Source code for sympy.core.mul

from __future__ import print_function, division

from collections import defaultdict
import operator

from .sympify import sympify
from .basic import Basic, C
from .singleton import S
from .operations import AssocOp
from .cache import cacheit
from .logic import fuzzy_not, _fuzzy_group
from .compatibility import cmp_to_key, reduce, xrange
from .expr import Expr

# 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)


[docs]class Mul(Expr, AssocOp): __slots__ = [] is_Mul = True #identity = S.One # cyclic import, so defined in numbers.py @classmethod
[docs] 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__. """ rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a assert not a is 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 rv = [cls(a*r, b, evaluate=False)], [], None elif b.is_commutative: if a is S.One: rv = [b], [], None else: r, b = b.as_coeff_Add() bargs = [_keep_coeff(a, bi) for bi in Add.make_args(b)] _addsort(bargs) ar = a*r if ar: bargs.insert(0, ar) bargs = [Add._from_args(bargs)] rv = bargs, [], 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 S.Zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number: # 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 o is S.ComplexInfinity: if not coeff: # 0 * zoo = NaN return [S.NaN], [], None if coeff is S.ComplexInfinity: # zoo * zoo = zoo return [S.ComplexInfinity], [], 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 elif b is S.ImaginaryUnit and e.is_Rational: neg1e += e/2 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): new_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) for b, e in common_b.items(): for t, c in e.items(): new_c_powers.append((b, c*t)) 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 for b, e in c_powers: if e is S.One: if b.is_Number: coeff *= b else: c_part.append(b) elif e is not S.Zero: c_part.append(Pow(b, e)) # 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 is S.Infinity) or (coeff is S.NegativeInfinity): def _handle_for_oo(c_part, coeff_sign): new_c_part = [] for t in c_part: if t.is_positive: continue if t.is_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 (c.is_nonzero and c.is_real is not None)] nc_part = [c for c in nc_part if not (c.is_nonzero and c.is_real is not None)] # 0 elif coeff is S.Zero: # we know for sure the result will be 0 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 (not nc_part and len(c_part) == 2 and c_part[0].is_Number 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(b, e): # don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B cargs, nc = b.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) p = Pow(b, 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 @cacheit
[docs] 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] >>> 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_coeff_mul(self, *deps): if deps: l1 = [] l2 = [] for f in self.args: if f.has(*deps): l2.append(f) else: l1.append(f) return self._new_rawargs(*l1), tuple(l2) args = self.args if args[0].is_Rational: return args[0], args[1:] elif args[0] is S.NegativeInfinity: return S.NegativeOne, (-args[0],) + args[1:] return S.One, args
[docs] 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 and not (rational and not coeff.is_Rational): if len(args) == 1: return coeff, args[0] else: return coeff, self._new_rawargs(*args) else: return S.One, self
def as_real_imag(self, deep=True, **hints): from sympy import expand_mul other = [] coeffr = [] coeffi = [] addterms = S.One for a in self.args: if a.is_real: coeffr.append(a) elif a.is_imaginary: coeffi.append(a) elif a.is_commutative: # search for complex conjugate pairs: for i, x in enumerate(other): if x == a.conjugate(): coeffr.append(C.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 = C.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*C.re(m), reco*C.im(m)) if addterms == 1: if m == 1: if imco is S.Zero: return (reco, S.Zero) else: return (S.Zero, reco*imco) if imco is S.Zero: return (r, i) return (-imco*i, imco*r) 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 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: 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): 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 xrange(len(args)): d = args[i].diff(s) if d: terms.append(self.func(*(args[:i] + [d] + args[i + 1:]))) return Add(*terms) 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={}, old=False): expr = sympify(expr) if self.is_commutative and expr.is_commutative: return AssocOp._matches_commutative(self, expr, repl_dict, old) elif self.is_commutative is not expr.is_commutative: return None c1, nc1 = self.args_cnc() c2, nc2 = expr.args_cnc() repl_dict = repl_dict.copy() if c1: if not c2: c2 = [1] a = self.func(*c1) if isinstance(a, AssocOp): repl_dict = a._matches_commutative(self.func(*c2), repl_dict, old) else: repl_dict = a.matches(self.func(*c2), repl_dict) if repl_dict: a = self.func(*nc1) if isinstance(a, self.func): repl_dict = a._matches(self.func(*nc2), repl_dict) else: repl_dict = a.matches(self.func(*nc2), repl_dict) return repl_dict or None def _matches(self, expr, repl_dict={}): # weed out negative one prefixes sign = 1 a, b = self.as_two_terms() if a is S.NegativeOne: if b.is_Mul: sign = -sign else: # the remainder, b, is not a Mul anymore return b.matches(-expr, repl_dict) expr = sympify(expr) if expr.is_Mul and expr.args[0] is S.NegativeOne: expr = -expr sign = -sign if not expr.is_Mul: # expr can only match if it matches b and a matches +/- 1 if len(self.args) == 2: # quickly test for equality if b == expr: return a.matches(Rational(sign), repl_dict) # do more expensive match dd = b.matches(expr, repl_dict) if dd is None: return None dd = a.matches(Rational(sign), dd) return dd return None d = repl_dict.copy() # weed out identical terms pp = list(self.args) ee = list(expr.args) for p in self.args: if p in expr.args: ee.remove(p) pp.remove(p) # only one symbol left in pattern -> match the remaining expression if len(pp) == 1 and isinstance(pp[0], C.Wild): if len(ee) == 1: d[pp[0]] = sign * ee[0] else: d[pp[0]] = sign * expr.func(*ee) return d if len(ee) != len(pp): return None for p, e in zip(pp, ee): d = p.xreplace(d).matches(e, d) if d is None: return None return d @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. """ 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 lhs.is_Mul and rhs.is_Mul: a = list(lhs.args) b = [1] for x in rhs.args: if x in a: a.remove(x) elif -x in a: a.remove(-x) b.append(-1) else: b.append(x) return lhs.func(*a)/rhs.func(*b) return lhs/rhs def as_powers_dict(self): d = defaultdict(int) for term in self.args: b, e = term.as_base_exp() 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_algebraic_expr(self, syms): return all(term._eval_is_algebraic_expr(syms) for term in self.args) _eval_is_finite = lambda self: _fuzzy_group( a.is_finite for a in self.args) _eval_is_commutative = lambda self: _fuzzy_group( a.is_commutative for a in self.args) _eval_is_complex = lambda self: _fuzzy_group( (a.is_complex for a in self.args), quick_exit=True) 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 def _eval_is_integer(self): is_rational = self.is_rational if is_rational: n, d = self.as_numer_denom() if d is S.One: return True elif d is S(2): return n.is_even elif is_rational is False: 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_real(self): real = True zero = one_neither = False for t in self.args: if not t.is_complex: return t.is_complex elif t.is_imaginary: real = not real elif t.is_real: 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_real is False: if one_neither: return # complex terms might cancel one_neither = True else: return if one_neither: # self is a+I*b or I*b if real: return zero # real*self is like self: neither is 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 elif z is False: return (S.ImaginaryUnit*self).is_real def _eval_is_hermitian(self): real = True 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 zero is False: zero = fuzzy_not(t.is_nonzero) if zero: return True 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 (S.ImaginaryUnit*self).is_hermitian 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 x.is_nonzero) is True for x in others): return True return if a is None: return return False def _eval_is_positive(self): """Return True if self is positive, False if not, and None if it cannot be determined. 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 """ sign = 1 saw_NON = False for t in self.args: if t.is_positive: continue elif t.is_negative: sign = -sign elif t.is_zero: return False elif t.is_nonpositive: sign = -sign saw_NON = True elif t.is_nonnegative: saw_NON = True else: return if sign == 1 and saw_NON is False: return True if sign < 0: return False def _eval_is_negative(self): return (-self).is_positive def _eval_is_odd(self): is_integer = self.is_integer if is_integer: r, acc = True, 1 for t in self.args: if not t.is_integer: return None elif t.is_even: r = False elif t.is_integer: if r is False: pass elif acc != 1 and (acc + t).is_odd: r = False elif t.is_odd is None: r = None acc = t return r # !integer -> !odd elif is_integer is False: return False def _eval_is_even(self): is_integer = self.is_integer if is_integer: return fuzzy_not(self.is_odd) elif is_integer is False: return False def _eval_subs(self, old, new): from sympy.functions.elementary.complexes import sign from sympy.ntheory.factor_ import multiplicity from sympy.simplify.simplify import powdenest, 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 if a.is_Pow or a.func is C.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) 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: 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 = self.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(self) (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 set(i[0] for i in old_nc).difference(set(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 inbetween. 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*self.func(*margs)*self.func(*nc) def _eval_nseries(self, x, n, logx): from sympy import powsimp terms = [t.nseries(x, n=n, logx=logx) for t in self.args] res = powsimp(self.func(*terms).expand(), combine='exp', deep=True) if res.has(C.Order): res += C.Order(x**n, x) return res def _eval_as_leading_term(self, x): return self.func(*[t.as_leading_term(x) 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 _sage_(self): s = 1 for x in self.args: s *= x._sage_() return s
[docs] def as_content_primitive(self, radical=False): """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)*(-sqrt(2) + 1)) See docstring of Expr.as_content_primitive for more examples. """ coef = S.One args = [] for i, a in enumerate(self.args): c, p = a.as_content_primitive(radical=radical) 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)
[docs] 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())
[docs]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 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: q = S(coeff.q) for i in factors.args: c, t = i.as_coeff_Mul() r = c/q if r == int(r): return coeff*factors return Mul._from_args((coeff, factors)) 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: return coeff*factors def expand_2arg(e): from sympy.simplify.simplify import bottom_up 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 Rational from .power import Pow from .add import Add, _addsort, _unevaluated_Add