Source code for diofant.core.add

from collections import defaultdict
from functools import reduce

from .cache import cacheit
from .compatibility import default_sort_key, is_sequence
from .logic import _fuzzy_group
from .numbers import I, igcd, ilcm, nan, oo, zoo
from .operations import AssocOp
from .singleton import S

[docs]class Add(AssocOp): is_Add = True
[docs] @classmethod def flatten(cls, seq): """ Takes the sequence "seq" of nested Adds and returns a flatten list. Returns: (commutative_part, noncommutative_part, order_symbols) Applies associativity, all terms are commutable with respect to addition. See also ======== diofant.core.mul.Mul.flatten """ from .mul import Mul from ..series.order import Order 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 # term -> coeff # e.g. x**2 -> 5 for ... + 5*x**2 + ... terms = {} # coefficient (Number or zoo) to always be in slot 0, e.g. 3 + ... coeff = S.Zero 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 nan or coeff is zoo and o.is_finite is False): # we know for sure the result will be nan return [nan], [], None if coeff.is_Number: coeff += o if coeff is nan: # we know for sure the result will be nan return [nan], [], None o # XXX "peephole" optimization, continue elif o is zoo: if coeff.is_finite is False: # we know for sure the result will be nan return [nan], [], None coeff = zoo continue # Add([...]) elif o.is_Add: # NB: here we assume Add is always commutative seq.extend(o.args) # TODO zerocopy? continue elif o.has(Order): c, s = S.One, o # Mul([...]) elif o.is_Mul: c, s = o.as_coeff_Mul() # check for unevaluated Pow, e.g. 2**3 or 2**(-1/2) elif o.is_Pow: b, e = o.as_base_exp() if b.is_Number and (e.is_Integer or (e.is_Rational and e.is_negative)): seq.append(b**e) continue c, s = S.One, o else: # everything else c = S.One s = o # now we have: # o = c*s, where # # c is a Number # s is an expression with number factor extracted # let's collect terms with the same s, so e.g. # 2*x**2 + 3*x**2 -> 5*x**2 if s in terms: terms[s] += c if terms[s] is nan: # we know for sure the result will be nan return [nan], [], None else: terms[s] = c # now let's construct new args: # [2*x**2, x**3, 7*x**4, pi, ...] newseq = [] noncommutative = False for s, c in terms.items(): # 0*s if c is 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 oo: newseq = [f for f in newseq if not (f.is_nonnegative or f.is_extended_real and f.is_finite)] elif coeff == -oo: newseq = [f for f in newseq if not (f.is_nonpositive or f.is_extended_real and f.is_finite)] if coeff is zoo: # zoo might be # infinite_real + finite_im # finite_real + infinite_im # infinite_real + infinite_im # addition of a finite real or imaginary number won't be able to # change the zoo nature; adding an infinite qualtity would result # in a NaN condition if it had sign opposite of the infinite # portion of zoo, e.g., infinite_real - infinite_real. newseq = [c for c in newseq if not (c.is_finite and (c.is_extended_real is not None or c.is_imaginary 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 newseq.sort(key=default_sort_key) # 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
[docs] @classmethod def class_key(cls): """Nice order of classes""" return 4, 1, cls.__name__
[docs] def as_coefficients_dict(self): """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 ======== >>> (3*x + x*y + 4).as_coefficients_dict() {1: 4, x: 3, x*y: 1} >>> _[y] 0 >>> (3*y*x).as_coefficients_dict() {x*y: 3} """ d = defaultdict(list) for ai in self.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
[docs] @cacheit def as_coeff_add(self, *deps): """ Returns a tuple (coeff, args) where self is treated as an Add and coeff is the Number term and args is a tuple of all other terms. Examples ======== >>> (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, rational=False): """Efficiently extract the coefficient of a summation. """ coeff, args = self.args[0], self.args[1:] if coeff.is_Number and not rational or coeff.is_Rational: return coeff, self._new_rawargs(*args) return S.Zero, self
# Note, we intentionally do not implement Add.as_coeff_mul(). Rather, we # let Expr.as_coeff_mul() just always return (S.One, self) for an Add. See # issue sympy/sympy#5524. @cacheit def _eval_derivative(self, s): return self.func(*[a.diff(s) for a 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={}): """Helper method for match(). See Also ======== diofant.core.basic.Basic.matches """ return AssocOp._matches_commutative(self, expr, repl_dict) @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 . import oo, I from ..simplify import signsimp if lhs == oo and rhs == oo or lhs == oo*I and rhs == oo*I: return S.Zero return signsimp(lhs - rhs)
[docs] @cacheit def as_two_terms(self): """Return head and tail of self. This is the most efficient way to get the head and tail of an expression. - if you want only the head, use self.args[0]; - if you want to process the arguments of the tail then use self.as_coef_add() which gives the head and a tuple containing the arguments of the tail when treated as an Add. - if you want the coefficient when self is treated as a Mul then use self.as_coeff_mul()[0] >>> (3*x*y).as_two_terms() (3, x*y) """ return self.args[0], self._new_rawargs(*self.args[1:])
def _eval_as_numer_denom(self): """expression -> a/b -> a, b See Also ======== diofant.core.expr.Expr.as_numer_denom """ from .mul import Mul, _keep_coeff # 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) # 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 def _eval_is_commutative(self): return _fuzzy_group((a.is_commutative for a in self.args), quick_exit=True) def _eval_is_extended_real(self): return _fuzzy_group((a.is_extended_real for a in self.args), quick_exit=True) def _eval_is_complex(self): return _fuzzy_group((a.is_complex for a in self.args), quick_exit=True) def _eval_is_antihermitian(self): return _fuzzy_group((a.is_antihermitian for a in self.args), quick_exit=True) def _eval_is_finite(self): return _fuzzy_group((a.is_finite for a in self.args), quick_exit=True) def _eval_is_hermitian(self): return _fuzzy_group((a.is_hermitian for a in self.args), quick_exit=True) def _eval_is_integer(self): return _fuzzy_group((a.is_integer for a in self.args), quick_exit=True) def _eval_is_rational(self): return _fuzzy_group((a.is_rational for a in self.args), quick_exit=True) def _eval_is_algebraic(self): return _fuzzy_group((a.is_algebraic for a in self.args), quick_exit=True) def _eval_is_imaginary(self): rv = _fuzzy_group((a.is_imaginary for a in self.args), quick_exit=True) if rv is False: return rv iargs = [a*I for a in self.args] return _fuzzy_group((a.is_real for a in iargs), quick_exit=True) def _eval_is_odd(self): l = [f for f in self.args if not f.is_even] 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: if all(x.is_rational for x in self.args if x != t): return True return if a is None: return return False def _eval_is_positive(self): if self.is_number: n = super()._eval_is_positive() if n is not None: return n if any(a.is_infinite for a in self.args): args = [a for a in self.args if not a.is_finite] else: args = self.args nonpos = nonneg = 0 for a in args: if a.is_positive: continue elif a.is_nonnegative: nonneg += 1 if a.is_zero: nonpos += 1 elif a.is_nonpositive: nonpos += 1 else: break else: if not nonpos and nonneg < len(args): return True elif nonpos == len(args): return False def _eval_is_negative(self): if self.is_number: n = super()._eval_is_negative() if n is not None: return n if any(a.is_infinite for a in self.args): args = [a for a in self.args if not a.is_finite] else: args = self.args nonneg = nonpos = 0 for a in args: if a.is_negative: continue elif a.is_nonpositive: nonpos += 1 if a.is_zero: nonneg += 1 elif a.is_nonnegative: nonneg += 1 else: break else: if not nonneg and nonpos < len(args): return True elif nonneg == len(args): return False def _eval_subs(self, old, new): if not old.is_Add: return 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])
[docs] def removeO(self): """Removes the additive O(..) symbol. See Also ======== diofant.core.expr.Expr.removeO """ args = [a for a in self.args if not a.is_Order] return self._new_rawargs(*args)
[docs] def getO(self): """Returns the additive O(..) symbol. See Also ======== diofant.core.expr.Expr.getO """ args = [a for a in self.args if a.is_Order] if args: return self._new_rawargs(*args)
[docs] @cacheit def extract_leading_order(self, symbols): """Returns the leading term and its order. Examples ======== >>> (x + 1 + 1/x**5).extract_leading_order(x) ((x**(-5), O(x**(-5))),) >>> (1 + x).extract_leading_order(x) ((1, O(1)),) >>> (x + x**2).extract_leading_order(x) ((x, O(x)),) """ from ..series import Order lst = [] symbols = list(symbols if is_sequence(symbols) else [symbols]) point = [0]*len(symbols) seq = [(f, Order(f, *zip(symbols, point))) for f in self.args] for ef, of in seq: for e, o in lst: if o.contains(of) and o != of: of = None break if of is None: continue new_lst = [(ef, of)] for e, o in lst: if of.contains(o) and o != of: continue new_lst.append((e, o)) lst = new_lst return tuple(lst)
[docs] def as_real_imag(self, deep=True, **hints): """ returns a tuple representing a complex number Examples ======== >>> (7 + 9*I).as_real_imag() (7, 9) >>> ((1 + I)/(1 - I)).as_real_imag() (0, 1) >>> ((1 + 2*I)*(1 + 3*I)).as_real_imag() (-5, 5) """ sargs = self.args re_part, im_part = [], [] for term in sargs: re, im = term.as_real_imag(deep=deep) re_part.append(re) im_part.append(im) return self.func(*re_part), self.func(*im_part)
def _eval_as_leading_term(self, x): from . import factor_terms expr = self.func(*[t.as_leading_term(x) for t in self.args]).removeO() if not expr: # simple leading term analysis gave us 0 but we have to send # back a term, so compute the leading term (via series) return self.compute_leading_term(x) elif not expr.is_Add: return expr else: plain = expr.func(*[s for s, _ in expr.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 (expr - 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])
[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 ======== >>> (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 ======== diofant.polys.polytools.primitive """ from .mul import _keep_coeff from .numbers import Rational 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 zoo terms.append((c.numerator, c.denominator, 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 for i, (p, q, term) in enumerate(terms): terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//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 zoo: c = terms.pop(0) else: c = None terms.sort(key=default_sort_key) 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 ======== >>> (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 Also ======== diofant.core.expr.Expr.as_content_primitive """ from .mul import Mul, _keep_coeff, prod from ..functions import root 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.denominator].append(abs(int(b))**e.numerator) if not term_rads: break if common_q is None: common_q = set(term_rads) else: common_q = common_q & set(term_rads) 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): 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(root(g, 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): return tuple(sorted(self.args, key=default_sort_key))