from typing import Tuple as tTuple
from collections import defaultdict
from functools import cmp_to_key, reduce
from operator import attrgetter
from .basic import Basic
from .parameters import global_parameters
from .logic import _fuzzy_group, fuzzy_or, fuzzy_not
from .singleton import S
from .operations import AssocOp, AssocOpDispatcher
from .cache import cacheit
from .numbers import ilcm, igcd, equal_valued
from .expr import Expr
from .kind import UndefinedKind
from sympy.utilities.iterables import is_sequence, sift
# Key for sorting commutative args in canonical order
_args_sortkey = cmp_to_key(Basic.compare)
def _could_extract_minus_sign(expr):
# assume expr is Add-like
# We choose the one with less arguments with minus signs
negative_args = sum(1 for i in expr.args
if i.could_extract_minus_sign())
positive_args = len(expr.args) - negative_args
if positive_args > negative_args:
return False
elif positive_args < negative_args:
return True
# choose based on .sort_key() to prefer
# x - 1 instead of 1 - x and
# 3 - sqrt(2) instead of -3 + sqrt(2)
return bool(expr.sort_key() < (-expr).sort_key())
def _addsort(args):
# in-place sorting of args
args.sort(key=_args_sortkey)
def _unevaluated_Add(*args):
"""Return a well-formed unevaluated Add: Numbers are collected and
put in slot 0 and args are sorted. Use this when args have changed
but you still want to return an unevaluated Add.
Examples
========
>>> from sympy.core.add import _unevaluated_Add as uAdd
>>> from sympy import S, Add
>>> from sympy.abc import x, y
>>> a = uAdd(*[S(1.0), x, S(2)])
>>> a.args[0]
3.00000000000000
>>> a.args[1]
x
Beyond the Number being in slot 0, there is no other assurance of
order for the arguments since they are hash sorted. So, for testing
purposes, output produced by this in some other function can only
be tested against the output of this function or as one of several
options:
>>> opts = (Add(x, y, evaluate=False), Add(y, x, evaluate=False))
>>> a = uAdd(x, y)
>>> assert a in opts and a == uAdd(x, y)
>>> uAdd(x + 1, x + 2)
x + x + 3
"""
args = list(args)
newargs = []
co = S.Zero
while args:
a = args.pop()
if a.is_Add:
# this will keep nesting from building up
# so that x + (x + 1) -> x + x + 1 (3 args)
args.extend(a.args)
elif a.is_Number:
co += a
else:
newargs.append(a)
_addsort(newargs)
if co:
newargs.insert(0, co)
return Add._from_args(newargs)
class Add(Expr, AssocOp):
"""
Expression representing addition operation for algebraic group.
.. deprecated:: 1.7
Using arguments that aren't subclasses of :class:`~.Expr` in core
operators (:class:`~.Mul`, :class:`~.Add`, and :class:`~.Pow`) is
deprecated. See :ref:`non-expr-args-deprecated` for details.
Every argument of ``Add()`` must be ``Expr``. Infix operator ``+``
on most scalar objects in SymPy calls this class.
Another use of ``Add()`` is to represent the structure of abstract
addition so that its arguments can be substituted to return different
class. Refer to examples section for this.
``Add()`` evaluates the argument unless ``evaluate=False`` is passed.
The evaluation logic includes:
1. Flattening
``Add(x, Add(y, z))`` -> ``Add(x, y, z)``
2. Identity removing
``Add(x, 0, y)`` -> ``Add(x, y)``
3. Coefficient collecting by ``.as_coeff_Mul()``
``Add(x, 2*x)`` -> ``Mul(3, x)``
4. Term sorting
``Add(y, x, 2)`` -> ``Add(2, x, y)``
If no argument is passed, identity element 0 is returned. If single
element is passed, that element is returned.
Note that ``Add(*args)`` is more efficient than ``sum(args)`` because
it flattens the arguments. ``sum(a, b, c, ...)`` recursively adds the
arguments as ``a + (b + (c + ...))``, which has quadratic complexity.
On the other hand, ``Add(a, b, c, d)`` does not assume nested
structure, making the complexity linear.
Since addition is group operation, every argument should have the
same :obj:`sympy.core.kind.Kind()`.
Examples
========
>>> from sympy import Add, I
>>> from sympy.abc import x, y
>>> Add(x, 1)
x + 1
>>> Add(x, x)
2*x
>>> 2*x**2 + 3*x + I*y + 2*y + 2*x/5 + 1.0*y + 1
2*x**2 + 17*x/5 + 3.0*y + I*y + 1
If ``evaluate=False`` is passed, result is not evaluated.
>>> Add(1, 2, evaluate=False)
1 + 2
>>> Add(x, x, evaluate=False)
x + x
``Add()`` also represents the general structure of addition operation.
>>> from sympy import MatrixSymbol
>>> A,B = MatrixSymbol('A', 2,2), MatrixSymbol('B', 2,2)
>>> expr = Add(x,y).subs({x:A, y:B})
>>> expr
A + B
>>> type(expr)
<class 'sympy.matrices.expressions.matadd.MatAdd'>
Note that the printers do not display in args order.
>>> Add(x, 1)
x + 1
>>> Add(x, 1).args
(1, x)
See Also
========
MatAdd
"""
__slots__ = ()
args: tTuple[Expr, ...]
is_Add = True
_args_type = Expr
@classmethod
def flatten(cls, seq):
"""
Takes the sequence "seq" of nested Adds and returns a flatten list.
Returns: (commutative_part, noncommutative_part, order_symbols)
Applies associativity, all terms are commutable with respect to
addition.
NB: the removal of 0 is already handled by AssocOp.__new__
See Also
========
sympy.core.mul.Mul.flatten
"""
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.matrices.expressions import MatrixExpr
from sympy.tensor.tensor import TensExpr
rv = None
if len(seq) == 2:
a, b = seq
if b.is_Rational:
a, b = b, a
if a.is_Rational:
if b.is_Mul:
rv = [a, b], [], None
if rv:
if all(s.is_commutative for s in rv[0]):
return rv
return [], rv[0], None
terms = {} # term -> coeff
# e.g. x**2 -> 5 for ... + 5*x**2 + ...
coeff = S.Zero # coefficient (Number or zoo) to always be in slot 0
# e.g. 3 + ...
order_factors = []
extra = []
for o in seq:
# O(x)
if o.is_Order:
if o.expr.is_zero:
continue
for o1 in order_factors:
if o1.contains(o):
o = None
break
if o is None:
continue
order_factors = [o] + [
o1 for o1 in order_factors if not o.contains(o1)]
continue
# 3 or NaN
elif o.is_Number:
if (o is S.NaN or coeff is S.ComplexInfinity and
o.is_finite is False) and not extra:
# we know for sure the result will be nan
return [S.NaN], [], None
if coeff.is_Number or isinstance(coeff, AccumBounds):
coeff += o
if coeff is S.NaN and not extra:
# we know for sure the result will be nan
return [S.NaN], [], None
continue
elif isinstance(o, AccumBounds):
coeff = o.__add__(coeff)
continue
elif isinstance(o, MatrixExpr):
# can't add 0 to Matrix so make sure coeff is not 0
extra.append(o)
continue
elif isinstance(o, TensExpr):
coeff = o.__add__(coeff) if coeff else o
continue
elif o is S.ComplexInfinity:
if coeff.is_finite is False and not extra:
# we know for sure the result will be nan
return [S.NaN], [], None
coeff = S.ComplexInfinity
continue
# Add([...])
elif o.is_Add:
# NB: here we assume Add is always commutative
seq.extend(o.args) # TODO zerocopy?
continue
# Mul([...])
elif o.is_Mul:
c, s = o.as_coeff_Mul()
# check for unevaluated Pow, e.g. 2**3 or 2**(-1/2)
elif o.is_Pow:
b, e = o.as_base_exp()
if b.is_Number and (e.is_Integer or
(e.is_Rational and e.is_negative)):
seq.append(b**e)
continue
c, s = S.One, o
else:
# everything else
c = S.One
s = o
# now we have:
# o = c*s, where
#
# c is a Number
# s is an expression with number factor extracted
# let's collect terms with the same s, so e.g.
# 2*x**2 + 3*x**2 -> 5*x**2
if s in terms:
terms[s] += c
if terms[s] is S.NaN and not extra:
# we know for sure the result will be nan
return [S.NaN], [], None
else:
terms[s] = c
# now let's construct new args:
# [2*x**2, x**3, 7*x**4, pi, ...]
newseq = []
noncommutative = False
for s, c in terms.items():
# 0*s
if c.is_zero:
continue
# 1*s
elif c is S.One:
newseq.append(s)
# c*s
else:
if s.is_Mul:
# Mul, already keeps its arguments in perfect order.
# so we can simply put c in slot0 and go the fast way.
cs = s._new_rawargs(*((c,) + s.args))
newseq.append(cs)
elif s.is_Add:
# we just re-create the unevaluated Mul
newseq.append(Mul(c, s, evaluate=False))
else:
# alternatively we have to call all Mul's machinery (slow)
newseq.append(Mul(c, s))
noncommutative = noncommutative or not s.is_commutative
# oo, -oo
if coeff is S.Infinity:
newseq = [f for f in newseq if not (f.is_extended_nonnegative or f.is_real)]
elif coeff is S.NegativeInfinity:
newseq = [f for f in newseq if not (f.is_extended_nonpositive or f.is_real)]
if coeff is S.ComplexInfinity:
# zoo might be
# infinite_real + finite_im
# finite_real + infinite_im
# infinite_real + infinite_im
# addition of a finite real or imaginary number won't be able to
# change the zoo nature; adding an infinite qualtity would result
# in a NaN condition if it had sign opposite of the infinite
# portion of zoo, e.g., infinite_real - infinite_real.
newseq = [c for c in newseq if not (c.is_finite and
c.is_extended_real is not None)]
# process O(x)
if order_factors:
newseq2 = []
for t in newseq:
for o in order_factors:
# x + O(x) -> O(x)
if o.contains(t):
t = None
break
# x + O(x**2) -> x + O(x**2)
if t is not None:
newseq2.append(t)
newseq = newseq2 + order_factors
# 1 + O(1) -> O(1)
for o in order_factors:
if o.contains(coeff):
coeff = S.Zero
break
# order args canonically
_addsort(newseq)
# current code expects coeff to be first
if coeff is not S.Zero:
newseq.insert(0, coeff)
if extra:
newseq += extra
noncommutative = True
# we are done
if noncommutative:
return [], newseq, None
else:
return newseq, [], None
@classmethod
def class_key(cls):
return 3, 1, cls.__name__
@property
def kind(self):
k = attrgetter('kind')
kinds = map(k, self.args)
kinds = frozenset(kinds)
if len(kinds) != 1:
# Since addition is group operator, kind must be same.
# We know that this is unexpected signature, so return this.
result = UndefinedKind
else:
result, = kinds
return result
def could_extract_minus_sign(self):
return _could_extract_minus_sign(self)
@cacheit
def as_coeff_add(self, *deps):
"""
Returns a tuple (coeff, args) where self is treated as an Add and coeff
is the Number term and args is a tuple of all other terms.
Examples
========
>>> from sympy.abc import x
>>> (7 + 3*x).as_coeff_add()
(7, (3*x,))
>>> (7*x).as_coeff_add()
(0, (7*x,))
"""
if deps:
l1, l2 = sift(self.args, lambda x: x.has_free(*deps), binary=True)
return self._new_rawargs(*l2), tuple(l1)
coeff, notrat = self.args[0].as_coeff_add()
if coeff is not S.Zero:
return coeff, notrat + self.args[1:]
return S.Zero, self.args
def as_coeff_Add(self, rational=False, deps=None):
"""
Efficiently extract the coefficient of a summation.
"""
coeff, args = self.args[0], self.args[1:]
if coeff.is_Number and not rational or coeff.is_Rational:
return coeff, self._new_rawargs(*args)
return S.Zero, self
# Note, we intentionally do not implement Add.as_coeff_mul(). Rather, we
# let Expr.as_coeff_mul() just always return (S.One, self) for an Add. See
# issue 5524.
def _eval_power(self, e):
from .evalf import pure_complex
from .relational import is_eq
if len(self.args) == 2 and any(_.is_infinite for _ in self.args):
if e.is_zero is False and is_eq(e, S.One) is False:
# looking for literal a + I*b
a, b = self.args
if a.coeff(S.ImaginaryUnit):
a, b = b, a
ico = b.coeff(S.ImaginaryUnit)
if ico and ico.is_extended_real and a.is_extended_real:
if e.is_extended_negative:
return S.Zero
if e.is_extended_positive:
return S.ComplexInfinity
return
if e.is_Rational and self.is_number:
ri = pure_complex(self)
if ri:
r, i = ri
if e.q == 2:
from sympy.functions.elementary.miscellaneous import sqrt
D = sqrt(r**2 + i**2)
if D.is_Rational:
from .exprtools import factor_terms
from sympy.functions.elementary.complexes import sign
from .function import expand_multinomial
# (r, i, D) is a Pythagorean triple
root = sqrt(factor_terms((D - r)/2))**e.p
return root*expand_multinomial((
# principle value
(D + r)/abs(i) + sign(i)*S.ImaginaryUnit)**e.p)
elif e == -1:
return _unevaluated_Mul(
r - i*S.ImaginaryUnit,
1/(r**2 + i**2))
elif e.is_Number and abs(e) != 1:
# handle the Float case: (2.0 + 4*x)**e -> 4**e*(0.5 + x)**e
c, m = zip(*[i.as_coeff_Mul() for i in self.args])
if any(i.is_Float for i in c): # XXX should this always be done?
big = -1
for i in c:
if abs(i) >= big:
big = abs(i)
if big > 0 and not equal_valued(big, 1):
from sympy.functions.elementary.complexes import sign
bigs = (big, -big)
c = [sign(i) if i in bigs else i/big for i in c]
addpow = Add(*[c*m for c, m in zip(c, m)])**e
return big**e*addpow
@cacheit
def _eval_derivative(self, s):
return self.func(*[a.diff(s) for a in self.args])
def _eval_nseries(self, x, n, logx, cdir=0):
terms = [t.nseries(x, n=n, logx=logx, cdir=cdir) for t in self.args]
return self.func(*terms)
def _matches_simple(self, expr, repl_dict):
# handle (w+3).matches('x+5') -> {w: x+2}
coeff, terms = self.as_coeff_add()
if len(terms) == 1:
return terms[0].matches(expr - coeff, repl_dict)
return
def matches(self, expr, repl_dict=None, old=False):
return self._matches_commutative(expr, repl_dict, old)
@staticmethod
def _combine_inverse(lhs, rhs):
"""
Returns lhs - rhs, but treats oo like a symbol so oo - oo
returns 0, instead of a nan.
"""
from sympy.simplify.simplify import signsimp
inf = (S.Infinity, S.NegativeInfinity)
if lhs.has(*inf) or rhs.has(*inf):
from .symbol import Dummy
oo = Dummy('oo')
reps = {
S.Infinity: oo,
S.NegativeInfinity: -oo}
ireps = {v: k for k, v in reps.items()}
eq = lhs.xreplace(reps) - rhs.xreplace(reps)
if eq.has(oo):
eq = eq.replace(
lambda x: x.is_Pow and x.base is oo,
lambda x: x.base)
rv = eq.xreplace(ireps)
else:
rv = lhs - rhs
srv = signsimp(rv)
return srv if srv.is_Number else rv
@cacheit
def as_two_terms(self):
"""Return head and tail of self.
This is the most efficient way to get the head and tail of an
expression.
- if you want only the head, use self.args[0];
- if you want to process the arguments of the tail then use
self.as_coef_add() which gives the head and a tuple containing
the arguments of the tail when treated as an Add.
- if you want the coefficient when self is treated as a Mul
then use self.as_coeff_mul()[0]
>>> from sympy.abc import x, y
>>> (3*x - 2*y + 5).as_two_terms()
(5, 3*x - 2*y)
"""
return self.args[0], self._new_rawargs(*self.args[1:])
def as_numer_denom(self):
"""
Decomposes an expression to its numerator part and its
denominator part.
Examples
========
>>> from sympy.abc import x, y, z
>>> (x*y/z).as_numer_denom()
(x*y, z)
>>> (x*(y + 1)/y**7).as_numer_denom()
(x*(y + 1), y**7)
See Also
========
sympy.core.expr.Expr.as_numer_denom
"""
# clear rational denominator
content, expr = self.primitive()
if not isinstance(expr, Add):
return Mul(content, expr, evaluate=False).as_numer_denom()
ncon, dcon = content.as_numer_denom()
# collect numerators and denominators of the terms
nd = defaultdict(list)
for f in expr.args:
ni, di = f.as_numer_denom()
nd[di].append(ni)
# check for quick exit
if len(nd) == 1:
d, n = nd.popitem()
return self.func(
*[_keep_coeff(ncon, ni) for ni in n]), _keep_coeff(dcon, d)
# sum up the terms having a common denominator
for d, n in nd.items():
if len(n) == 1:
nd[d] = n[0]
else:
nd[d] = self.func(*n)
# assemble single numerator and denominator
denoms, numers = [list(i) for i in zip(*iter(nd.items()))]
n, d = self.func(*[Mul(*(denoms[:i] + [numers[i]] + denoms[i + 1:]))
for i in range(len(numers))]), Mul(*denoms)
return _keep_coeff(ncon, n), _keep_coeff(dcon, d)
def _eval_is_polynomial(self, syms):
return all(term._eval_is_polynomial(syms) for term in self.args)
def _eval_is_rational_function(self, syms):
return all(term._eval_is_rational_function(syms) for term in self.args)
def _eval_is_meromorphic(self, x, a):
return _fuzzy_group((arg.is_meromorphic(x, a) for arg in self.args),
quick_exit=True)
def _eval_is_algebraic_expr(self, syms):
return all(term._eval_is_algebraic_expr(syms) for term in self.args)
# assumption methods
_eval_is_real = lambda self: _fuzzy_group(
(a.is_real for a in self.args), quick_exit=True)
_eval_is_extended_real = lambda self: _fuzzy_group(
(a.is_extended_real for a in self.args), quick_exit=True)
_eval_is_complex = lambda self: _fuzzy_group(
(a.is_complex for a in self.args), quick_exit=True)
_eval_is_antihermitian = lambda self: _fuzzy_group(
(a.is_antihermitian for a in self.args), quick_exit=True)
_eval_is_finite = lambda self: _fuzzy_group(
(a.is_finite for a in self.args), quick_exit=True)
_eval_is_hermitian = lambda self: _fuzzy_group(
(a.is_hermitian for a in self.args), quick_exit=True)
_eval_is_integer = lambda self: _fuzzy_group(
(a.is_integer for a in self.args), quick_exit=True)
_eval_is_rational = lambda self: _fuzzy_group(
(a.is_rational for a in self.args), quick_exit=True)
_eval_is_algebraic = lambda self: _fuzzy_group(
(a.is_algebraic for a in self.args), quick_exit=True)
_eval_is_commutative = lambda self: _fuzzy_group(
a.is_commutative for a in self.args)
def _eval_is_infinite(self):
sawinf = False
for a in self.args:
ainf = a.is_infinite
if ainf is None:
return None
elif ainf is True:
# infinite+infinite might not be infinite
if sawinf is True:
return None
sawinf = True
return sawinf
def _eval_is_imaginary(self):
nz = []
im_I = []
for a in self.args:
if a.is_extended_real:
if a.is_zero:
pass
elif a.is_zero is False:
nz.append(a)
else:
return
elif a.is_imaginary:
im_I.append(a*S.ImaginaryUnit)
elif a.is_Mul and S.ImaginaryUnit in a.args:
coeff, ai = a.as_coeff_mul(S.ImaginaryUnit)
if ai == (S.ImaginaryUnit,) and coeff.is_extended_real:
im_I.append(-coeff)
else:
return
else:
return
b = self.func(*nz)
if b != self:
if b.is_zero:
return fuzzy_not(self.func(*im_I).is_zero)
elif b.is_zero is False:
return False
def _eval_is_zero(self):
if self.is_commutative is False:
# issue 10528: there is no way to know if a nc symbol
# is zero or not
return
nz = []
z = 0
im_or_z = False
im = 0
for a in self.args:
if a.is_extended_real:
if a.is_zero:
z += 1
elif a.is_zero is False:
nz.append(a)
else:
return
elif a.is_imaginary:
im += 1
elif a.is_Mul and S.ImaginaryUnit in a.args:
coeff, ai = a.as_coeff_mul(S.ImaginaryUnit)
if ai == (S.ImaginaryUnit,) and coeff.is_extended_real:
im_or_z = True
else:
return
else:
return
if z == len(self.args):
return True
if len(nz) in [0, len(self.args)]:
return None
b = self.func(*nz)
if b.is_zero:
if not im_or_z:
if im == 0:
return True
elif im == 1:
return False
if b.is_zero is False:
return False
def _eval_is_odd(self):
l = [f for f in self.args if not (f.is_even is True)]
if not l:
return False
if l[0].is_odd:
return self._new_rawargs(*l[1:]).is_even
def _eval_is_irrational(self):
for t in self.args:
a = t.is_irrational
if a:
others = list(self.args)
others.remove(t)
if all(x.is_rational is True for x in others):
return True
return None
if a is None:
return
return False
def _all_nonneg_or_nonppos(self):
nn = np = 0
for a in self.args:
if a.is_nonnegative:
if np:
return False
nn = 1
elif a.is_nonpositive:
if nn:
return False
np = 1
else:
break
else:
return True
def _eval_is_extended_positive(self):
if self.is_number:
return super()._eval_is_extended_positive()
c, a = self.as_coeff_Add()
if not c.is_zero:
from .exprtools import _monotonic_sign
v = _monotonic_sign(a)
if v is not None:
s = v + c
if s != self and s.is_extended_positive and a.is_extended_nonnegative:
return True
if len(self.free_symbols) == 1:
v = _monotonic_sign(self)
if v is not None and v != self and v.is_extended_positive:
return True
pos = nonneg = nonpos = unknown_sign = False
saw_INF = set()
args = [a for a in self.args if not a.is_zero]
if not args:
return False
for a in args:
ispos = a.is_extended_positive
infinite = a.is_infinite
if infinite:
saw_INF.add(fuzzy_or((ispos, a.is_extended_nonnegative)))
if True in saw_INF and False in saw_INF:
return
if ispos:
pos = True
continue
elif a.is_extended_nonnegative:
nonneg = True
continue
elif a.is_extended_nonpositive:
nonpos = True
continue
if infinite is None:
return
unknown_sign = True
if saw_INF:
if len(saw_INF) > 1:
return
return saw_INF.pop()
elif unknown_sign:
return
elif not nonpos and not nonneg and pos:
return True
elif not nonpos and pos:
return True
elif not pos and not nonneg:
return False
def _eval_is_extended_nonnegative(self):
if not self.is_number:
c, a = self.as_coeff_Add()
if not c.is_zero and a.is_extended_nonnegative:
from .exprtools import _monotonic_sign
v = _monotonic_sign(a)
if v is not None:
s = v + c
if s != self and s.is_extended_nonnegative:
return True
if len(self.free_symbols) == 1:
v = _monotonic_sign(self)
if v is not None and v != self and v.is_extended_nonnegative:
return True
def _eval_is_extended_nonpositive(self):
if not self.is_number:
c, a = self.as_coeff_Add()
if not c.is_zero and a.is_extended_nonpositive:
from .exprtools import _monotonic_sign
v = _monotonic_sign(a)
if v is not None:
s = v + c
if s != self and s.is_extended_nonpositive:
return True
if len(self.free_symbols) == 1:
v = _monotonic_sign(self)
if v is not None and v != self and v.is_extended_nonpositive:
return True
def _eval_is_extended_negative(self):
if self.is_number:
return super()._eval_is_extended_negative()
c, a = self.as_coeff_Add()
if not c.is_zero:
from .exprtools import _monotonic_sign
v = _monotonic_sign(a)
if v is not None:
s = v + c
if s != self and s.is_extended_negative and a.is_extended_nonpositive:
return True
if len(self.free_symbols) == 1:
v = _monotonic_sign(self)
if v is not None and v != self and v.is_extended_negative:
return True
neg = nonpos = nonneg = unknown_sign = False
saw_INF = set()
args = [a for a in self.args if not a.is_zero]
if not args:
return False
for a in args:
isneg = a.is_extended_negative
infinite = a.is_infinite
if infinite:
saw_INF.add(fuzzy_or((isneg, a.is_extended_nonpositive)))
if True in saw_INF and False in saw_INF:
return
if isneg:
neg = True
continue
elif a.is_extended_nonpositive:
nonpos = True
continue
elif a.is_extended_nonnegative:
nonneg = True
continue
if infinite is None:
return
unknown_sign = True
if saw_INF:
if len(saw_INF) > 1:
return
return saw_INF.pop()
elif unknown_sign:
return
elif not nonneg and not nonpos and neg:
return True
elif not nonneg and neg:
return True
elif not neg and not nonpos:
return False
def _eval_subs(self, old, new):
if not old.is_Add:
if old is S.Infinity and -old in self.args:
# foo - oo is foo + (-oo) internally
return self.xreplace({-old: -new})
return None
coeff_self, terms_self = self.as_coeff_Add()
coeff_old, terms_old = old.as_coeff_Add()
if coeff_self.is_Rational and coeff_old.is_Rational:
if terms_self == terms_old: # (2 + a).subs( 3 + a, y) -> -1 + y
return self.func(new, coeff_self, -coeff_old)
if terms_self == -terms_old: # (2 + a).subs(-3 - a, y) -> -1 - y
return self.func(-new, coeff_self, coeff_old)
if coeff_self.is_Rational and coeff_old.is_Rational \
or coeff_self == coeff_old:
args_old, args_self = self.func.make_args(
terms_old), self.func.make_args(terms_self)
if len(args_old) < len(args_self): # (a+b+c).subs(b+c,x) -> a+x
self_set = set(args_self)
old_set = set(args_old)
if old_set < self_set:
ret_set = self_set - old_set
return self.func(new, coeff_self, -coeff_old,
*[s._subs(old, new) for s in ret_set])
args_old = self.func.make_args(
-terms_old) # (a+b+c+d).subs(-b-c,x) -> a-x+d
old_set = set(args_old)
if old_set < self_set:
ret_set = self_set - old_set
return self.func(-new, coeff_self, coeff_old,
*[s._subs(old, new) for s in ret_set])
def removeO(self):
args = [a for a in self.args if not a.is_Order]
return self._new_rawargs(*args)
def getO(self):
args = [a for a in self.args if a.is_Order]
if args:
return self._new_rawargs(*args)
@cacheit
def extract_leading_order(self, symbols, point=None):
"""
Returns the leading term and its order.
Examples
========
>>> from sympy.abc import x
>>> (x + 1 + 1/x**5).extract_leading_order(x)
((x**(-5), O(x**(-5))),)
>>> (1 + x).extract_leading_order(x)
((1, O(1)),)
>>> (x + x**2).extract_leading_order(x)
((x, O(x)),)
"""
from sympy.series.order import Order
lst = []
symbols = list(symbols if is_sequence(symbols) else [symbols])
if not point:
point = [0]*len(symbols)
seq = [(f, Order(f, *zip(symbols, point))) for f in self.args]
for ef, of in seq:
for e, o in lst:
if o.contains(of) and o != of:
of = None
break
if of is None:
continue
new_lst = [(ef, of)]
for e, o in lst:
if of.contains(o) and o != of:
continue
new_lst.append((e, o))
lst = new_lst
return tuple(lst)
def as_real_imag(self, deep=True, **hints):
"""
Return a tuple representing a complex number.
Examples
========
>>> from sympy import I
>>> (7 + 9*I).as_real_imag()
(7, 9)
>>> ((1 + I)/(1 - I)).as_real_imag()
(0, 1)
>>> ((1 + 2*I)*(1 + 3*I)).as_real_imag()
(-5, 5)
"""
sargs = self.args
re_part, im_part = [], []
for term in sargs:
re, im = term.as_real_imag(deep=deep)
re_part.append(re)
im_part.append(im)
return (self.func(*re_part), self.func(*im_part))
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.core.symbol import Dummy, Symbol
from sympy.series.order import Order
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold
from .function import expand_mul
o = self.getO()
if o is None:
o = Order(0)
old = self.removeO()
if old.has(Piecewise):
old = piecewise_fold(old)
# This expansion is the last part of expand_log. expand_log also calls
# expand_mul with factor=True, which would be more expensive
if any(isinstance(a, log) for a in self.args):
logflags = {"deep": True, "log": True, "mul": False, "power_exp": False,
"power_base": False, "multinomial": False, "basic": False, "force": False,
"factor": False}
old = old.expand(**logflags)
expr = expand_mul(old)
if not expr.is_Add:
return expr.as_leading_term(x, logx=logx, cdir=cdir)
infinite = [t for t in expr.args if t.is_infinite]
_logx = Dummy('logx') if logx is None else logx
leading_terms = [t.as_leading_term(x, logx=_logx, cdir=cdir) for t in expr.args]
min, new_expr = Order(0), 0
try:
for term in leading_terms:
order = Order(term, x)
if not min or order not in min:
min = order
new_expr = term
elif min in order:
new_expr += term
except TypeError:
return expr
if logx is None:
new_expr = new_expr.subs(_logx, log(x))
is_zero = new_expr.is_zero
if is_zero is None:
new_expr = new_expr.trigsimp().cancel()
is_zero = new_expr.is_zero
if is_zero is True:
# simple leading term analysis gave us cancelled terms but we have to send
# back a term, so compute the leading term (via series)
try:
n0 = min.getn()
except NotImplementedError:
n0 = S.One
if n0.has(Symbol):
n0 = S.One
res = Order(1)
incr = S.One
while res.is_Order:
res = old._eval_nseries(x, n=n0+incr, logx=logx, cdir=cdir).cancel().powsimp().trigsimp()
incr *= 2
return res.as_leading_term(x, logx=logx, cdir=cdir)
elif new_expr is S.NaN:
return old.func._from_args(infinite) + o
else:
return new_expr
def _eval_adjoint(self):
return self.func(*[t.adjoint() for t in self.args])
def _eval_conjugate(self):
return self.func(*[t.conjugate() for t in self.args])
def _eval_transpose(self):
return self.func(*[t.transpose() for t in self.args])
def primitive(self):
"""
Return ``(R, self/R)`` where ``R``` is the Rational GCD of ``self```.
``R`` is collected only from the leading coefficient of each term.
Examples
========
>>> from sympy.abc import x, y
>>> (2*x + 4*y).primitive()
(2, x + 2*y)
>>> (2*x/3 + 4*y/9).primitive()
(2/9, 3*x + 2*y)
>>> (2*x/3 + 4.2*y).primitive()
(1/3, 2*x + 12.6*y)
No subprocessing of term factors is performed:
>>> ((2 + 2*x)*x + 2).primitive()
(1, x*(2*x + 2) + 2)
Recursive processing can be done with the ``as_content_primitive()``
method:
>>> ((2 + 2*x)*x + 2).as_content_primitive()
(2, x*(x + 1) + 1)
See also: primitive() function in polytools.py
"""
terms = []
inf = False
for a in self.args:
c, m = a.as_coeff_Mul()
if not c.is_Rational:
c = S.One
m = a
inf = inf or m is S.ComplexInfinity
terms.append((c.p, c.q, m))
if not inf:
ngcd = reduce(igcd, [t[0] for t in terms], 0)
dlcm = reduce(ilcm, [t[1] for t in terms], 1)
else:
ngcd = reduce(igcd, [t[0] for t in terms if t[1]], 0)
dlcm = reduce(ilcm, [t[1] for t in terms if t[1]], 1)
if ngcd == dlcm == 1:
return S.One, self
if not inf:
for i, (p, q, term) in enumerate(terms):
terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term)
else:
for i, (p, q, term) in enumerate(terms):
if q:
terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term)
else:
terms[i] = _keep_coeff(Rational(p, q), term)
# we don't need a complete re-flattening since no new terms will join
# so we just use the same sort as is used in Add.flatten. When the
# coefficient changes, the ordering of terms may change, e.g.
# (3*x, 6*y) -> (2*y, x)
#
# We do need to make sure that term[0] stays in position 0, however.
#
if terms[0].is_Number or terms[0] is S.ComplexInfinity:
c = terms.pop(0)
else:
c = None
_addsort(terms)
if c:
terms.insert(0, c)
return Rational(ngcd, dlcm), self._new_rawargs(*terms)
def as_content_primitive(self, radical=False, clear=True):
"""Return the tuple (R, self/R) where R is the positive Rational
extracted from self. If radical is True (default is False) then
common radicals will be removed and included as a factor of the
primitive expression.
Examples
========
>>> from sympy import sqrt
>>> (3 + 3*sqrt(2)).as_content_primitive()
(3, 1 + sqrt(2))
Radical content can also be factored out of the primitive:
>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))
See docstring of Expr.as_content_primitive for more examples.
"""
con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(
radical=radical, clear=clear)) for a in self.args]).primitive()
if not clear and not con.is_Integer and prim.is_Add:
con, d = con.as_numer_denom()
_p = prim/d
if any(a.as_coeff_Mul()[0].is_Integer for a in _p.args):
prim = _p
else:
con /= d
if radical and prim.is_Add:
# look for common radicals that can be removed
args = prim.args
rads = []
common_q = None
for m in args:
term_rads = defaultdict(list)
for ai in Mul.make_args(m):
if ai.is_Pow:
b, e = ai.as_base_exp()
if e.is_Rational and b.is_Integer:
term_rads[e.q].append(abs(int(b))**e.p)
if not term_rads:
break
if common_q is None:
common_q = set(term_rads.keys())
else:
common_q = common_q & set(term_rads.keys())
if not common_q:
break
rads.append(term_rads)
else:
# process rads
# keep only those in common_q
for r in rads:
for q in list(r.keys()):
if q not in common_q:
r.pop(q)
for q in r:
r[q] = Mul(*r[q])
# find the gcd of bases for each q
G = []
for q in common_q:
g = reduce(igcd, [r[q] for r in rads], 0)
if g != 1:
G.append(g**Rational(1, q))
if G:
G = Mul(*G)
args = [ai/G for ai in args]
prim = G*prim.func(*args)
return con, prim
@property
def _sorted_args(self):
from .sorting import default_sort_key
return tuple(sorted(self.args, key=default_sort_key))
def _eval_difference_delta(self, n, step):
from sympy.series.limitseq import difference_delta as dd
return self.func(*[dd(a, n, step) for a in self.args])
@property
def _mpc_(self):
"""
Convert self to an mpmath mpc if possible
"""
from .numbers import Float
re_part, rest = self.as_coeff_Add()
im_part, imag_unit = rest.as_coeff_Mul()
if not imag_unit == S.ImaginaryUnit:
# ValueError may seem more reasonable but since it's a @property,
# we need to use AttributeError to keep from confusing things like
# hasattr.
raise AttributeError("Cannot convert Add to mpc. Must be of the form Number + Number*I")
return (Float(re_part)._mpf_, Float(im_part)._mpf_)
def __neg__(self):
if not global_parameters.distribute:
return super().__neg__()
return Mul(S.NegativeOne, self)
add = AssocOpDispatcher('add')
from .mul import Mul, _keep_coeff, _unevaluated_Mul
from .numbers import Rational