from sympy.core import Basic, S, sympify, Expr, Rational, Symbol
from sympy.core import Add, Mul
from sympy.core.cache import cacheit
from sympy.core.compatibility import cmp_to_key
[docs]class Order(Expr):
""" Represents the limiting behavior of some function
The order of a function characterizes the function based on the limiting
behavior of the function as it goes to some limit. Only taking the limit
point to be 0 is currently supported. This is expressed in big O notation
[1]_.
The formal definition for the order of a function `g(x)` about a point `a`
is such that `g(x) = O(f(x))` as `x \\rightarrow a` if and only if for any
`\delta > 0` there exists a `M > 0` such that `|g(x)| \leq M|f(x)|` for
`|x-a| < \delta`. This is equivalent to `\lim_{x \\rightarrow a}
|g(x)/f(x)| < \infty`.
Let's illustrate it on the following example by taking the expansion of
`\sin(x)` about 0:
.. math ::
\sin(x) = x - x^3/3! + O(x^5)
where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the definition
of `O`, for any `\delta > 0` there is an `M` such that:
.. math ::
|x^5/5! - x^7/7! + ....| <= M|x^5| \\text{ for } |x| < \delta
or by the alternate definition:
.. math ::
\lim_{x \\rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| < \infty
which surely is true, because
.. math ::
\lim_{x \\rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5!
As it is usually used, the order of a function can be intuitively thought
of representing all terms of powers greater than the one specified. For
example, `O(x^3)` corresponds to any terms proportional to `x^3,
x^4,\ldots` and any higher power. For a polynomial, this leaves terms
proportional to `x^2`, `x` and constants.
Examples
========
>>> from sympy import O
>>> from sympy.abc import x
>>> O(x)
O(x)
>>> O(x)*x
O(x**2)
>>> O(x)-O(x)
O(x)
References
==========
.. [1] `Big O notation <http://en.wikipedia.org/wiki/Big_O_notation>`_
Notes
=====
In ``O(f(x), x)`` the expression ``f(x)`` is assumed to have a leading
term. ``O(f(x), x)`` is automatically transformed to
``O(f(x).as_leading_term(x),x)``.
``O(expr*f(x), x)`` is ``O(f(x), x)``
``O(expr, x)`` is ``O(1)``
``O(0, x)`` is 0.
Multivariate O is also supported:
``O(f(x, y), x, y)`` is transformed to
``O(f(x, y).as_leading_term(x,y).as_leading_term(y), x, y)``
In the multivariate case, it is assumed the limits w.r.t. the various
symbols commute.
If no symbols are passed then all symbols in the expression are used:
"""
is_Order = True
__slots__ = []
@cacheit
def __new__(cls, expr, *symbols, **assumptions):
expr = sympify(expr).expand()
if expr is S.NaN:
return S.NaN
if symbols:
symbols = map(sympify, symbols)
if not all(isinstance(s, Symbol) for s in symbols):
raise NotImplementedError(
'Order at points other than 0 not supported.')
else:
symbols = list(expr.free_symbols)
if expr.is_Order:
new_symbols = list(expr.variables)
for s in symbols:
if s not in new_symbols:
new_symbols.append(s)
if len(new_symbols) == len(expr.variables):
return expr
symbols = new_symbols
elif symbols:
if expr.is_Add:
lst = expr.extract_leading_order(*symbols)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
if len(symbols) > 1 or expr.is_commutative is False:
# TODO
# We cannot use compute_leading_term because that only
# works in one symbol.
expr = expr.as_leading_term(*symbols)
else:
expr = expr.compute_leading_term(symbols[0])
terms = expr.as_coeff_mul(*symbols)[1]
s = set(symbols)
expr = Mul(*[t for t in terms if s & t.free_symbols])
if expr is S.Zero:
return expr
elif not expr.has(*symbols):
expr = S.One
# create Order instance:
symbols.sort(key=cmp_to_key(Basic.compare))
obj = Expr.__new__(cls, expr, *symbols, **assumptions)
return obj
def _hashable_content(self):
return self.args
def oseries(self, order):
return self
def _eval_nseries(self, x, n, logx):
return self
@property
def expr(self):
return self._args[0]
@property
def variables(self):
return self._args[1:]
@property
def free_symbols(self):
return self.expr.free_symbols
def _eval_power(b, e):
if e.is_Number:
return Order(b.expr ** e, *b.variables)
return
def as_expr_variables(self, order_symbols):
if order_symbols is None:
order_symbols = self.variables
else:
for s in self.variables:
if s not in order_symbols:
order_symbols = order_symbols + (s,)
return self.expr, order_symbols
def removeO(self):
return S.Zero
def getO(self):
return self
@cacheit
[docs] def contains(self, expr):
"""
Return True if expr belongs to Order(self.expr, \*self.variables).
Return False if self belongs to expr.
Return None if the inclusion relation cannot be determined
(e.g. when self and expr have different symbols).
"""
# NOTE: when multiplying out series a lot of queries like
# O(...).contains(a*x**b) with many a and few b are made.
# Separating out the independent part allows for better caching.
c, m = expr.as_coeff_mul(*self.variables)
if m != ():
return self._contains(Mul(*m))
else:
# Mul(*m) == 1, and O(1) treatment is somewhat peculiar ...
# some day this else should not be necessary
return self._contains(expr)
@cacheit
def _contains(self, expr):
from sympy import powsimp, limit
if expr is S.Zero:
return True
if expr is S.NaN:
return False
if expr.is_Order:
if self.variables and expr.variables:
common_symbols = tuple(
[s for s in self.variables if s in expr.variables])
elif self.variables:
common_symbols = self.variables
else:
common_symbols = expr.variables
if not common_symbols:
if not (self.variables or expr.variables): # O(1),O(1)
return True
return None
r = None
for s in common_symbols:
l = limit(powsimp(self.expr/expr.expr, deep=True,
combine='exp'), s, 0) != 0
if r is None:
r = l
else:
if r != l:
return
return r
obj = Order(expr, *self.variables)
return self.contains(obj)
def _eval_subs(self, old, new):
if old.is_Symbol and old in self.variables:
i = list(self.variables).index(old)
if isinstance(new, Symbol):
return Order(self.expr._subs(old, new), *(self.variables[:i] + (new,) + self.variables[i + 1:]))
return Order(self.expr._subs(old, new), *(self.variables[:i] + self.variables[i + 1:]))
return Order(self.expr._subs(old, new), *self.variables)
def _eval_conjugate(self):
expr = self.expr._eval_conjugate()
if expr is not None:
return self.func(expr, *self.variables)
def _eval_derivative(self, x):
return self.func(self.expr.diff(x), *self.variables) or self
def _eval_transpose(self):
expr = self.expr._eval_transpose()
if expr is not None:
return self.func(expr, *self.variables)
def _sage_(self):
#XXX: SAGE doesn't have Order yet. Let's return 0 instead.
return Rational(0)._sage_()
O = Order