from typing import Tuple as tTuple
from sympy.calculus.singularities import is_decreasing
from sympy.calculus.accumulationbounds import AccumulationBounds
from .expr_with_intlimits import ExprWithIntLimits
from .expr_with_limits import AddWithLimits
from .gosper import gosper_sum
from sympy.core.expr import Expr
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import Derivative, expand
from sympy.core.mul import Mul
from sympy.core.numbers import Float, _illegal
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy, Wild, Symbol, symbols
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.combinatorial.numbers import bernoulli, harmonic
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cot, csc
from sympy.functions.special.hyper import hyper
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.zeta_functions import zeta
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import And
from sympy.polys.partfrac import apart
from sympy.polys.polyerrors import PolynomialError, PolificationFailed
from sympy.polys.polytools import parallel_poly_from_expr, Poly, factor
from sympy.polys.rationaltools import together
from sympy.series.limitseq import limit_seq
from sympy.series.order import O
from sympy.series.residues import residue
from sympy.sets.sets import FiniteSet, Interval
from sympy.utilities.iterables import sift
import itertools
class Sum(AddWithLimits, ExprWithIntLimits):
r"""
Represents unevaluated summation.
Explanation
===========
``Sum`` represents a finite or infinite series, with the first argument
being the general form of terms in the series, and the second argument
being ``(dummy_variable, start, end)``, with ``dummy_variable`` taking
all integer values from ``start`` through ``end``. In accordance with
long-standing mathematical convention, the end term is included in the
summation.
Finite sums
===========
For finite sums (and sums with symbolic limits assumed to be finite) we
follow the summation convention described by Karr [1], especially
definition 3 of section 1.4. The sum:
.. math::
\sum_{m \leq i < n} f(i)
has *the obvious meaning* for `m < n`, namely:
.. math::
\sum_{m \leq i < n} f(i) = f(m) + f(m+1) + \ldots + f(n-2) + f(n-1)
with the upper limit value `f(n)` excluded. The sum over an empty set is
zero if and only if `m = n`:
.. math::
\sum_{m \leq i < n} f(i) = 0 \quad \mathrm{for} \quad m = n
Finally, for all other sums over empty sets we assume the following
definition:
.. math::
\sum_{m \leq i < n} f(i) = - \sum_{n \leq i < m} f(i) \quad \mathrm{for} \quad m > n
It is important to note that Karr defines all sums with the upper
limit being exclusive. This is in contrast to the usual mathematical notation,
but does not affect the summation convention. Indeed we have:
.. math::
\sum_{m \leq i < n} f(i) = \sum_{i = m}^{n - 1} f(i)
where the difference in notation is intentional to emphasize the meaning,
with limits typeset on the top being inclusive.
Examples
========
>>> from sympy.abc import i, k, m, n, x
>>> from sympy import Sum, factorial, oo, IndexedBase, Function
>>> Sum(k, (k, 1, m))
Sum(k, (k, 1, m))
>>> Sum(k, (k, 1, m)).doit()
m**2/2 + m/2
>>> Sum(k**2, (k, 1, m))
Sum(k**2, (k, 1, m))
>>> Sum(k**2, (k, 1, m)).doit()
m**3/3 + m**2/2 + m/6
>>> Sum(x**k, (k, 0, oo))
Sum(x**k, (k, 0, oo))
>>> Sum(x**k, (k, 0, oo)).doit()
Piecewise((1/(1 - x), Abs(x) < 1), (Sum(x**k, (k, 0, oo)), True))
>>> Sum(x**k/factorial(k), (k, 0, oo)).doit()
exp(x)
Here are examples to do summation with symbolic indices. You
can use either Function of IndexedBase classes:
>>> f = Function('f')
>>> Sum(f(n), (n, 0, 3)).doit()
f(0) + f(1) + f(2) + f(3)
>>> Sum(f(n), (n, 0, oo)).doit()
Sum(f(n), (n, 0, oo))
>>> f = IndexedBase('f')
>>> Sum(f[n]**2, (n, 0, 3)).doit()
f[0]**2 + f[1]**2 + f[2]**2 + f[3]**2
An example showing that the symbolic result of a summation is still
valid for seemingly nonsensical values of the limits. Then the Karr
convention allows us to give a perfectly valid interpretation to
those sums by interchanging the limits according to the above rules:
>>> S = Sum(i, (i, 1, n)).doit()
>>> S
n**2/2 + n/2
>>> S.subs(n, -4)
6
>>> Sum(i, (i, 1, -4)).doit()
6
>>> Sum(-i, (i, -3, 0)).doit()
6
An explicit example of the Karr summation convention:
>>> S1 = Sum(i**2, (i, m, m+n-1)).doit()
>>> S1
m**2*n + m*n**2 - m*n + n**3/3 - n**2/2 + n/6
>>> S2 = Sum(i**2, (i, m+n, m-1)).doit()
>>> S2
-m**2*n - m*n**2 + m*n - n**3/3 + n**2/2 - n/6
>>> S1 + S2
0
>>> S3 = Sum(i, (i, m, m-1)).doit()
>>> S3
0
See Also
========
summation
Product, sympy.concrete.products.product
References
==========
.. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM,
Volume 28 Issue 2, April 1981, Pages 305-350
https://dl.acm.org/doi/10.1145/322248.322255
.. [2] https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation
.. [3] https://en.wikipedia.org/wiki/Empty_sum
"""
__slots__ = ()
limits: tTuple[tTuple[Symbol, Expr, Expr]]
def __new__(cls, function, *symbols, **assumptions):
obj = AddWithLimits.__new__(cls, function, *symbols, **assumptions)
if not hasattr(obj, 'limits'):
return obj
if any(len(l) != 3 or None in l for l in obj.limits):
raise ValueError('Sum requires values for lower and upper bounds.')
return obj
def _eval_is_zero(self):
# a Sum is only zero if its function is zero or if all terms
# cancel out. This only answers whether the summand is zero; if
# not then None is returned since we don't analyze whether all
# terms cancel out.
if self.function.is_zero or self.has_empty_sequence:
return True
def _eval_is_extended_real(self):
if self.has_empty_sequence:
return True
return self.function.is_extended_real
def _eval_is_positive(self):
if self.has_finite_limits and self.has_reversed_limits is False:
return self.function.is_positive
def _eval_is_negative(self):
if self.has_finite_limits and self.has_reversed_limits is False:
return self.function.is_negative
def _eval_is_finite(self):
if self.has_finite_limits and self.function.is_finite:
return True
def doit(self, **hints):
if hints.get('deep', True):
f = self.function.doit(**hints)
else:
f = self.function
# first make sure any definite limits have summation
# variables with matching assumptions
reps = {}
for xab in self.limits:
d = _dummy_with_inherited_properties_concrete(xab)
if d:
reps[xab[0]] = d
if reps:
undo = {v: k for k, v in reps.items()}
did = self.xreplace(reps).doit(**hints)
if isinstance(did, tuple): # when separate=True
did = tuple([i.xreplace(undo) for i in did])
elif did is not None:
did = did.xreplace(undo)
else:
did = self
return did
if self.function.is_Matrix:
expanded = self.expand()
if self != expanded:
return expanded.doit()
return _eval_matrix_sum(self)
for n, limit in enumerate(self.limits):
i, a, b = limit
dif = b - a
if dif == -1:
# Any summation over an empty set is zero
return S.Zero
if dif.is_integer and dif.is_negative:
a, b = b + 1, a - 1
f = -f
newf = eval_sum(f, (i, a, b))
if newf is None:
if f == self.function:
zeta_function = self.eval_zeta_function(f, (i, a, b))
if zeta_function is not None:
return zeta_function
return self
else:
return self.func(f, *self.limits[n:])
f = newf
if hints.get('deep', True):
# eval_sum could return partially unevaluated
# result with Piecewise. In this case we won't
# doit() recursively.
if not isinstance(f, Piecewise):
return f.doit(**hints)
return f
def eval_zeta_function(self, f, limits):
"""
Check whether the function matches with the zeta function.
If it matches, then return a `Piecewise` expression because
zeta function does not converge unless `s > 1` and `q > 0`
"""
i, a, b = limits
w, y, z = Wild('w', exclude=[i]), Wild('y', exclude=[i]), Wild('z', exclude=[i])
result = f.match((w * i + y) ** (-z))
if result is not None and b is S.Infinity:
coeff = 1 / result[w] ** result[z]
s = result[z]
q = result[y] / result[w] + a
return Piecewise((coeff * zeta(s, q), And(q > 0, s > 1)), (self, True))
def _eval_derivative(self, x):
"""
Differentiate wrt x as long as x is not in the free symbols of any of
the upper or lower limits.
Explanation
===========
Sum(a*b*x, (x, 1, a)) can be differentiated wrt x or b but not `a`
since the value of the sum is discontinuous in `a`. In a case
involving a limit variable, the unevaluated derivative is returned.
"""
# diff already confirmed that x is in the free symbols of self, but we
# don't want to differentiate wrt any free symbol in the upper or lower
# limits
# XXX remove this test for free_symbols when the default _eval_derivative is in
if isinstance(x, Symbol) and x not in self.free_symbols:
return S.Zero
# get limits and the function
f, limits = self.function, list(self.limits)
limit = limits.pop(-1)
if limits: # f is the argument to a Sum
f = self.func(f, *limits)
_, a, b = limit
if x in a.free_symbols or x in b.free_symbols:
return None
df = Derivative(f, x, evaluate=True)
rv = self.func(df, limit)
return rv
def _eval_difference_delta(self, n, step):
k, _, upper = self.args[-1]
new_upper = upper.subs(n, n + step)
if len(self.args) == 2:
f = self.args[0]
else:
f = self.func(*self.args[:-1])
return Sum(f, (k, upper + 1, new_upper)).doit()
def _eval_simplify(self, **kwargs):
function = self.function
if kwargs.get('deep', True):
function = function.simplify(**kwargs)
# split the function into adds
terms = Add.make_args(expand(function))
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
if term.has(Sum):
# if there is an embedded sum here
# it is of the form x * (Sum(whatever))
# hence we make a Mul out of it, and simplify all interior sum terms
subterms = Mul.make_args(expand(term))
out_terms = []
for subterm in subterms:
# go through each term
if isinstance(subterm, Sum):
# if it's a sum, simplify it
out_terms.append(subterm._eval_simplify(**kwargs))
else:
# otherwise, add it as is
out_terms.append(subterm)
# turn it back into a Mul
s_t.append(Mul(*out_terms))
else:
o_t.append(term)
# next try to combine any interior sums for further simplification
from sympy.simplify.simplify import factor_sum, sum_combine
result = Add(sum_combine(s_t), *o_t)
return factor_sum(result, limits=self.limits)
def is_convergent(self):
r"""
Checks for the convergence of a Sum.
Explanation
===========
We divide the study of convergence of infinite sums and products in
two parts.
First Part:
One part is the question whether all the terms are well defined, i.e.,
they are finite in a sum and also non-zero in a product. Zero
is the analogy of (minus) infinity in products as
:math:`e^{-\infty} = 0`.
Second Part:
The second part is the question of convergence after infinities,
and zeros in products, have been omitted assuming that their number
is finite. This means that we only consider the tail of the sum or
product, starting from some point after which all terms are well
defined.
For example, in a sum of the form:
.. math::
\sum_{1 \leq i < \infty} \frac{1}{n^2 + an + b}
where a and b are numbers. The routine will return true, even if there
are infinities in the term sequence (at most two). An analogous
product would be:
.. math::
\prod_{1 \leq i < \infty} e^{\frac{1}{n^2 + an + b}}
This is how convergence is interpreted. It is concerned with what
happens at the limit. Finding the bad terms is another independent
matter.
Note: It is responsibility of user to see that the sum or product
is well defined.
There are various tests employed to check the convergence like
divergence test, root test, integral test, alternating series test,
comparison tests, Dirichlet tests. It returns true if Sum is convergent
and false if divergent and NotImplementedError if it cannot be checked.
References
==========
.. [1] https://en.wikipedia.org/wiki/Convergence_tests
Examples
========
>>> from sympy import factorial, S, Sum, Symbol, oo
>>> n = Symbol('n', integer=True)
>>> Sum(n/(n - 1), (n, 4, 7)).is_convergent()
True
>>> Sum(n/(2*n + 1), (n, 1, oo)).is_convergent()
False
>>> Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent()
False
>>> Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent()
True
See Also
========
Sum.is_absolutely_convergent
sympy.concrete.products.Product.is_convergent
"""
p, q, r = symbols('p q r', cls=Wild)
sym = self.limits[0][0]
lower_limit = self.limits[0][1]
upper_limit = self.limits[0][2]
sequence_term = self.function.simplify()
if len(sequence_term.free_symbols) > 1:
raise NotImplementedError("convergence checking for more than one symbol "
"containing series is not handled")
if lower_limit.is_finite and upper_limit.is_finite:
return S.true
# transform sym -> -sym and swap the upper_limit = S.Infinity
# and lower_limit = - upper_limit
if lower_limit is S.NegativeInfinity:
if upper_limit is S.Infinity:
return Sum(sequence_term, (sym, 0, S.Infinity)).is_convergent() and \
Sum(sequence_term, (sym, S.NegativeInfinity, 0)).is_convergent()
from sympy.simplify.simplify import simplify
sequence_term = simplify(sequence_term.xreplace({sym: -sym}))
lower_limit = -upper_limit
upper_limit = S.Infinity
sym_ = Dummy(sym.name, integer=True, positive=True)
sequence_term = sequence_term.xreplace({sym: sym_})
sym = sym_
interval = Interval(lower_limit, upper_limit)
# Piecewise function handle
if sequence_term.is_Piecewise:
for func, cond in sequence_term.args:
# see if it represents something going to oo
if cond == True or cond.as_set().sup is S.Infinity:
s = Sum(func, (sym, lower_limit, upper_limit))
return s.is_convergent()
return S.true
### -------- Divergence test ----------- ###
try:
lim_val = limit_seq(sequence_term, sym)
if lim_val is not None and lim_val.is_zero is False:
return S.false
except NotImplementedError:
pass
try:
lim_val_abs = limit_seq(abs(sequence_term), sym)
if lim_val_abs is not None and lim_val_abs.is_zero is False:
return S.false
except NotImplementedError:
pass
order = O(sequence_term, (sym, S.Infinity))
### --------- p-series test (1/n**p) ---------- ###
p_series_test = order.expr.match(sym**p)
if p_series_test is not None:
if p_series_test[p] < -1:
return S.true
if p_series_test[p] >= -1:
return S.false
### ------------- comparison test ------------- ###
# 1/(n**p*log(n)**q*log(log(n))**r) comparison
n_log_test = (order.expr.match(1/(sym**p*log(1/sym)**q*log(-log(1/sym))**r)) or
order.expr.match(1/(sym**p*(-log(1/sym))**q*log(-log(1/sym))**r)))
if n_log_test is not None:
if (n_log_test[p] > 1 or
(n_log_test[p] == 1 and n_log_test[q] > 1) or
(n_log_test[p] == n_log_test[q] == 1 and n_log_test[r] > 1)):
return S.true
return S.false
### ------------- Limit comparison test -----------###
# (1/n) comparison
try:
lim_comp = limit_seq(sym*sequence_term, sym)
if lim_comp is not None and lim_comp.is_number and lim_comp > 0:
return S.false
except NotImplementedError:
pass
### ----------- ratio test ---------------- ###
next_sequence_term = sequence_term.xreplace({sym: sym + 1})
from sympy.simplify.combsimp import combsimp
from sympy.simplify.powsimp import powsimp
ratio = combsimp(powsimp(next_sequence_term/sequence_term))
try:
lim_ratio = limit_seq(ratio, sym)
if lim_ratio is not None and lim_ratio.is_number:
if abs(lim_ratio) > 1:
return S.false
if abs(lim_ratio) < 1:
return S.true
except NotImplementedError:
lim_ratio = None
### ---------- Raabe's test -------------- ###
if lim_ratio == 1: # ratio test inconclusive
test_val = sym*(sequence_term/
sequence_term.subs(sym, sym + 1) - 1)
test_val = test_val.gammasimp()
try:
lim_val = limit_seq(test_val, sym)
if lim_val is not None and lim_val.is_number:
if lim_val > 1:
return S.true
if lim_val < 1:
return S.false
except NotImplementedError:
pass
### ----------- root test ---------------- ###
# lim = Limit(abs(sequence_term)**(1/sym), sym, S.Infinity)
try:
lim_evaluated = limit_seq(abs(sequence_term)**(1/sym), sym)
if lim_evaluated is not None and lim_evaluated.is_number:
if lim_evaluated < 1:
return S.true
if lim_evaluated > 1:
return S.false
except NotImplementedError:
pass
### ------------- alternating series test ----------- ###
dict_val = sequence_term.match(S.NegativeOne**(sym + p)*q)
if not dict_val[p].has(sym) and is_decreasing(dict_val[q], interval):
return S.true
### ------------- integral test -------------- ###
check_interval = None
from sympy.solvers.solveset import solveset
maxima = solveset(sequence_term.diff(sym), sym, interval)
if not maxima:
check_interval = interval
elif isinstance(maxima, FiniteSet) and maxima.sup.is_number:
check_interval = Interval(maxima.sup, interval.sup)
if (check_interval is not None and
(is_decreasing(sequence_term, check_interval) or
is_decreasing(-sequence_term, check_interval))):
integral_val = Integral(
sequence_term, (sym, lower_limit, upper_limit))
try:
integral_val_evaluated = integral_val.doit()
if integral_val_evaluated.is_number:
return S(integral_val_evaluated.is_finite)
except NotImplementedError:
pass
### ----- Dirichlet and bounded times convergent tests ----- ###
# TODO
#
# Dirichlet_test
# https://en.wikipedia.org/wiki/Dirichlet%27s_test
#
# Bounded times convergent test
# It is based on comparison theorems for series.
# In particular, if the general term of a series can
# be written as a product of two terms a_n and b_n
# and if a_n is bounded and if Sum(b_n) is absolutely
# convergent, then the original series Sum(a_n * b_n)
# is absolutely convergent and so convergent.
#
# The following code can grows like 2**n where n is the
# number of args in order.expr
# Possibly combined with the potentially slow checks
# inside the loop, could make this test extremely slow
# for larger summation expressions.
if order.expr.is_Mul:
args = order.expr.args
argset = set(args)
### -------------- Dirichlet tests -------------- ###
m = Dummy('m', integer=True)
def _dirichlet_test(g_n):
try:
ing_val = limit_seq(Sum(g_n, (sym, interval.inf, m)).doit(), m)
if ing_val is not None and ing_val.is_finite:
return S.true
except NotImplementedError:
pass
### -------- bounded times convergent test ---------###
def _bounded_convergent_test(g1_n, g2_n):
try:
lim_val = limit_seq(g1_n, sym)
if lim_val is not None and (lim_val.is_finite or (
isinstance(lim_val, AccumulationBounds)
and (lim_val.max - lim_val.min).is_finite)):
if Sum(g2_n, (sym, lower_limit, upper_limit)).is_absolutely_convergent():
return S.true
except NotImplementedError:
pass
for n in range(1, len(argset)):
for a_tuple in itertools.combinations(args, n):
b_set = argset - set(a_tuple)
a_n = Mul(*a_tuple)
b_n = Mul(*b_set)
if is_decreasing(a_n, interval):
dirich = _dirichlet_test(b_n)
if dirich is not None:
return dirich
bc_test = _bounded_convergent_test(a_n, b_n)
if bc_test is not None:
return bc_test
_sym = self.limits[0][0]
sequence_term = sequence_term.xreplace({sym: _sym})
raise NotImplementedError("The algorithm to find the Sum convergence of %s "
"is not yet implemented" % (sequence_term))
def is_absolutely_convergent(self):
"""
Checks for the absolute convergence of an infinite series.
Same as checking convergence of absolute value of sequence_term of
an infinite series.
References
==========
.. [1] https://en.wikipedia.org/wiki/Absolute_convergence
Examples
========
>>> from sympy import Sum, Symbol, oo
>>> n = Symbol('n', integer=True)
>>> Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent()
False
>>> Sum((-1)**n/n**2, (n, 1, oo)).is_absolutely_convergent()
True
See Also
========
Sum.is_convergent
"""
return Sum(abs(self.function), self.limits).is_convergent()
def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True):
"""
Return an Euler-Maclaurin approximation of self, where m is the
number of leading terms to sum directly and n is the number of
terms in the tail.
With m = n = 0, this is simply the corresponding integral
plus a first-order endpoint correction.
Returns (s, e) where s is the Euler-Maclaurin approximation
and e is the estimated error (taken to be the magnitude of
the first omitted term in the tail):
>>> from sympy.abc import k, a, b
>>> from sympy import Sum
>>> Sum(1/k, (k, 2, 5)).doit().evalf()
1.28333333333333
>>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
>>> s
-log(2) + 7/20 + log(5)
>>> from sympy import sstr
>>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
(1.26629073187415, 0.0175000000000000)
The endpoints may be symbolic:
>>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
>>> s
-log(a) + log(b) + 1/(2*b) + 1/(2*a)
>>> e
Abs(1/(12*b**2) - 1/(12*a**2))
If the function is a polynomial of degree at most 2n+1, the
Euler-Maclaurin formula becomes exact (and e = 0 is returned):
>>> Sum(k, (k, 2, b)).euler_maclaurin()
(b**2/2 + b/2 - 1, 0)
>>> Sum(k, (k, 2, b)).doit()
b**2/2 + b/2 - 1
With a nonzero eps specified, the summation is ended
as soon as the remainder term is less than the epsilon.
"""
m = int(m)
n = int(n)
f = self.function
if len(self.limits) != 1:
raise ValueError("More than 1 limit")
i, a, b = self.limits[0]
if (a > b) == True:
if a - b == 1:
return S.Zero, S.Zero
a, b = b + 1, a - 1
f = -f
s = S.Zero
if m:
if b.is_Integer and a.is_Integer:
m = min(m, b - a + 1)
if not eps or f.is_polynomial(i):
s = Add(*[f.subs(i, a + k) for k in range(m)])
else:
term = f.subs(i, a)
if term:
test = abs(term.evalf(3)) < eps
if test == True:
return s, abs(term)
elif not (test == False):
# a symbolic Relational class, can't go further
return term, S.Zero
s = term
for k in range(1, m):
term = f.subs(i, a + k)
if abs(term.evalf(3)) < eps and term != 0:
return s, abs(term)
s += term
if b - a + 1 == m:
return s, S.Zero
a += m
x = Dummy('x')
I = Integral(f.subs(i, x), (x, a, b))
if eval_integral:
I = I.doit()
s += I
def fpoint(expr):
if b is S.Infinity:
return expr.subs(i, a), 0
return expr.subs(i, a), expr.subs(i, b)
fa, fb = fpoint(f)
iterm = (fa + fb)/2
g = f.diff(i)
for k in range(1, n + 2):
ga, gb = fpoint(g)
term = bernoulli(2*k)/factorial(2*k)*(gb - ga)
if k > n:
break
if eps and term:
term_evalf = term.evalf(3)
if term_evalf is S.NaN:
return S.NaN, S.NaN
if abs(term_evalf) < eps:
break
s += term
g = g.diff(i, 2, simplify=False)
return s + iterm, abs(term)
def reverse_order(self, *indices):
"""
Reverse the order of a limit in a Sum.
Explanation
===========
``reverse_order(self, *indices)`` reverses some limits in the expression
``self`` which can be either a ``Sum`` or a ``Product``. The selectors in
the argument ``indices`` specify some indices whose limits get reversed.
These selectors are either variable names or numerical indices counted
starting from the inner-most limit tuple.
Examples
========
>>> from sympy import Sum
>>> from sympy.abc import x, y, a, b, c, d
>>> Sum(x, (x, 0, 3)).reverse_order(x)
Sum(-x, (x, 4, -1))
>>> Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(x, y)
Sum(x*y, (x, 6, 0), (y, 7, -1))
>>> Sum(x, (x, a, b)).reverse_order(x)
Sum(-x, (x, b + 1, a - 1))
>>> Sum(x, (x, a, b)).reverse_order(0)
Sum(-x, (x, b + 1, a - 1))
While one should prefer variable names when specifying which limits
to reverse, the index counting notation comes in handy in case there
are several symbols with the same name.
>>> S = Sum(x**2, (x, a, b), (x, c, d))
>>> S
Sum(x**2, (x, a, b), (x, c, d))
>>> S0 = S.reverse_order(0)
>>> S0
Sum(-x**2, (x, b + 1, a - 1), (x, c, d))
>>> S1 = S0.reverse_order(1)
>>> S1
Sum(x**2, (x, b + 1, a - 1), (x, d + 1, c - 1))
Of course we can mix both notations:
>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
See Also
========
sympy.concrete.expr_with_intlimits.ExprWithIntLimits.index, reorder_limit,
sympy.concrete.expr_with_intlimits.ExprWithIntLimits.reorder
References
==========
.. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM,
Volume 28 Issue 2, April 1981, Pages 305-350
https://dl.acm.org/doi/10.1145/322248.322255
"""
l_indices = list(indices)
for i, indx in enumerate(l_indices):
if not isinstance(indx, int):
l_indices[i] = self.index(indx)
e = 1
limits = []
for i, limit in enumerate(self.limits):
l = limit
if i in l_indices:
e = -e
l = (limit[0], limit[2] + 1, limit[1] - 1)
limits.append(l)
return Sum(e * self.function, *limits)
def _eval_rewrite_as_Product(self, *args, **kwargs):
from sympy.concrete.products import Product
if self.function.is_extended_real:
return log(Product(exp(self.function), *self.limits))
def summation(f, *symbols, **kwargs):
r"""
Compute the summation of f with respect to symbols.
Explanation
===========
The notation for symbols is similar to the notation used in Integral.
summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
i.e.,
::
b
____
\ `
summation(f, (i, a, b)) = ) f
/___,
i = a
If it cannot compute the sum, it returns an unevaluated Sum object.
Repeated sums can be computed by introducing additional symbols tuples::
Examples
========
>>> from sympy import summation, oo, symbols, log
>>> i, n, m = symbols('i n m', integer=True)
>>> summation(2*i - 1, (i, 1, n))
n**2
>>> summation(1/2**i, (i, 0, oo))
2
>>> summation(1/log(n)**n, (n, 2, oo))
Sum(log(n)**(-n), (n, 2, oo))
>>> summation(i, (i, 0, n), (n, 0, m))
m**3/6 + m**2/2 + m/3
>>> from sympy.abc import x
>>> from sympy import factorial
>>> summation(x**n/factorial(n), (n, 0, oo))
exp(x)
See Also
========
Sum
Product, sympy.concrete.products.product
"""
return Sum(f, *symbols, **kwargs).doit(deep=False)
def telescopic_direct(L, R, n, limits):
"""
Returns the direct summation of the terms of a telescopic sum
Explanation
===========
L is the term with lower index
R is the term with higher index
n difference between the indexes of L and R
Examples
========
>>> from sympy.concrete.summations import telescopic_direct
>>> from sympy.abc import k, a, b
>>> telescopic_direct(1/k, -1/(k+2), 2, (k, a, b))
-1/(b + 2) - 1/(b + 1) + 1/(a + 1) + 1/a
"""
(i, a, b) = limits
return Add(*[L.subs(i, a + m) + R.subs(i, b - m) for m in range(n)])
def telescopic(L, R, limits):
'''
Tries to perform the summation using the telescopic property.
Return None if not possible.
'''
(i, a, b) = limits
if L.is_Add or R.is_Add:
return None
# We want to solve(L.subs(i, i + m) + R, m)
# First we try a simple match since this does things that
# solve doesn't do, e.g. solve(cos(k+m)-cos(k), m) gives
# a more complicated solution than m == 0.
k = Wild("k")
sol = (-R).match(L.subs(i, i + k))
s = None
if sol and k in sol:
s = sol[k]
if not (s.is_Integer and L.subs(i, i + s) + R == 0):
# invalid match or match didn't work
s = None
# But there are things that match doesn't do that solve
# can do, e.g. determine that 1/(x + m) = 1/(1 - x) when m = 1
if s is None:
m = Dummy('m')
try:
from sympy.solvers.solvers import solve
sol = solve(L.subs(i, i + m) + R, m) or []
except NotImplementedError:
return None
sol = [si for si in sol if si.is_Integer and
(L.subs(i, i + si) + R).expand().is_zero]
if len(sol) != 1:
return None
s = sol[0]
if s < 0:
return telescopic_direct(R, L, abs(s), (i, a, b))
elif s > 0:
return telescopic_direct(L, R, s, (i, a, b))
def eval_sum(f, limits):
(i, a, b) = limits
if f.is_zero:
return S.Zero
if i not in f.free_symbols:
return f*(b - a + 1)
if a == b:
return f.subs(i, a)
if isinstance(f, Piecewise):
if not any(i in arg.args[1].free_symbols for arg in f.args):
# Piecewise conditions do not depend on the dummy summation variable,
# therefore we can fold: Sum(Piecewise((e, c), ...), limits)
# --> Piecewise((Sum(e, limits), c), ...)
newargs = []
for arg in f.args:
newexpr = eval_sum(arg.expr, limits)
if newexpr is None:
return None
newargs.append((newexpr, arg.cond))
return f.func(*newargs)
if f.has(KroneckerDelta):
from .delta import deltasummation, _has_simple_delta
f = f.replace(
lambda x: isinstance(x, Sum),
lambda x: x.factor()
)
if _has_simple_delta(f, limits[0]):
return deltasummation(f, limits)
dif = b - a
definite = dif.is_Integer
# Doing it directly may be faster if there are very few terms.
if definite and (dif < 100):
return eval_sum_direct(f, (i, a, b))
if isinstance(f, Piecewise):
return None
# Try to do it symbolically. Even when the number of terms is
# known, this can save time when b-a is big.
value = eval_sum_symbolic(f.expand(), (i, a, b))
if value is not None:
return value
# Do it directly
if definite:
return eval_sum_direct(f, (i, a, b))
def eval_sum_direct(expr, limits):
"""
Evaluate expression directly, but perform some simple checks first
to possibly result in a smaller expression and faster execution.
"""
(i, a, b) = limits
dif = b - a
# Linearity
if expr.is_Mul:
# Try factor out everything not including i
without_i, with_i = expr.as_independent(i)
if without_i != 1:
s = eval_sum_direct(with_i, (i, a, b))
if s:
r = without_i*s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = expr.as_two_terms()
if not L.has(i):
sR = eval_sum_direct(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_direct(L, (i, a, b))
if sL:
return sL*R
# do this whether its an Add or Mul
# e.g. apart(1/(25*i**2 + 45*i + 14)) and
# apart(1/((5*i + 2)*(5*i + 7))) ->
# -1/(5*(5*i + 7)) + 1/(5*(5*i + 2))
try:
expr = apart(expr, i) # see if it becomes an Add
except PolynomialError:
pass
if expr.is_Add:
# Try factor out everything not including i
without_i, with_i = expr.as_independent(i)
if without_i != 0:
s = eval_sum_direct(with_i, (i, a, b))
if s:
r = without_i*(dif + 1) + s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = expr.as_two_terms()
lsum = eval_sum_direct(L, (i, a, b))
rsum = eval_sum_direct(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if r is not S.NaN:
return r
return Add(*[expr.subs(i, a + j) for j in range(dif + 1)])
def eval_sum_symbolic(f, limits):
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
# Try factor out everything not including i
without_i, with_i = f.as_independent(i)
if without_i != 1:
s = eval_sum_symbolic(with_i, (i, a, b))
if s:
r = without_i*s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return sL*R
# do this whether its an Add or Mul
# e.g. apart(1/(25*i**2 + 45*i + 14)) and
# apart(1/((5*i + 2)*(5*i + 7))) ->
# -1/(5*(5*i + 7)) + 1/(5*(5*i + 2))
try:
f = apart(f, i)
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
# Try factor out everything not including i
without_i, with_i = f.as_independent(i)
if without_i != 0:
s = eval_sum_symbolic(with_i, (i, a, b))
if s:
r = without_i*(b - a + 1) + s
if r is not S.NaN:
return r
else:
# Try term by term
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if r is not S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and a is not S.NegativeInfinity) or \
(a is S.NegativeInfinity and b is not S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1 ** wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2*i + c3)
if e_exp is not None:
e.update(e_exp)
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if isinstance(r, (Mul,Add)):
from sympy.simplify.radsimp import denom
from sympy.solvers.solvers import solve
non_limit = r.free_symbols - Tuple(*limits[1:]).free_symbols
den = denom(together(r))
den_sym = non_limit & den.free_symbols
args = []
for v in ordered(den_sym):
try:
s = solve(den, v)
m = Eq(v, s[0]) if s else S.false
if m != False:
args.append((Sum(f_orig.subs(*m.args), limits).doit(), m))
break
except NotImplementedError:
continue
args.append((r, True))
return Piecewise(*args)
if r not in (None, S.NaN):
return r
h = eval_sum_hyper(f_orig, (i, a, b))
if h is not None:
return h
r = eval_sum_residue(f_orig, (i, a, b))
if r is not None:
return r
factored = f_orig.factor()
if factored != f_orig:
return eval_sum_symbolic(factored, (i, a, b))
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
from sympy.simplify.simplify import simplify
if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
return S.Zero, True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
from sympy.simplify.simplify import hypersimp
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
from sympy.simplify.simplify import nsimplify
hs = nsimplify(hs)
from sympy.simplify.combsimp import combsimp
from sympy.simplify.hyperexpand import hyperexpand
from sympy.simplify.radsimp import fraction
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m**mul
params[k] += [n/m]*mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0]/ab[1]
h = hyper(ap, bq, x)
f = combsimp(f)
return f.subs(i, 0)*hyperexpand(h), h.convergence_statement
def eval_sum_hyper(f, i_a_b):
i, a, b = i_a_b
if f.is_hypergeometric(i) is False:
return
if (b - a).is_Integer:
# We are never going to do better than doing the sum in the obvious way
return None
old_sum = Sum(f, (i, a, b))
if b != S.Infinity:
if a is S.NegativeInfinity:
res = _eval_sum_hyper(f.subs(i, -i), i, -b)
if res is not None:
return Piecewise(res, (old_sum, True))
else:
n_illegal = lambda x: sum(x.count(_) for _ in _illegal)
had = n_illegal(f)
# check that no extra illegals are introduced
res1 = _eval_sum_hyper(f, i, a)
if res1 is None or n_illegal(res1) > had:
return
res2 = _eval_sum_hyper(f, i, b + 1)
if res2 is None or n_illegal(res2) > had:
return
(res1, cond1), (res2, cond2) = res1, res2
cond = And(cond1, cond2)
if cond == False:
return None
return Piecewise((res1 - res2, cond), (old_sum, True))
if a is S.NegativeInfinity:
res1 = _eval_sum_hyper(f.subs(i, -i), i, 1)
res2 = _eval_sum_hyper(f, i, 0)
if res1 is None or res2 is None:
return None
res1, cond1 = res1
res2, cond2 = res2
cond = And(cond1, cond2)
if cond == False or cond.as_set() == S.EmptySet:
return None
return Piecewise((res1 + res2, cond), (old_sum, True))
# Now b == oo, a != -oo
res = _eval_sum_hyper(f, i, a)
if res is not None:
r, c = res
if c == False:
if r.is_number:
f = f.subs(i, Dummy('i', integer=True, positive=True) + a)
if f.is_positive or f.is_zero:
return S.Infinity
elif f.is_negative:
return S.NegativeInfinity
return None
return Piecewise(res, (old_sum, True))
def eval_sum_residue(f, i_a_b):
r"""Compute the infinite summation with residues
Notes
=====
If $f(n), g(n)$ are polynomials with $\deg(g(n)) - \deg(f(n)) \ge 2$,
some infinite summations can be computed by the following residue
evaluations.
.. math::
\sum_{n=-\infty, g(n) \ne 0}^{\infty} \frac{f(n)}{g(n)} =
-\pi \sum_{\alpha|g(\alpha)=0}
\text{Res}(\cot(\pi x) \frac{f(x)}{g(x)}, \alpha)
.. math::
\sum_{n=-\infty, g(n) \ne 0}^{\infty} (-1)^n \frac{f(n)}{g(n)} =
-\pi \sum_{\alpha|g(\alpha)=0}
\text{Res}(\csc(\pi x) \frac{f(x)}{g(x)}, \alpha)
Examples
========
>>> from sympy import Sum, oo, Symbol
>>> x = Symbol('x')
Doubly infinite series of rational functions.
>>> Sum(1 / (x**2 + 1), (x, -oo, oo)).doit()
pi/tanh(pi)
Doubly infinite alternating series of rational functions.
>>> Sum((-1)**x / (x**2 + 1), (x, -oo, oo)).doit()
pi/sinh(pi)
Infinite series of even rational functions.
>>> Sum(1 / (x**2 + 1), (x, 0, oo)).doit()
1/2 + pi/(2*tanh(pi))
Infinite series of alternating even rational functions.
>>> Sum((-1)**x / (x**2 + 1), (x, 0, oo)).doit()
pi/(2*sinh(pi)) + 1/2
This also have heuristics to transform arbitrarily shifted summand or
arbitrarily shifted summation range to the canonical problem the
formula can handle.
>>> Sum(1 / (x**2 + 2*x + 2), (x, -1, oo)).doit()
1/2 + pi/(2*tanh(pi))
>>> Sum(1 / (x**2 + 4*x + 5), (x, -2, oo)).doit()
1/2 + pi/(2*tanh(pi))
>>> Sum(1 / (x**2 + 1), (x, 1, oo)).doit()
-1/2 + pi/(2*tanh(pi))
>>> Sum(1 / (x**2 + 1), (x, 2, oo)).doit()
-1 + pi/(2*tanh(pi))
References
==========
.. [#] http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf
.. [#] Asmar N.H., Grafakos L. (2018) Residue Theory.
In: Complex Analysis with Applications.
Undergraduate Texts in Mathematics. Springer, Cham.
https://doi.org/10.1007/978-3-319-94063-2_5
"""
i, a, b = i_a_b
def is_even_function(numer, denom):
"""Test if the rational function is an even function"""
numer_even = all(i % 2 == 0 for (i,) in numer.monoms())
denom_even = all(i % 2 == 0 for (i,) in denom.monoms())
numer_odd = all(i % 2 == 1 for (i,) in numer.monoms())
denom_odd = all(i % 2 == 1 for (i,) in denom.monoms())
return (numer_even and denom_even) or (numer_odd and denom_odd)
def match_rational(f, i):
numer, denom = f.as_numer_denom()
try:
(numer, denom), opt = parallel_poly_from_expr((numer, denom), i)
except (PolificationFailed, PolynomialError):
return None
return numer, denom
def get_poles(denom):
roots = denom.sqf_part().all_roots()
roots = sift(roots, lambda x: x.is_integer)
if None in roots:
return None
int_roots, nonint_roots = roots[True], roots[False]
return int_roots, nonint_roots
def get_shift(denom):
n = denom.degree(i)
a = denom.coeff_monomial(i**n)
b = denom.coeff_monomial(i**(n-1))
shift = - b / a / n
return shift
#Need a dummy symbol with no assumptions set for get_residue_factor
z = Dummy('z')
def get_residue_factor(numer, denom, alternating):
residue_factor = (numer.as_expr() / denom.as_expr()).subs(i, z)
if not alternating:
residue_factor *= cot(S.Pi * z)
else:
residue_factor *= csc(S.Pi * z)
return residue_factor
# We don't know how to deal with symbolic constants in summand
if f.free_symbols - {i}:
return None
if not (a.is_Integer or a in (S.Infinity, S.NegativeInfinity)):
return None
if not (b.is_Integer or b in (S.Infinity, S.NegativeInfinity)):
return None
# Quick exit heuristic for the sums which doesn't have infinite range
if a != S.NegativeInfinity and b != S.Infinity:
return None
match = match_rational(f, i)
if match:
alternating = False
numer, denom = match
else:
match = match_rational(f / S.NegativeOne**i, i)
if match:
alternating = True
numer, denom = match
else:
return None
if denom.degree(i) - numer.degree(i) < 2:
return None
if (a, b) == (S.NegativeInfinity, S.Infinity):
poles = get_poles(denom)
if poles is None:
return None
int_roots, nonint_roots = poles
if int_roots:
return None
residue_factor = get_residue_factor(numer, denom, alternating)
residues = [residue(residue_factor, z, root) for root in nonint_roots]
return -S.Pi * sum(residues)
if not (a.is_finite and b is S.Infinity):
return None
if not is_even_function(numer, denom):
# Try shifting summation and check if the summand can be made
# and even function from the origin.
# Sum(f(n), (n, a, b)) => Sum(f(n + s), (n, a - s, b - s))
shift = get_shift(denom)
if not shift.is_Integer:
return None
if shift == 0:
return None
numer = numer.shift(shift)
denom = denom.shift(shift)
if not is_even_function(numer, denom):
return None
if alternating:
f = S.NegativeOne**i * (S.NegativeOne**shift * numer.as_expr() / denom.as_expr())
else:
f = numer.as_expr() / denom.as_expr()
return eval_sum_residue(f, (i, a-shift, b-shift))
poles = get_poles(denom)
if poles is None:
return None
int_roots, nonint_roots = poles
if int_roots:
int_roots = [int(root) for root in int_roots]
int_roots_max = max(int_roots)
int_roots_min = min(int_roots)
# Integer valued poles must be next to each other
# and also symmetric from origin (Because the function is even)
if not len(int_roots) == int_roots_max - int_roots_min + 1:
return None
# Check whether the summation indices contain poles
if a <= max(int_roots):
return None
residue_factor = get_residue_factor(numer, denom, alternating)
residues = [residue(residue_factor, z, root) for root in int_roots + nonint_roots]
full_sum = -S.Pi * sum(residues)
if not int_roots:
# Compute Sum(f, (i, 0, oo)) by adding a extraneous evaluation
# at the origin.
half_sum = (full_sum + f.xreplace({i: 0})) / 2
# Add and subtract extraneous evaluations
extraneous_neg = [f.xreplace({i: i0}) for i0 in range(int(a), 0)]
extraneous_pos = [f.xreplace({i: i0}) for i0 in range(0, int(a))]
result = half_sum + sum(extraneous_neg) - sum(extraneous_pos)
return result
# Compute Sum(f, (i, min(poles) + 1, oo))
half_sum = full_sum / 2
# Subtract extraneous evaluations
extraneous = [f.xreplace({i: i0}) for i0 in range(max(int_roots) + 1, int(a))]
result = half_sum - sum(extraneous)
return result
def _eval_matrix_sum(expression):
f = expression.function
for n, limit in enumerate(expression.limits):
i, a, b = limit
dif = b - a
if dif.is_Integer:
if (dif < 0) == True:
a, b = b + 1, a - 1
f = -f
newf = eval_sum_direct(f, (i, a, b))
if newf is not None:
return newf.doit()
def _dummy_with_inherited_properties_concrete(limits):
"""
Return a Dummy symbol that inherits as many assumptions as possible
from the provided symbol and limits.
If the symbol already has all True assumption shared by the limits
then return None.
"""
x, a, b = limits
l = [a, b]
assumptions_to_consider = ['extended_nonnegative', 'nonnegative',
'extended_nonpositive', 'nonpositive',
'extended_positive', 'positive',
'extended_negative', 'negative',
'integer', 'rational', 'finite',
'zero', 'real', 'extended_real']
assumptions_to_keep = {}
assumptions_to_add = {}
for assum in assumptions_to_consider:
assum_true = x._assumptions.get(assum, None)
if assum_true:
assumptions_to_keep[assum] = True
elif all(getattr(i, 'is_' + assum) for i in l):
assumptions_to_add[assum] = True
if assumptions_to_add:
assumptions_to_keep.update(assumptions_to_add)
return Dummy('d', **assumptions_to_keep)