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from typing import Tuple as tTuplefrom collections import defaultdictfrom functools import cmp_to_key, reducefrom itertools import productimport operator
from .sympify import sympifyfrom .basic import Basicfrom .singleton import Sfrom .operations import AssocOp, AssocOpDispatcherfrom .cache import cacheitfrom .logic import fuzzy_not, _fuzzy_groupfrom .expr import Exprfrom .parameters import global_parametersfrom .kind import KindDispatcherfrom .traversal import bottom_up
from sympy.utilities.iterables import sift
# internal marker to indicate:# "there are still non-commutative objects -- don't forget to process them"class NC_Marker: is_Order = False is_Mul = False is_Number = False is_Poly = False
is_commutative = False
# Key for sorting commutative args in canonical order_args_sortkey = cmp_to_key(Basic.compare)def _mulsort(args): # in-place sorting of args args.sort(key=_args_sortkey)
def _unevaluated_Mul(*args): """Return a well-formed unevaluated Mul: Numbers are collected and
put in slot 0, any arguments that are Muls will be flattened, and args are sorted. Use this when args have changed but you still want to return an unevaluated Mul.
Examples ========
>>> from sympy.core.mul import _unevaluated_Mul as uMul >>> from sympy import S, sqrt, Mul >>> from sympy.abc import x >>> a = uMul(*[S(3.0), x, S(2)]) >>> a.args[0] 6.00000000000000 >>> a.args[1] x
Two unevaluated Muls with the same arguments will always compare as equal during testing:
>>> m = uMul(sqrt(2), sqrt(3)) >>> m == uMul(sqrt(3), sqrt(2)) True >>> u = Mul(sqrt(3), sqrt(2), evaluate=False) >>> m == uMul(u) True >>> m == Mul(*m.args) False
"""
args = list(args) newargs = [] ncargs = [] co = S.One while args: a = args.pop() if a.is_Mul: c, nc = a.args_cnc() args.extend(c) if nc: ncargs.append(Mul._from_args(nc)) elif a.is_Number: co *= a else: newargs.append(a) _mulsort(newargs) if co is not S.One: newargs.insert(0, co) if ncargs: newargs.append(Mul._from_args(ncargs)) return Mul._from_args(newargs)
class Mul(Expr, AssocOp): """
Expression representing multiplication operation for algebraic field.
.. 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 ``Mul()`` must be ``Expr``. Infix operator ``*`` on most scalar objects in SymPy calls this class.
Another use of ``Mul()`` is to represent the structure of abstract multiplication so that its arguments can be substituted to return different class. Refer to examples section for this.
``Mul()`` evaluates the argument unless ``evaluate=False`` is passed. The evaluation logic includes:
1. Flattening ``Mul(x, Mul(y, z))`` -> ``Mul(x, y, z)``
2. Identity removing ``Mul(x, 1, y)`` -> ``Mul(x, y)``
3. Exponent collecting by ``.as_base_exp()`` ``Mul(x, x**2)`` -> ``Pow(x, 3)``
4. Term sorting ``Mul(y, x, 2)`` -> ``Mul(2, x, y)``
Since multiplication can be vector space operation, arguments may have the different :obj:`sympy.core.kind.Kind()`. Kind of the resulting object is automatically inferred.
Examples ========
>>> from sympy import Mul >>> from sympy.abc import x, y >>> Mul(x, 1) x >>> Mul(x, x) x**2
If ``evaluate=False`` is passed, result is not evaluated.
>>> Mul(1, 2, evaluate=False) 1*2 >>> Mul(x, x, evaluate=False) x*x
``Mul()`` also represents the general structure of multiplication operation.
>>> from sympy import MatrixSymbol >>> A = MatrixSymbol('A', 2,2) >>> expr = Mul(x,y).subs({y:A}) >>> expr x*A >>> type(expr) <class 'sympy.matrices.expressions.matmul.MatMul'>
See Also ========
MatMul
"""
__slots__ = ()
args: tTuple[Expr]
is_Mul = True
_args_type = Expr _kind_dispatcher = KindDispatcher("Mul_kind_dispatcher", commutative=True)
@property def kind(self): arg_kinds = (a.kind for a in self.args) return self._kind_dispatcher(*arg_kinds)
def could_extract_minus_sign(self): if self == (-self): return False # e.g. zoo*x == -zoo*x c = self.args[0] return c.is_Number and c.is_extended_negative
def __neg__(self): c, args = self.as_coeff_mul() if args[0] is not S.ComplexInfinity: c = -c if c is not S.One: if args[0].is_Number: args = list(args) if c is S.NegativeOne: args[0] = -args[0] else: args[0] *= c else: args = (c,) + args return self._from_args(args, self.is_commutative)
@classmethod def flatten(cls, seq): """Return commutative, noncommutative and order arguments by
combining related terms.
Notes ===== * In an expression like ``a*b*c``, Python process this through SymPy as ``Mul(Mul(a, b), c)``. This can have undesirable consequences.
- Sometimes terms are not combined as one would like: {c.f. https://github.com/sympy/sympy/issues/4596}
>>> from sympy import Mul, sqrt >>> from sympy.abc import x, y, z >>> 2*(x + 1) # this is the 2-arg Mul behavior 2*x + 2 >>> y*(x + 1)*2 2*y*(x + 1) >>> 2*(x + 1)*y # 2-arg result will be obtained first y*(2*x + 2) >>> Mul(2, x + 1, y) # all 3 args simultaneously processed 2*y*(x + 1) >>> 2*((x + 1)*y) # parentheses can control this behavior 2*y*(x + 1)
Powers with compound bases may not find a single base to combine with unless all arguments are processed at once. Post-processing may be necessary in such cases. {c.f. https://github.com/sympy/sympy/issues/5728}
>>> a = sqrt(x*sqrt(y)) >>> a**3 (x*sqrt(y))**(3/2) >>> Mul(a,a,a) (x*sqrt(y))**(3/2) >>> a*a*a x*sqrt(y)*sqrt(x*sqrt(y)) >>> _.subs(a.base, z).subs(z, a.base) (x*sqrt(y))**(3/2)
- If more than two terms are being multiplied then all the previous terms will be re-processed for each new argument. So if each of ``a``, ``b`` and ``c`` were :class:`Mul` expression, then ``a*b*c`` (or building up the product with ``*=``) will process all the arguments of ``a`` and ``b`` twice: once when ``a*b`` is computed and again when ``c`` is multiplied.
Using ``Mul(a, b, c)`` will process all arguments once.
* The results of Mul are cached according to arguments, so flatten will only be called once for ``Mul(a, b, c)``. If you can structure a calculation so the arguments are most likely to be repeats then this can save time in computing the answer. For example, say you had a Mul, M, that you wished to divide by ``d[i]`` and multiply by ``n[i]`` and you suspect there are many repeats in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the product, ``M*n[i]`` will be returned without flattening -- the cached value will be returned. If you divide by the ``d[i]`` first (and those are more unique than the ``n[i]``) then that will create a new Mul, ``M/d[i]`` the args of which will be traversed again when it is multiplied by ``n[i]``.
{c.f. https://github.com/sympy/sympy/issues/5706}
This consideration is moot if the cache is turned off.
NB -- The validity of the above notes depends on the implementation details of Mul and flatten which may change at any time. Therefore, you should only consider them when your code is highly performance sensitive.
Removal of 1 from the sequence is already handled by AssocOp.__new__. """
from sympy.calculus.accumulationbounds import AccumBounds from sympy.matrices.expressions import MatrixExpr rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a seq = [a, b] assert a is not S.One if not a.is_zero and a.is_Rational: r, b = b.as_coeff_Mul() if b.is_Add: if r is not S.One: # 2-arg hack # leave the Mul as a Mul? ar = a*r if ar is S.One: arb = b else: arb = cls(a*r, b, evaluate=False) rv = [arb], [], None elif global_parameters.distribute and b.is_commutative: newb = Add(*[_keep_coeff(a, bi) for bi in b.args]) rv = [newb], [], None if rv: return rv
# apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors
nc_seq = []
coeff = S.One # standalone term # e.g. 3 * ...
c_powers = [] # (base,exp) n # e.g. (x,n) for x
num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ...
neg1e = S.Zero # exponent on -1 extracted from Number-based Pow and I
pnum_rat = {} # (num-base, Rat-exp) 1/2 # e.g. (3, 1/2) for ... * 3 * ...
order_symbols = None
# --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # o neg1e # o pnum_rat # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_variables(order_symbols)
# Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy?
else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q)
# append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker)
continue
# 3 elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o.is_zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number or isinstance(coeff, AccumBounds): # it could be zoo coeff *= o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue
elif isinstance(o, AccumBounds): coeff = o.__mul__(coeff) continue
elif o is S.ComplexInfinity: if not coeff: # 0 * zoo = NaN return [S.NaN], [], None coeff = S.ComplexInfinity continue
elif o is S.ImaginaryUnit: neg1e += S.Half continue
elif o.is_commutative: # e # o = b b, e = o.as_base_exp()
# y # 3 if o.is_Pow: if b.is_Number:
# get all the factors with numeric base so they can be # combined below, but don't combine negatives unless # the exponent is an integer if e.is_Rational: if e.is_Integer: coeff *= Pow(b, e) # it is an unevaluated power continue elif e.is_negative: # also a sign of an unevaluated power seq.append(Pow(b, e)) continue elif b.is_negative: neg1e += e b = -b if b is not S.One: pnum_rat.setdefault(b, []).append(e) continue elif b.is_positive or e.is_integer: num_exp.append((b, e)) continue
c_powers.append((b, e))
# NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o)
# process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue
# b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1, e1 = o1.as_base_exp() b2, e2 = o.as_base_exp() new_exp = e1 + e2 # Only allow powers to combine if the new exponent is # not an Add. This allow things like a**2*b**3 == a**5 # if a.is_commutative == False, but prohibits # a**x*a**y and x**a*x**b from combining (x,y commute). if b1 == b2 and (not new_exp.is_Add): o12 = b1 ** new_exp
# now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12)
else: nc_part.append(o1) nc_part.append(o)
# We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine if two exponents have the same term by using # as_coeff_Mul. # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so things like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down.
# gather exponents of common bases... def _gather(c_powers): common_b = {} # b:e for b, e in c_powers: co = e.as_coeff_Mul() common_b.setdefault(b, {}).setdefault( co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) new_c_powers = [] for b, e in common_b.items(): new_c_powers.extend([(b, c*t) for t, c in e.items()]) return new_c_powers
# in c_powers c_powers = _gather(c_powers)
# and in num_exp num_exp = _gather(num_exp)
# --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base
# ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # - pnum_rat: {(1/3, [1/3, 2/3, 1/4])}
# 0 1 # x -> 1 x -> x
# this should only need to run twice; if it fails because # it needs to be run more times, perhaps this should be # changed to a "while True" loop -- the only reason it # isn't such now is to allow a less-than-perfect result to # be obtained rather than raising an error or entering an # infinite loop for i in range(2): new_c_powers = [] changed = False for b, e in c_powers: if e.is_zero: # canceling out infinities yields NaN if (b.is_Add or b.is_Mul) and any(infty in b.args for infty in (S.ComplexInfinity, S.Infinity, S.NegativeInfinity)): return [S.NaN], [], None continue if e is S.One: if b.is_Number: coeff *= b continue p = b if e is not S.One: p = Pow(b, e) # check to make sure that the base doesn't change # after exponentiation; to allow for unevaluated # Pow, we only do so if b is not already a Pow if p.is_Pow and not b.is_Pow: bi = b b, e = p.as_base_exp() if b != bi: changed = True c_part.append(p) new_c_powers.append((b, e)) # there might have been a change, but unless the base # matches some other base, there is nothing to do if changed and len({ b for b, e in new_c_powers}) != len(new_c_powers): # start over again c_part = [] c_powers = _gather(new_c_powers) else: break
# x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: inv_exp_dict.setdefault(e, []).append(b) for e, b in inv_exp_dict.items(): inv_exp_dict[e] = cls(*b) c_part.extend([Pow(b, e) for e, b in inv_exp_dict.items() if e])
# b, e -> e' = sum(e), b # {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])} comb_e = {} for b, e in pnum_rat.items(): comb_e.setdefault(Add(*e), []).append(b) del pnum_rat # process them, reducing exponents to values less than 1 # and updating coeff if necessary else adding them to # num_rat for further processing num_rat = [] for e, b in comb_e.items(): b = cls(*b) if e.q == 1: coeff *= Pow(b, e) continue if e.p > e.q: e_i, ep = divmod(e.p, e.q) coeff *= Pow(b, e_i) e = Rational(ep, e.q) num_rat.append((b, e)) del comb_e
# extract gcd of bases in num_rat # 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4) pnew = defaultdict(list) i = 0 # steps through num_rat which may grow while i < len(num_rat): bi, ei = num_rat[i] grow = [] for j in range(i + 1, len(num_rat)): bj, ej = num_rat[j] g = bi.gcd(bj) if g is not S.One: # 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2 # this might have a gcd with something else e = ei + ej if e.q == 1: coeff *= Pow(g, e) else: if e.p > e.q: e_i, ep = divmod(e.p, e.q) # change e in place coeff *= Pow(g, e_i) e = Rational(ep, e.q) grow.append((g, e)) # update the jth item num_rat[j] = (bj/g, ej) # update bi that we are checking with bi = bi/g if bi is S.One: break if bi is not S.One: obj = Pow(bi, ei) if obj.is_Number: coeff *= obj else: # changes like sqrt(12) -> 2*sqrt(3) for obj in Mul.make_args(obj): if obj.is_Number: coeff *= obj else: assert obj.is_Pow bi, ei = obj.args pnew[ei].append(bi)
num_rat.extend(grow) i += 1
# combine bases of the new powers for e, b in pnew.items(): pnew[e] = cls(*b)
# handle -1 and I if neg1e: # treat I as (-1)**(1/2) and compute -1's total exponent p, q = neg1e.as_numer_denom() # if the integer part is odd, extract -1 n, p = divmod(p, q) if n % 2: coeff = -coeff # if it's a multiple of 1/2 extract I if q == 2: c_part.append(S.ImaginaryUnit) elif p: # see if there is any positive base this power of # -1 can join neg1e = Rational(p, q) for e, b in pnew.items(): if e == neg1e and b.is_positive: pnew[e] = -b break else: # keep it separate; we've already evaluated it as # much as possible so evaluate=False c_part.append(Pow(S.NegativeOne, neg1e, evaluate=False))
# add all the pnew powers c_part.extend([Pow(b, e) for e, b in pnew.items()])
# oo, -oo if coeff in (S.Infinity, S.NegativeInfinity): def _handle_for_oo(c_part, coeff_sign): new_c_part = [] for t in c_part: if t.is_extended_positive: continue if t.is_extended_negative: coeff_sign *= -1 continue new_c_part.append(t) return new_c_part, coeff_sign c_part, coeff_sign = _handle_for_oo(c_part, 1) nc_part, coeff_sign = _handle_for_oo(nc_part, coeff_sign) coeff *= coeff_sign
# zoo if coeff is S.ComplexInfinity: # zoo might be # infinite_real + bounded_im # bounded_real + infinite_im # infinite_real + infinite_im # and non-zero real or imaginary will not change that status. c_part = [c for c in c_part if not (fuzzy_not(c.is_zero) and c.is_extended_real is not None)] nc_part = [c for c in nc_part if not (fuzzy_not(c.is_zero) and c.is_extended_real is not None)]
# 0 elif coeff.is_zero: # we know for sure the result will be 0 except the multiplicand # is infinity or a matrix if any(isinstance(c, MatrixExpr) for c in nc_part): return [coeff], nc_part, order_symbols if any(c.is_finite == False for c in c_part): return [S.NaN], [], order_symbols return [coeff], [], order_symbols
# check for straggling Numbers that were produced _new = [] for i in c_part: if i.is_Number: coeff *= i else: _new.append(i) c_part = _new
# order commutative part canonically _mulsort(c_part)
# current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff)
# we are done if (global_parameters.distribute and not nc_part and len(c_part) == 2 and c_part[0].is_Number and c_part[0].is_finite and c_part[1].is_Add): # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff*f for f in c_part[1].args])]
return c_part, nc_part, order_symbols
def _eval_power(self, e):
# don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B cargs, nc = self.args_cnc(split_1=False)
if e.is_Integer: return Mul(*[Pow(b, e, evaluate=False) for b in cargs]) * \ Pow(Mul._from_args(nc), e, evaluate=False) if e.is_Rational and e.q == 2: if self.is_imaginary: a = self.as_real_imag()[1] if a.is_Rational: from .power import integer_nthroot n, d = abs(a/2).as_numer_denom() n, t = integer_nthroot(n, 2) if t: d, t = integer_nthroot(d, 2) if t: from sympy.functions.elementary.complexes import sign r = sympify(n)/d return _unevaluated_Mul(r**e.p, (1 + sign(a)*S.ImaginaryUnit)**e.p)
p = Pow(self, e, evaluate=False)
if e.is_Rational or e.is_Float: return p._eval_expand_power_base()
return p
@classmethod def class_key(cls): return 3, 0, cls.__name__
def _eval_evalf(self, prec): c, m = self.as_coeff_Mul() if c is S.NegativeOne: if m.is_Mul: rv = -AssocOp._eval_evalf(m, prec) else: mnew = m._eval_evalf(prec) if mnew is not None: m = mnew rv = -m else: rv = AssocOp._eval_evalf(self, prec) if rv.is_number: return rv.expand() return rv
@property def _mpc_(self): """
Convert self to an mpmath mpc if possible """
from .numbers import Float im_part, imag_unit = self.as_coeff_Mul() if imag_unit is not 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 Mul to mpc. Must be of the form Number*I")
return (Float(0)._mpf_, Float(im_part)._mpf_)
@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_mul() which gives the head and a tuple containing the arguments of the tail when treated as a Mul. - if you want the coefficient when self is treated as an Add then use self.as_coeff_add()[0]
Examples ========
>>> from sympy.abc import x, y >>> (3*x*y).as_two_terms() (3, x*y) """
args = self.args
if len(args) == 1: return S.One, self elif len(args) == 2: return args
else: return args[0], self._new_rawargs(*args[1:])
@cacheit def as_coefficients_dict(self): """Return a dictionary mapping terms to their coefficient.
Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. The dictionary is considered to have a single term.
Examples ========
>>> from sympy.abc import a, x >>> (3*a*x).as_coefficients_dict() {a*x: 3} >>> _[a] 0 """
d = defaultdict(int) args = self.args
if len(args) == 1 or not args[0].is_Number: d[self] = S.One else: d[self._new_rawargs(*args[1:])] = args[0]
return d
@cacheit def as_coeff_mul(self, *deps, rational=True, **kwargs): if deps: l1, l2 = sift(self.args, lambda x: x.has(*deps), binary=True) return self._new_rawargs(*l2), tuple(l1) args = self.args if args[0].is_Number: if not rational or args[0].is_Rational: return args[0], args[1:] elif args[0].is_extended_negative: return S.NegativeOne, (-args[0],) + args[1:] return S.One, args
def as_coeff_Mul(self, rational=False): """
Efficiently extract the coefficient of a product. """
coeff, args = self.args[0], self.args[1:]
if coeff.is_Number: if not rational or coeff.is_Rational: if len(args) == 1: return coeff, args[0] else: return coeff, self._new_rawargs(*args) elif coeff.is_extended_negative: return S.NegativeOne, self._new_rawargs(*((-coeff,) + args)) return S.One, self
def as_real_imag(self, deep=True, **hints): from sympy.functions.elementary.complexes import Abs, im, re other = [] coeffr = [] coeffi = [] addterms = S.One for a in self.args: r, i = a.as_real_imag() if i.is_zero: coeffr.append(r) elif r.is_zero: coeffi.append(i*S.ImaginaryUnit) elif a.is_commutative: # search for complex conjugate pairs: for i, x in enumerate(other): if x == a.conjugate(): coeffr.append(Abs(x)**2) del other[i] break else: if a.is_Add: addterms *= a else: other.append(a) else: other.append(a) m = self.func(*other) if hints.get('ignore') == m: return if len(coeffi) % 2: imco = im(coeffi.pop(0)) # all other pairs make a real factor; they will be # put into reco below else: imco = S.Zero reco = self.func(*(coeffr + coeffi)) r, i = (reco*re(m), reco*im(m)) if addterms == 1: if m == 1: if imco.is_zero: return (reco, S.Zero) else: return (S.Zero, reco*imco) if imco is S.Zero: return (r, i) return (-imco*i, imco*r) from .function import expand_mul addre, addim = expand_mul(addterms, deep=False).as_real_imag() if imco is S.Zero: return (r*addre - i*addim, i*addre + r*addim) else: r, i = -imco*i, imco*r return (r*addre - i*addim, r*addim + i*addre)
@staticmethod def _expandsums(sums): """
Helper function for _eval_expand_mul.
sums must be a list of instances of Basic. """
L = len(sums) if L == 1: return sums[0].args terms = [] left = Mul._expandsums(sums[:L//2]) right = Mul._expandsums(sums[L//2:])
terms = [Mul(a, b) for a in left for b in right] added = Add(*terms) return Add.make_args(added) # it may have collapsed down to one term
def _eval_expand_mul(self, **hints): from sympy.simplify.radsimp import fraction
# Handle things like 1/(x*(x + 1)), which are automatically converted # to 1/x*1/(x + 1) expr = self n, d = fraction(expr) if d.is_Mul: n, d = [i._eval_expand_mul(**hints) if i.is_Mul else i for i in (n, d)] expr = n/d if not expr.is_Mul: return expr
plain, sums, rewrite = [], [], False for factor in expr.args: if factor.is_Add: sums.append(factor) rewrite = True else: if factor.is_commutative: plain.append(factor) else: sums.append(Basic(factor)) # Wrapper
if not rewrite: return expr else: plain = self.func(*plain) if sums: deep = hints.get("deep", False) terms = self.func._expandsums(sums) args = [] for term in terms: t = self.func(plain, term) if t.is_Mul and any(a.is_Add for a in t.args) and deep: t = t._eval_expand_mul() args.append(t) return Add(*args) else: return plain
@cacheit def _eval_derivative(self, s): args = list(self.args) terms = [] for i in range(len(args)): d = args[i].diff(s) if d: # Note: reduce is used in step of Mul as Mul is unable to # handle subtypes and operation priority: terms.append(reduce(lambda x, y: x*y, (args[:i] + [d] + args[i + 1:]), S.One)) return Add.fromiter(terms)
@cacheit def _eval_derivative_n_times(self, s, n): from .function import AppliedUndef from .symbol import Symbol, symbols, Dummy if not isinstance(s, (AppliedUndef, Symbol)): # other types of s may not be well behaved, e.g. # (cos(x)*sin(y)).diff([[x, y, z]]) return super()._eval_derivative_n_times(s, n) from .numbers import Integer args = self.args m = len(args) if isinstance(n, (int, Integer)): # https://en.wikipedia.org/wiki/General_Leibniz_rule#More_than_two_factors terms = [] from sympy.ntheory.multinomial import multinomial_coefficients_iterator for kvals, c in multinomial_coefficients_iterator(m, n): p = prod([arg.diff((s, k)) for k, arg in zip(kvals, args)]) terms.append(c * p) return Add(*terms) from sympy.concrete.summations import Sum from sympy.functions.combinatorial.factorials import factorial from sympy.functions.elementary.miscellaneous import Max kvals = symbols("k1:%i" % m, cls=Dummy) klast = n - sum(kvals) nfact = factorial(n) e, l = (# better to use the multinomial? nfact/prod(map(factorial, kvals))/factorial(klast)*\ prod([args[t].diff((s, kvals[t])) for t in range(m-1)])*\ args[-1].diff((s, Max(0, klast))), [(k, 0, n) for k in kvals]) return Sum(e, *l)
def _eval_difference_delta(self, n, step): from sympy.series.limitseq import difference_delta as dd arg0 = self.args[0] rest = Mul(*self.args[1:]) return (arg0.subs(n, n + step) * dd(rest, n, step) + dd(arg0, n, step) * rest)
def _matches_simple(self, expr, repl_dict): # handle (w*3).matches('x*5') -> {w: x*5/3} coeff, terms = self.as_coeff_Mul() terms = Mul.make_args(terms) if len(terms) == 1: newexpr = self.__class__._combine_inverse(expr, coeff) return terms[0].matches(newexpr, repl_dict) return
def matches(self, expr, repl_dict=None, old=False): expr = sympify(expr) if self.is_commutative and expr.is_commutative: return self._matches_commutative(expr, repl_dict, old) elif self.is_commutative is not expr.is_commutative: return None
# Proceed only if both both expressions are non-commutative c1, nc1 = self.args_cnc() c2, nc2 = expr.args_cnc() c1, c2 = [c or [1] for c in [c1, c2]]
# TODO: Should these be self.func? comm_mul_self = Mul(*c1) comm_mul_expr = Mul(*c2)
repl_dict = comm_mul_self.matches(comm_mul_expr, repl_dict, old)
# If the commutative arguments didn't match and aren't equal, then # then the expression as a whole doesn't match if not repl_dict and c1 != c2: return None
# Now match the non-commutative arguments, expanding powers to # multiplications nc1 = Mul._matches_expand_pows(nc1) nc2 = Mul._matches_expand_pows(nc2)
repl_dict = Mul._matches_noncomm(nc1, nc2, repl_dict)
return repl_dict or None
@staticmethod def _matches_expand_pows(arg_list): new_args = [] for arg in arg_list: if arg.is_Pow and arg.exp > 0: new_args.extend([arg.base] * arg.exp) else: new_args.append(arg) return new_args
@staticmethod def _matches_noncomm(nodes, targets, repl_dict=None): """Non-commutative multiplication matcher.
`nodes` is a list of symbols within the matcher multiplication expression, while `targets` is a list of arguments in the multiplication expression being matched against. """
if repl_dict is None: repl_dict = dict() else: repl_dict = repl_dict.copy()
# List of possible future states to be considered agenda = [] # The current matching state, storing index in nodes and targets state = (0, 0) node_ind, target_ind = state # Mapping between wildcard indices and the index ranges they match wildcard_dict = {}
while target_ind < len(targets) and node_ind < len(nodes): node = nodes[node_ind]
if node.is_Wild: Mul._matches_add_wildcard(wildcard_dict, state)
states_matches = Mul._matches_new_states(wildcard_dict, state, nodes, targets) if states_matches: new_states, new_matches = states_matches agenda.extend(new_states) if new_matches: for match in new_matches: repl_dict[match] = new_matches[match] if not agenda: return None else: state = agenda.pop() node_ind, target_ind = state
return repl_dict
@staticmethod def _matches_add_wildcard(dictionary, state): node_ind, target_ind = state if node_ind in dictionary: begin, end = dictionary[node_ind] dictionary[node_ind] = (begin, target_ind) else: dictionary[node_ind] = (target_ind, target_ind)
@staticmethod def _matches_new_states(dictionary, state, nodes, targets): node_ind, target_ind = state node = nodes[node_ind] target = targets[target_ind]
# Don't advance at all if we've exhausted the targets but not the nodes if target_ind >= len(targets) - 1 and node_ind < len(nodes) - 1: return None
if node.is_Wild: match_attempt = Mul._matches_match_wilds(dictionary, node_ind, nodes, targets) if match_attempt: # If the same node has been matched before, don't return # anything if the current match is diverging from the previous # match other_node_inds = Mul._matches_get_other_nodes(dictionary, nodes, node_ind) for ind in other_node_inds: other_begin, other_end = dictionary[ind] curr_begin, curr_end = dictionary[node_ind]
other_targets = targets[other_begin:other_end + 1] current_targets = targets[curr_begin:curr_end + 1]
for curr, other in zip(current_targets, other_targets): if curr != other: return None
# A wildcard node can match more than one target, so only the # target index is advanced new_state = [(node_ind, target_ind + 1)] # Only move on to the next node if there is one if node_ind < len(nodes) - 1: new_state.append((node_ind + 1, target_ind + 1)) return new_state, match_attempt else: # If we're not at a wildcard, then make sure we haven't exhausted # nodes but not targets, since in this case one node can only match # one target if node_ind >= len(nodes) - 1 and target_ind < len(targets) - 1: return None
match_attempt = node.matches(target)
if match_attempt: return [(node_ind + 1, target_ind + 1)], match_attempt elif node == target: return [(node_ind + 1, target_ind + 1)], None else: return None
@staticmethod def _matches_match_wilds(dictionary, wildcard_ind, nodes, targets): """Determine matches of a wildcard with sub-expression in `target`.""" wildcard = nodes[wildcard_ind] begin, end = dictionary[wildcard_ind] terms = targets[begin:end + 1] # TODO: Should this be self.func? mult = Mul(*terms) if len(terms) > 1 else terms[0] return wildcard.matches(mult)
@staticmethod def _matches_get_other_nodes(dictionary, nodes, node_ind): """Find other wildcards that may have already been matched.""" other_node_inds = [] for ind in dictionary: if nodes[ind] == nodes[node_ind]: other_node_inds.append(ind) return other_node_inds
@staticmethod def _combine_inverse(lhs, rhs): """
Returns lhs/rhs, but treats arguments like symbols, so things like oo/oo return 1 (instead of a nan) and ``I`` behaves like a symbol instead of sqrt(-1). """
from sympy.simplify.simplify import signsimp from .symbol import Dummy if lhs == rhs: return S.One
def check(l, r): if l.is_Float and r.is_comparable: # if both objects are added to 0 they will share the same "normalization" # and are more likely to compare the same. Since Add(foo, 0) will not allow # the 0 to pass, we use __add__ directly. return l.__add__(0) == r.evalf().__add__(0) return False if check(lhs, rhs) or check(rhs, lhs): return S.One if any(i.is_Pow or i.is_Mul for i in (lhs, rhs)): # gruntz and limit wants a literal I to not combine # with a power of -1 d = Dummy('I') _i = {S.ImaginaryUnit: d} i_ = {d: S.ImaginaryUnit} a = lhs.xreplace(_i).as_powers_dict() b = rhs.xreplace(_i).as_powers_dict() blen = len(b) for bi in tuple(b.keys()): if bi in a: a[bi] -= b.pop(bi) if not a[bi]: a.pop(bi) if len(b) != blen: lhs = Mul(*[k**v for k, v in a.items()]).xreplace(i_) rhs = Mul(*[k**v for k, v in b.items()]).xreplace(i_) rv = lhs/rhs srv = signsimp(rv) return srv if srv.is_Number else rv
def as_powers_dict(self): d = defaultdict(int) for term in self.args: for b, e in term.as_powers_dict().items(): d[b] += e return d
def as_numer_denom(self): # don't use _from_args to rebuild the numerators and denominators # as the order is not guaranteed to be the same once they have # been separated from each other numers, denoms = list(zip(*[f.as_numer_denom() for f in self.args])) return self.func(*numers), self.func(*denoms)
def as_base_exp(self): e1 = None bases = [] nc = 0 for m in self.args: b, e = m.as_base_exp() if not b.is_commutative: nc += 1 if e1 is None: e1 = e elif e != e1 or nc > 1: return self, S.One bases.append(b) return self.func(*bases), e1
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)
_eval_is_commutative = lambda self: _fuzzy_group( a.is_commutative for a in self.args)
def _eval_is_complex(self): comp = _fuzzy_group(a.is_complex for a in self.args) if comp is False: if any(a.is_infinite for a in self.args): if any(a.is_zero is not False for a in self.args): return None return False return comp
def _eval_is_finite(self): if all(a.is_finite for a in self.args): return True if any(a.is_infinite for a in self.args): if all(a.is_zero is False for a in self.args): return False
def _eval_is_infinite(self): if any(a.is_infinite for a in self.args): if any(a.is_zero for a in self.args): return S.NaN.is_infinite if any(a.is_zero is None for a in self.args): return None return True
def _eval_is_rational(self): r = _fuzzy_group((a.is_rational for a in self.args), quick_exit=True) if r: return r elif r is False: return self.is_zero
def _eval_is_algebraic(self): r = _fuzzy_group((a.is_algebraic for a in self.args), quick_exit=True) if r: return r elif r is False: return self.is_zero
def _eval_is_zero(self): zero = infinite = False for a in self.args: z = a.is_zero if z: if infinite: return # 0*oo is nan and nan.is_zero is None zero = True else: if not a.is_finite: if zero: return # 0*oo is nan and nan.is_zero is None infinite = True if zero is False and z is None: # trap None zero = None return zero
# without involving odd/even checks this code would suffice: #_eval_is_integer = lambda self: _fuzzy_group( # (a.is_integer for a in self.args), quick_exit=True) def _eval_is_integer(self): from sympy.ntheory.factor_ import trailing is_rational = self._eval_is_rational() if is_rational is False: return False
numerators = [] denominators = [] unknown = False for a in self.args: hit = False if a.is_integer: if abs(a) is not S.One: numerators.append(a) elif a.is_Rational: n, d = a.as_numer_denom() if abs(n) is not S.One: numerators.append(n) if d is not S.One: denominators.append(d) elif a.is_Pow: b, e = a.as_base_exp() if not b.is_integer or not e.is_integer: hit = unknown = True if e.is_negative: denominators.append(2 if a is S.Half else Pow(a, S.NegativeOne)) elif not hit: # int b and pos int e: a = b**e is integer assert not e.is_positive # for rational self and e equal to zero: a = b**e is 1 assert not e.is_zero return # sign of e unknown -> self.is_integer unknown else: return
if not denominators and not unknown: return True
allodd = lambda x: all(i.is_odd for i in x) alleven = lambda x: all(i.is_even for i in x) anyeven = lambda x: any(i.is_even for i in x)
from .relational import is_gt if not numerators and denominators and all(is_gt(_, S.One) for _ in denominators): return False elif unknown: return elif allodd(numerators) and anyeven(denominators): return False elif anyeven(numerators) and denominators == [2]: return True elif alleven(numerators) and allodd(denominators ) and (Mul(*denominators, evaluate=False) - 1 ).is_positive: return False if len(denominators) == 1: d = denominators[0] if d.is_Integer and d.is_even: # if minimal power of 2 in num vs den is not # negative then we have an integer if (Add(*[i.as_base_exp()[1] for i in numerators if i.is_even]) - trailing(d.p) ).is_nonnegative: return True if len(numerators) == 1: n = numerators[0] if n.is_Integer and n.is_even: # if minimal power of 2 in den vs num is positive # then we have have a non-integer if (Add(*[i.as_base_exp()[1] for i in denominators if i.is_even]) - trailing(n.p) ).is_positive: return False
def _eval_is_polar(self): has_polar = any(arg.is_polar for arg in self.args) return has_polar and \ all(arg.is_polar or arg.is_positive for arg in self.args)
def _eval_is_extended_real(self): return self._eval_real_imag(True)
def _eval_real_imag(self, real): zero = False t_not_re_im = None
for t in self.args: if (t.is_complex or t.is_infinite) is False and t.is_extended_real is False: return False elif t.is_imaginary: # I real = not real elif t.is_extended_real: # 2 if not zero: z = t.is_zero if not z and zero is False: zero = z elif z: if all(a.is_finite for a in self.args): return True return elif t.is_extended_real is False: # symbolic or literal like `2 + I` or symbolic imaginary if t_not_re_im: return # complex terms might cancel t_not_re_im = t elif t.is_imaginary is False: # symbolic like `2` or `2 + I` if t_not_re_im: return # complex terms might cancel t_not_re_im = t else: return
if t_not_re_im: if t_not_re_im.is_extended_real is False: if real: # like 3 return zero # 3*(smthng like 2 + I or i) is not real if t_not_re_im.is_imaginary is False: # symbolic 2 or 2 + I if not real: # like I return zero # I*(smthng like 2 or 2 + I) is not real elif zero is False: return real # can't be trumped by 0 elif real: return real # doesn't matter what zero is
def _eval_is_imaginary(self): z = self.is_zero if z: return False if self.is_finite is False: return False elif z is False and self.is_finite is True: return self._eval_real_imag(False)
def _eval_is_hermitian(self): return self._eval_herm_antiherm(True)
def _eval_herm_antiherm(self, real): one_nc = zero = one_neither = False
for t in self.args: if not t.is_commutative: if one_nc: return one_nc = True
if t.is_antihermitian: real = not real elif t.is_hermitian: if not zero: z = t.is_zero if not z and zero is False: zero = z elif z: if all(a.is_finite for a in self.args): return True return elif t.is_hermitian is False: if one_neither: return one_neither = True else: return
if one_neither: if real: return zero elif zero is False or real: return real
def _eval_is_antihermitian(self): z = self.is_zero if z: return False elif z is False: return self._eval_herm_antiherm(False)
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 and fuzzy_not(x.is_zero)) is True for x in others): return True return if a is None: return if all(x.is_real for x in self.args): return False
def _eval_is_extended_positive(self): """Return True if self is positive, False if not, and None if it
cannot be determined.
Explanation ===========
This algorithm is non-recursive and works by keeping track of the sign which changes when a negative or nonpositive is encountered. Whether a nonpositive or nonnegative is seen is also tracked since the presence of these makes it impossible to return True, but possible to return False if the end result is nonpositive. e.g.
pos * neg * nonpositive -> pos or zero -> None is returned pos * neg * nonnegative -> neg or zero -> False is returned """
return self._eval_pos_neg(1)
def _eval_pos_neg(self, sign): saw_NON = saw_NOT = False for t in self.args: if t.is_extended_positive: continue elif t.is_extended_negative: sign = -sign elif t.is_zero: if all(a.is_finite for a in self.args): return False return elif t.is_extended_nonpositive: sign = -sign saw_NON = True elif t.is_extended_nonnegative: saw_NON = True # FIXME: is_positive/is_negative is False doesn't take account of # Symbol('x', infinite=True, extended_real=True) which has # e.g. is_positive is False but has uncertain sign. elif t.is_positive is False: sign = -sign if saw_NOT: return saw_NOT = True elif t.is_negative is False: if saw_NOT: return saw_NOT = True else: return if sign == 1 and saw_NON is False and saw_NOT is False: return True if sign < 0: return False
def _eval_is_extended_negative(self): return self._eval_pos_neg(-1)
def _eval_is_odd(self): is_integer = self.is_integer if is_integer: if self.is_zero: return False from sympy.simplify.radsimp import fraction n, d = fraction(self) if d.is_Integer and d.is_even: from sympy.ntheory.factor_ import trailing # if minimal power of 2 in num vs den is # positive then we have an even number if (Add(*[i.as_base_exp()[1] for i in Mul.make_args(n) if i.is_even]) - trailing(d.p) ).is_positive: return False return r, acc = True, 1 for t in self.args: if abs(t) is S.One: continue assert t.is_integer if t.is_even: return False if r is False: pass elif acc != 1 and (acc + t).is_odd: r = False elif t.is_even is None: r = None acc = t return r return is_integer # !integer -> !odd
def _eval_is_even(self): is_integer = self.is_integer
if is_integer: return fuzzy_not(self.is_odd)
from sympy.simplify.radsimp import fraction n, d = fraction(self) if n.is_Integer and n.is_even: # if minimal power of 2 in den vs num is not # negative then this is not an integer and # can't be even from sympy.ntheory.factor_ import trailing if (Add(*[i.as_base_exp()[1] for i in Mul.make_args(d) if i.is_even]) - trailing(n.p) ).is_nonnegative: return False return is_integer
def _eval_is_composite(self): """
Here we count the number of arguments that have a minimum value greater than two. If there are more than one of such a symbol then the result is composite. Else, the result cannot be determined. """
number_of_args = 0 # count of symbols with minimum value greater than one for arg in self.args: if not (arg.is_integer and arg.is_positive): return None if (arg-1).is_positive: number_of_args += 1
if number_of_args > 1: return True
def _eval_subs(self, old, new): from sympy.functions.elementary.complexes import sign from sympy.ntheory.factor_ import multiplicity from sympy.simplify.powsimp import powdenest from sympy.simplify.radsimp import fraction
if not old.is_Mul: return None
# try keep replacement literal so -2*x doesn't replace 4*x if old.args[0].is_Number and old.args[0] < 0: if self.args[0].is_Number: if self.args[0] < 0: return self._subs(-old, -new) return None
def base_exp(a): # if I and -1 are in a Mul, they get both end up with # a -1 base (see issue 6421); all we want here are the # true Pow or exp separated into base and exponent from sympy.functions.elementary.exponential import exp if a.is_Pow or isinstance(a, exp): return a.as_base_exp() return a, S.One
def breakup(eq): """break up powers of eq when treated as a Mul:
b**(Rational*e) -> b**e, Rational commutatives come back as a dictionary {b**e: Rational} noncommutatives come back as a list [(b**e, Rational)] """
(c, nc) = (defaultdict(int), list()) for a in Mul.make_args(eq): a = powdenest(a) (b, e) = base_exp(a) if e is not S.One: (co, _) = e.as_coeff_mul() b = Pow(b, e/co) e = co if a.is_commutative: c[b] += e else: nc.append([b, e]) return (c, nc)
def rejoin(b, co): """
Put rational back with exponent; in general this is not ok, but since we took it from the exponent for analysis, it's ok to put it back. """
(b, e) = base_exp(b) return Pow(b, e*co)
def ndiv(a, b): """if b divides a in an extractive way (like 1/4 divides 1/2
but not vice versa, and 2/5 does not divide 1/3) then return the integer number of times it divides, else return 0. """
if not b.q % a.q or not a.q % b.q: return int(a/b) return 0
# give Muls in the denominator a chance to be changed (see issue 5651) # rv will be the default return value rv = None n, d = fraction(self) self2 = self if d is not S.One: self2 = n._subs(old, new)/d._subs(old, new) if not self2.is_Mul: return self2._subs(old, new) if self2 != self: rv = self2
# Now continue with regular substitution.
# handle the leading coefficient and use it to decide if anything # should even be started; we always know where to find the Rational # so it's a quick test
co_self = self2.args[0] co_old = old.args[0] co_xmul = None if co_old.is_Rational and co_self.is_Rational: # if coeffs are the same there will be no updating to do # below after breakup() step; so skip (and keep co_xmul=None) if co_old != co_self: co_xmul = co_self.extract_multiplicatively(co_old) elif co_old.is_Rational: return rv
# break self and old into factors
(c, nc) = breakup(self2) (old_c, old_nc) = breakup(old)
# update the coefficients if we had an extraction # e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5 # then co_self in c is replaced by (3/5)**2 and co_residual # is 2*(1/7)**2
if co_xmul and co_xmul.is_Rational and abs(co_old) != 1: mult = S(multiplicity(abs(co_old), co_self)) c.pop(co_self) if co_old in c: c[co_old] += mult else: c[co_old] = mult co_residual = co_self/co_old**mult else: co_residual = 1
# do quick tests to see if we can't succeed
ok = True if len(old_nc) > len(nc): # more non-commutative terms ok = False elif len(old_c) > len(c): # more commutative terms ok = False elif {i[0] for i in old_nc}.difference({i[0] for i in nc}): # unmatched non-commutative bases ok = False elif set(old_c).difference(set(c)): # unmatched commutative terms ok = False elif any(sign(c[b]) != sign(old_c[b]) for b in old_c): # differences in sign ok = False if not ok: return rv
if not old_c: cdid = None else: rat = [] for (b, old_e) in old_c.items(): c_e = c[b] rat.append(ndiv(c_e, old_e)) if not rat[-1]: return rv cdid = min(rat)
if not old_nc: ncdid = None for i in range(len(nc)): nc[i] = rejoin(*nc[i]) else: ncdid = 0 # number of nc replacements we did take = len(old_nc) # how much to look at each time limit = cdid or S.Infinity # max number that we can take failed = [] # failed terms will need subs if other terms pass i = 0 while limit and i + take <= len(nc): hit = False
# the bases must be equivalent in succession, and # the powers must be extractively compatible on the # first and last factor but equal in between.
rat = [] for j in range(take): if nc[i + j][0] != old_nc[j][0]: break elif j == 0: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif j == take - 1: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif nc[i + j][1] != old_nc[j][1]: break else: rat.append(1) j += 1 else: ndo = min(rat) if ndo: if take == 1: if cdid: ndo = min(cdid, ndo) nc[i] = Pow(new, ndo)*rejoin(nc[i][0], nc[i][1] - ndo*old_nc[0][1]) else: ndo = 1
# the left residual
l = rejoin(nc[i][0], nc[i][1] - ndo* old_nc[0][1])
# eliminate all middle terms
mid = new
# the right residual (which may be the same as the middle if take == 2)
ir = i + take - 1 r = (nc[ir][0], nc[ir][1] - ndo* old_nc[-1][1]) if r[1]: if i + take < len(nc): nc[i:i + take] = [l*mid, r] else: r = rejoin(*r) nc[i:i + take] = [l*mid*r] else:
# there was nothing left on the right
nc[i:i + take] = [l*mid]
limit -= ndo ncdid += ndo hit = True if not hit:
# do the subs on this failing factor
failed.append(i) i += 1 else:
if not ncdid: return rv
# although we didn't fail, certain nc terms may have # failed so we rebuild them after attempting a partial # subs on them
failed.extend(range(i, len(nc))) for i in failed: nc[i] = rejoin(*nc[i]).subs(old, new)
# rebuild the expression
if cdid is None: do = ncdid elif ncdid is None: do = cdid else: do = min(ncdid, cdid)
margs = [] for b in c: if b in old_c:
# calculate the new exponent
e = c[b] - old_c[b]*do margs.append(rejoin(b, e)) else: margs.append(rejoin(b.subs(old, new), c[b])) if cdid and not ncdid:
# in case we are replacing commutative with non-commutative, # we want the new term to come at the front just like the # rest of this routine
margs = [Pow(new, cdid)] + margs return co_residual*self2.func(*margs)*self2.func(*nc)
def _eval_nseries(self, x, n, logx, cdir=0): from .function import PoleError from sympy.functions.elementary.integers import ceiling from sympy.series.order import Order
def coeff_exp(term, x): lt = term.as_coeff_exponent(x) if lt[0].has(x): try: lt = term.leadterm(x) except ValueError: return term, S.Zero return lt
ords = []
try: for t in self.args: coeff, exp = t.leadterm(x, logx=logx) if not coeff.has(x): ords.append((t, exp)) else: raise ValueError
n0 = sum(t[1] for t in ords if t[1].is_number) facs = [] for t, m in ords: n1 = ceiling(n - n0 + (m if m.is_number else 0)) s = t.nseries(x, n=n1, logx=logx, cdir=cdir) ns = s.getn() if ns is not None: if ns < n1: # less than expected n -= n1 - ns # reduce n facs.append(s)
except (ValueError, NotImplementedError, TypeError, AttributeError, PoleError): n0 = sympify(sum(t[1] for t in ords if t[1].is_number)) if n0.is_nonnegative: n0 = S.Zero facs = [t.nseries(x, n=ceiling(n-n0), logx=logx, cdir=cdir) for t in self.args] from sympy.simplify.powsimp import powsimp res = powsimp(self.func(*facs).expand(), combine='exp', deep=True) if res.has(Order): res += Order(x**n, x) return res
res = S.Zero ords2 = [Add.make_args(factor) for factor in facs]
for fac in product(*ords2): ords3 = [coeff_exp(term, x) for term in fac] coeffs, powers = zip(*ords3) power = sum(powers) if (power - n).is_negative: res += Mul(*coeffs)*(x**power)
def max_degree(e, x): if e is x: return S.One if e.is_Atom: return S.Zero if e.is_Add: return max(max_degree(a, x) for a in e.args) if e.is_Mul: return Add(*[max_degree(a, x) for a in e.args]) if e.is_Pow: return max_degree(e.base, x)*e.exp return S.Zero
if self.is_polynomial(x): from sympy.polys.polyerrors import PolynomialError from sympy.polys.polytools import degree try: if max_degree(self, x) >= n or degree(self, x) != degree(res, x): res += Order(x**n, x) except PolynomialError: pass else: return res
if res != self: res += Order(x**n, x) return res
def _eval_as_leading_term(self, x, logx=None, cdir=0): return self.func(*[t.as_leading_term(x, logx=logx, cdir=cdir) 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[::-1]])
def _eval_adjoint(self): return self.func(*[t.adjoint() for t in self.args[::-1]])
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 sqrt >>> (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive() (6, -sqrt(2)*(1 - sqrt(2)))
See docstring of Expr.as_content_primitive for more examples. """
coef = S.One args = [] for a in self.args: c, p = a.as_content_primitive(radical=radical, clear=clear) coef *= c if p is not S.One: args.append(p) # don't use self._from_args here to reconstruct args # since there may be identical args now that should be combined # e.g. (2+2*x)*(3+3*x) should be (6, (1 + x)**2) not (6, (1+x)*(1+x)) return coef, self.func(*args)
def as_ordered_factors(self, order=None): """Transform an expression into an ordered list of factors.
Examples ========
>>> from sympy import sin, cos >>> from sympy.abc import x, y
>>> (2*x*y*sin(x)*cos(x)).as_ordered_factors() [2, x, y, sin(x), cos(x)]
"""
cpart, ncpart = self.args_cnc() cpart.sort(key=lambda expr: expr.sort_key(order=order)) return cpart + ncpart
@property def _sorted_args(self): return tuple(self.as_ordered_factors())
mul = AssocOpDispatcher('mul')
def prod(a, start=1): """Return product of elements of a. Start with int 1 so if only
ints are included then an int result is returned.
Examples ========
>>> from sympy import prod, S >>> prod(range(3)) 0 >>> type(_) is int True >>> prod([S(2), 3]) 6 >>> _.is_Integer True
You can start the product at something other than 1:
>>> prod([1, 2], 3) 6
"""
return reduce(operator.mul, a, start)
def _keep_coeff(coeff, factors, clear=True, sign=False): """Return ``coeff*factors`` unevaluated if necessary.
If ``clear`` is False, do not keep the coefficient as a factor if it can be distributed on a single factor such that one or more terms will still have integer coefficients.
If ``sign`` is True, allow a coefficient of -1 to remain factored out.
Examples ========
>>> from sympy.core.mul import _keep_coeff >>> from sympy.abc import x, y >>> from sympy import S
>>> _keep_coeff(S.Half, x + 2) (x + 2)/2 >>> _keep_coeff(S.Half, x + 2, clear=False) x/2 + 1 >>> _keep_coeff(S.Half, (x + 2)*y, clear=False) y*(x + 2)/2 >>> _keep_coeff(S(-1), x + y) -x - y >>> _keep_coeff(S(-1), x + y, sign=True) -(x + y) """
if not coeff.is_Number: if factors.is_Number: factors, coeff = coeff, factors else: return coeff*factors if factors is S.One: return coeff if coeff is S.One: return factors elif coeff is S.NegativeOne and not sign: return -factors elif factors.is_Add: if not clear and coeff.is_Rational and coeff.q != 1: args = [i.as_coeff_Mul() for i in factors.args] args = [(_keep_coeff(c, coeff), m) for c, m in args] if any(c.is_Integer for c, _ in args): return Add._from_args([Mul._from_args( i[1:] if i[0] == 1 else i) for i in args]) return Mul(coeff, factors, evaluate=False) elif factors.is_Mul: margs = list(factors.args) if margs[0].is_Number: margs[0] *= coeff if margs[0] == 1: margs.pop(0) else: margs.insert(0, coeff) return Mul._from_args(margs) else: m = coeff*factors if m.is_Number and not factors.is_Number: m = Mul._from_args((coeff, factors)) return m
def expand_2arg(e): def do(e): if e.is_Mul: c, r = e.as_coeff_Mul() if c.is_Number and r.is_Add: return _unevaluated_Add(*[c*ri for ri in r.args]) return e return bottom_up(e, do)
from .numbers import Rationalfrom .power import Powfrom .add import Add, _unevaluated_Add
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