You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
258 lines
7.6 KiB
258 lines
7.6 KiB
"""Limits of sequences"""
|
|
|
|
from sympy.calculus.accumulationbounds import AccumulationBounds
|
|
from sympy.core.add import Add
|
|
from sympy.core.function import PoleError
|
|
from sympy.core.power import Pow
|
|
from sympy.core.singleton import S
|
|
from sympy.core.symbol import Dummy
|
|
from sympy.core.sympify import sympify
|
|
from sympy.functions.combinatorial.numbers import fibonacci
|
|
from sympy.functions.combinatorial.factorials import factorial, subfactorial
|
|
from sympy.functions.special.gamma_functions import gamma
|
|
from sympy.functions.elementary.complexes import Abs
|
|
from sympy.functions.elementary.miscellaneous import Max, Min
|
|
from sympy.functions.elementary.trigonometric import cos, sin
|
|
from sympy.series.limits import Limit
|
|
|
|
|
|
def difference_delta(expr, n=None, step=1):
|
|
"""Difference Operator.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
Discrete analog of differential operator. Given a sequence x[n],
|
|
returns the sequence x[n + step] - x[n].
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import difference_delta as dd
|
|
>>> from sympy.abc import n
|
|
>>> dd(n*(n + 1), n)
|
|
2*n + 2
|
|
>>> dd(n*(n + 1), n, 2)
|
|
4*n + 6
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://reference.wolfram.com/language/ref/DifferenceDelta.html
|
|
"""
|
|
expr = sympify(expr)
|
|
|
|
if n is None:
|
|
f = expr.free_symbols
|
|
if len(f) == 1:
|
|
n = f.pop()
|
|
elif len(f) == 0:
|
|
return S.Zero
|
|
else:
|
|
raise ValueError("Since there is more than one variable in the"
|
|
" expression, a variable must be supplied to"
|
|
" take the difference of %s" % expr)
|
|
step = sympify(step)
|
|
if step.is_number is False or step.is_finite is False:
|
|
raise ValueError("Step should be a finite number.")
|
|
|
|
if hasattr(expr, '_eval_difference_delta'):
|
|
result = expr._eval_difference_delta(n, step)
|
|
if result:
|
|
return result
|
|
|
|
return expr.subs(n, n + step) - expr
|
|
|
|
|
|
def dominant(expr, n):
|
|
"""Finds the dominant term in a sum, that is a term that dominates
|
|
every other term.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If limit(a/b, n, oo) is oo then a dominates b.
|
|
If limit(a/b, n, oo) is 0 then b dominates a.
|
|
Otherwise, a and b are comparable.
|
|
|
|
If there is no unique dominant term, then returns ``None``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Sum
|
|
>>> from sympy.series.limitseq import dominant
|
|
>>> from sympy.abc import n, k
|
|
>>> dominant(5*n**3 + 4*n**2 + n + 1, n)
|
|
5*n**3
|
|
>>> dominant(2**n + Sum(k, (k, 0, n)), n)
|
|
2**n
|
|
|
|
See Also
|
|
========
|
|
|
|
sympy.series.limitseq.dominant
|
|
"""
|
|
terms = Add.make_args(expr.expand(func=True))
|
|
term0 = terms[-1]
|
|
comp = [term0] # comparable terms
|
|
for t in terms[:-1]:
|
|
r = term0/t
|
|
e = r.gammasimp()
|
|
if e == r:
|
|
e = r.factor()
|
|
l = limit_seq(e, n)
|
|
if l is None:
|
|
return None
|
|
elif l.is_zero:
|
|
term0 = t
|
|
comp = [term0]
|
|
elif l not in [S.Infinity, S.NegativeInfinity]:
|
|
comp.append(t)
|
|
if len(comp) > 1:
|
|
return None
|
|
return term0
|
|
|
|
|
|
def _limit_inf(expr, n):
|
|
try:
|
|
return Limit(expr, n, S.Infinity).doit(deep=False)
|
|
except (NotImplementedError, PoleError):
|
|
return None
|
|
|
|
|
|
def _limit_seq(expr, n, trials):
|
|
from sympy.concrete.summations import Sum
|
|
|
|
for i in range(trials):
|
|
if not expr.has(Sum):
|
|
result = _limit_inf(expr, n)
|
|
if result is not None:
|
|
return result
|
|
|
|
num, den = expr.as_numer_denom()
|
|
if not den.has(n) or not num.has(n):
|
|
result = _limit_inf(expr.doit(), n)
|
|
if result is not None:
|
|
return result
|
|
return None
|
|
|
|
num, den = (difference_delta(t.expand(), n) for t in [num, den])
|
|
expr = (num / den).gammasimp()
|
|
|
|
if not expr.has(Sum):
|
|
result = _limit_inf(expr, n)
|
|
if result is not None:
|
|
return result
|
|
|
|
num, den = expr.as_numer_denom()
|
|
|
|
num = dominant(num, n)
|
|
if num is None:
|
|
return None
|
|
|
|
den = dominant(den, n)
|
|
if den is None:
|
|
return None
|
|
|
|
expr = (num / den).gammasimp()
|
|
|
|
|
|
def limit_seq(expr, n=None, trials=5):
|
|
"""Finds the limit of a sequence as index ``n`` tends to infinity.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
expr : Expr
|
|
SymPy expression for the ``n-th`` term of the sequence
|
|
n : Symbol, optional
|
|
The index of the sequence, an integer that tends to positive
|
|
infinity. If None, inferred from the expression unless it has
|
|
multiple symbols.
|
|
trials: int, optional
|
|
The algorithm is highly recursive. ``trials`` is a safeguard from
|
|
infinite recursion in case the limit is not easily computed by the
|
|
algorithm. Try increasing ``trials`` if the algorithm returns ``None``.
|
|
|
|
Admissible Terms
|
|
================
|
|
|
|
The algorithm is designed for sequences built from rational functions,
|
|
indefinite sums, and indefinite products over an indeterminate n. Terms of
|
|
alternating sign are also allowed, but more complex oscillatory behavior is
|
|
not supported.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import limit_seq, Sum, binomial
|
|
>>> from sympy.abc import n, k, m
|
|
>>> limit_seq((5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5), n)
|
|
5/3
|
|
>>> limit_seq(binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)), n)
|
|
3/4
|
|
>>> limit_seq(Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n), n)
|
|
4
|
|
|
|
See Also
|
|
========
|
|
|
|
sympy.series.limitseq.dominant
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] Computing Limits of Sequences - Manuel Kauers
|
|
"""
|
|
|
|
from sympy.concrete.summations import Sum
|
|
if n is None:
|
|
free = expr.free_symbols
|
|
if len(free) == 1:
|
|
n = free.pop()
|
|
elif not free:
|
|
return expr
|
|
else:
|
|
raise ValueError("Expression has more than one variable. "
|
|
"Please specify a variable.")
|
|
elif n not in expr.free_symbols:
|
|
return expr
|
|
|
|
expr = expr.rewrite(fibonacci, S.GoldenRatio)
|
|
expr = expr.rewrite(factorial, subfactorial, gamma)
|
|
n_ = Dummy("n", integer=True, positive=True)
|
|
n1 = Dummy("n", odd=True, positive=True)
|
|
n2 = Dummy("n", even=True, positive=True)
|
|
|
|
# If there is a negative term raised to a power involving n, or a
|
|
# trigonometric function, then consider even and odd n separately.
|
|
powers = (p.as_base_exp() for p in expr.atoms(Pow))
|
|
if (any(b.is_negative and e.has(n) for b, e in powers) or
|
|
expr.has(cos, sin)):
|
|
L1 = _limit_seq(expr.xreplace({n: n1}), n1, trials)
|
|
if L1 is not None:
|
|
L2 = _limit_seq(expr.xreplace({n: n2}), n2, trials)
|
|
if L1 != L2:
|
|
if L1.is_comparable and L2.is_comparable:
|
|
return AccumulationBounds(Min(L1, L2), Max(L1, L2))
|
|
else:
|
|
return None
|
|
else:
|
|
L1 = _limit_seq(expr.xreplace({n: n_}), n_, trials)
|
|
if L1 is not None:
|
|
return L1
|
|
else:
|
|
if expr.is_Add:
|
|
limits = [limit_seq(term, n, trials) for term in expr.args]
|
|
if any(result is None for result in limits):
|
|
return None
|
|
else:
|
|
return Add(*limits)
|
|
# Maybe the absolute value is easier to deal with (though not if
|
|
# it has a Sum). If it tends to 0, the limit is 0.
|
|
elif not expr.has(Sum):
|
|
lim = _limit_seq(Abs(expr.xreplace({n: n_})), n_, trials)
|
|
if lim is not None and lim.is_zero:
|
|
return S.Zero
|