Source code for sympy.core.expr

from __future__ import print_function, division

from .core import C
from .sympify import sympify
from .basic import Basic, Atom
from .singleton import S
from .evalf import EvalfMixin, pure_complex
from .decorators import _sympifyit, call_highest_priority
from .cache import cacheit
from .compatibility import reduce, as_int, default_sort_key, xrange
from sympy.mpmath.libmp import mpf_log, prec_to_dps

from collections import defaultdict

[docs] def is_number(self): """Returns True if 'self' has no free symbols. It will be faster than if not self.free_symbols, however, since is_number will fail as soon as it hits a free symbol. Examples ======== >>> from sympy import log, Integral >>> from sympy.abc import x >>> x.is_number False >>> (2*x).is_number False >>> (2 + log(2)).is_number True >>> (2 + Integral(2, x)).is_number False >>> (2 + Integral(2, (x, 1, 2))).is_number True """ if not self.args: return False return all(obj.is_number for obj in self.iter_basic_args())
def _random(self, n=None, re_min=-1, im_min=-1, re_max=1, im_max=1): """Return self evaluated, if possible, replacing free symbols with random complex values, if necessary. The random complex value for each free symbol is generated by the random_complex_number routine giving real and imaginary parts in the range given by the re_min, re_max, im_min, and im_max values. The returned value is evaluated to a precision of n (if given) else the maximum of 15 and the precision needed to get more than 1 digit of precision. If the expression could not be evaluated to a number, or could not be evaluated to more than 1 digit of precision, then None is returned. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x, y >>> x._random() # doctest: +SKIP 0.0392918155679172 + 0.916050214307199*I >>> x._random(2) # doctest: +SKIP -0.77 - 0.87*I >>> (x + y/2)._random(2) # doctest: +SKIP -0.57 + 0.16*I >>> sqrt(2)._random(2) 1.4 See Also ======== sympy.utilities.randtest.random_complex_number """ free = self.free_symbols prec = 1 if free: from sympy.utilities.randtest import random_complex_number a, c, b, d = re_min, re_max, im_min, im_max reps = dict(list(zip(free, [random_complex_number(a, b, c, d, rational=True) for zi in free]))) try: nmag = abs(self.evalf(2, subs=reps)) except TypeError: # if an out of range value resulted in evalf problems # then return None -- XXX is there a way to know how to # select a good random number for a given expression? # e.g. when calculating n! negative values for n should not # be used return None else: reps = {} nmag = abs(self.evalf(2)) if not hasattr(nmag, '_prec'): # e.g. exp_polar(2*I*pi) doesn't evaluate but is_number is True return None if nmag._prec == 1: # increase the precision up to the default maximum # precision to see if we can get any significance # get the prec steps (patterned after giant_steps in # libintmath) which approximately doubles the prec # each step from sympy.core.evalf import DEFAULT_MAXPREC as target L = [target] start = 2 while 1: Li = L[-1]//2 + 2 if Li >= L[-1] or Li < start: if L[-1] != start: L.append(start) break L.append(Li) L = L[::-1] # evaluate for prec in L: nmag = abs(self.evalf(prec, subs=reps)) if nmag._prec != 1: break if nmag._prec != 1: if n is None: n = max(prec, 15) return self.evalf(n, subs=reps) # never got any significance return None
[docs] def is_constant(self, *wrt, **flags): """Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, two strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples ======== >>> from sympy import cos, sin, Sum, S, pi >>> from sympy.abc import a, n, x, y >>> x.is_constant() False >>> S(2).is_constant() True >>> Sum(x, (x, 1, 10)).is_constant() True >>> Sum(x, (x, 1, n)).is_constant() False >>> Sum(x, (x, 1, n)).is_constant(y) True >>> Sum(x, (x, 1, n)).is_constant(n) False >>> Sum(x, (x, 1, n)).is_constant(x) True >>> eq = a*cos(x)**2 + a*sin(x)**2 - a >>> eq.is_constant() True >>> eq.subs({x:pi, a:2}) == eq.subs({x:pi, a:3}) == 0 True >>> (0**x).is_constant() False >>> x.is_constant() False >>> (x**x).is_constant() False >>> one = cos(x)**2 + sin(x)**2 >>> one.is_constant() True >>> ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True """ simplify = flags.get('simplify', True) # Except for expressions that contain units, only one of these should # be necessary since if something is # known to be a number it should also know that there are no # free symbols. But is_number quits as soon as it hits a non-number # whereas free_symbols goes until all free symbols have been collected, # thus is_number should be faster. But a double check on free symbols # is made just in case there is a discrepancy between the two. free = self.free_symbols if self.is_number or not free: # if the following assertion fails then that object's free_symbols # method needs attention: if an expression is a number it cannot # have free symbols assert not free return True # if we are only interested in some symbols and they are not in the # free symbols then this expression is constant wrt those symbols wrt = set(wrt) if wrt and not wrt & free: return True wrt = wrt or free # simplify unless this has already been done if simplify: self = self.simplify() # is_zero should be a quick assumptions check; it can be wrong for # numbers (see test_is_not_constant test), giving False when it # shouldn't, but hopefully it will never give True unless it is sure. if self.is_zero: return True # try numerical evaluation to see if we get two different values failing_number = None if wrt == free: # try 0 and 1 a = self.subs(list(zip(free, [0]*len(free)))) if a is S.NaN: a = self._random(None, 0, 0, 0, 0) if a is not None and a is not S.NaN: b = self.subs(list(zip(free, [1]*len(free)))) if b is S.NaN: b = self._random(None, 1, 0, 1, 0) if b is not None and b is not S.NaN: if b.equals(a) is False: return False # try random real b = self._random(None, -1, 0, 1, 0) if b is not None and b is not S.NaN and b.equals(a) is False: return False # try random complex b = self._random() if b is not None and b is not S.NaN: if a != b: return False failing_number = a if a.is_number else b # now we will test each wrt symbol (or all free symbols) to see if the # expression depends on them or not using differentiation. This is # not sufficient for all expressions, however, so we don't return # False if we get a derivative other than 0 with free symbols. for w in wrt: deriv = self.diff(w) if simplify: deriv = deriv.simplify() if deriv != 0: if not (deriv.is_Number or pure_complex(deriv)): if flags.get('failing_number', False): return failing_number elif deriv.free_symbols: # dead line provided _random returns None in such cases return None return False return True
[docs] def equals(self, other, failing_expression=False): """Return True if self == other, False if it doesn't, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. """ from sympy.simplify.simplify import nsimplify, simplify from sympy.solvers.solvers import solve from sympy.polys.polyerrors import NotAlgebraic from sympy.polys.numberfields import minimal_polynomial other = sympify(other) if self == other: return True # they aren't the same so see if we can make the difference 0; # don't worry about doing simplification steps one at a time # because if the expression ever goes to 0 then the subsequent # simplification steps that are done will be very fast. diff = factor_terms((self - other).simplify(), radical=True) if not diff: return True if not diff.has(Add): # if there is no expanding to be done after simplifying # then this can't be a zero return False constant = diff.is_constant(simplify=False, failing_number=True) if constant is False: return False if constant is None and (diff.free_symbols or not diff.is_number): # e.g. unless the right simplification is done, a symbolic # zero is possible (see expression of issue 3730: without # simplification constant will be None). return if constant is True: ndiff = diff._random() if ndiff: return False # sometimes we can use a simplified result to give a clue as to # what the expression should be; if the expression is *not* zero # then we should have been able to compute that and so now # we can just consider the cases where the approximation appears # to be zero -- we try to prove it via minimal_polynomial. if diff.is_number: approx = diff.nsimplify() if not approx: # try to prove via self-consistency surds = [s for s in diff.atoms(Pow) if s.args[0].is_Integer] # it seems to work better to try big ones first surds.sort(key=lambda x: -x.args[0]) for s in surds: try: # simplify is False here -- this expression has already # been identified as being hard to identify as zero; # we will handle the checking ourselves using nsimplify # to see if we are in the right ballpark or not and if so # *then* the simplification will be attempted. sol = solve(diff, s, check=False, simplify=False) if sol: if s in sol: return True if any(nsimplify(si, [s]) == s and simplify(si) == s for si in sol): return True except NotImplementedError: pass # try to prove with minimal_polynomial but know when # *not* to use this or else it can take a long time. # Pernici noted the following: # >>> q = -73*sqrt(3) + 1 + 128*sqrt(5) + 1315*sqrt(2) # >>> p = expand(q**3)**Rational(1, 3) # >>> minimal_polynomial(p - q) # hangs for at least 15 minutes if False: # change False to condition that assures non-hang try: mp = minimal_polynomial(diff) if mp.is_Symbol: return True return False except NotAlgebraic: pass # diff has not simplified to zero; constant is either None, True # or the number with significance (prec != 1) that was randomly # calculated twice as the same value. if constant not in (True, None) and constant != 0: return False if failing_expression: return diff return None
def _eval_is_positive(self): if self.is_number: if self.is_real is False: return False try: # check to see that we can get a value n2 = self._eval_evalf(2) if n2 is None: raise AttributeError if n2._prec == 1: # no significance raise AttributeError except (AttributeError, ValueError): return None n, i = self.evalf(2).as_real_imag() if not i.is_Number or not n.is_Number: return False if i: if i._prec != 1: return False elif n._prec != 1: if n > 0: return True return False def _eval_is_negative(self): if self.is_number: if self.is_real is False: return False try: # check to see that we can get a value n2 = self._eval_evalf(2) if n2 is None: raise AttributeError if n2._prec == 1: # no significance raise AttributeError except (AttributeError, ValueError): return None n, i = self.evalf(2).as_real_imag() if not i.is_Number or not n.is_Number: return False if i: if i._prec != 1: return False elif n._prec != 1: if n < 0: return True return False def _eval_interval(self, x, a, b): """ Returns evaluation over an interval. For most functions this is: self.subs(x, b) - self.subs(x, a), possibly using limit() if NaN is returned from subs. If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x), respectively. """ from sympy.series import limit if (a is None and b is None): raise ValueError('Both interval ends cannot be None.') if a is None: A = 0 else: A = self.subs(x, a) if A.has(S.NaN) or A.has(S.Infinity): A = limit(self, x, a) if A is S.NaN: return A if b is None: B = 0 else: B = self.subs(x, b) if B.has(S.NaN) or B.has(S.Infinity): B = limit(self, x, b) return B - A def _eval_power(self, other): # subclass to compute self**other for cases when # other is not NaN, 0, or 1 return None def _eval_conjugate(self): if self.is_real: return self elif self.is_imaginary: return -self def conjugate(self): from sympy.functions.elementary.complexes import conjugate as c return c(self) def _eval_transpose(self): from sympy.functions.elementary.complexes import conjugate if self.is_complex: return self elif self.is_hermitian: return conjugate(self) elif self.is_antihermitian: return -conjugate(self) def transpose(self): from sympy.functions.elementary.complexes import transpose return transpose(self) def _eval_adjoint(self): from sympy.functions.elementary.complexes import conjugate, transpose if self.is_hermitian: return self elif self.is_antihermitian: return -self obj = self._eval_conjugate() if obj is not None: return transpose(obj) obj = self._eval_transpose() if obj is not None: return conjugate(obj) def adjoint(self): from sympy.functions.elementary.complexes import adjoint return adjoint(self) @classmethod def _parse_order(cls, order): """Parse and configure the ordering of terms. """ from sympy.polys.orderings import monomial_key try: reverse = order.startswith('rev-') except AttributeError: reverse = False else: if reverse: order = order[4:] monom_key = monomial_key(order) def neg(monom): result = [] for m in monom: if isinstance(m, tuple): result.append(neg(m)) else: result.append(-m) return tuple(result) def key(term): _, ((re, im), monom, ncpart) = term monom = neg(monom_key(monom)) ncpart = tuple([ e.sort_key(order=order) for e in ncpart ]) coeff = ((bool(im), im), (re, im)) return monom, ncpart, coeff return key, reverse
[docs] def as_ordered_factors(self, order=None): """Return list of ordered factors (if Mul) else [self].""" return [self]
[docs] def as_ordered_terms(self, order=None, data=False): """ Transform an expression to an ordered list of terms. Examples ======== >>> from sympy import sin, cos >>> from sympy.abc import x >>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms() [sin(x)**2*cos(x), sin(x)**2, 1] """ key, reverse = self._parse_order(order) terms, gens = self.as_terms() if not any(term.is_Order for term, _ in terms): ordered = sorted(terms, key=key, reverse=reverse) else: _terms, _order = [], [] for term, repr in terms: if not term.is_Order: _terms.append((term, repr)) else: _order.append((term, repr)) ordered = sorted(_terms, key=key, reverse=True) \ + sorted(_order, key=key, reverse=True) if data: return ordered, gens else: return [ term for term, _ in ordered ]
[docs] def as_terms(self): """Transform an expression to a list of terms. """ from sympy.core import Add, Mul, S from sympy.core.exprtools import decompose_power gens, terms = set([]), [] for term in Add.make_args(self): coeff, _term = term.as_coeff_Mul() coeff = complex(coeff) cpart, ncpart = {}, [] if _term is not S.One: for factor in Mul.make_args(_term): if factor.is_number: try: coeff *= complex(factor) except TypeError: pass else: continue if factor.is_commutative: base, exp = decompose_power(factor) cpart[base] = exp gens.add(base) else: ncpart.append(factor) coeff = coeff.real, coeff.imag ncpart = tuple(ncpart) terms.append((term, (coeff, cpart, ncpart))) gens = sorted(gens, key=default_sort_key) k, indices = len(gens), {} for i, g in enumerate(gens): indices[g] = i result = [] for term, (coeff, cpart, ncpart) in terms: monom = [0]*k for base, exp in cpart.items(): monom[indices[base]] = exp result.append((term, (coeff, tuple(monom), ncpart))) return result, gens
[docs] def removeO(self): """Removes the additive O(..) symbol if there is one""" return self
[docs] def getO(self): """Returns the additive O(..) symbol if there is one, else None.""" return None
[docs] def getn(self): """ Returns the order of the expression. The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples ======== >>> from sympy import O >>> from sympy.abc import x >>> (1 + x + O(x**2)).getn() 2 >>> (1 + x).getn() """ o = self.getO() if o is None: return None elif o.is_Order: o = o.expr if o is S.One: return S.Zero if o.is_Symbol: return S.One if o.is_Pow: return o.args[1] if o.is_Mul: # x**n*log(x)**n or x**n/log(x)**n for oi in o.args: if oi.is_Symbol: return S.One if oi.is_Pow: syms = oi.atoms(C.Symbol) if len(syms) == 1: x = syms.pop() oi = oi.subs(x, C.Dummy('x', positive=True)) if oi.base.is_Symbol and oi.exp.is_Rational: return abs(oi.exp) raise NotImplementedError('not sure of order of %s' % o)
[docs] def count_ops(self, visual=None): """wrapper for count_ops that returns the operation count.""" from .function import count_ops return count_ops(self, visual)
[docs] def args_cnc(self, cset=False, warn=True, split_1=True): """Return [commutative factors, non-commutative factors] of self. self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly supressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. >>> from sympy import symbols, oo >>> A, B = symbols('A B', commutative=0) >>> x, y = symbols('x y') >>> (-2*x*y).args_cnc() [[-1, 2, x, y], []] >>> (-2.5*x).args_cnc() [[-1, 2.5, x], []] >>> (-2*x*A*B*y).args_cnc() [[-1, 2, x, y], [A, B]] >>> (-2*x*A*B*y).args_cnc(split_1=False) [[-2, x, y], [A, B]] >>> (-2*x*y).args_cnc(cset=True) [set([-1, 2, x, y]), []] The arg is always treated as a Mul: >>> (-2 + x + A).args_cnc() [[], [x - 2 + A]] >>> (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] """ if self.is_Mul: args = list(self.args) else: args = [self] for i, mi in enumerate(args): if not mi.is_commutative: c = args[:i] nc = args[i:] break else: c = args nc = [] if c and split_1 and ( c[0].is_Number and c[0].is_negative and c[0] is not S.NegativeOne): c[:1] = [S.NegativeOne, -c[0]] if cset: clen = len(c) c = set(c) if clen and warn and len(c) != clen: raise ValueError('repeated commutative arguments: %s' % [ci for ci in c if list(self.args).count(ci) > 1]) return [c, nc]
[docs] def coeff(self, x, n=1, right=False): """ Returns the coefficient from the term(s) containing x**n or None. If n is zero then all terms independent of x will be returned. When x is noncommutative, the coeff to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. See Also ======== as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.nth: like coeff_monomial but powers of monomial terms are used Examples ======== >>> from sympy import symbols >>> from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: >>> (-x + 2*y).coeff(-1) x >>> (x - 2*y).coeff(-1) 2*y You can select terms with no Rational coefficient: >>> (x + 2*y).coeff(1) x >>> (3 + 2*x + 4*x**2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): >>> (3 + 2*x + 4*x**2).coeff(x, 0) 3 >>> eq = ((x + 1)**3).expand() + 1 >>> eq x**3 + 3*x**2 + 3*x + 2 >>> [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] >>> eq -= 2 >>> [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: >>> (-x - 2*y).coeff(2) -y >>> from sympy import sqrt >>> (x + sqrt(2)*x).coeff(sqrt(2)) x The matching is exact: >>> (3 + 2*x + 4*x**2).coeff(x) 2 >>> (3 + 2*x + 4*x**2).coeff(x**2) 4 >>> (3 + 2*x + 4*x**2).coeff(x**3) 0 >>> (z*(x + y)**2).coeff((x + y)**2) z >>> (z*(x + y)**2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: >>> (x + z*(x + x*y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: >>> from sympy import factor_terms >>> factor_terms(x + z*(x + x*y)).coeff(x) z*(y + 1) + 1 >>> n, m, o = symbols('n m o', commutative=False) >>> n.coeff(n) 1 >>> (3*n).coeff(n) 3 >>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m 1 + m >>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m m If there is more than one possible coefficient 0 is returned: >>> (n*m + m*n).coeff(n) 0 If there is only one possible coefficient, it is returned: >>> (n*m + x*m*n).coeff(m*n) x >>> (n*m + x*m*n).coeff(m*n, right=1) 1 """ x = sympify(x) if not isinstance(x, Basic): return S.Zero n = as_int(n) if not x: return S.Zero if x == self: if n == 1: return S.One return S.Zero if x is S.One: co = [a for a in Add.make_args(self) if a.as_coeff_Mul()[0] is S.One] if not co: return S.Zero return Add(*co) if n == 0: if x.is_Add and self.is_Add: c = self.coeff(x, right=right) if not c: return S.Zero if not right: return self - Add(*[a*x for a in Add.make_args(c)]) return self - Add(*[x*a for a in Add.make_args(c)]) return self.as_independent(x, as_Add=not self.is_Mul)[0] # continue with the full method, looking for this power of x: x = x**n def incommon(l1, l2): if not l1 or not l2: return [] n = min(len(l1), len(l2)) for i in xrange(n): if l1[i] != l2[i]: return l1[:i] return l1[:] def find(l, sub, first=True): """ Find where list sub appears in list l. When first is True the first occurance from the left is returned, else the last occurance is returned. Return None if sub is not in l. >> l = range(5)*2 >> find(l, [2, 3]) 2 >> find(l, [2, 3], first=0) 7 >> find(l, [2, 4]) None """ if not sub or not l or len(sub) > len(l): return None n = len(sub) if not first: l.reverse() sub.reverse() for i in xrange(0, len(l) - n + 1): if all(l[i + j] == sub[j] for j in range(n)): break else: i = None if not first: l.reverse() sub.reverse() if i is not None and not first: i = len(l) - (i + n) return i co = [] args = Add.make_args(self) self_c = self.is_commutative x_c = x.is_commutative if self_c and not x_c: return S.Zero if self_c: xargs = x.args_cnc(cset=True, warn=False)[0] for a in args: margs = a.args_cnc(cset=True, warn=False)[0] if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): co.append(Mul(*resid)) if co == []: return S.Zero elif co: return Add(*co) elif x_c: xargs = x.args_cnc(cset=True, warn=False)[0] for a in args: margs, nc = a.args_cnc(cset=True) if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): co.append(Mul(*(list(resid) + nc))) if co == []: return S.Zero elif co: return Add(*co) else: # both nc xargs, nx = x.args_cnc(cset=True) # find the parts that pass the commutative terms for a in args: margs, nc = a.args_cnc(cset=True) if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): co.append((resid, nc)) # now check the non-comm parts if not co: return S.Zero if all(n == co[0][1] for r, n in co): ii = find(co[0][1], nx, right) if ii is not None: if not right: return Mul(Add(*[Mul(*r) for r, c in co]), Mul(*co[0][1][:ii])) else: return Mul(*co[0][1][ii + len(nx):]) beg = reduce(incommon, (n[1] for n in co)) if beg: ii = find(beg, nx, right) if ii is not None: if not right: gcdc = co[0][0] for i in xrange(1, len(co)): gcdc = gcdc.intersection(co[i][0]) if not gcdc: break return Mul(*(list(gcdc) + beg[:ii])) else: m = ii + len(nx) return Add(*[Mul(*(list(r) + n[m:])) for r, n in co]) end = list(reversed( reduce(incommon, (list(reversed(n[1])) for n in co)))) if end: ii = find(end, nx, right) if ii is not None: if not right: return Add(*[Mul(*(list(r) + n[:-len(end) + ii])) for r, n in co]) else: return Mul(*end[ii + len(nx):]) # look for single match hit = None for i, (r, n) in enumerate(co): ii = find(n, nx, right) if ii is not None: if not hit: hit = ii, r, n else: break else: if hit: ii, r, n = hit if not right: return Mul(*(list(r) + n[:ii])) else: return Mul(*n[ii + len(nx):]) return S.Zero
[docs] def as_expr(self, *gens): """ Convert a polynomial to a SymPy expression. Examples ======== >>> from sympy import sin >>> from sympy.abc import x, y >>> f = (x**2 + x*y).as_poly(x, y) >>> f.as_expr() x**2 + x*y >>> sin(x).as_expr() sin(x) """ return self
[docs] def as_coefficient(self, expr): """ Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples ======== >>> from sympy import E, pi, sin, I, Poly >>> from sympy.abc import x >>> E.as_coefficient(E) 1 >>> (2*E).as_coefficient(E) 2 >>> (2*sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) >>> (2*E + x*E).as_coefficient(E) x + 2 >>> _.args[0] # just want the exact match 2 >>> p = Poly(2*E + x*E); p Poly(x*E + 2*E, x, E, domain='ZZ') >>> p.coeff_monomial(E) 2 >>> p.nth(0,1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) >>> (2*E*x + x).as_coefficient(E) >>> (2*E*x + x).coeff(E) 2*x >>> (E*(x + 1) + x).as_coefficient(E) >>> (2*pi*I).as_coefficient(pi*I) 2 >>> (2*I).as_coefficient(pi*I) See Also ======== coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.nth: like coeff_monomial but powers of monomial terms are used """ r = self.extract_multiplicatively(expr) if r and not r.has(expr): return r
[docs] def as_real_imag(self, deep=True, **hints): """Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method can't be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. >>> from sympy import symbols, I >>> x, y = symbols('x,y', real=True) >>> (x + y*I).as_real_imag() (x, y) >>> from sympy.abc import z, w >>> (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) """ if hints.get('ignore') == self: return None else: return (C.re(self), C.im(self))
[docs] def as_powers_dict(self): """Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.""" d = defaultdict(int) d.update(dict([self.as_base_exp()])) return d
[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 ======== >>> 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} """ c, m = self.as_coeff_Mul() if not c.is_Rational: c = S.One m = self d = defaultdict(int) d.update({m: c}) return d
def as_base_exp(self): # a -> b ** e return self, S.One
[docs] def as_coeff_mul(self, *deps): """Return the tuple (c, args) where self is written as a Mul, m. c should be a Rational multiplied by any terms of the Mul that are independent of deps. args should be a tuple of all other terms of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you don't know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. - if you know self is a Mul and want only the head, use self.args[0]; - if you don't want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; - if you want to split self into an independent and dependent parts use self.as_independent(*deps) >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_mul() (3, ()) >>> (3*x*y).as_coeff_mul() (3, (x, y)) >>> (3*x*y).as_coeff_mul(x) (3*y, (x,)) >>> (3*y).as_coeff_mul(x) (3*y, ()) """ if deps: if not self.has(*deps): return self, tuple() return S.One, (self,)
[docs] def as_coeff_add(self, *deps): """Return the tuple (c, args) where self is written as an Add, a. c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you don't know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. - if you know self is an Add and want only the head, use self.args[0]; - if you don't want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. - if you want to split self into an independent and dependent parts use self.as_independent(*deps) >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_add() (3, ()) >>> (3 + x).as_coeff_add() (3, (x,)) >>> (3 + x + y).as_coeff_add(x) (y + 3, (x,)) >>> (3 + y).as_coeff_add(x) (y + 3, ()) """ if deps: if not self.has(*deps): return self, tuple() return S.Zero, (self,)
[docs] def primitive(self): """Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples ======== >>> from sympy.abc import x >>> (3*(x + 1)**2).primitive() (3, (x + 1)**2) >>> a = (6*x + 2); a.primitive() (2, 3*x + 1) >>> b = (x/2 + 3); b.primitive() (1/2, x + 6) >>> (a*b).primitive() == (1, a*b) True """ if not self: return S.One, S.Zero c, r = self.as_coeff_Mul(rational=True) if c.is_negative: c, r = -c, -r return c, r
[docs] def as_content_primitive(self, radical=False): """This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo. The primitive need no be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x, y, z >>> eq = 2 + 2*x + 2*y*(3 + 3*y) The as_content_primitive function is recursive and retains structure: >>> eq.as_content_primitive() (2, x + 3*y*(y + 1) + 1) Integer powers will have Rationals extracted from the base: >>> ((2 + 6*x)**2).as_content_primitive() (4, (3*x + 1)**2) >>> ((2 + 6*x)**(2*y)).as_content_primitive() (1, (2*(3*x + 1))**(2*y)) Terms may end up joining once their as_content_primitives are added: >>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() (11, x*(y + 1)) >>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() (9, x*(y + 1)) >>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive() (1, 6.0*x*(y + 1) + 3*z*(y + 1)) >>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() (121, x**2*(y + 1)**2) >>> ((5*(x*(1 + y)) + 2.0*x*(3 + 3*y))**2).as_content_primitive() (1, 121.0*x**2*(y + 1)**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))) """ return S.One, self
[docs] def as_numer_denom(self): """ expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also ======== normal: return a/b instead of a, b """ return self, S.One
def normal(self): n, d = self.as_numer_denom() if d is S.One: return n return n/d
[docs] def extract_multiplicatively(self, c): """Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. >>> from sympy import symbols, Rational >>> x, y = symbols('x,y', real=True) >>> ((x*y)**3).extract_multiplicatively(x**2 * y) x*y**2 >>> ((x*y)**3).extract_multiplicatively(x**4 * y) >>> (2*x).extract_multiplicatively(2) x >>> (2*x).extract_multiplicatively(3) >>> (Rational(1,2)*x).extract_multiplicatively(3) x/6 """ c = sympify(c) if c is S.One: return self elif c == self: return S.One if c.is_Add: cc, pc = c.primitive() if cc is not S.One: c = Mul(cc, pc, evaluate=False) if c.is_Mul: a, b = c.as_two_terms() x = self.extract_multiplicatively(a) if x is not None: return x.extract_multiplicatively(b) quotient = self / c if self.is_Number: if self is S.Infinity: if c.is_positive: return S.Infinity elif self is S.NegativeInfinity: if c.is_negative: return S.Infinity elif c.is_positive: return S.NegativeInfinity elif self is S.ComplexInfinity: if not c.is_zero: return S.ComplexInfinity elif self is S.NaN: return S.NaN elif self.is_Integer: if not quotient.is_Integer: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_Rational: if not quotient.is_Rational: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_Float: if not quotient.is_Float: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_NumberSymbol or self.is_Symbol or self is S.ImaginaryUnit: if quotient.is_Mul and len(quotient.args) == 2: if quotient.args[0].is_Integer and quotient.args[0].is_positive and quotient.args[1] == self: return quotient elif quotient.is_Integer: return quotient elif self.is_Add: cs, ps = self.primitive() if cs is not S.One: return Mul(cs, ps, evaluate=False).extract_multiplicatively(c) newargs = [] for arg in self.args: newarg = arg.extract_multiplicatively(c) if newarg is not None: newargs.append(newarg) else: return None return Add(*newargs) elif self.is_Mul: args = list(self.args) for i, arg in enumerate(args): newarg = arg.extract_multiplicatively(c) if newarg is not None: args[i] = newarg return Mul(*args) elif self.is_Pow: if c.is_Pow and c.base == self.base: new_exp = self.exp.extract_additively(c.exp) if new_exp is not None: return self.base ** (new_exp) elif c == self.base: new_exp = self.exp.extract_additively(1) if new_exp is not None: return self.base ** (new_exp)
[docs] def could_extract_minus_sign(self): """Canonical way to choose an element in the set {e, -e} where e is any expression. If the canonical element is e, we have e.could_extract_minus_sign() == True, else e.could_extract_minus_sign() == False. For any expression, the set {e.could_extract_minus_sign(), (-e).could_extract_minus_sign()} must be {True, False}. >>> from sympy.abc import x, y >>> (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign() True """ negative_self = -self self_has_minus = (self.extract_multiplicatively(-1) is not None) negative_self_has_minus = ( (negative_self).extract_multiplicatively(-1) is not None) if self_has_minus != negative_self_has_minus: return self_has_minus else: if self.is_Add: # We choose the one with less arguments with minus signs all_args = len(self.args) negative_args = len([False for arg in self.args if arg.could_extract_minus_sign()]) positive_args = all_args - negative_args if positive_args > negative_args: return False elif positive_args < negative_args: return True elif self.is_Mul: # We choose the one with an odd number of minus signs num, den = self.as_numer_denom() args = Mul.make_args(num) + Mul.make_args(den) arg_signs = [arg.could_extract_minus_sign() for arg in args] negative_args = list(filter(None, arg_signs)) return len(negative_args) % 2 == 1 # As a last resort, we choose the one with greater value of .sort_key() return self.sort_key() < negative_self.sort_key()
[docs] def extract_branch_factor(self, allow_half=False): """ Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). >>> from sympy import exp_polar, I, pi >>> from sympy.abc import x, y >>> exp_polar(I*pi).extract_branch_factor() (exp_polar(I*pi), 0) >>> exp_polar(2*I*pi).extract_branch_factor() (1, 1) >>> exp_polar(-pi*I).extract_branch_factor() (exp_polar(I*pi), -1) >>> exp_polar(3*pi*I + x).extract_branch_factor() (exp_polar(x + I*pi), 1) >>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor() (y*exp_polar(2*pi*x), -1) >>> exp_polar(-I*pi/2).extract_branch_factor() (exp_polar(-I*pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): >>> exp_polar(I*pi).extract_branch_factor(allow_half=True) (1, 1/2) >>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True) (1, 1) >>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True) (1, 3/2) >>> exp_polar(-I*pi).extract_branch_factor(allow_half=True) (1, -1/2) """ from sympy import exp_polar, pi, I, ceiling, Add n = S(0) res = S(1) args = Mul.make_args(self) exps = [] for arg in args: if arg.func is exp_polar: exps += [arg.exp] else: res *= arg piimult = S(0) extras = [] while exps: exp = exps.pop() if exp.is_Add: exps += exp.args continue if exp.is_Mul: coeff = exp.as_coefficient(pi*I) if coeff is not None: piimult += coeff continue extras += [exp] if not piimult.free_symbols: coeff = piimult tail = () else: coeff, tail = piimult.as_coeff_add(*piimult.free_symbols) # round down to nearest multiple of 2 branchfact = ceiling(coeff/2 - S(1)/2)*2 n += branchfact/2 c = coeff - branchfact if allow_half: nc = c.extract_additively(1) if nc is not None: n += S(1)/2 c = nc newexp = pi*I*Add(*((c, ) + tail)) + Add(*extras) if newexp != 0: res *= exp_polar(newexp) return res, n
def _eval_is_polynomial(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
[docs] def is_polynomial(self, *syms): """ Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are "polynomials" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, \*syms) should work if and only if expr.is_polynomial(\*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples ======== >>> from sympy import Symbol >>> x = Symbol('x') >>> ((x**2 + 1)**4).is_polynomial(x) True >>> ((x**2 + 1)**4).is_polynomial() True >>> (2**x + 1).is_polynomial(x) False >>> n = Symbol('n', nonnegative=True, integer=True) >>> (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. >>> from sympy import sqrt, factor, cancel >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1) >>> a.is_polynomial(y) False >>> factor(a) y + 1 >>> factor(a).is_polynomial(y) True >>> b = (y**2 + 2*y + 1)/(y + 1) >>> b.is_polynomial(y) False >>> cancel(b) y + 1 >>> cancel(b).is_polynomial(y) True See also .is_rational_function() """ if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set([]): # constant polynomial return True else: return self._eval_is_polynomial(syms)
def _eval_is_rational_function(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
[docs] def is_rational_function(self, *syms): """ Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "rational functions" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples ======== >>> from sympy import Symbol, sin >>> from sympy.abc import x, y >>> (x/y).is_rational_function() True >>> (x**2).is_rational_function() True >>> (x/sin(y)).is_rational_function(y) False >>> n = Symbol('n', integer=True) >>> (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. >>> from sympy import sqrt, factor >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1)/y >>> a.is_rational_function(y) False >>> factor(a) (y + 1)/y >>> factor(a).is_rational_function(y) True See also is_algebraic_expr(). """ if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set([]): # constant rational function return True else: return self._eval_is_rational_function(syms)
def _eval_is_algebraic_expr(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
[docs] def is_algebraic_expr(self, *syms): ''' This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "algebraic expressions" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples ======== >>> from sympy import Symbol, sqrt >>> x = Symbol('x') >>> sqrt(1 + x).is_rational_function() False >>> sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. >>> from sympy import sin, factor >>> a = sqrt(sin(x)**2 + 2*sin(x) + 1)/(sin(x) + 1) >>> a.is_algebraic_expr(x) False >>> factor(a).is_algebraic_expr() True See Also ======== is_rational_function() References ========== - http://en.wikipedia.org/wiki/Algebraic_expression ''' if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set([]): # constant algebraic expression return True else: return self._eval_is_algebraic_expr(syms) ################################################################################### ##################### SERIES, LEADING TERM, LIMIT, ORDER METHODS ################## ###################################################################################
[docs] def series(self, x=None, x0=0, n=6, dir="+", logx=None): """ Series expansion of "self" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Note: when n != None, if an O() term is returned then the x in the in it and the entire expression represents x - x0, the displacement from x0. (If there is no O() term then the series was exact and x has it's normal meaning.) This is currently necessary since sympy's O() can only represent terms at x0=0. So instead of:: cos(x).series(x0=1, n=2) --> (1 - x)*sin(1) + cos(1) + O((x - 1)**2) which graphically looks like this:: | .|. . . . | \ . . ---+---------------------- | . . . . | \ x=0 the following is returned instead:: -x*sin(1) + cos(1) + O(x**2) whose graph is this:: \ | . .| . . . \ . . -----+\------------------. | . . . . | \ x=0 which is identical to cos(x + 1).series(n=2). Usage: Returns the series expansion of "self" around the point x = x0 with respect to x up to O(x**n) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. >>> from sympy import cos, exp >>> from sympy.abc import x, y >>> cos(x).series() 1 - x**2/2 + x**4/24 + O(x**6) >>> cos(x).series(n=4) 1 - x**2/2 + O(x**4) >>> e = cos(x + exp(y)) >>> e.series(y, n=2) cos(x + 1) - y*sin(x + 1) + O(y**2) >>> e.series(x, n=2) cos(exp(y)) - x*sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. >>> term=cos(x).series(n=None) >>> [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. >>> abs(x).series(dir="+") x >>> abs(x).series(dir="-") -x """ from sympy import collect if x is None: syms = self.atoms(C.Symbol) if len(syms) > 1: raise ValueError('x must be given for multivariate functions.') x = syms.pop() if not self.has(x): if n is None: return (s for s in [self]) else: return self if len(dir) != 1 or dir not in '+-': raise ValueError("Dir must be '+' or '-'") if x0 in [S.Infinity, S.NegativeInfinity]: dir = {S.Infinity: '+', S.NegativeInfinity: '-'}[x0] s = self.subs(x, 1/x).series(x, n=n, dir=dir) if n is None: return (si.subs(x, 1/x) for si in s) return s.subs(x, 1/x) # use rep to shift origin to x0 and change sign (if dir is negative) # and undo the process with rep2 if x0 or dir == '-': if dir == '-': rep = -x + x0 rep2 = -x rep2b = x0 else: rep = x + x0 rep2 = x rep2b = -x0 s = self.subs(x, rep).series(x, x0=0, n=n, dir='+', logx=logx) if n is None: # lseries... return (si.subs(x, rep2 + rep2b) for si in s) # nseries... o = s.getO() or S.Zero s = s.removeO() if o and x0: rep2b = 0 # when O() can handle x0 != 0 this can be removed return s.subs(x, rep2 + rep2b) + o # from here on it's x0=0 and dir='+' handling if x.is_positive is x.is_negative is None: # replace x with an x that has a positive assumption xpos = C.Dummy('x', positive=True, bounded=True) rv = self.subs(x, xpos).series(xpos, x0, n, dir, logx=logx) if n is None: return (s.subs(xpos, x) for s in rv) else: return rv.subs(xpos, x) if n is not None: # nseries handling s1 = self._eval_nseries(x, n=n, logx=logx) o = s1.getO() or S.Zero if o: # make sure the requested order is returned ngot = o.getn() if ngot > n: # leave o in its current form (e.g. with x*log(x)) so # it eats terms properly, then replace it below if n != 0: s1 += o.subs(x, x**C.Rational(n, ngot)) else: s1 += C.Order(1, x) elif ngot < n: # increase the requested number of terms to get the desired # number keep increasing (up to 9) until the received order # is different than the original order and then predict how # many additional terms are needed for more in range(1, 9): s1 = self._eval_nseries(x, n=n + more, logx=logx) newn = s1.getn() if newn != ngot: ndo = n + (n - ngot)*more/(newn - ngot) s1 = self._eval_nseries(x, n=ndo, logx=logx) while s1.getn() < n: s1 = self._eval_nseries(x, n=ndo, logx=logx) ndo += 1 break else: raise ValueError('Could not calculate %s terms for %s' % (str(n), self)) s1 += C.Order(x**n, x) o = s1.getO() s1 = s1.removeO() else: o = C.Order(x**n, x) if (s1 + o).removeO() == s1: o = S.Zero try: return collect(s1, x) + o except NotImplementedError: return s1 + o else: # lseries handling def yield_lseries(s): """Return terms of lseries one at a time.""" for si in s: if not si.is_Add: yield si continue # yield terms 1 at a time if possible # by increasing order until all the # terms have been returned yielded = 0 o = C.Order(si, x)*x ndid = 0 ndo = len(si.args) while 1: do = (si - yielded + o).removeO() o *= x if not do or do.is_Order: continue if do.is_Add: ndid += len(do.args) else: ndid += 1 yield do if ndid == ndo: break yielded += do return yield_lseries(self.removeO()._eval_lseries(x, logx=logx))
[docs] def taylor_term(self, n, x, *previous_terms): """General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the "previous_terms". """ x = sympify(x) _x = C.Dummy('x') return self.subs(x, _x).diff(_x, n).subs(_x, x).subs(x, 0) * x**n / C.factorial(n)
[docs] def lseries(self, x=None, x0=0, dir='+', logx=None): """ Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you don't know how many you should ask for in nseries() using the "n" parameter. See also nseries(). """ return self.series(x, x0, n=None, dir=dir, logx=logx)
def _eval_lseries(self, x, logx=None): # default implementation of lseries is using nseries(), and adaptively # increasing the "n". As you can see, it is not very efficient, because # we are calculating the series over and over again. Subclasses should # override this method and implement much more efficient yielding of # terms. n = 0 series = self._eval_nseries(x, n=n, logx=logx) if not series.is_Order: if series.is_Add: yield series.removeO() else: yield series raise StopIteration while series.is_Order: n += 1 series = self._eval_nseries(x, n=n, logx=logx) e = series.removeO() yield e while 1: while 1: n += 1 series = self._eval_nseries(x, n=n, logx=logx).removeO() if e != series: break yield series - e e = series
[docs] def nseries(self, x=None, x0=0, n=6, dir='+', logx=None): """ Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates "n" terms in the innermost expressions and then builds up the final series just by "cross-multiplying" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we don't have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples ======== >>> from sympy import sin, log, Symbol >>> from sympy.abc import x, y >>> sin(x).nseries(x, 0, 6) x - x**3/6 + x**5/120 + O(x**6) >>> log(x+1).nseries(x, 0, 5) x - x**2/2 + x**3/3 - x**4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): >>> e = sin(log(x)) >>> e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... >>> logx = Symbol('logx') >>> e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but gives only an Order term unless the logx parameter is used: >>> e = x**y >>> e.nseries(x, 0, 2) O(log(x)**2) >>> e.nseries(x, 0, 2, logx=logx) exp(logx*y) """ if x and not x in self.free_symbols: return self if x is None or x0 or dir != '+': # {see XPOS above} or (x.is_positive == x.is_negative == None): return self.series(x, x0, n, dir) else: return self._eval_nseries(x, n=n, logx=logx)
def _eval_nseries(self, x, n, logx): """ Return terms of series for self up to O(x**n) at x=0 from the positive direction. This is a method that should be overridden in subclasses. Users should never call this method directly (use .nseries() instead), so you don't have to write docstrings for _eval_nseries(). """ from sympy.utilities.misc import filldedent raise NotImplementedError(filldedent(""" The _eval_nseries method should be added to %s to give terms up to O(x**n) at x=0 from the positive direction so it is available when nseries calls it.""" % self.func) )
[docs] def limit(self, x, xlim, dir='+'): """ Compute limit x->xlim. """ from sympy.series.limits import limit return limit(self, x, xlim, dir)
[docs] def compute_leading_term(self, x, logx=None): """ as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. """ from sympy.series.gruntz import calculate_series if self.removeO() == 0: return self if logx is None: d = C.Dummy('logx') s = calculate_series(self, x, d).subs(d, C.log(x)) else: s = calculate_series(self, x, logx) return s.as_leading_term(x)
@cacheit
[docs] def as_leading_term(self, *symbols): """ Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples ======== >>> from sympy.abc import x >>> (1 + x + x**2).as_leading_term(x) 1 >>> (1/x**2 + x + x**2).as_leading_term(x) x**(-2) """ from sympy import powsimp if len(symbols) > 1: c = self for x in symbols: c = c.as_leading_term(x) return c elif not symbols: return self x = sympify(symbols[0]) if not x.is_Symbol: raise ValueError('expecting a Symbol but got %s' % x) if x not in self.free_symbols: return self obj = self._eval_as_leading_term(x) if obj is not None: return powsimp(obj, deep=True, combine='exp') raise NotImplementedError('as_leading_term(%s, %s)' % (self, x))
[docs] def as_coeff_exponent(self, x): """ c*x**e -> c,e where x can be any symbolic expression. """ from sympy import collect s = collect(self, x) c, p = s.as_coeff_mul(x) if len(p) == 1: b, e = p[0].as_base_exp() if b == x: return c, e return s, S.Zero
[docs] def leadterm(self, x): """ Returns the leading term a*x**b as a tuple (a, b). Examples ======== >>> from sympy.abc import x >>> (1+x+x**2).leadterm(x) (1, 0) >>> (1/x**2+x+x**2).leadterm(x) (1, -2) """ c, e = self.as_leading_term(x).as_coeff_exponent(x) if x in c.free_symbols: from sympy.utilities.misc import filldedent raise ValueError(filldedent(""" cannot compute leadterm(%s, %s). The coefficient should have been free of x but got %s""" % (self, x, c))) return c, e
[docs] def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """ return S.One, self
[docs] def as_coeff_Add(self): """Efficiently extract the coefficient of a summation. """ return S.Zero, self ################################################################################### ##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS #################### ###################################################################################
def diff(self, *symbols, **assumptions): new_symbols = list(map(sympify, symbols)) # e.g. x, 2, y, z assumptions.setdefault("evaluate", True) return Derivative(self, *new_symbols, **assumptions) ########################################################################### ###################### EXPRESSION EXPANSION METHODS ####################### ########################################################################### # Relevant subclasses should override _eval_expand_hint() methods. See # the docstring of expand() for more info. def _eval_expand_complex(self, **hints): real, imag = self.as_real_imag(**hints) return real + S.ImaginaryUnit*imag @staticmethod def _expand_hint(expr, hint, deep=True, **hints): """ Helper for expand(). Recursively calls expr._eval_expand_hint(). Returns (expr, hit), where expr is the (possibly) expanded expr and hit is True if expr was truly expanded and False otherwise. """ hit = False # XXX: Hack to support non-Basic args # | # V if deep and getattr(expr, 'args', ()) and not expr.is_Atom: sargs = [] for arg in expr.args: arg, arghit = Expr._expand_hint(arg, hint, **hints) hit |= arghit sargs.append(arg) if hit: expr = expr.func(*sargs) if hasattr(expr, hint): newexpr = getattr(expr, hint)(**hints) if newexpr != expr: return (newexpr, True) return (expr, hit) @cacheit
[docs] def expand(self, deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints): """ Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. """ from sympy.simplify.simplify import fraction hints.update(power_base=power_base, power_exp=power_exp, mul=mul, log=log, multinomial=multinomial, basic=basic) expr = self if hints.pop('frac', False): n, d = [a.expand(deep=deep, modulus=modulus, **hints) for a in fraction(self)] return n/d elif hints.pop('denom', False): n, d = fraction(self) return n/d.expand(deep=deep, modulus=modulus, **hints) elif hints.pop('numer', False): n, d = fraction(self) return n.expand(deep=deep, modulus=modulus, **hints)/d # Although the hints are sorted here, an earlier hint may get applied # at a given node in the expression tree before another because of how # the hints are applied. e.g. expand(log(x*(y + z))) -> log(x*y + # x*z) because while applying log at the top level, log and mul are # applied at the deeper level in the tree so that when the log at the # upper level gets applied, the mul has already been applied at the # lower level. # Additionally, because hints are only applied once, the expression # may not be expanded all the way. For example, if mul is applied # before multinomial, x*(x + 1)**2 won't be expanded all the way. For # now, we just use a special case to make multinomial run before mul, # so that at least polynomials will be expanded all the way. In the # future, smarter heuristics should be applied. # TODO: Smarter heuristics def _expand_hint_key(hint): """Make multinomial come before mul""" if hint == 'mul': return 'mulz' return hint for hint in sorted(hints.keys(), key=_expand_hint_key): use_hint = hints[hint] if use_hint: hint = '_eval_expand_' + hint expr, hit = Expr._expand_hint(expr, hint, deep=deep, **hints) while True: was = expr if hints.get('multinomial', False): expr, _ = Expr._expand_hint( expr, '_eval_expand_multinomial', deep=deep, **hints) if hints.get('mul', False): expr, _ = Expr._expand_hint( expr, '_eval_expand_mul', deep=deep, **hints) if hints.get('log', False): expr, _ = Expr._expand_hint( expr, '_eval_expand_log', deep=deep, **hints) if expr == was: break if modulus is not None: modulus = sympify(modulus) if not modulus.is_Integer or modulus <= 0: raise ValueError( "modulus must be a positive integer, got %s" % modulus) terms = [] for term in Add.make_args(expr): coeff, tail = term.as_coeff_Mul(rational=True) coeff %= modulus if coeff: terms.append(coeff*tail) expr = Add(*terms) return expr ########################################################################### ################### GLOBAL ACTION VERB WRAPPER METHODS #################### ###########################################################################
[docs] def integrate(self, *args, **kwargs): """See the integrate function in sympy.integrals""" from sympy.integrals import integrate return integrate(self, *args, **kwargs)
[docs] def simplify(self, ratio=1.7, measure=None): """See the simplify function in sympy.simplify""" from sympy.simplify import simplify from sympy.core.function import count_ops measure = measure or count_ops return simplify(self, ratio, measure)
[docs] def nsimplify(self, constants=[], tolerance=None, full=False): """See the nsimplify function in sympy.simplify""" from sympy.simplify import nsimplify return nsimplify(self, constants, tolerance, full)
[docs] def separate(self, deep=False, force=False): """See the separate function in sympy.simplify""" from sympy.simplify import separate return separate(self, deep)
[docs] def collect(self, syms, func=None, evaluate=True, exact=False, distribute_order_term=True): """See the collect function in sympy.simplify""" from sympy.simplify import collect return collect(self, syms, func, evaluate, exact, distribute_order_term)
[docs] def together(self, *args, **kwargs): """See the together function in sympy.polys""" from sympy.polys import together return together(self, *args, **kwargs)
[docs] def apart(self, x=None, **args): """See the apart function in sympy.polys""" from sympy.polys import apart return apart(self, x, **args)
[docs] def ratsimp(self): """See the ratsimp function in sympy.simplify""" from sympy.simplify import ratsimp return ratsimp(self)
[docs] def trigsimp(self, **args): """See the trigsimp function in sympy.simplify""" from sympy.simplify import trigsimp return trigsimp(self, **args)
def _mag(x): """Return integer i such that .1 <= x/10**i < 1 Examples ======== >>> from sympy.core.expr import _mag >>> from sympy import Float >>> _mag(Float(.1)) 0 >>> _mag(Float(.01)) -1 >>> _mag(Float(1234)) 4 """ from math import log10, ceil, log xpos = abs(x.n()) if not xpos: return S.Zero try: mag_first_dig = int(ceil(log10(xpos))) except (ValueError, OverflowError): mag_first_dig = int(ceil(C.Float(mpf_log(xpos._mpf_, 53))/log(10))) # check that we aren't off by 1 if (xpos/10**mag_first_dig) >= 1: assert 1 <= (xpos/10**mag_first_dig) < 10 mag_first_dig += 1 return mag_first_dig from .mul import Mul from .add import Add from .power import Pow from .function import Derivative, expand_mul from .mod import Mod from .exprtools import factor_terms from .numbers import Integer, Rational