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from __future__ import print_function, division
from .sympify import sympify, _sympify, SympifyError 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, range from mpmath.libmp import mpf_log, prec_to_dps
from collections import defaultdict
class Expr(Basic, EvalfMixin): """ Base class for algebraic expressions.
Everything that requires arithmetic operations to be defined should subclass this class, instead of Basic (which should be used only for argument storage and expression manipulation, i.e. pattern matching, substitutions, etc).
See Also ========
sympy.core.basic.Basic """
__slots__ = []
@property def _diff_wrt(self): """Is it allowed to take derivative wrt to this instance.
This determines if it is allowed to take derivatives wrt this object. Subclasses such as Symbol, Function and Derivative should return True to enable derivatives wrt them. The implementation in Derivative separates the Symbol and non-Symbol _diff_wrt=True variables and temporarily converts the non-Symbol vars in Symbols when performing the differentiation.
Note, see the docstring of Derivative for how this should work mathematically. In particular, note that expr.subs(yourclass, Symbol) should be well-defined on a structural level, or this will lead to inconsistent results.
Examples ========
>>> from sympy import Expr >>> e = Expr() >>> e._diff_wrt False >>> class MyClass(Expr): ... _diff_wrt = True ... >>> (2*MyClass()).diff(MyClass()) 2 """ return False
@cacheit def sort_key(self, order=None):
else:
else: else:
[ default_sort_key(arg, order=order) for arg in args ])
# *************** # * Arithmetics * # *************** # Expr and its sublcasses use _op_priority to determine which object # passed to a binary special method (__mul__, etc.) will handle the # operation. In general, the 'call_highest_priority' decorator will choose # the object with the highest _op_priority to handle the call. # Custom subclasses that want to define their own binary special methods # should set an _op_priority value that is higher than the default. # # **NOTE**: # This is a temporary fix, and will eventually be replaced with # something better and more powerful. See issue 5510. _op_priority = 10.0
def __pos__(self): return self
def __neg__(self):
def __abs__(self):
@_sympifyit('other', NotImplemented) @call_highest_priority('__radd__') def __add__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__add__') def __radd__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__rsub__') def __sub__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__sub__') def __rsub__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__rmul__') def __mul__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__mul__') def __rmul__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__rpow__') def __pow__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__pow__') def __rpow__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__rdiv__') def __div__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__div__') def __rdiv__(self, other):
__truediv__ = __div__ __rtruediv__ = __rdiv__
@_sympifyit('other', NotImplemented) @call_highest_priority('__rmod__') def __mod__(self, other):
@_sympifyit('other', NotImplemented) @call_highest_priority('__mod__') def __rmod__(self, other): return Mod(other, self)
@_sympifyit('other', NotImplemented) @call_highest_priority('__rfloordiv__') def __floordiv__(self, other): from sympy.functions.elementary.integers import floor return floor(self / other)
@_sympifyit('other', NotImplemented) @call_highest_priority('__floordiv__') def __rfloordiv__(self, other): from sympy.functions.elementary.integers import floor return floor(self / other)
def __int__(self): # Although we only need to round to the units position, we'll # get one more digit so the extra testing below can be avoided # unless the rounded value rounded to an integer, e.g. if an # expression were equal to 1.9 and we rounded to the unit position # we would get a 2 and would not know if this rounded up or not # without doing a test (as done below). But if we keep an extra # digit we know that 1.9 is not the same as 1 and there is no # need for further testing: our int value is correct. If the value # were 1.99, however, this would round to 2.0 and our int value is # off by one. So...if our round value is the same as the int value # (regardless of how much extra work we do to calculate extra decimal # places) we need to test whether we are off by one. raise TypeError("can't convert %s to int" % r) return 0 # off-by-one check isign = 1 if i > 0 else -1 x = Dummy() # in the following (self - i).evalf(2) will not always work while # (self - r).evalf(2) and the use of subs does; if the test that # was added when this comment was added passes, it might be safe # to simply use sign to compute this rather than doing this by hand: diff_sign = 1 if (self - x).evalf(2, subs={x: i}) > 0 else -1 if diff_sign != isign: i -= isign __long__ = __int__
def __float__(self): # Don't bother testing if it's a number; if it's not this is going # to fail, and if it is we still need to check that it evalf'ed to # a number. return float(result)
def __complex__(self):
def __ge__(self, other): except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) me.has(S.ComplexInfinity): raise TypeError("Invalid comparison of complex %s" % me) raise TypeError("Invalid NaN comparison") dif.is_nonnegative is not dif.is_negative:
def __le__(self, other): except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) me.has(S.ComplexInfinity): raise TypeError("Invalid comparison of complex %s" % me) raise TypeError("Invalid NaN comparison") dif.is_nonpositive is not dif.is_positive:
def __gt__(self, other): except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) me.has(S.ComplexInfinity): raise TypeError("Invalid comparison of complex %s" % me) raise TypeError("Invalid NaN comparison") dif.is_positive is not dif.is_nonpositive:
def __lt__(self, other): except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) me.has(S.ComplexInfinity): raise TypeError("Invalid comparison of complex %s" % me) raise TypeError("Invalid NaN comparison") dif.is_negative is not dif.is_nonnegative:
@staticmethod def _from_mpmath(x, prec): else: raise TypeError("expected mpmath number (mpf or mpc)")
@property 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
"""
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 """
for zi in free]))) except (ValueError, 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:
# e.g. exp_polar(2*I*pi) doesn't evaluate but is_number is True return None
# increase the precision up to the default maximum # precision to see if we can get any significance
from mpmath.libmp.libintmath import giant_steps from sympy.core.evalf import DEFAULT_MAXPREC as target
# evaluate for prec in giant_steps(2, target): nmag = abs(self.evalf(prec, subs=reps)) if nmag._prec != 1: break
# never got any significance return None
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 """
# 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. # 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
# if we are only interested in some symbols and they are not in the # free symbols then this expression is constant wrt those symbols
# simplify unless this has already been done
# 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. return True
# try numerical evaluation to see if we get two different values # try 0 (for a) and 1 (for b) simultaneous=True) # evaluation may succeed when substitution fails except ZeroDivisionError: a = None simultaneous=True) # evaluation may succeed when substitution fails b = expr._random(None, 1, 0, 1, 0) except ZeroDivisionError: b = None # try random real b = expr._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 = expr._random() if b is not None and b is not S.NaN: if b.equals(a) is False: 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 = expr.diff(w) if simplify: deriv = deriv.simplify() if deriv != 0: if not (pure_complex(deriv, or_real=True)): 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
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.
"""
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.
return True
# if there is no expanding to be done after simplifying # then this can't be a zero
# e.g. unless the right simplification is done, a symbolic # zero is possible (see expression of issue 6829: without # simplification constant will be None). return
# 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. if s.is_Symbol: sol = list(solveset(diff, s)) else: sol = [s] if sol: if s in sol: return True if s.is_real: 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. e.g. issue 8354 if True: # change True to condition that assures non-hang try: mp = minimal_polynomial(diff) if mp.is_Symbol: return True return False except (NotAlgebraic, NotImplementedError): 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): # check to see that we can get a value raise AttributeError raise AttributeError return False self.is_algebraic and not self.has(Function): except (NotAlgebraic, NotImplementedError): pass
def _eval_is_negative(self): # check to see that we can get a value raise AttributeError raise AttributeError return False self.is_algebraic and not self.has(Function): except (NotAlgebraic, NotImplementedError): pass
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, 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, S.Infinity, S.NegativeInfinity, S.ComplexInfinity): A = limit(self, x, a) if A is S.NaN: return A if isinstance(A, Limit): raise NotImplementedError("Could not compute limit")
if b is None: B = 0 else: B = self.subs(x, b) if B.has(S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity): B = limit(self, x, b) if isinstance(B, Limit): raise NotImplementedError("Could not compute limit")
return B - A
def _eval_power(self, other): # subclass to compute self**other for cases when # other is not NaN, 0, or 1
def _eval_conjugate(self): return -self
def conjugate(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. """
else: order = order[4:]
result.append(neg(m)) else:
def as_ordered_factors(self, order=None): """Return list of ordered factors (if Mul) else [self]."""
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]
"""
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)
return ordered, gens else:
def as_terms(self): """Transform an expression to a list of terms. """
else: continue
else: ncpart.append(factor)
def removeO(self): """Removes the additive O(..) symbol if there is one"""
def getO(self): """Returns the additive O(..) symbol if there is one, else None."""
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()
""" return None return S.Zero return o.args[1] for oi in o.args: if oi.is_Symbol: return S.One if oi.is_Pow: syms = oi.atoms(Symbol) if len(syms) == 1: x = syms.pop() oi = oi.subs(x, 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)
def count_ops(self, visual=None): """wrapper for count_ops that returns the operation count."""
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], []] """
else: c = args[:i] nc = args[i:] break else:
c[0].is_Number and c[0].is_negative and c[0] is not S.NegativeOne):
raise ValueError('repeated commutative arguments: %s' % [ci for ci in c if list(self.args).count(ci) > 1])
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
""" return S.Zero
return S.Zero
return S.Zero
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 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=True)[0]
# continue with the full method, looking for this power of x:
if not l1 or not l2: return [] n = min(len(l1), len(l2)) for i in range(n): if l1[i] != l2[i]: return l1[:i] return l1[:]
""" 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 range(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
return S.Zero
continue 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 range(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
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)
"""
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
"""
def as_independent(self, *deps, **hint): """ A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.:
* separatevars() to change Mul, Add and Pow (including exp) into Mul * .expand(mul=True) to change Add or Mul into Add * .expand(log=True) to change log expr into an Add
The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for `self` of zero regardless of hints.
For nonzero `self`, the returned tuple (i, d) has the following interpretation:
* i will has no variable that appears in deps * d will be 1 or else have terms that contain variables that are in deps * if self is an Add then self = i + d * if self is a Mul then self = i*d * otherwise (self, S.One) or (S.One, self) is returned.
To force the expression to be treated as an Add, use the hint as_Add=True
Examples ========
-- self is an Add
>>> from sympy import sin, cos, exp >>> from sympy.abc import x, y, z
>>> (x + x*y).as_independent(x) (0, x*y + x) >>> (x + x*y).as_independent(y) (x, x*y) >>> (2*x*sin(x) + y + x + z).as_independent(x) (y + z, 2*x*sin(x) + x) >>> (2*x*sin(x) + y + x + z).as_independent(x, y) (z, 2*x*sin(x) + x + y)
-- self is a Mul
>>> (x*sin(x)*cos(y)).as_independent(x) (cos(y), x*sin(x))
non-commutative terms cannot always be separated out when self is a Mul
>>> from sympy import symbols >>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False) >>> (n1 + n1*n2).as_independent(n2) (n1, n1*n2) >>> (n2*n1 + n1*n2).as_independent(n2) (0, n1*n2 + n2*n1) >>> (n1*n2*n3).as_independent(n1) (1, n1*n2*n3) >>> (n1*n2*n3).as_independent(n2) (n1, n2*n3) >>> ((x-n1)*(x-y)).as_independent(x) (1, (x - y)*(x - n1))
-- self is anything else:
>>> (sin(x)).as_independent(x) (1, sin(x)) >>> (sin(x)).as_independent(y) (sin(x), 1) >>> exp(x+y).as_independent(x) (1, exp(x + y))
-- force self to be treated as an Add:
>>> (3*x).as_independent(x, as_Add=True) (0, 3*x)
-- force self to be treated as a Mul:
>>> (3+x).as_independent(x, as_Add=False) (1, x + 3) >>> (-3+x).as_independent(x, as_Add=False) (1, x - 3)
Note how the below differs from the above in making the constant on the dep term positive.
>>> (y*(-3+x)).as_independent(x) (y, x - 3)
-- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols
>>> from sympy import Integral >>> I = Integral(x, (x, 1, 2)) >>> I.has(x) True >>> x in I.free_symbols False >>> I.as_independent(x) == (I, 1) True >>> (I + x).as_independent(x) == (I, x) True
Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values
>>> from sympy import separatevars, log >>> separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) >>> (x + x*y).as_independent(y) (x, x*y) >>> separatevars(x + x*y).as_independent(y) (x, y + 1) >>> (x*(1 + y)).as_independent(y) (x, y + 1) >>> (x*(1 + y)).expand(mul=True).as_independent(y) (x, x*y) >>> a, b=symbols('a b', positive=True) >>> (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))
See Also ======== .separatevars(), .expand(log=True), Add.as_two_terms(), Mul.as_two_terms(), .as_coeff_add(), .as_coeff_mul() """
return S.Zero, S.Zero
else:
# sift out deps into symbolic and other and ignore # all symbols but those that are in the free symbols else:
"""return the standard has() if there are no literal symbols, else check to see that symbol-deps are in the free symbols."""
func is not Add and func is not Mul): else: else: else:
else: # handle noncommutative by stopping at first dependent term if has(n): depend.extend(nc[i:]) break indep.append(n) Mul(*depend, evaluate=False) if nc else _unevaluated_Mul(*depend))
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))
""" else:
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."""
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
def as_coeff_mul(self, *deps, **kwargs): """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, ()) """ return self, tuple()
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 not self.has(*deps): return self, tuple()
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 """ return S.One, S.Zero c, r = -c, -r
def as_content_primitive(self, radical=False, clear=True): """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)))
If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients.
>>> (x/2 + y).as_content_primitive() (1/2, x + 2*y) >>> (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) """
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 """
def normal(self):
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
""" return None return self cc, pc = c.primitive() if cc is not S.One: c = Mul(cc, pc, evaluate=False) a, b = c.as_two_terms() x = self.extract_multiplicatively(a) if x is not None: return x.extract_multiplicatively(b) if c.is_positive: return S.Infinity if c.is_negative: return S.Infinity elif c.is_positive: return S.NegativeInfinity if not c.is_zero: return S.ComplexInfinity else: else: return None else: return quotient return quotient else: new_exp = self.exp.extract_additively(c.exp) if new_exp is not None: return self.base ** (new_exp) return self.base ** (new_exp)
def extract_additively(self, c): """Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.
Examples ========
>>> from sympy.abc import x, y >>> e = 2*x + 3 >>> e.extract_additively(x + 1) x + 2 >>> e.extract_additively(3*x) >>> e.extract_additively(4) >>> (y*(x + 1)).extract_additively(x + 1) >>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1) (x + 1)*(x + 2*y) + 3
Sometimes auto-expansion will return a less simplified result than desired; gcd_terms might be used in such cases:
>>> from sympy import gcd_terms >>> (4*x*(y + 1) + y).extract_additively(x) 4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y >>> gcd_terms(_) x*(4*y + 3) + y
See Also ======== extract_multiplicatively coeff as_coefficient
"""
return None return self return S.Zero return None
return None # XXX should we match types? i.e should 3 - .1 succeed? co < 0 and diff < 0 and diff > co): return diff
if c.is_Number: co, t = self.as_coeff_Add() xa = co.extract_additively(c) if xa is None: return None return xa + t
# handle the args[0].is_Number case separately # since we will have trouble looking for the coeff of # a number. if c.is_Add and c.args[0].is_Number: # whole term as a term factor co = self.coeff(c) xa0 = (co.extract_additively(1) or 0)*c if xa0: diff = self - co*c return (xa0 + (diff.extract_additively(c) or diff)) or None # term-wise h, t = c.as_coeff_Add() sh, st = self.as_coeff_Add() xa = sh.extract_additively(h) if xa is None: return None xa2 = st.extract_additively(t) if xa2 is None: return None return xa + xa2
# whole term as a term factor co = self.coeff(c) xa0 = (co.extract_additively(1) or 0)*c if xa0: diff = self - co*c return (xa0 + (diff.extract_additively(c) or diff)) or None # term-wise coeffs = [] for a in Add.make_args(c): ac, at = a.as_coeff_Mul() co = self.coeff(at) if not co: return None coc, cot = co.as_coeff_Add() xa = coc.extract_additively(ac) if xa is None: return None self -= co*at coeffs.append((cot + xa)*at) coeffs.append(self) return Add(*coeffs)
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).extract_multiplicatively(-1) is not None) else: # We choose the one with less arguments with minus signs # 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()
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):
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()
""" else:
# constant polynomial else:
def _eval_is_rational_function(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
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().
""" return False
else: syms = self.free_symbols
# constant rational function return True else:
def _eval_is_algebraic_expr(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
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', real=True) >>> 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 exp, factor >>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(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 ################## ###################################################################################
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.
Returns the series expansion of "self" around the point ``x = x0`` with respect to ``x`` up to ``O((x - x0)**n, x, x0)`` (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) >>> cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1)) >>> 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
""" syms = self.atoms(Symbol) if not syms: return self elif len(syms) > 1: raise ValueError('x must be given for multivariate functions.') x = syms.pop()
if n is None: return (s for s in [self]) else: return self
raise ValueError("Dir must be '+' or '-'")
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 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) return s.subs(x, rep2 + rep2b)
# from here on it's x0=0 and dir='+' handling
# replace x with an x that has a positive assumption xpos = Dummy('x', positive=True, finite=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)
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**Rational(n, ngot)) else: s1 += 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 += Order(x**n, x) o = s1.getO() s1 = s1.removeO() else: o = Order(x**n, x) s1done = s1.doit() if (s1done + o).removeO() == s1done: o = S.Zero
try: return collect(s1, x) + o except NotImplementedError: return s1 + o
else: # lseries handling """Return terms of lseries one at a time.""" continue # yield terms 1 at a time if possible # by increasing order until all the # terms have been returned continue ndid += len(do.args) else: if ndid == ndo: break yielded += do
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". """ from sympy import Dummy, factorial x = sympify(x) _x = Dummy('x') return self.subs(x, _x).diff(_x, n).subs(_x, x).subs(x, 0) * x**n / factorial(n)
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(). """
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. else: return
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
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)
""" return self.series(x, x0, n, dir) else:
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) )
def limit(self, x, xlim, dir='+'): """ Compute limit x->xlim. """
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 import Dummy, log from sympy.series.gruntz import calculate_series
if self.removeO() == 0: return self
if logx is None: d = Dummy('logx') s = calculate_series(self, x, d).subs(d, log(x)) else: s = calculate_series(self, x, logx)
return s.as_leading_term(x)
@cacheit 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)
""" c = self for x in symbols: c = c.as_leading_term(x) return c return self raise ValueError('expecting a Symbol but got %s' % x) raise NotImplementedError('as_leading_term(%s, %s)' % (self, x))
def _eval_as_leading_term(self, x):
def as_coeff_exponent(self, x): """ ``c*x**e -> c,e`` where x can be any symbolic expression. """
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)
""" l = l.subs(log(x), d) 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)))
def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """
def as_coeff_Add(self, rational=False): """Efficiently extract the coefficient of a summation. """
def fps(self, x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False): """ Compute formal power power series of self.
See the docstring of the :func:`fps` function in sympy.series.formal for more information. """ from sympy.series.formal import fps
return fps(self, x, x0, dir, hyper, order, rational, full)
def fourier_series(self, limits=None): """Compute fourier sine/cosine series of self.
See the docstring of the :func:`fourier_series` in sympy.series.fourier for more information. """ from sympy.series.fourier import fourier_series
return fourier_series(self, limits)
################################################################################### ##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS #################### ###################################################################################
def diff(self, *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):
@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. """ # XXX: Hack to support non-Basic args # | # V
@cacheit 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.
"""
log=log, multinomial=multinomial, basic=basic)
n, d = [a.expand(deep=deep, modulus=modulus, **hints) for a in fraction(self)] return n/d n, d = fraction(self) return n/d.expand(deep=deep, modulus=modulus, **hints) 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
"""Make multinomial come before mul"""
expr, '_eval_expand_multinomial', deep=deep, **hints) expr, '_eval_expand_mul', deep=deep, **hints) expr, '_eval_expand_log', deep=deep, **hints)
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)
########################################################################### ################### GLOBAL ACTION VERB WRAPPER METHODS #################### ###########################################################################
def integrate(self, *args, **kwargs): """See the integrate function in sympy.integrals""" from sympy.integrals import integrate return integrate(self, *args, **kwargs)
def simplify(self, ratio=1.7, measure=None): """See the simplify function in sympy.simplify"""
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)
def separate(self, deep=False, force=False): """See the separate function in sympy.simplify""" from sympy.core.function import expand_power_base return expand_power_base(self, deep=deep, force=force)
def collect(self, syms, func=None, evaluate=True, exact=False, distribute_order_term=True): """See the collect function in sympy.simplify"""
def together(self, *args, **kwargs): """See the together function in sympy.polys""" from sympy.polys import together return together(self, *args, **kwargs)
def apart(self, x=None, **args): """See the apart function in sympy.polys""" from sympy.polys import apart return apart(self, x, **args)
def ratsimp(self): """See the ratsimp function in sympy.simplify""" from sympy.simplify import ratsimp return ratsimp(self)
def trigsimp(self, **args): """See the trigsimp function in sympy.simplify""" from sympy.simplify import trigsimp return trigsimp(self, **args)
def radsimp(self, **kwargs): """See the radsimp function in sympy.simplify""" from sympy.simplify import radsimp return radsimp(self, **kwargs)
def powsimp(self, *args, **kwargs): """See the powsimp function in sympy.simplify""" from sympy.simplify import powsimp return powsimp(self, *args, **kwargs)
def combsimp(self): """See the combsimp function in sympy.simplify""" from sympy.simplify import combsimp return combsimp(self)
def factor(self, *gens, **args): """See the factor() function in sympy.polys.polytools""" from sympy.polys import factor return factor(self, *gens, **args)
def refine(self, assumption=True): """See the refine function in sympy.assumptions""" from sympy.assumptions import refine return refine(self, assumption)
def cancel(self, *gens, **args): """See the cancel function in sympy.polys"""
def invert(self, g, *gens, **args): """Return the multiplicative inverse of ``self`` mod ``g`` where ``self`` (and ``g``) may be symbolic expressions).
See Also ======== sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert """ from sympy.polys.polytools import invert from sympy.core.numbers import mod_inverse if self.is_number and getattr(g, 'is_number', True): return mod_inverse(self, g) return invert(self, g, *gens, **args)
def round(self, p=0): """Return x rounded to the given decimal place.
If a complex number would results, apply round to the real and imaginary components of the number.
Examples ========
>>> from sympy import pi, E, I, S, Add, Mul, Number >>> S(10.5).round() 11. >>> pi.round() 3. >>> pi.round(2) 3.14 >>> (2*pi + E*I).round() 6. + 3.*I
The round method has a chopping effect:
>>> (2*pi + I/10).round() 6. >>> (pi/10 + 2*I).round() 2.*I >>> (pi/10 + E*I).round(2) 0.31 + 2.72*I
Notes =====
Do not confuse the Python builtin function, round, with the SymPy method of the same name. The former always returns a float (or raises an error if applied to a complex value) while the latter returns either a Number or a complex number:
>>> isinstance(round(S(123), -2), Number) False >>> isinstance(S(123).round(-2), Number) True >>> isinstance((3*I).round(), Mul) True >>> isinstance((1 + 3*I).round(), Add) True
""" raise TypeError("can't round symbolic expression") raise TypeError('Expected a number but got %s:' % getattr(getattr(x,'func', x), '__name__', type(x))) return x
allow = dps x = xwas - 1/(2*mag) # should have gone the other way i10 = 10*mag*x.n((dps if dps is not None else digits_needed) + 1) rv = -(Integer(-i10)//10) else: elif p < 0: rv /= mag # use str or else it won't be a float return Float(str(rv), digits_needed) else: allow += 1
class AtomicExpr(Atom, Expr): """ A parent class for object which are both atoms and Exprs.
For example: Symbol, Number, Rational, Integer, ... But not: Add, Mul, Pow, ... """ is_number = False is_Atom = True
__slots__ = []
def _eval_derivative(self, s): return S.Zero
def _eval_is_polynomial(self, syms):
def _eval_is_rational_function(self, syms):
def _eval_is_algebraic_expr(self, syms): return True
def _eval_nseries(self, x, n, logx):
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 """ return S.Zero except (ValueError, OverflowError): mag_first_dig = int(ceil(Float(mpf_log(xpos._mpf_, 53))/log(10))) # check that we aren't off by 1 assert 1 <= (xpos/10**mag_first_dig) < 10 mag_first_dig += 1
from .mul import Mul from .add import Add from .power import Pow from .function import Derivative, Function from .mod import Mod from .exprtools import factor_terms from .numbers import Integer, Rational |