Source code for sympy.core.mul

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
from .compatibility import cmp_to_key
from functools import reduce

# 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


[docs]class Mul(AssocOp): __slots__ = [] is_Mul = True #identity = S.One # cyclic import, so defined in numbers.py # Key for sorting commutative args in canonical order _args_sortkey = cmp_to_key(Basic.compare) @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. http://code.google.com/p/sympy/issues/detail?id=1497} >>> 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. http://code.google.com/p/sympy/issues/detail?id=2629} >>> 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. http://code.google.com/p/sympy/issues/detail?id=2607} 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 a and a.is_Rational: r, b = b.as_coeff_Mul() a *= r if b.is_Mul: bargs, nc = b.args_cnc() rv = bargs, nc, None if a is not S.One: bargs.insert(0, a) elif b.is_Add and 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)] bargs.sort(key=hash) 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 * ... iu = [] # ImaginaryUnits, I c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... neg1e = 0 # exponent on -1 extracted from Number-based Pow 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: iu.append(o) continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow and 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) # handle the ImaginaryUnits if iu: if len(iu) == 1: c_powers.append((iu[0], S.One)) else: # a product of I's has one of 4 values; select that value # based on the length of iu: # len(iu) % 4 of (0, 1, 2, 3) has a corresponding value of # (1, I,-1,-I) niu = len(iu) % 4 if niu % 2: c_powers.append((S.ImaginaryUnit, S.One)) if niu in (2, 3): coeff = -coeff # 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 list(common_b.items()): for di, li in list(d.items()): d[di] = Add(*li) for b, e in list(common_b.items()): for t, c in list(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 list(inv_exp_dict.items()): inv_exp_dict[e] = Mul(*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 = Mul(*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 = {} 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 = igcd(bi, bj) if g != 1: # 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: if obj.is_Mul: # sqrt(12) -> 2*sqrt(3) c, obj = obj.args # expecting only 2 args coeff *= c assert obj.is_Pow bi, ei = obj.args pnew.setdefault(ei, []).append(bi) num_rat.extend(grow) i += 1 # combine bases of the new powers for e, b in pnew.items(): pnew[e] = Mul(*b) # see if there is a base with matching coefficient # that the -1 can be joined with if neg1e: p = Pow(S.NegativeOne, neg1e) if p.is_Number: coeff *= p else: c, p = p.as_coeff_Mul() coeff *= c if p.is_Pow and p.base is S.NegativeOne: neg1e = p.exp for e, b in pnew.items(): if e == neg1e and b.is_positive: pnew[e] = -b break else: c_part.append(p) # 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 # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_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 # order commutative part canonically c_part.sort(key=cls._args_sortkey) # 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 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 coeff, b = b.as_coeff_Mul() bc, bnc = b.args_cnc() done = [Pow(Mul._from_args(bnc), e, evaluate=False) if bnc else S.One] if e.is_Number: if e.is_Integer: # (a*b)**2 -> a**2 * b**2 return Mul(*([s**e for s in [coeff] + bc] + done)) if e.is_Rational or e.is_Float: unk = [] nonneg = [] neg = [] iu = [] for bi in bc: if bi.is_polar: nonneg.append(bi) elif bi.is_negative is not None: if bi.is_negative: neg.append(bi) elif bi.is_nonnegative: nonneg.append(bi) elif bi is S.ImaginaryUnit: iu.append(bi) else: unk.append(bi) else: unk.append(bi) if iu: niu = len(iu) % 4 i = S.ImaginaryUnit if niu % 2 else S.One if niu in (2, 3): coeff = -coeff if i is S.ImaginaryUnit: if unk or e.is_Float: unk.append(i) else: if coeff.is_negative and e.is_Rational: coeff = -coeff ie = Rational(4*e.q - e.p, 2*e.q) done.append(Pow(-1, ie)) else: done.append(i**e) if len(unk) == len(bc) or len(neg) == len(bc) == 1: # if all terms were unknown there is nothing to pull # out except maybe the coeff; if there is a single # negative term, this is the base case which cannot # be processed further if coeff.is_negative: coeff *= -1 bc[0] = -bc[0] if coeff is S.One: return None return Mul(*([Pow(coeff, e), Pow(Mul(*bc), e)] + done)) # otherwise return the new expression expanding out the known # terms; those that are not known can be expanded out with # expand_power_base() but this will introduce a lot of # "garbage" that is needed to keep one on the same branch as # the unexpanded expression. The negatives are brought out # with a negative sign added and a negative left behind in the # unexpanded terms if there were an odd number of negatives. if coeff.is_negative: coeff = -coeff neg.append(S.NegativeOne) if neg: neg = [-w for w in neg] if len(neg) % 2: unk.append(S.NegativeOne) done.extend([Pow(s, e) for s in nonneg + neg + [coeff, Mul(*unk)]]) return Mul(*done) if e.is_even and coeff.is_negative: return Pow(-coeff, e)*Pow(b, e) @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): other = [] coeff = S(1) for a in self.args: if a.is_real: coeff *= a else: other.append(a) m = Mul(*other) if hints.get('ignore') == m: return None else: return (coeff*C.re(m), coeff*C.im(m)) @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, expand_mul # 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: expr = n/d._eval_expand_mul(**hints) if not expr.is_Mul: return expand_mul(expr, deep=False) 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 = Mul(*plain) if sums: terms = Mul._expandsums(sums) args = [] for term in terms: t = Mul(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 def _eval_derivative(self, s): terms = list(self.args) factors = [] for i in range(len(terms)): t = terms[i].diff(s) if t is S.Zero: continue factors.append(Mul(*(terms[:i]+[t]+terms[i+1:]))) return Add(*factors) 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={}): expr = sympify(expr) if self.is_commutative and expr.is_commutative: return AssocOp._matches_commutative(self, expr, repl_dict) # todo for commutative parts, until then use the default matches method for non-commutative products return self._matches(expr, repl_dict) 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 == 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) else: b.append(x) return Mul(*a)/Mul(*b) return lhs/rhs def as_powers_dict(self): d = defaultdict(list) for term in self.args: b, e = term.as_base_exp() d[b].append(e) for b, e in d.items(): if len(e) == 1: e = e[0] else: e = Add(*e) 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 Mul(*numers), Mul(*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 Mul(*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) _eval_is_bounded = lambda self: self._eval_template_is_attr('is_bounded') _eval_is_integer = lambda self: self._eval_template_is_attr('is_integer', when_multiple=None) _eval_is_commutative = lambda self: self._eval_template_is_attr('is_commutative') 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) # I*I -> R, I*I*I -> -I def _eval_is_real(self): im_count = 0 is_neither = False for t in self.args: if t.is_imaginary: im_count += 1 continue t_real = t.is_real if t_real: continue elif t_real is False: if is_neither: return None else: is_neither = True else: return None if is_neither: return False return (im_count % 2 == 0) def _eval_is_imaginary(self): im_count = 0 is_neither = False for t in self.args: if t.is_imaginary: im_count += 1 continue t_real = t.is_real if t_real: continue elif t_real is False: if is_neither: return None else: is_neither = True else: return None if is_neither: return False return (im_count % 2 == 1) def _eval_is_hermitian(self): nc_count = 0 im_count = 0 is_neither = False for t in self.args: if not t.is_commutative: nc_count += 1 if nc_count > 1: return None if t.is_antihermitian: im_count += 1 continue t_real = t.is_hermitian if t_real: continue elif t_real is False: if is_neither: return None else: is_neither = True else: return None if is_neither: return False return (im_count % 2 == 0) def _eval_is_antihermitian(self): nc_count = 0 im_count = 0 is_neither = False for t in self.args: if not t.is_commutative: nc_count += 1 if nc_count > 1: return None if t.is_antihermitian: im_count += 1 continue t_real = t.is_hermitian if t_real: continue elif t_real is False: if is_neither: return None else: is_neither = True else: return None if is_neither: return False return (im_count % 2 == 1) 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_zero(self): zero = None for a in self.args: if a.is_zero: zero = True continue bound = a.is_bounded if not bound: return bound if zero: return True 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 True if self is negative, 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 nonnegative. e.g. pos * neg * nonpositive -> pos or zero -> False is returned pos * neg * nonnegative -> neg or zero -> None 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_odd(self): is_integer = self.is_integer if is_integer: r = True for t in self.args: if t.is_even: return False if t.is_odd is None: r = None return r # !integer -> !odd elif is_integer == False: return False def _eval_is_even(self): is_integer = self.is_integer if is_integer: return fuzzy_not(self._eval_is_odd()) elif is_integer == False: return False def _eval_subs(self, old, new): from sympy import sign, Dummy, multiplicity from sympy.simplify.simplify import powdenest, fraction if not old.is_Mul: 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 3322); 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 2552) # 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: n_old, d_old = co_old.as_numer_denom() n_self, d_self = co_self.as_numer_denom() def _multiplicity(p, n): p = abs(p) if p is S.One: return S.Infinity return multiplicity(p, abs(n)) mult = S(min(_multiplicity(n_old, n_self), _multiplicity(d_old, d_self))) c.pop(co_self) 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 ( # more non-commutative terms len(old_nc) > len(nc)): ok = False elif ( # more commutative terms len(old_c) > len(c)): ok = False elif ( # unmatched non-commutative bases set(_[0] for _ in old_nc).difference(set(_[0] for _ in nc))): ok = False elif ( # unmatched commutative terms set(old_c).difference(set(c))): ok = False elif ( # differences in sign any(sign(c[b]) != sign(old_c[b]) for b in old_c)): ok = False if not ok: return rv if not old_c: cdid = None else: rat = [] for (b, old_e) in list(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(list(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*Mul(*margs)*Mul(*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] return powsimp(Mul(*terms).expand(), combine='exp', deep=True) def _eval_as_leading_term(self, x): return Mul(*[t.as_leading_term(x) for t in self.args]) def _eval_conjugate(self): return Mul(*[t.conjugate() for t in self.args]) def _eval_transpose(self): return Mul(*[t.transpose() for t in self.args[::-1]]) def _eval_adjoint(self): return Mul(*[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, Mul(*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 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(list(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): """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. 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 """ if not coeff.is_Number: if factors.is_Number: factors, coeff = coeff, factors else: return coeff*factors if coeff == 1: return factors elif coeff == -1: # don't keep 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 from .numbers import Rational, igcd from .power import Pow from .add import Add