if n starts like a floating point number."""
return bool(_floatpat.match(f))
# (a,b) -> gcd(a,b)
# TODO caching with decorator, but not to degrade performance
@lru_cache(1024)
def igcd(*args):
"""Computes nonnegative integer greatest common divisor.
Explanation
===========
The algorithm is based on the well known Euclid's algorithm [1]_. To
improve speed, ``igcd()`` has its own caching mechanism.
Examples
========
>>> from sympy import igcd
>>> igcd(2, 4)
2
>>> igcd(5, 10, 15)
5
References
==========
.. [1] https://en.wikipedia.org/wiki/Euclidean_algorithm
"""
if len(args) < 2:
raise TypeError(
'igcd() takes at least 2 arguments (%s given)' % len(args))
args_temp = [abs(as_int(i)) for i in args]
if 1 in args_temp:
return 1
a = args_temp.pop()
if HAS_GMPY: # Using gmpy if present to speed up.
for b in args_temp:
a = gmpy.gcd(a, b) if b else a
return as_int(a)
for b in args_temp:
a = math.gcd(a, b)
return a
igcd2 = math.gcd
def igcd_lehmer(a, b):
r"""Computes greatest common divisor of two integers.
Explanation
===========
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
$$ a = q \times b + r$$,
where the quotient $q$ and the remainder $r$ are integers
and $0 \le r < b$. 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 [1]_ 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*sys.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 consecutive 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
def ilcm(*args):
"""Computes integer least common multiple.
Examples
========
>>> from sympy 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 // igcd(a, b) * b # since gcd(a,b) | a
return a
def igcdex(a, b):
"""Returns x, y, g such that g = x*a + y*b = gcd(a, b).
Examples
========
>>> 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):
r"""
Return the number $c$ such that, $a \times c = 1 \pmod{m}$
where $c$ has the same sign as $m$. If no such value exists,
a ValueError is raised.
Examples
========
>>> from sympy import mod_inverse, S
Suppose we wish to find multiplicative inverse $x$ of
3 modulo 11. This is the same as finding $x$ such
that $3x = 1 \pmod{11}$. One value of x that satisfies
this congruence is 4. Because $3 \times 4 = 12$ and $12 = 1 \pmod{11}$.
This is the value returned by ``mod_inverse``:
>>> mod_inverse(3, 11)
4
>>> mod_inverse(-3, 11)
7
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
==========
.. [1] https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
.. [2] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
"""
c = None
try:
a, m = as_int(a), as_int(m)
if m != 1 and m != -1:
x, _, g = igcdex(a, m)
if g == 1:
c = x % 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 big not in (S.true, 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
class Number(AtomicExpr):
"""Represents atomic numbers in SymPy.
Explanation
===========
Floating point numbers are represented by the Float class.
Rational numbers (of any size) are represented by the Rational class.
Integer numbers (of any size) are represented by the Integer class.
Float and Rational are subclasses of Number; Integer is a subclass
of Rational.
For example, ``2/3`` is represented as ``Rational(2, 3)`` which is
a different object from the floating point number obtained with
Python division ``2/3``. Even for numbers that are exactly
represented in binary, there is a difference between how two forms,
such as ``Rational(1, 2)`` and ``Float(0.5)``, are used in SymPy.
The rational form is to be preferred in symbolic computations.
Other kinds of numbers, such as algebraic numbers ``sqrt(2)`` or
complex numbers ``3 + 4*I``, are not instances of Number class as
they are not atomic.
See Also
========
Float, Integer, Rational
"""
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
kind = NumberKind
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, str):
_obj = obj.lower() # float('INF') == float('inf')
if _obj == 'nan':
return S.NaN
elif _obj == 'inf':
return S.Infinity
elif _obj == '+inf':
return S.Infinity
elif _obj == '-inf':
return S.NegativeInfinity
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 could_extract_minus_sign(self):
return bool(self.is_extended_negative)
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 sympy.functions.elementary.complexes import sign
try:
other = Number(other)
if self.is_infinite or S.NaN in (self, other):
return (S.NaN, S.NaN)
except TypeError:
return NotImplemented
if not other:
raise ZeroDivisionError('modulo by zero')
if self.is_Integer and other.is_Integer:
return Tuple(*divmod(self.p, other.p))
elif isinstance(other, Float):
rat = self/Rational(other)
else:
rat = self/other
if other.is_finite:
w = int(rat) if rat >= 0 else int(rat) - 1
r = self - other*w
else:
w = 0 if not self or (sign(self) == sign(other)) else -1
r = other if w else self
return Tuple(w, r)
def __rdivmod__(self, other):
try:
other = Number(other)
except TypeError:
return NotImplemented
return divmod(other, self)
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 __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
def _eval_conjugate(self):
return self
def _eval_order(self, *symbols):
from sympy.series.order 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
@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_parameters.evaluate:
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_parameters.evaluate:
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_parameters.evaluate:
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 __truediv__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NaN:
return S.NaN
elif other in (S.Infinity, S.NegativeInfinity):
return S.Zero
return AtomicExpr.__truediv__(self, other)
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().__hash__()
def is_constant(self, *wrt, **flags):
return True
def as_coeff_mul(self, *deps, rational=True, **kwargs):
# a -> c*t
if self.is_Rational or not rational:
return self, ()
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, ()
return S.Zero, (self,)
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)
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
def gcd(self, other):
"""Compute GCD of `self` and `other`. """
from sympy.polys.polytools import gcd
return gcd(self, other)
def lcm(self, other):
"""Compute LCM of `self` and `other`. """
from sympy.polys.polytools import lcm
return lcm(self, other)
def cofactors(self, other):
"""Compute GCD and cofactors of `self` and `other`. """
from sympy.polys.polytools import cofactors
return cofactors(self, other)
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; spaces or underscores are also allowed. (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.
In SymPy, a Float is a number that can be computed with arbitrary
precision. Although floating point 'inf' and 'nan' are not such
numbers, Float can create these numbers:
>>> Float('-inf')
-oo
>>> _.is_Float
False
Zero in Float only has a single value. Values are not separate for
positive and negative zeroes.
"""
__slots__ = ('_mpf_', '_prec')
_mpf_: tuple[int, int, int, int]
# A Float represents many real numbers,
# both rational and irrational.
is_rational = None
is_irrational = None
is_number = True
is_real = True
is_extended_real = True
is_Float = True
def __new__(cls, num, dps=None, precision=None):
if dps is not None and precision is not None:
raise ValueError('Both decimal and binary precision supplied. '
'Supply only one. ')
if isinstance(num, str):
# Float accepts spaces as digit separators
num = num.replace(' ', '').lower()
if num.startswith('.') and len(num) > 1:
num = '0' + num
elif num.startswith('-.') and len(num) > 2:
num = '-0.' + num[2:]
elif num in ('inf', '+inf'):
return S.Infinity
elif num == '-inf':
return S.NegativeInfinity
elif isinstance(num, float) and num == 0:
num = '0'
elif isinstance(num, float) and num == float('inf'):
return S.Infinity
elif isinstance(num, float) and num == float('-inf'):
return S.NegativeInfinity
elif isinstance(num, float) and math.isnan(num):
return S.NaN
elif isinstance(num, (SYMPY_INTS, Integer)):
num = str(num)
elif num is S.Infinity:
return num
elif num is S.NegativeInfinity:
return num
elif num is S.NaN:
return num
elif _is_numpy_instance(num): # 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, str) 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 = dps_to_prec(dps)
elif precision == '' and dps is None or precision is None and dps == '':
if not isinstance(num, str):
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 = 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 = dps_to_prec(dps)
precision = int(precision)
if isinstance(num, float):
_mpf_ = mlib.from_float(num, precision, rnd)
elif isinstance(num, str):
_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():
return S.NaN
elif num.is_infinite():
if num > 0:
return S.Infinity
return S.NegativeInfinity
else:
raise ValueError("unexpected decimal value %s" % str(num))
elif isinstance(num, tuple) and len(num) in (3, 4):
if isinstance(num[1], str):
# it's a hexadecimal (coming from a pickled object)
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]
# Strip leading '0x' - gmpy2 only documents such inputs
# with base prefix as valid when the 2nd argument (base) is 0.
# When mpmath uses Sage as the backend, however, it
# ends up including '0x' when preparing the picklable tuple.
# See issue #19690.
if num[1].startswith('0x'):
num[1] = num[1][2:]
# Now we can assume that it is in standard form
num[1] = MPZ(num[1], 16)
_mpf_ = tuple(num)
else:
if len(num) == 4:
# handle normalization hack
return Float._new(num, precision)
else:
if not all((
num[0] in (0, 1),
num[1] >= 0,
all(type(i) in (int, int) for i in num)
)):
raise ValueError('malformed mpf: %s' % (num,))
# don't compute number or else it may
# over/underflow
return Float._new(
(num[0], num[1], num[2], bitcount(num[1])),
precision)
else:
try:
_mpf_ = num._as_mpf_val(precision)
except (NotImplementedError, AttributeError):
_mpf_ = mpmath.mpf(num, prec=precision)._mpf_
return cls._new(_mpf_, precision, zero=False)
@classmethod
def _new(cls, _mpf_, _prec, zero=True):
# special cases
if zero and _mpf_ == fzero:
return S.Zero # Float(0) -> 0.0; Float._new((0,0,0,0)) -> 0
elif _mpf_ == _mpf_nan:
return S.NaN
elif _mpf_ == _mpf_inf:
return S.Infinity
elif _mpf_ == _mpf_ninf:
return S.NegativeInfinity
obj = Expr.__new__(cls)
obj._mpf_ = mpf_norm(_mpf_, _prec)
obj._prec = _prec
return obj
# mpz can't be pickled
def __getnewargs_ex__(self):
return ((mlib.to_pickable(self._mpf_),), {'precision': 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))))
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
@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_ == fzero
def _eval_is_negative(self):
if self._mpf_ in (_mpf_ninf, _mpf_inf):
return False
return self.num < 0
def _eval_is_positive(self):
if self._mpf_ in (_mpf_ninf, _mpf_inf):
return False
return self.num > 0
def _eval_is_extended_negative(self):
if self._mpf_ == _mpf_ninf:
return True
if self._mpf_ == _mpf_inf:
return False
return self.num < 0
def _eval_is_extended_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_ == fzero
def __bool__(self):
return self._mpf_ != fzero
def __neg__(self):
if not self:
return self
return Float._new(mlib.mpf_neg(self._mpf_), self._prec)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
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_parameters.evaluate:
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_parameters.evaluate:
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 __truediv__(self, other):
if isinstance(other, Number) and other != 0 and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_div(self._mpf_, rhs, prec, rnd), prec)
return Number.__truediv__(self, other)
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if isinstance(other, Rational) and other.q != 1 and global_parameters.evaluate:
# calculate mod with Rationals, *then* round the result
return Float(Rational.__mod__(Rational(self), other),
precision=self._prec)
if isinstance(other, Float) and global_parameters.evaluate:
r = self/other
if r == int(r):
return Float(0, precision=max(self._prec, other._prec))
if isinstance(other, Number) and global_parameters.evaluate:
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_parameters.evaluate:
return other.__mod__(self)
if isinstance(other, Number) and global_parameters.evaluate:
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 equal_valued(self, 0):
if expt.is_extended_positive:
return self
if expt.is_extended_negative:
return S.ComplexInfinity
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, fzero), (expt, 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):
if self._mpf_ == fzero:
return 0
return int(mlib.to_int(self._mpf_)) # uses round_fast = round_down
def __eq__(self, other):
from sympy.logic.boolalg import Boolean
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if isinstance(other, Boolean):
return False
if other.is_NumberSymbol:
if other.is_irrational:
return False
return other.__eq__(self)
if other.is_Float:
# comparison is exact
# so Float(.1, 3) != Float(.1, 33)
return self._mpf_ == other._mpf_
if other.is_Rational:
return other.__eq__(self)
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))
if not self:
return not other
return False # Float != non-Number
def __ne__(self, other):
return not self == other
def _Frel(self, other, op):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Rational:
# test self*other.q <?> other.p without losing precision
'''
>>> f = Float(.1,2)
>>> i = 1234567890
>>> (f*i)._mpf_
(0, 471, 18, 9)
>>> mlib.mpf_mul(f._mpf_, mlib.from_int(i))
(0, 505555550955, -12, 39)
'''
smpf = mlib.mpf_mul(self._mpf_, mlib.from_int(other.q))
ompf = mlib.from_int(other.p)
return _sympify(bool(op(smpf, ompf)))
elif other.is_Float:
return _sympify(bool(
op(self._mpf_, other._mpf_)))
elif other.is_comparable and other not in (
S.Infinity, S.NegativeInfinity):
other = other.evalf(prec_to_dps(self._prec))
if other._prec > 1:
if other.is_Number:
return _sympify(bool(
op(self._mpf_, other._as_mpf_val(self._prec))))
def __gt__(self, other):
if isinstance(other, NumberSymbol):
return other.__lt__(self)
rv = self._Frel(other, mlib.mpf_gt)
if rv is None:
return Expr.__gt__(self, other)
return rv
def __ge__(self, other):
if isinstance(other, NumberSymbol):
return other.__le__(self)
rv = self._Frel(other, mlib.mpf_ge)
if rv is None:
return Expr.__ge__(self, other)
return rv
def __lt__(self, other):
if isinstance(other, NumberSymbol):
return other.__gt__(self)
rv = self._Frel(other, mlib.mpf_lt)
if rv is None:
return Expr.__lt__(self, other)
return rv
def __le__(self, other):
if isinstance(other, NumberSymbol):
return other.__ge__(self)
rv = self._Frel(other, mlib.mpf_le)
if rv is None:
return Expr.__le__(self, other)
return rv
def __hash__(self):
return super().__hash__()
def epsilon_eq(self, other, epsilon="1e-15"):
return abs(self - other) < Float(epsilon)
def __format__(self, format_spec):
return format(decimal.Decimal(str(self)), format_spec)
# Add sympify converters
_sympy_converter[float] = _sympy_converter[decimal.Decimal] = Float
# this is here to work nicely in Sage
RealNumber = Float
class Rational(Number):
"""Represents rational numbers (p/q) of any size.
Examples
========
>>> from sympy import Rational, nsimplify, S, pi
>>> 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
If an unevaluated Rational is desired, ``gcd=1`` can be passed and
this will keep common divisors of the numerator and denominator
from being eliminated. It is not possible, however, to leave a
negative value in the denominator.
>>> Rational(2, 4, gcd=1)
2/4
>>> Rational(2, -4, gcd=1).q
4
See Also
========
sympy.core.sympify.sympify, sympy.simplify.simplify.nsimplify
"""
is_real = True
is_integer = False
is_rational = True
is_number = True
__slots__ = ('p', 'q')
p: int
q: int
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, SYMPY_INTS):
pass
else:
if isinstance(p, (float, Float)):
return Rational(*_as_integer_ratio(p))
if not isinstance(p, str):
try:
p = sympify(p)
except (SympifyError, SyntaxError):
pass # error will raise below
else:
if p.count('/') > 1:
raise TypeError('invalid input: %s' % p)
p = p.replace(' ', '')
pq = p.rsplit('/', 1)
if len(pq) == 2:
p, q = pq
fp = fractions.Fraction(p)
fq = fractions.Fraction(q)
p = fp/fq
try:
p = fractions.Fraction(p)
except ValueError:
pass # error will raise below
else:
return Rational(p.numerator, p.denominator, 1)
if not isinstance(p, Rational):
raise TypeError('invalid input: %s' % p)
q = 1
gcd = 1
Q = 1
if not isinstance(p, SYMPY_INTS):
p = Rational(p)
Q *= p.q
p = p.p
else:
p = int(p)
if not isinstance(q, SYMPY_INTS):
q = Rational(q)
p *= q.q
Q *= q.p
else:
Q *= int(q)
q = Q
# p and q are now ints
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
def limit_denominator(self, max_denominator=1000000):
"""Closest Rational to self with denominator at most max_denominator.
Examples
========
>>> 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_parameters.evaluate:
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_parameters.evaluate:
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_parameters.evaluate:
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_parameters.evaluate:
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 __truediv__(self, other):
if global_parameters.evaluate:
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.__truediv__(self, other)
return Number.__truediv__(self, other)
@_sympifyit('other', NotImplemented)
def __rtruediv__(self, other):
if global_parameters.evaluate:
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.__rtruediv__(self, other)
return Number.__rtruediv__(self, other)
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if global_parameters.evaluate:
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_extended_negative:
# (3/4)**-2 -> (4/3)**2
ne = -expt
if (ne is S.One):
return Rational(self.q, self.p)
if self.is_negative:
return S.NegativeOne**expt*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):
intpart = expt.p // expt.q
if intpart:
intpart += 1
remfracpart = intpart*expt.q - expt.p
ratfracpart = Rational(remfracpart, expt.q)
if self.p != 1:
return Integer(self.p)**expt*Integer(self.q)**ratfracpart*Rational(1, self.q**intpart, 1)
return Integer(self.q)**ratfracpart*Rational(1, self.q**intpart, 1)
else:
remfracpart = expt.q - expt.p
ratfracpart = Rational(remfracpart, expt.q)
if self.p != 1:
return Integer(self.p)**expt*Integer(self.q)**ratfracpart*Rational(1, self.q, 1)
return Integer(self.q)**ratfracpart*Rational(1, self.q, 1)
if self.is_extended_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)
def floor(self):
return Integer(self.p // self.q)
def ceiling(self):
return -Integer(-self.p // self.q)
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
def __eq__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if not isinstance(other, Number):
# S(0) == S.false is False
# S(0) == False is True
return False
if not self:
return not other
if other.is_NumberSymbol:
if other.is_irrational:
return False
return other.__eq__(self)
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:
# all Floats have a denominator that is a power of 2
# so if self doesn't, it can't be equal to other
if self.q & (self.q - 1):
return False
s, m, t = other._mpf_[:3]
if s:
m = -m
if not t:
# other is an odd integer
if not self.is_Integer or self.is_even:
return False
return m == self.p
from .power import integer_log
if t > 0:
# other is an even integer
if not self.is_Integer:
return False
# does m*2**t == self.p
return self.p and not self.p % m and \
integer_log(self.p//m, 2) == (t, True)
# does non-integer s*m/2**-t = p/q?
if self.is_Integer:
return False
return m == self.p and integer_log(self.q, 2) == (-t, True)
return False
def __ne__(self, other):
return not self == other
def _Rrel(self, other, attr):
# if you want self < other, pass self, other, __gt__
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Number:
op = None
s, o = self, other
if other.is_NumberSymbol:
op = getattr(o, attr)
elif other.is_Float:
op = getattr(o, attr)
elif other.is_Rational:
s, o = Integer(s.p*o.q), Integer(s.q*o.p)
op = getattr(o, attr)
if op:
return op(s)
if o.is_number and o.is_extended_real:
return Integer(s.p), s.q*o
def __gt__(self, other):
rv = self._Rrel(other, '__lt__')
if rv is None:
rv = self, other
elif not isinstance(rv, tuple):
return rv
return Expr.__gt__(*rv)
def __ge__(self, other):
rv = self._Rrel(other, '__le__')
if rv is None:
rv = self, other
elif not isinstance(rv, tuple):
return rv
return Expr.__ge__(*rv)
def __lt__(self, other):
rv = self._Rrel(other, '__gt__')
if rv is None:
rv = self, other
elif not isinstance(rv, tuple):
return rv
return Expr.__lt__(*rv)
def __le__(self, other):
rv = self._Rrel(other, '__ge__')
if rv is None:
rv = self, other
elif not isinstance(rv, tuple):
return rv
return Expr.__le__(*rv)
def __hash__(self):
return super().__hash__()
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.factor_ import factorrat
return factorrat(self, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose).copy()
@property
def numerator(self):
return self.p
@property
def denominator(self):
return self.q
@_sympifyit('other', NotImplemented)
def gcd(self, other):
if isinstance(other, Rational):
if other == S.Zero:
return other
return Rational(
igcd(self.p, other.p),
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 // igcd(self.p, other.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 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
def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product."""
return self, S.One
def as_coeff_Add(self, rational=False):
"""Efficiently extract the coefficient of a summation."""
return self, S.Zero
class Integer(Rational):
"""Represents integer numbers of any size.
Examples
========
>>> from sympy import Integer
>>> Integer(3)
3
If a float or a rational is passed to Integer, the fractional part
will be discarded; the effect is of rounding toward zero.
>>> Integer(3.8)
3
>>> Integer(-3.8)
-3
A string is acceptable input if it can be parsed as an integer:
>>> Integer("9" * 20)
99999999999999999999
It is rarely needed to explicitly instantiate an Integer, because
Python integers are automatically converted to Integer when they
are used in SymPy expressions.
"""
q = 1
is_integer = True
is_number = True
is_Integer = True
__slots__ = ()
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))
@cacheit
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.
try:
ival = int(i)
except TypeError:
raise TypeError(
"Argument of Integer should be of numeric type, got %s." % i)
# We only work with well-behaved integer types. This converts, for
# example, numpy.int32 instances.
if ival == 1:
return S.One
if ival == -1:
return S.NegativeOne
if ival == 0:
return S.Zero
obj = Expr.__new__(cls)
obj.p = ival
return obj
def __getnewargs__(self):
return (self.p,)
# Arithmetic operations are here for efficiency
def __int__(self):
return self.p
def floor(self):
return Integer(self.p)
def ceiling(self):
return Integer(self.p)
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
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):
if isinstance(other, Integer) and global_parameters.evaluate:
return Tuple(*(divmod(self.p, other.p)))
else:
return Number.__divmod__(self, other)
def __rdivmod__(self, other):
if isinstance(other, int) and global_parameters.evaluate:
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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_parameters.evaluate:
if isinstance(other, int):
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, 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 == other
def __gt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
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:
return NotImplemented
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:
return NotImplemented
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:
return NotImplemented
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.
Explanation
===========
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.ntheory.factor_ 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, 1)**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()._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:
return S.NegativeOne**expt*Rational(1, -self, 1)**ne
else:
return Rational(1, self.p, 1)**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(b_pos).factors(limit=2**15)
# now process the dict of factors
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, 1))
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 == b_pos and out_int == 1 and out_rad == 1:
result = None
else:
result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q))
if self.is_negative:
result *= Pow(S.NegativeOne, expt)
return result
def _eval_is_prime(self):
from sympy.ntheory.primetest 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
@_sympifyit('other', NotImplemented)
def __floordiv__(self, other):
if not isinstance(other, Expr):
return NotImplemented
if isinstance(other, Integer):
return Integer(self.p // other)
return divmod(self, other)[0]
def __rfloordiv__(self, other):
return Integer(Integer(other).p // self.p)
# These bitwise operations (__lshift__, __rlshift__, ..., __invert__) are defined
# for Integer only and not for general SymPy expressions. This is to achieve
# compatibility with the numbers.Integral ABC which only defines these operations
# among instances of numbers.Integral. Therefore, these methods check explicitly for
# integer types rather than using sympify because they should not accept arbitrary
# symbolic expressions and there is no symbolic analogue of numbers.Integral's
# bitwise operations.
def __lshift__(self, other):
if isinstance(other, (int, Integer, numbers.Integral)):
return Integer(self.p << int(other))
else:
return NotImplemented
def __rlshift__(self, other):
if isinstance(other, (int, numbers.Integral)):
return Integer(int(other) << self.p)
else:
return NotImplemented
def __rshift__(self, other):
if isinstance(other, (int, Integer, numbers.Integral)):
return Integer(self.p >> int(other))
else:
return NotImplemented
def __rrshift__(self, other):
if isinstance(other, (int, numbers.Integral)):
return Integer(int(other) >> self.p)
else:
return NotImplemented
def __and__(self, other):
if isinstance(other, (int, Integer, numbers.Integral)):
return Integer(self.p & int(other))
else:
return NotImplemented
def __rand__(self, other):
if isinstance(other, (int, numbers.Integral)):
return Integer(int(other) & self.p)
else:
return NotImplemented
def __xor__(self, other):
if isinstance(other, (int, Integer, numbers.Integral)):
return Integer(self.p ^ int(other))
else:
return NotImplemented
def __rxor__(self, other):
if isinstance(other, (int, numbers.Integral)):
return Integer(int(other) ^ self.p)
else:
return NotImplemented
def __or__(self, other):
if isinstance(other, (int, Integer, numbers.Integral)):
return Integer(self.p | int(other))
else:
return NotImplemented
def __ror__(self, other):
if isinstance(other, (int, numbers.Integral)):
return Integer(int(other) | self.p)
else:
return NotImplemented
def __invert__(self):
return Integer(~self.p)
# Add sympify converters
_sympy_converter[int] = Integer
class AlgebraicNumber(Expr):
r"""
Class for representing algebraic numbers in SymPy.
Symbolically, an instance of this class represents an element
$\alpha \in \mathbb{Q}(\theta) \hookrightarrow \mathbb{C}$. That is, the
algebraic number $\alpha$ is represented as an element of a particular
number field $\mathbb{Q}(\theta)$, with a particular embedding of this
field into the complex numbers.
Formally, the primitive element $\theta$ is given by two data points: (1)
its minimal polynomial (which defines $\mathbb{Q}(\theta)$), and (2) a
particular complex number that is a root of this polynomial (which defines
the embedding $\mathbb{Q}(\theta) \hookrightarrow \mathbb{C}$). Finally,
the algebraic number $\alpha$ which we represent is then given by the
coefficients of a polynomial in $\theta$.
"""
__slots__ = ('rep', 'root', 'alias', 'minpoly', '_own_minpoly')
is_AlgebraicNumber = True
is_algebraic = True
is_number = True
kind = NumberKind
# Optional alias symbol is not free.
# Actually, alias should be a Str, but some methods
# expect that it be an instance of Expr.
free_symbols: set[Basic] = set()
def __new__(cls, expr, coeffs=None, alias=None, **args):
r"""
Construct a new algebraic number $\alpha$ belonging to a number field
$k = \mathbb{Q}(\theta)$.
There are four instance attributes to be determined:
=========== ============================================================================
Attribute Type/Meaning
=========== ============================================================================
``root`` :py:class:`~.Expr` for $\theta$ as a complex number
``minpoly`` :py:class:`~.Poly`, the minimal polynomial of $\theta$
``rep`` :py:class:`~sympy.polys.polyclasses.DMP` giving $\alpha$ as poly in $\theta$
``alias`` :py:class:`~.Symbol` for $\theta$, or ``None``
=========== ============================================================================
See Parameters section for how they are determined.
Parameters
==========
expr : :py:class:`~.Expr`, or pair $(m, r)$
There are three distinct modes of construction, depending on what
is passed as *expr*.
**(1)** *expr* is an :py:class:`~.AlgebraicNumber`:
In this case we begin by copying all four instance attributes from
*expr*. If *coeffs* were also given, we compose the two coeff
polynomials (see below). If an *alias* was given, it overrides.
**(2)** *expr* is any other type of :py:class:`~.Expr`:
Then ``root`` will equal *expr*. Therefore it
must express an algebraic quantity, and we will compute its
``minpoly``.
**(3)** *expr* is an ordered pair $(m, r)$ giving the
``minpoly`` $m$, and a ``root`` $r$ thereof, which together
define $\theta$. In this case $m$ may be either a univariate
:py:class:`~.Poly` or any :py:class:`~.Expr` which represents the
same, while $r$ must be some :py:class:`~.Expr` representing a
complex number that is a root of $m$, including both explicit
expressions in radicals, and instances of
:py:class:`~.ComplexRootOf` or :py:class:`~.AlgebraicNumber`.
coeffs : list, :py:class:`~.ANP`, None, optional (default=None)
This defines ``rep``, giving the algebraic number $\alpha$ as a
polynomial in $\theta$.
If a list, the elements should be integers or rational numbers.
If an :py:class:`~.ANP`, we take its coefficients (using its
:py:meth:`~.ANP.to_list()` method). If ``None``, then the list of
coefficients defaults to ``[1, 0]``, meaning that $\alpha = \theta$
is the primitive element of the field.
If *expr* was an :py:class:`~.AlgebraicNumber`, let $g(x)$ be its
``rep`` polynomial, and let $f(x)$ be the polynomial defined by
*coeffs*. Then ``self.rep`` will represent the composition
$(f \circ g)(x)$.
alias : str, :py:class:`~.Symbol`, None, optional (default=None)
This is a way to provide a name for the primitive element. We
described several ways in which the *expr* argument can define the
value of the primitive element, but none of these methods gave it
a name. Here, for example, *alias* could be set as
``Symbol('theta')``, in order to make this symbol appear when
$\alpha$ is printed, or rendered as a polynomial, using the
:py:meth:`~.as_poly()` method.
Examples
========
Recall that we are constructing an algebraic number as a field element
$\alpha \in \mathbb{Q}(\theta)$.
>>> from sympy import AlgebraicNumber, sqrt, CRootOf, S
>>> from sympy.abc import x
Example (1): $\alpha = \theta = \sqrt{2}$
>>> a1 = AlgebraicNumber(sqrt(2))
>>> a1.minpoly_of_element().as_expr(x)
x**2 - 2
>>> a1.evalf(10)
1.414213562
Example (2): $\alpha = 3 \sqrt{2} - 5$, $\theta = \sqrt{2}$. We can
either build on the last example:
>>> a2 = AlgebraicNumber(a1, [3, -5])
>>> a2.as_expr()
-5 + 3*sqrt(2)
or start from scratch:
>>> a2 = AlgebraicNumber(sqrt(2), [3, -5])
>>> a2.as_expr()
-5 + 3*sqrt(2)
Example (3): $\alpha = 6 \sqrt{2} - 11$, $\theta = \sqrt{2}$. Again we
can build on the previous example, and we see that the coeff polys are
composed:
>>> a3 = AlgebraicNumber(a2, [2, -1])
>>> a3.as_expr()
-11 + 6*sqrt(2)
reflecting the fact that $(2x - 1) \circ (3x - 5) = 6x - 11$.
Example (4): $\alpha = \sqrt{2}$, $\theta = \sqrt{2} + \sqrt{3}$. The
easiest way is to use the :py:func:`~.to_number_field()` function:
>>> from sympy import to_number_field
>>> a4 = to_number_field(sqrt(2), sqrt(2) + sqrt(3))
>>> a4.minpoly_of_element().as_expr(x)
x**2 - 2
>>> a4.to_root()
sqrt(2)
>>> a4.primitive_element()
sqrt(2) + sqrt(3)
>>> a4.coeffs()
[1/2, 0, -9/2, 0]
but if you already knew the right coefficients, you could construct it
directly:
>>> a4 = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1)/2, 0, S(-9)/2, 0])
>>> a4.to_root()
sqrt(2)
>>> a4.primitive_element()
sqrt(2) + sqrt(3)
Example (5): Construct the Golden Ratio as an element of the 5th
cyclotomic field, supposing we already know its coefficients. This time
we introduce the alias $\zeta$ for the primitive element of the field:
>>> from sympy import cyclotomic_poly
>>> from sympy.abc import zeta
>>> a5 = AlgebraicNumber(CRootOf(cyclotomic_poly(5), -1),
... [-1, -1, 0, 0], alias=zeta)
>>> a5.as_poly().as_expr()
-zeta**3 - zeta**2
>>> a5.evalf()
1.61803398874989
(The index ``-1`` to ``CRootOf`` selects the complex root with the
largest real and imaginary parts, which in this case is
$\mathrm{e}^{2i\pi/5}$. See :py:class:`~.ComplexRootOf`.)
Example (6): Building on the last example, construct the number
$2 \phi \in \mathbb{Q}(\phi)$, where $\phi$ is the Golden Ratio:
>>> from sympy.abc import phi
>>> a6 = AlgebraicNumber(a5.to_root(), coeffs=[2, 0], alias=phi)
>>> a6.as_poly().as_expr()
2*phi
>>> a6.primitive_element().evalf()
1.61803398874989
Note that we needed to use ``a5.to_root()``, since passing ``a5`` as
the first argument would have constructed the number $2 \phi$ as an
element of the field $\mathbb{Q}(\zeta)$:
>>> a6_wrong = AlgebraicNumber(a5, coeffs=[2, 0])
>>> a6_wrong.as_poly().as_expr()
-2*zeta**3 - 2*zeta**2
>>> a6_wrong.primitive_element().evalf()
0.309016994374947 + 0.951056516295154*I
"""
from sympy.polys.polyclasses import ANP, DMP
from sympy.polys.numberfields import minimal_polynomial
expr = sympify(expr)
rep0 = None
alias0 = None
if isinstance(expr, (tuple, Tuple)):
minpoly, root = expr
if not minpoly.is_Poly:
from sympy.polys.polytools import Poly
minpoly = Poly(minpoly)
elif expr.is_AlgebraicNumber:
minpoly, root, rep0, alias0 = (expr.minpoly, expr.root,
expr.rep, expr.alias)
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())
else:
rep = DMP.from_list([1, 0], 0, dom)
scoeffs = Tuple(1, 0)
if rep0 is not None:
from sympy.polys.densetools import dup_compose
c = dup_compose(rep.rep, rep0.rep, dom)
rep = DMP.from_list(c, 0, dom)
scoeffs = Tuple(*c)
if rep.degree() >= minpoly.degree():
rep = rep.rem(minpoly.rep)
sargs = (root, scoeffs)
alias = alias or alias0
if alias is not None:
from .symbol import Symbol
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
obj._own_minpoly = None
return obj
def __hash__(self):
return super().__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
def as_poly(self, x=None):
"""Create a Poly instance from ``self``. """
from sympy.polys.polytools import 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:
from .symbol import Dummy
return PurePoly.new(self.rep, Dummy('x'))
def as_expr(self, x=None):
"""Create a Basic expression from ``self``. """
return self.as_poly(x or self.root).as_expr().expand()
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() ]
def native_coeffs(self):
"""Returns all native coefficients of an algebraic number. """
return self.rep.all_coeffs()
def to_algebraic_integer(self):
"""Convert ``self`` to an algebraic integer. """
from sympy.polys.polytools 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, **kwargs):
from sympy.polys.rootoftools import CRootOf
from sympy.polys import minpoly
measure, ratio = kwargs['measure'], kwargs['ratio']
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
def field_element(self, coeffs):
r"""
Form another element of the same number field.
Explanation
===========
If we represent $\alpha \in \mathbb{Q}(\theta)$, form another element
$\beta \in \mathbb{Q}(\theta)$ of the same number field.
Parameters
==========
coeffs : list, :py:class:`~.ANP`
Like the *coeffs* arg to the class
:py:meth:`constructor<.AlgebraicNumber.__new__>`, defines the
new element as a polynomial in the primitive element.
If a list, the elements should be integers or rational numbers.
If an :py:class:`~.ANP`, we take its coefficients (using its
:py:meth:`~.ANP.to_list()` method).
Examples
========
>>> from sympy import AlgebraicNumber, sqrt
>>> a = AlgebraicNumber(sqrt(5), [-1, 1])
>>> b = a.field_element([3, 2])
>>> print(a)
1 - sqrt(5)
>>> print(b)
2 + 3*sqrt(5)
>>> print(b.primitive_element() == a.primitive_element())
True
See Also
========
AlgebraicNumber
"""
return AlgebraicNumber(
(self.minpoly, self.root), coeffs=coeffs, alias=self.alias)
@property
def is_primitive_element(self):
r"""
Say whether this algebraic number $\alpha \in \mathbb{Q}(\theta)$ is
equal to the primitive element $\theta$ for its field.
"""
c = self.coeffs()
# Second case occurs if self.minpoly is linear:
return c == [1, 0] or c == [self.root]
def primitive_element(self):
r"""
Get the primitive element $\theta$ for the number field
$\mathbb{Q}(\theta)$ to which this algebraic number $\alpha$ belongs.
Returns
=======
AlgebraicNumber
"""
if self.is_primitive_element:
return self
return self.field_element([1, 0])
def to_primitive_element(self, radicals=True):
r"""
Convert ``self`` to an :py:class:`~.AlgebraicNumber` instance that is
equal to its own primitive element.
Explanation
===========
If we represent $\alpha \in \mathbb{Q}(\theta)$, $\alpha \neq \theta$,
construct a new :py:class:`~.AlgebraicNumber` that represents
$\alpha \in \mathbb{Q}(\alpha)$.
Examples
========
>>> from sympy import sqrt, to_number_field
>>> from sympy.abc import x
>>> a = to_number_field(sqrt(2), sqrt(2) + sqrt(3))
The :py:class:`~.AlgebraicNumber` ``a`` represents the number
$\sqrt{2}$ in the field $\mathbb{Q}(\sqrt{2} + \sqrt{3})$. Rendering
``a`` as a polynomial,
>>> a.as_poly().as_expr(x)
x**3/2 - 9*x/2
reflects the fact that $\sqrt{2} = \theta^3/2 - 9 \theta/2$, where
$\theta = \sqrt{2} + \sqrt{3}$.
``a`` is not equal to its own primitive element. Its minpoly
>>> a.minpoly.as_poly().as_expr(x)
x**4 - 10*x**2 + 1
is that of $\theta$.
Converting to a primitive element,
>>> a_prim = a.to_primitive_element()
>>> a_prim.minpoly.as_poly().as_expr(x)
x**2 - 2
we obtain an :py:class:`~.AlgebraicNumber` whose ``minpoly`` is that of
the number itself.
Parameters
==========
radicals : boolean, optional (default=True)
If ``True``, then we will try to return an
:py:class:`~.AlgebraicNumber` whose ``root`` is an expression
in radicals. If that is not possible (or if *radicals* is
``False``), ``root`` will be a :py:class:`~.ComplexRootOf`.
Returns
=======
AlgebraicNumber
See Also
========
is_primitive_element
"""
if self.is_primitive_element:
return self
m = self.minpoly_of_element()
r = self.to_root(radicals=radicals)
return AlgebraicNumber((m, r))
def minpoly_of_element(self):
r"""
Compute the minimal polynomial for this algebraic number.
Explanation
===========
Recall that we represent an element $\alpha \in \mathbb{Q}(\theta)$.
Our instance attribute ``self.minpoly`` is the minimal polynomial for
our primitive element $\theta$. This method computes the minimal
polynomial for $\alpha$.
"""
if self._own_minpoly is None:
if self.is_primitive_element:
self._own_minpoly = self.minpoly
else:
from sympy.polys.numberfields.minpoly import minpoly
theta = self.primitive_element()
self._own_minpoly = minpoly(self.as_expr(theta), polys=True)
return self._own_minpoly
def to_root(self, radicals=True, minpoly=None):
"""
Convert to an :py:class:`~.Expr` that is not an
:py:class:`~.AlgebraicNumber`, specifically, either a
:py:class:`~.ComplexRootOf`, or, optionally and where possible, an
expression in radicals.
Parameters
==========
radicals : boolean, optional (default=True)
If ``True``, then we will try to return the root as an expression
in radicals. If that is not possible, we will return a
:py:class:`~.ComplexRootOf`.
minpoly : :py:class:`~.Poly`
If the minimal polynomial for `self` has been pre-computed, it can
be passed in order to save time.
"""
if self.is_primitive_element and not isinstance(self.root, AlgebraicNumber):
return self.root
m = minpoly or self.minpoly_of_element()
roots = m.all_roots(radicals=radicals)
if len(roots) == 1:
return roots[0]
ex = self.as_expr()
for b in roots:
if m.same_root(b, ex):
return b
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):
"""The number zero.
Zero is a singleton, and can be accessed by ``S.Zero``
Examples
========
>>> from sympy import S, Integer
>>> Integer(0) is S.Zero
True
>>> 1/S.Zero
zoo
References
==========
.. [1] https://en.wikipedia.org/wiki/Zero
"""
p = 0
q = 1
is_positive = False
is_negative = False
is_zero = True
is_number = True
is_comparable = True
__slots__ = ()
def __getnewargs__(self):
return ()
@staticmethod
def __abs__():
return S.Zero
@staticmethod
def __neg__():
return S.Zero
def _eval_power(self, expt):
if expt.is_extended_positive:
return self
if expt.is_extended_negative:
return S.ComplexInfinity
if expt.is_extended_real is False:
return S.NaN
if expt.is_zero:
return S.One
# infinities are already handled with pos and neg
# tests above; now throw away leading numbers on Mul
# exponent since 0**-x = zoo**x even when x == 0
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 __bool__(self):
return False
class One(IntegerConstant, metaclass=Singleton):
"""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] https://en.wikipedia.org/wiki/1_%28number%29
"""
is_number = True
is_positive = True
p = 1
q = 1
__slots__ = ()
def __getnewargs__(self):
return ()
@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 {}
class NegativeOne(IntegerConstant, metaclass=Singleton):
"""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] https://en.wikipedia.org/wiki/%E2%88%921_%28number%29
"""
is_number = True
p = -1
q = 1
__slots__ = ()
def __getnewargs__(self):
return ()
@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 in (S.Infinity, 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):
"""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] https://en.wikipedia.org/wiki/One_half
"""
is_number = True
p = 1
q = 2
__slots__ = ()
def __getnewargs__(self):
return ()
@staticmethod
def __abs__():
return S.Half
class Infinity(Number, metaclass=Singleton):
r"""Positive infinite quantity.
Explanation
===========
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] https://en.wikipedia.org/wiki/Infinity
"""
is_commutative = True
is_number = True
is_complex = False
is_extended_real = True
is_infinite = True
is_comparable = True
is_extended_positive = 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
def _eval_evalf(self, prec=None):
return Float('inf')
def evalf(self, prec=None, **options):
return self._eval_evalf(prec)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other in (S.NegativeInfinity, S.NaN):
return S.NaN
return self
return Number.__add__(self, other)
__radd__ = __add__
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other in (S.Infinity, S.NaN):
return S.NaN
return self
return Number.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __rsub__(self, other):
return (-self).__add__(other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other.is_zero or other is S.NaN:
return S.NaN
if other.is_extended_positive:
return self
return S.NegativeInfinity
return Number.__mul__(self, other)
__rmul__ = __mul__
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.Infinity or \
other is S.NegativeInfinity or \
other is S.NaN:
return S.NaN
if other.is_extended_nonnegative:
return self
return S.NegativeInfinity
return Number.__truediv__(self, other)
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
"""
if expt.is_extended_positive:
return S.Infinity
if expt.is_extended_negative:
return S.Zero
if expt is S.NaN:
return S.NaN
if expt is S.ComplexInfinity:
return S.NaN
if expt.is_extended_real is False and expt.is_number:
from sympy.functions.elementary.complexes import re
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 __hash__(self):
return super().__hash__()
def __eq__(self, other):
return other is S.Infinity or other == float('inf')
def __ne__(self, other):
return other is not S.Infinity and other != float('inf')
__gt__ = Expr.__gt__
__ge__ = Expr.__ge__
__lt__ = Expr.__lt__
__le__ = Expr.__le__
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if not isinstance(other, Expr):
return NotImplemented
return S.NaN
__rmod__ = __mod__
def floor(self):
return self
def ceiling(self):
return self
oo = S.Infinity
class NegativeInfinity(Number, metaclass=Singleton):
"""Negative infinite quantity.
NegativeInfinity is a singleton, and can be accessed
by ``S.NegativeInfinity``.
See Also
========
Infinity
"""
is_extended_real = True
is_complex = False
is_commutative = True
is_infinite = True
is_comparable = True
is_extended_negative = 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
def _eval_evalf(self, prec=None):
return Float('-inf')
def evalf(self, prec=None, **options):
return self._eval_evalf(prec)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other in (S.Infinity, S.NaN):
return S.NaN
return self
return Number.__add__(self, other)
__radd__ = __add__
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other in (S.NegativeInfinity, S.NaN):
return S.NaN
return self
return Number.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __rsub__(self, other):
return (-self).__add__(other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other.is_zero or other is S.NaN:
return S.NaN
if other.is_extended_positive:
return self
return S.Infinity
return Number.__mul__(self, other)
__rmul__ = __mul__
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.Infinity or \
other is S.NegativeInfinity or \
other is S.NaN:
return S.NaN
if other.is_extended_nonnegative:
return self
return S.Infinity
return Number.__truediv__(self, other)
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_extended_positive:
if expt.is_odd:
return S.NegativeInfinity
else:
return S.Infinity
inf_part = S.Infinity**expt
s_part = S.NegativeOne**expt
if inf_part == 0 and s_part.is_finite:
return inf_part
if (inf_part is S.ComplexInfinity and
s_part.is_finite and not s_part.is_zero):
return S.ComplexInfinity
return s_part*inf_part
def _as_mpf_val(self, prec):
return mlib.fninf
def __hash__(self):
return super().__hash__()
def __eq__(self, other):
return other is S.NegativeInfinity or other == float('-inf')
def __ne__(self, other):
return other is not S.NegativeInfinity and other != float('-inf')
__gt__ = Expr.__gt__
__ge__ = Expr.__ge__
__lt__ = Expr.__lt__
__le__ = Expr.__le__
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if not isinstance(other, Expr):
return NotImplemented
return S.NaN
__rmod__ = __mod__
def floor(self):
return self
def ceiling(self):
return self
def as_powers_dict(self):
return {S.NegativeOne: 1, S.Infinity: 1}
class NaN(Number, metaclass=Singleton):
"""
Not a Number.
Explanation
===========
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
contrast 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] https://en.wikipedia.org/wiki/NaN
"""
is_commutative = True
is_extended_real = None
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"\text{NaN}"
def __neg__(self):
return self
@_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 __truediv__(self, other):
return self
def floor(self):
return self
def ceiling(self):
return self
def _as_mpf_val(self, prec):
return _mpf_nan
def __hash__(self):
return super().__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
# Expr will _sympify and raise TypeError
__gt__ = Expr.__gt__
__ge__ = Expr.__ge__
__lt__ = Expr.__lt__
__le__ = Expr.__le__
nan = S.NaN
@dispatch(NaN, Expr) # type:ignore
def _eval_is_eq(a, b): # noqa:F811
return False
class ComplexInfinity(AtomicExpr, metaclass=Singleton):
r"""Complex infinity.
Explanation
===========
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
>>> 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 = False
is_extended_real = False
kind = NumberKind
__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_zero:
return S.NaN
else:
if expt.is_positive:
return S.ComplexInfinity
else:
return S.Zero
zoo = S.ComplexInfinity
class NumberSymbol(AtomicExpr):
is_commutative = True
is_finite = True
is_number = True
__slots__ = ()
is_NumberSymbol = True
kind = NumberKind
def __new__(cls):
return AtomicExpr.__new__(cls)
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 __hash__(self):
return super().__hash__()
class Exp1(NumberSymbol, metaclass=Singleton):
r"""The `e` constant.
Explanation
===========
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] https://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):
if global_parameters.exp_is_pow:
return self._eval_power_exp_is_pow(expt)
else:
from sympy.functions.elementary.exponential import exp
return exp(expt)
def _eval_power_exp_is_pow(self, arg):
if arg.is_Number:
if arg is oo:
return oo
elif arg == -oo:
return S.Zero
from sympy.functions.elementary.exponential import log
if isinstance(arg, log):
return arg.args[0]
# don't autoexpand Pow or Mul (see the issue 3351):
elif not arg.is_Add:
Ioo = I*oo
if arg in [Ioo, -Ioo]:
return nan
coeff = arg.coeff(pi*I)
if coeff:
if (2*coeff).is_integer:
if coeff.is_even:
return S.One
elif coeff.is_odd:
return S.NegativeOne
elif (coeff + S.Half).is_even:
return -I
elif (coeff + S.Half).is_odd:
return I
elif coeff.is_Rational:
ncoeff = coeff % 2 # restrict to [0, 2pi)
if ncoeff > 1: # restrict to (-pi, pi]
ncoeff -= 2
if ncoeff != coeff:
return S.Exp1**(ncoeff*S.Pi*S.ImaginaryUnit)
# Warning: code in risch.py will be very sensitive to changes
# in this (see DifferentialExtension).
# look for a single log factor
coeff, terms = arg.as_coeff_Mul()
# but it can't be multiplied by oo
if coeff in (oo, -oo):
return
coeffs, log_term = [coeff], None
for term in Mul.make_args(terms):
if isinstance(term, log):
if log_term is None:
log_term = term.args[0]
else:
return
elif term.is_comparable:
coeffs.append(term)
else:
return
return log_term**Mul(*coeffs) if log_term else None
elif arg.is_Add:
out = []
add = []
argchanged = False
for a in arg.args:
if a is S.One:
add.append(a)
continue
newa = self**a
if isinstance(newa, Pow) and newa.base is self:
if newa.exp != a:
add.append(newa.exp)
argchanged = True
else:
add.append(a)
else:
out.append(newa)
if out or argchanged:
return Mul(*out)*Pow(self, Add(*add), evaluate=False)
elif arg.is_Matrix:
return arg.exp()
def _eval_rewrite_as_sin(self, **kwargs):
from sympy.functions.elementary.trigonometric import sin
return sin(I + S.Pi/2) - I*sin(I)
def _eval_rewrite_as_cos(self, **kwargs):
from sympy.functions.elementary.trigonometric import cos
return cos(I) + I*cos(I + S.Pi/2)
E = S.Exp1
class Pi(NumberSymbol, metaclass=Singleton):
r"""The `\pi` constant.
Explanation
===========
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] https://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, 1), Rational(22, 7, 1))
pi = S.Pi
class GoldenRatio(NumberSymbol, metaclass=Singleton):
r"""The golden ratio, `\phi`.
Explanation
===========
`\phi = \frac{1 + \sqrt{5}}{2}` is an 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] https://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.functions.elementary.miscellaneous 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
_eval_rewrite_as_sqrt = _eval_expand_func
class TribonacciConstant(NumberSymbol, metaclass=Singleton):
r"""The tribonacci constant.
Explanation
===========
The tribonacci numbers are like the Fibonacci numbers, but instead
of starting with two predetermined terms, the sequence starts with
three predetermined terms and each term afterwards is the sum of the
preceding three terms.
The tribonacci constant is the ratio toward which adjacent tribonacci
numbers tend. It is a root of the polynomial `x^3 - x^2 - x - 1 = 0`,
and also satisfies the equation `x + x^{-3} = 2`.
TribonacciConstant is a singleton, and can be accessed
by ``S.TribonacciConstant``.
Examples
========
>>> from sympy import S
>>> S.TribonacciConstant > 1
True
>>> S.TribonacciConstant.expand(func=True)
1/3 + (19 - 3*sqrt(33))**(1/3)/3 + (3*sqrt(33) + 19)**(1/3)/3
>>> S.TribonacciConstant.is_irrational
True
>>> S.TribonacciConstant.n(20)
1.8392867552141611326
References
==========
.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
"""
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"\text{TribonacciConstant}"
def __int__(self):
return 1
def _eval_evalf(self, prec):
rv = self._eval_expand_func(function=True)._eval_evalf(prec + 4)
return Float(rv, precision=prec)
def _eval_expand_func(self, **hints):
from sympy.functions.elementary.miscellaneous import cbrt, sqrt
return (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (S.One, Rational(2))
elif issubclass(number_cls, Rational):
pass
_eval_rewrite_as_sqrt = _eval_expand_func
class EulerGamma(NumberSymbol, metaclass=Singleton):
r"""The Euler-Mascheroni constant.
Explanation
===========
`\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] https://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, 1))
class Catalan(NumberSymbol, metaclass=Singleton):
r"""Catalan's constant.
Explanation
===========
$G = 0.91596559\ldots$ is given by the infinite series
.. math:: G = \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] https://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, 1), S.One)
def _eval_rewrite_as_Sum(self, k_sym=None, symbols=None):
if (k_sym is not None) or (symbols is not None):
return self
from .symbol import Dummy
from sympy.concrete.summations import Sum
k = Dummy('k', integer=True, nonnegative=True)
return Sum(S.NegativeOne**k / (2*k+1)**2, (k, 0, S.Infinity))
def _latex(self, printer):
return "G"
class ImaginaryUnit(AtomicExpr, metaclass=Singleton):
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] https://en.wikipedia.org/wiki/Imaginary_unit
"""
is_commutative = True
is_imaginary = True
is_finite = True
is_number = True
is_algebraic = True
is_transcendental = False
kind = NumberKind
__slots__ = ()
def _latex(self, printer):
return printer._settings['imaginary_unit_latex']
@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, Integer):
expt = expt % 4
if expt == 0:
return S.One
elif expt == 1:
return S.ImaginaryUnit
elif expt == 2:
return S.NegativeOne
elif expt == 3:
return -S.ImaginaryUnit
if isinstance(expt, Rational):
i, r = divmod(expt, 2)
rv = Pow(S.ImaginaryUnit, r, evaluate=False)
if i % 2:
return Mul(S.NegativeOne, rv, evaluate=False)
return rv
def as_base_exp(self):
return S.NegativeOne, S.Half
@property
def _mpc_(self):
return (Float(0)._mpf_, Float(1)._mpf_)
I = S.ImaginaryUnit
def equal_valued(x, y):
"""Compare expressions treating plain floats as rationals.
Examples
========
>>> from sympy import S, symbols, Rational, Float
>>> from sympy.core.numbers import equal_valued
>>> equal_valued(1, 2)
False
>>> equal_valued(1, 1)
True
In SymPy expressions with Floats compare unequal to corresponding
expressions with rationals:
>>> x = symbols('x')
>>> x**2 == x**2.0
False
However an individual Float compares equal to a Rational:
>>> Rational(1, 2) == Float(0.5)
True
In a future version of SymPy this might change so that Rational and Float
compare unequal. This function provides the behavior currently expected of
``==`` so that it could still be used if the behavior of ``==`` were to
change in future.
>>> equal_valued(1, 1.0) # Float vs Rational
True
>>> equal_valued(S(1).n(3), S(1).n(5)) # Floats of different precision
True
Explanation
===========
In future SymPy verions Float and Rational might compare unequal and floats
with different precisions might compare unequal. In that context a function
is needed that can check if a number is equal to 1 or 0 etc. The idea is
that instead of testing ``if x == 1:`` if we want to accept floats like
``1.0`` as well then the test can be written as ``if equal_valued(x, 1):``
or ``if equal_valued(x, 2):``. Since this function is intended to be used
in situations where one or both operands are expected to be concrete
numbers like 1 or 0 the function does not recurse through the args of any
compound expression to compare any nested floats.
References
==========
.. [1] https://github.com/sympy/sympy/pull/20033
"""
x = _sympify(x)
y = _sympify(y)
# Handle everything except Float/Rational first
if not x.is_Float and not y.is_Float:
return x == y
elif x.is_Float and y.is_Float:
# Compare values without regard for precision
return x._mpf_ == y._mpf_
elif x.is_Float:
x, y = y, x
if not x.is_Rational:
return False
# Now y is Float and x is Rational. A simple approach at this point would
# just be x == Rational(y) but if y has a large exponent creating a
# Rational could be prohibitively expensive.
sign, man, exp, _ = y._mpf_
p, q = x.p, x.q
if sign:
man = -man
if exp == 0:
# y odd integer
return q == 1 and man == p
elif exp > 0:
# y even integer
if q != 1:
return False
if p.bit_length() != man.bit_length() + exp:
return False
return man << exp == p
else:
# y non-integer. Need p == man and q == 2**-exp
if p != man:
return False
neg_exp = -exp
if q.bit_length() - 1 != neg_exp:
return False
return (1 << neg_exp) == q
@dispatch(Tuple, Number) # type:ignore
def _eval_is_eq(self, other): # noqa: F811
return False
def sympify_fractions(f):
return Rational(f.numerator, f.denominator, 1)
_sympy_converter[fractions.Fraction] = sympify_fractions
if HAS_GMPY:
def sympify_mpz(x):
return Integer(int(x))
# XXX: The sympify_mpq function here was never used because it is
# overridden by the other sympify_mpq function below. Maybe it should just
# be removed or maybe it should be used for something...
def sympify_mpq(x):
return Rational(int(x.numerator), int(x.denominator))
_sympy_converter[type(gmpy.mpz(1))] = sympify_mpz
_sympy_converter[type(gmpy.mpq(1, 2))] = sympify_mpq
def sympify_mpmath_mpq(x):
p, q = x._mpq_
return Rational(p, q, 1)
_sympy_converter[type(mpmath.rational.mpq(1, 2))] = sympify_mpmath_mpq
def sympify_mpmath(x):
return Expr._from_mpmath(x, x.context.prec)
_sympy_converter[mpnumeric] = sympify_mpmath
def sympify_complex(a):
real, imag = list(map(sympify, (a.real, a.imag)))
return real + S.ImaginaryUnit*imag
_sympy_converter[complex] = sympify_complex
from .power import Pow, integer_nthroot
from .mul import Mul
Mul.identity = One()
from .add import Add
Add.identity = Zero()
def _register_classes():
numbers.Number.register(Number)
numbers.Real.register(Float)
numbers.Rational.register(Rational)
numbers.Integral.register(Integer)
_register_classes()
_illegal = (S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity)