Source code for sympy.core.add

from __future__ import print_function, division

from collections import defaultdict

from sympy.core.core import C
from sympy.core.compatibility import reduce, is_sequence
from sympy.core.singleton import S
from sympy.core.operations import AssocOp
from sympy.core.cache import cacheit
from sympy.core.numbers import ilcm, igcd
from sympy.core.expr import Expr


def _addsort(args):
    # in-place sorting of args

    # Currently we sort things using hashes, as it is quite fast. A better
    # solution is not to sort things at all - but this needs some more
    # fixing.
    args.sort(key=hash)


def _unevaluated_Add(*args):
    """Return a well-formed unevaluated Add: Numbers are collected and
    put in slot 0 and args are sorted. Use this when args have changed
    but you still want to return an unevaluated Add.

    Examples
    ========

    >>> from sympy.core.add import _unevaluated_Add as uAdd
    >>> from sympy import S, Add
    >>> from sympy.abc import x, y
    >>> a = uAdd(*[S(1.0), x, S(2)])
    >>> a.args[0]
    3.00000000000000
    >>> a.args[1]
    x

    Beyond the Number being in slot 0, there is no other assurance of
    order for the arguments since they are hash sorted. So, for testing
    purposes, output produced by this in some other function can only
    be tested against the output of this function or as one of several
    options:

    >>> opts = (Add(x, y, evaluated=False), Add(y, x, evaluated=False))
    >>> a = uAdd(x, y)
    >>> assert a in opts and a == uAdd(x, y)

    """
    args = list(args)
    newargs = []
    co = S.Zero
    while args:
        a = args.pop()
        if a.is_Add:
            # this will keep nesting from building up
            # so that x + (x + 1) -> x + x + 1 (3 args)
            args.extend(a.args)
        elif a.is_Number:
            co += a
        else:
            newargs.append(a)
    _addsort(newargs)
    if co:
        newargs.insert(0, co)
    return Add._from_args(newargs)


[docs]class Add(Expr, AssocOp): __slots__ = [] is_Add = True #identity = S.Zero # cyclic import, so defined in numbers.py @classmethod
[docs] def flatten(cls, seq): """ Takes the sequence "seq" of nested Adds and returns a flatten list. Returns: (commutative_part, noncommutative_part, order_symbols) Applies associativity, all terms are commutable with respect to addition. NB: the removal of 0 is already handled by AssocOp.__new__ See also ======== sympy.core.mul.Mul.flatten """ rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a if a.is_Rational: if b.is_Mul: rv = [a, b], [], None if rv: if all(s.is_commutative for s in rv[0]): return rv return [], rv[0], None terms = {} # term -> coeff # e.g. x**2 -> 5 for ... + 5*x**2 + ... coeff = S.Zero # coefficient (Number or zoo) to always be in slot 0 # e.g. 3 + ... order_factors = [] for o in seq: # O(x) if o.is_Order: for o1 in order_factors: if o1.contains(o): o = None break if o is None: continue order_factors = [o] + [ o1 for o1 in order_factors if not o.contains(o1)] continue # 3 or NaN elif o.is_Number: if (o is S.NaN or coeff is S.ComplexInfinity and o.is_bounded is False): # we know for sure the result will be nan return [S.NaN], [], None if coeff.is_Number: 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 coeff.is_bounded is False: # we know for sure the result will be nan return [S.NaN], [], None coeff = S.ComplexInfinity continue # Add([...]) elif o.is_Add: # NB: here we assume Add is always commutative seq.extend(o.args) # TODO zerocopy? continue # Mul([...]) elif o.is_Mul: c, s = o.as_coeff_Mul() # check for unevaluated Pow, e.g. 2**3 or 2**(-1/2) elif o.is_Pow: b, e = o.as_base_exp() if b.is_Number and (e.is_Integer or (e.is_Rational and e.is_negative)): seq.append(b**e) continue c, s = S.One, o else: # everything else c = S.One s = o # now we have: # o = c*s, where # # c is a Number # s is an expression with number factor extracted # let's collect terms with the same s, so e.g. # 2*x**2 + 3*x**2 -> 5*x**2 if s in terms: terms[s] += c else: terms[s] = c # now let's construct new args: # [2*x**2, x**3, 7*x**4, pi, ...] newseq = [] noncommutative = False for s, c in terms.items(): # 0*s if c is S.Zero: continue # 1*s elif c is S.One: newseq.append(s) # c*s else: if s.is_Mul: # Mul, already keeps its arguments in perfect order. # so we can simply put c in slot0 and go the fast way. cs = s._new_rawargs(*((c,) + s.args)) newseq.append(cs) elif s.is_Add: # we just re-create the unevaluated Mul newseq.append(Mul(c, s, evaluate=False)) else: # alternatively we have to call all Mul's machinery (slow) newseq.append(Mul(c, s)) noncommutative = noncommutative or not s.is_commutative # oo, -oo if coeff is S.Infinity: newseq = [f for f in newseq if not (f.is_nonnegative or f.is_real and (f.is_bounded or f.is_infinitesimal))] elif coeff is S.NegativeInfinity: newseq = [f for f in newseq if not (f.is_nonpositive or f.is_real and (f.is_bounded or f.is_infinitesimal))] if coeff is S.ComplexInfinity: # zoo might be # unbounded_real + bounded_im # bounded_real + unbounded_im # unbounded_real + unbounded_im # addition of a bounded real or imaginary number won't be able to # change the zoo nature; if unbounded a NaN condition could result # if the unbounded symbol had sign opposite of the unbounded # portion of zoo, e.g., unbounded_real - unbounded_real. newseq = [c for c in newseq if not (c.is_bounded and c.is_real is not None)] # process O(x) if order_factors: newseq2 = [] for t in newseq: for o in order_factors: # x + O(x) -> O(x) if o.contains(t): t = None break # x + O(x**2) -> x + O(x**2) if t is not None: newseq2.append(t) newseq = newseq2 + order_factors # 1 + O(1) -> O(1) for o in order_factors: if o.contains(coeff): coeff = S.Zero break # order args canonically _addsort(newseq) # current code expects coeff to be first if coeff is not S.Zero: newseq.insert(0, coeff) # we are done if noncommutative: return [], newseq, None else: return newseq, [], None
@classmethod
[docs] def class_key(cls): """Nice order of classes""" return 3, 1, cls.__name__
[docs] def as_coefficients_dict(a): """Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term. Examples ======== >>> from sympy.abc import a, x >>> (3*x + a*x + 4).as_coefficients_dict() {1: 4, x: 3, a*x: 1} >>> _[a] 0 >>> (3*a*x).as_coefficients_dict() {a*x: 3} """ d = defaultdict(list) for ai in a.args: c, m = ai.as_coeff_Mul() d[m].append(c) for k, v in d.items(): if len(v) == 1: d[k] = v[0] else: d[k] = Add(*v) di = defaultdict(int) di.update(d) return di
@cacheit
[docs] def as_coeff_add(self, *deps): """ Returns a tuple (coeff, args) where self is treated as an Add and coeff is the Number term and args is a tuple of all other terms. Examples ======== >>> from sympy.abc import x >>> (7 + 3*x).as_coeff_add() (7, (3*x,)) >>> (7*x).as_coeff_add() (0, (7*x,)) """ if deps: l1 = [] l2 = [] for f in self.args: if f.has(*deps): l2.append(f) else: l1.append(f) return self._new_rawargs(*l1), tuple(l2) coeff, notrat = self.args[0].as_coeff_add() if coeff is not S.Zero: return coeff, notrat + self.args[1:] return S.Zero, self.args
[docs] def as_coeff_Add(self): """Efficiently extract the coefficient of a summation. """ coeff, args = self.args[0], self.args[1:] if coeff.is_Number: if len(args) == 1: return coeff, args[0] else: return coeff, self._new_rawargs(*args) else: return S.Zero, self # Note, we intentionally do not implement Add.as_coeff_mul(). Rather, we # let Expr.as_coeff_mul() just always return (S.One, self) for an Add. See # issue 2425.
def _eval_derivative(self, s): return self.func(*[f.diff(s) for f in self.args]) def _eval_nseries(self, x, n, logx): terms = [t.nseries(x, n=n, logx=logx) for t in self.args] return self.func(*terms) def _matches_simple(self, expr, repl_dict): # handle (w+3).matches('x+5') -> {w: x+2} coeff, terms = self.as_coeff_add() if len(terms) == 1: return terms[0].matches(expr - coeff, repl_dict) return def matches(self, expr, repl_dict={}, old=False): return AssocOp._matches_commutative(self, expr, repl_dict, old) @staticmethod def _combine_inverse(lhs, rhs): """ Returns lhs - rhs, but treats arguments like symbols, so things like oo - oo return 0, instead of a nan. """ from sympy import oo, I, expand_mul if lhs == oo and rhs == oo or lhs == oo*I and rhs == oo*I: return S.Zero return expand_mul(lhs - rhs) @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_add() which gives the head and a tuple containing the arguments of the tail when treated as an Add. - if you want the coefficient when self is treated as a Mul then use self.as_coeff_mul()[0] >>> from sympy.abc import x, y >>> (3*x*y).as_two_terms() (3, x*y) """ if len(self.args) == 1: return S.Zero, self return self.args[0], self._new_rawargs(*self.args[1:])
def as_numer_denom(self): # clear rational denominator content, expr = self.primitive() ncon, dcon = content.as_numer_denom() # collect numerators and denominators of the terms nd = defaultdict(list) for f in expr.args: ni, di = f.as_numer_denom() nd[di].append(ni) # put infinity in the numerator if S.Zero in nd: n = nd.pop(S.Zero) assert len(n) == 1 n = n[0] nd[S.One].append(n/S.Zero) # check for quick exit if len(nd) == 1: d, n = nd.popitem() return self.func( *[_keep_coeff(ncon, ni) for ni in n]), _keep_coeff(dcon, d) # sum up the terms having a common denominator for d, n in nd.items(): if len(n) == 1: nd[d] = n[0] else: nd[d] = self.func(*n) # assemble single numerator and denominator denoms, numers = [list(i) for i in zip(*iter(nd.items()))] n, d = self.func(*[Mul(*(denoms[:i] + [numers[i]] + denoms[i + 1:])) for i in range(len(numers))]), Mul(*denoms) return _keep_coeff(ncon, n), _keep_coeff(dcon, d) def _eval_is_polynomial(self, syms): return all(term._eval_is_polynomial(syms) for term in self.args) def _eval_is_rational_function(self, syms): return all(term._eval_is_rational_function(syms) for term in self.args) def _eval_is_algebraic_expr(self, syms): return all(term._eval_is_algebraic_expr(syms) for term in self.args) # assumption methods _eval_is_real = lambda self: self._eval_template_is_attr( 'is_real', when_multiple=None) _eval_is_antihermitian = lambda self: self._eval_template_is_attr( 'is_antihermitian', when_multiple=None) _eval_is_bounded = lambda self: self._eval_template_is_attr( 'is_bounded', when_multiple=None) _eval_is_hermitian = lambda self: self._eval_template_is_attr( 'is_hermitian', when_multiple=None) _eval_is_imaginary = lambda self: self._eval_template_is_attr( 'is_imaginary', when_multiple=None) _eval_is_integer = lambda self: self._eval_template_is_attr( 'is_integer', when_multiple=None) _eval_is_rational = lambda self: self._eval_template_is_attr( 'is_rational', when_multiple=None) _eval_is_commutative = lambda self: self._eval_template_is_attr( 'is_commutative') def _eval_is_odd(self): l = [f for f in self.args if not (f.is_even is True)] if not l: return False if l[0].is_odd: return self._new_rawargs(*l[1:]).is_even def _eval_is_irrational(self): for t in self.args: a = t.is_irrational if a: others = list(self.args) others.remove(t) if all(x.is_rational is True for x in others): return True return None if a is None: return return False def _eval_is_positive(self): if self.is_number: return super(Add, self)._eval_is_positive() pos = nonneg = nonpos = unknown_sign = False unbounded = set() args = [a for a in self.args if not a.is_zero] if not args: return False for a in args: ispos = a.is_positive ubound = a.is_unbounded if ubound: unbounded.add(ispos) if len(unbounded) > 1: return None if ispos: pos = True continue elif a.is_nonnegative: nonneg = True continue elif a.is_nonpositive: nonpos = True continue elif a.is_zero: continue if ubound is None: # sign is unknown; if we don't know the boundedness # we're done: we don't know. That is technically true, # but the only option is that we have something like # oo - oo which is NaN and it really doesn't matter # what sign we apply to that because it (when finally # computed) will trump any sign. So instead of returning # None, we pass. pass else: return None unknown_sign = True if unbounded: return unbounded.pop() elif unknown_sign: return None elif not nonpos and not nonneg and pos: return True elif not nonpos and pos: return True elif not pos and not nonneg: return False def _eval_is_negative(self): if self.is_number: return super(Add, self)._eval_is_negative() neg = nonpos = nonneg = unknown_sign = False unbounded = set() args = [a for a in self.args if not a.is_zero] if not args: return False for a in args: isneg = a.is_negative ubound = a.is_unbounded if ubound: unbounded.add(isneg) if len(unbounded) > 1: return None if isneg: neg = True continue elif a.is_nonpositive: nonpos = True continue elif a.is_nonnegative: nonneg = True continue elif a.is_zero: continue if ubound is None: # sign is unknown; if we don't know the boundedness # we're done: we don't know. That is technically true, # but the only option is that we have something like # oo - oo which is NaN and it really doesn't matter # what sign we apply to that because it (when finally # computed) will trump any sign. So instead of returning # None, we pass. pass unknown_sign = True if unbounded: return unbounded.pop() elif unknown_sign: return None elif not nonneg and not nonpos and neg: return True elif not nonneg and neg: return True elif not neg and not nonpos: return False def _eval_subs(self, old, new): if not old.is_Add: return None coeff_self, terms_self = self.as_coeff_Add() coeff_old, terms_old = old.as_coeff_Add() if coeff_self.is_Rational and coeff_old.is_Rational: if terms_self == terms_old: # (2 + a).subs( 3 + a, y) -> -1 + y return self.func(new, coeff_self, -coeff_old) if terms_self == -terms_old: # (2 + a).subs(-3 - a, y) -> -1 - y return self.func(-new, coeff_self, coeff_old) if coeff_self.is_Rational and coeff_old.is_Rational \ or coeff_self == coeff_old: args_old, args_self = self.func.make_args( terms_old), self.func.make_args(terms_self) if len(args_old) < len(args_self): # (a+b+c).subs(b+c,x) -> a+x self_set = set(args_self) old_set = set(args_old) if old_set < self_set: ret_set = self_set - old_set return self.func(new, coeff_self, -coeff_old, *[s._subs(old, new) for s in ret_set]) args_old = self.func.make_args( -terms_old) # (a+b+c+d).subs(-b-c,x) -> a-x+d old_set = set(args_old) if old_set < self_set: ret_set = self_set - old_set return self.func(-new, coeff_self, coeff_old, *[s._subs(old, new) for s in ret_set]) def removeO(self): args = [a for a in self.args if not a.is_Order] return self._new_rawargs(*args) def getO(self): args = [a for a in self.args if a.is_Order] if args: return self._new_rawargs(*args) @cacheit
[docs] def extract_leading_order(self, symbols, point=None): """ Returns the leading term and it's order. Examples ======== >>> from sympy.abc import x >>> (x + 1 + 1/x**5).extract_leading_order(x) ((x**(-5), O(x**(-5))),) >>> (1 + x).extract_leading_order(x) ((1, O(1)),) >>> (x + x**2).extract_leading_order(x) ((x, O(x)),) """ lst = [] symbols = list(symbols if is_sequence(symbols) else [symbols]) if not point: point = [0]*len(symbols) seq = [(f, C.Order(f, *zip(symbols, point))) for f in self.args] for ef, of in seq: for e, o in lst: if o.contains(of) and o != of: of = None break if of is None: continue new_lst = [(ef, of)] for e, o in lst: if of.contains(o) and o != of: continue new_lst.append((e, o)) lst = new_lst return tuple(lst)
[docs] def as_real_imag(self, deep=True, **hints): """ returns a tuple represeting a complex numbers Examples ======== >>> from sympy import I >>> (7 + 9*I).as_real_imag() (7, 9) >>> ((1 + I)/(1 - I)).as_real_imag() (0, 1) >>> ((1 + 2*I)*(1 + 3*I)).as_real_imag() (-5, 5) """ sargs, terms = self.args, [] re_part, im_part = [], [] for term in sargs: re, im = term.as_real_imag(deep=deep) re_part.append(re) im_part.append(im) return (self.func(*re_part), self.func(*im_part))
def _eval_as_leading_term(self, x): from sympy import expand_mul, factor_terms old = self self = expand_mul(self) if not self.is_Add: return self.as_leading_term(x) unbounded = [t for t in self.args if t.is_unbounded] self = self.func(*[t.as_leading_term(x) for t in self.args]).removeO() if not self: # simple leading term analysis gave us 0 but we have to send # back a term, so compute the leading term (via series) return old.compute_leading_term(x) elif self is S.NaN: return old.func._from_args(unbounded) elif not self.is_Add: return self else: plain = self.func(*[s for s, _ in self.extract_leading_order(x)]) rv = factor_terms(plain, fraction=False) rv_simplify = rv.simplify() # if it simplifies to an x-free expression, return that; # tests don't fail if we don't but it seems nicer to do this if x not in rv_simplify.free_symbols: if rv_simplify.is_zero and plain.is_zero is not True: return (self - plain)._eval_as_leading_term(x) return rv_simplify return rv def _eval_adjoint(self): return self.func(*[t.adjoint() for t in self.args]) def _eval_conjugate(self): return self.func(*[t.conjugate() for t in self.args]) def _eval_transpose(self): return self.func(*[t.transpose() for t in self.args]) def __neg__(self): return self.func(*[-t for t in self.args]) def _sage_(self): s = 0 for x in self.args: s += x._sage_() return s
[docs] def primitive(self): """ Return ``(R, self/R)`` where ``R``` is the Rational GCD of ``self```. ``R`` is collected only from the leading coefficient of each term. Examples ======== >>> from sympy.abc import x, y >>> (2*x + 4*y).primitive() (2, x + 2*y) >>> (2*x/3 + 4*y/9).primitive() (2/9, 3*x + 2*y) >>> (2*x/3 + 4.2*y).primitive() (1/3, 2*x + 12.6*y) No subprocessing of term factors is performed: >>> ((2 + 2*x)*x + 2).primitive() (1, x*(2*x + 2) + 2) Recursive subprocessing can be done with the as_content_primitive() method: >>> ((2 + 2*x)*x + 2).as_content_primitive() (2, x*(x + 1) + 1) See also: primitive() function in polytools.py """ terms = [] inf = False for a in self.args: c, m = a.as_coeff_Mul() if not c.is_Rational: c = S.One m = a inf = inf or m is S.ComplexInfinity terms.append((c.p, c.q, m)) if not inf: ngcd = reduce(igcd, [t[0] for t in terms], 0) dlcm = reduce(ilcm, [t[1] for t in terms], 1) else: ngcd = reduce(igcd, [t[0] for t in terms if t[1]], 0) dlcm = reduce(ilcm, [t[1] for t in terms if t[1]], 1) if ngcd == dlcm == 1: return S.One, self if not inf: for i, (p, q, term) in enumerate(terms): terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term) else: for i, (p, q, term) in enumerate(terms): if q: terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term) else: terms[i] = _keep_coeff(Rational(p, q), term) # we don't need a complete re-flattening since no new terms will join # so we just use the same sort as is used in Add.flatten. When the # coefficient changes, the ordering of terms may change, e.g. # (3*x, 6*y) -> (2*y, x) # # We do need to make sure that term[0] stays in position 0, however. # if terms[0].is_Number or terms[0] is S.ComplexInfinity: c = terms.pop(0) else: c = None _addsort(terms) if c: terms.insert(0, c) return Rational(ngcd, dlcm), self._new_rawargs(*terms)
[docs] def as_content_primitive(self, radical=False): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. If radical is True (default is False) then common radicals will be removed and included as a factor of the primitive expression. Examples ======== >>> from sympy import sqrt >>> (3 + 3*sqrt(2)).as_content_primitive() (3, 1 + sqrt(2)) Radical content can also be factored out of the primitive: >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2)*(1 + 2*sqrt(5))) See docstring of Expr.as_content_primitive for more examples. """ con, prim = self.func(*[_keep_coeff(*a.as_content_primitive( radical=radical)) for a in self.args]).primitive() if radical and prim.is_Add: # look for common radicals that can be removed args = prim.args rads = [] common_q = None for m in args: term_rads = defaultdict(list) for ai in Mul.make_args(m): if ai.is_Pow: b, e = ai.as_base_exp() if e.is_Rational and b.is_Integer: term_rads[e.q].append(abs(int(b))**e.p) if not term_rads: break if common_q is None: common_q = set(term_rads.keys()) else: common_q = common_q & set(term_rads.keys()) if not common_q: break rads.append(term_rads) else: # process rads # keep only those in common_q for r in rads: for q in list(r.keys()): if q not in common_q: r.pop(q) for q in r: r[q] = prod(r[q]) # find the gcd of bases for each q G = [] for q in common_q: g = reduce(igcd, [r[q] for r in rads], 0) if g != 1: G.append(g**Rational(1, q)) if G: G = Mul(*G) args = [ai/G for ai in args] prim = G*prim.func(*args) return con, prim
@property def _sorted_args(self): from sympy.core.compatibility import default_sort_key return sorted(self.args, key=lambda w: default_sort_key(w))
from .mul import Mul, _keep_coeff, prod from sympy.core.numbers import Rational