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from typing import Tuple as tTuplefrom collections import defaultdictfrom functools import cmp_to_key, reducefrom operator import attrgetterfrom .basic import Basicfrom .parameters import global_parametersfrom .logic import _fuzzy_group, fuzzy_or, fuzzy_notfrom .singleton import Sfrom .operations import AssocOp, AssocOpDispatcherfrom .cache import cacheitfrom .numbers import ilcm, igcdfrom .expr import Exprfrom .kind import UndefinedKindfrom 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): """Nice order of classes""" 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)
def as_coefficients_dict(a): """Return a dictionary mapping terms to their Rational coefficient.
Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term.
Examples ========
>>> from sympy.abc import a, x >>> (3*x + a*x + 4).as_coefficients_dict() {1: 4, x: 3, a*x: 1} >>> _[a] 0 >>> (3*a*x).as_coefficients_dict() {a*x: 3} """
d = defaultdict(list) for ai in a.args: c, m = ai.as_coeff_Mul() d[m].append(c) for k, v in d.items(): if len(v) == 1: d[k] = v[0] else: d[k] = Add(*v) di = defaultdict(int) di.update(d) return di
@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 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 (S.ImaginaryUnit*a).is_extended_real: im_I.append(a*S.ImaginaryUnit) else: return b = self.func(*nz) 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 (S.ImaginaryUnit*a).is_extended_real: im_or_z = True 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 _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): """
returns 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.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
old = self
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 = dict(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]
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
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) n0 = min.getn() res = Order(1) incr = S.One while res.is_Order: res = old._eval_nseries(x, n=n0+incr, logx=None, 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)
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] = prod(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 Add(*[-i for i in self.args])
add = AssocOpDispatcher('add')
from .mul import Mul, _keep_coeff, prod, _unevaluated_Mulfrom .numbers import Rational
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