from __future__ import print_function, division
from sympy.core.compatibility import range
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
from sympy.core.exprtools import factor_terms
from sympy.functions.elementary.exponential import exp, log
from sympy.polys import quo, roots
from sympy.simplify import powsimp
[docs]class Product(ExprWithIntLimits):
r"""Represents unevaluated products.
``Product`` represents a finite or infinite product, 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 product.
Finite products
===============
For finite products (and products with symbolic limits assumed to be finite)
we follow the analogue of the summation convention described by Karr [1],
especially definition 3 of section 1.4. The product:
.. math::
\prod_{m \leq i < n} f(i)
has *the obvious meaning* for `m < n`, namely:
.. math::
\prod_{m \leq i < n} f(i) = f(m) f(m+1) \cdot \ldots \cdot f(n-2) f(n-1)
with the upper limit value `f(n)` excluded. The product over an empty set is
one if and only if `m = n`:
.. math::
\prod_{m \leq i < n} f(i) = 1 \quad \mathrm{for} \quad m = n
Finally, for all other products over empty sets we assume the following
definition:
.. math::
\prod_{m \leq i < n} f(i) = \frac{1}{\prod_{n \leq i < m} f(i)} \quad \mathrm{for} \quad m > n
It is important to note that above we define all products with the upper
limit being exclusive. This is in contrast to the usual mathematical notation,
but does not affect the product convention. Indeed we have:
.. math::
\prod_{m \leq i < n} f(i) = \prod_{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 a, b, i, k, m, n, x
>>> from sympy import Product, factorial, oo
>>> Product(k, (k, 1, m))
Product(k, (k, 1, m))
>>> Product(k, (k, 1, m)).doit()
factorial(m)
>>> Product(k**2,(k, 1, m))
Product(k**2, (k, 1, m))
>>> Product(k**2,(k, 1, m)).doit()
factorial(m)**2
Wallis' product for pi:
>>> W = Product(2*i/(2*i-1) * 2*i/(2*i+1), (i, 1, oo))
>>> W
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, oo))
Direct computation currently fails:
>>> W.doit()
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, oo))
But we can approach the infinite product by a limit of finite products:
>>> from sympy import limit
>>> W2 = Product(2*i/(2*i-1)*2*i/(2*i+1), (i, 1, n))
>>> W2
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, n))
>>> W2e = W2.doit()
>>> W2e
2**(-2*n)*4**n*factorial(n)**2/(RisingFactorial(1/2, n)*RisingFactorial(3/2, n))
>>> limit(W2e, n, oo)
pi/2
By the same formula we can compute sin(pi/2):
>>> from sympy import pi, gamma, simplify
>>> P = pi * x * Product(1 - x**2/k**2, (k, 1, n))
>>> P = P.subs(x, pi/2)
>>> P
pi**2*Product(1 - pi**2/(4*k**2), (k, 1, n))/2
>>> Pe = P.doit()
>>> Pe
pi**2*RisingFactorial(1 - pi/2, n)*RisingFactorial(1 + pi/2, n)/(2*factorial(n)**2)
>>> Pe = Pe.rewrite(gamma)
>>> Pe
pi**2*gamma(n + 1 + pi/2)*gamma(n - pi/2 + 1)/(2*gamma(1 - pi/2)*gamma(1 + pi/2)*gamma(n + 1)**2)
>>> Pe = simplify(Pe)
>>> Pe
sin(pi**2/2)*gamma(n + 1 + pi/2)*gamma(n - pi/2 + 1)/gamma(n + 1)**2
>>> limit(Pe, n, oo)
sin(pi**2/2)
Products with the lower limit being larger than the upper one:
>>> Product(1/i, (i, 6, 1)).doit()
120
>>> Product(i, (i, 2, 5)).doit()
120
The empty product:
>>> Product(i, (i, n, n-1)).doit()
1
An example showing that the symbolic result of a product is still
valid for seemingly nonsensical values of the limits. Then the Karr
convention allows us to give a perfectly valid interpretation to
those products by interchanging the limits according to the above rules:
>>> P = Product(2, (i, 10, n)).doit()
>>> P
2**(n - 9)
>>> P.subs(n, 5)
1/16
>>> Product(2, (i, 10, 5)).doit()
1/16
>>> 1/Product(2, (i, 6, 9)).doit()
1/16
An explicit example of the Karr summation convention applied to products:
>>> P1 = Product(x, (i, a, b)).doit()
>>> P1
x**(-a + b + 1)
>>> P2 = Product(x, (i, b+1, a-1)).doit()
>>> P2
x**(a - b - 1)
>>> simplify(P1 * P2)
1
And another one:
>>> P1 = Product(i, (i, b, a)).doit()
>>> P1
RisingFactorial(b, a - b + 1)
>>> P2 = Product(i, (i, a+1, b-1)).doit()
>>> P2
RisingFactorial(a + 1, -a + b - 1)
>>> P1 * P2
RisingFactorial(b, a - b + 1)*RisingFactorial(a + 1, -a + b - 1)
>>> simplify(P1 * P2)
1
See Also
========
Sum, summation
product
References
==========
.. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM,
Volume 28 Issue 2, April 1981, Pages 305-350
http://dl.acm.org/citation.cfm?doid=322248.322255
.. [2] https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation
.. [3] https://en.wikipedia.org/wiki/Empty_product
"""
__slots__ = ['is_commutative']
def __new__(cls, function, *symbols, **assumptions):
obj = ExprWithIntLimits.__new__(cls, function, *symbols, **assumptions)
return obj
def _eval_rewrite_as_Sum(self, *args, **kwargs):
from sympy.concrete.summations import Sum
return exp(Sum(log(self.function), *self.limits))
@property
def term(self):
return self._args[0]
function = term
def _eval_is_zero(self):
# a Product is zero only if its term is zero.
return self.term.is_zero
def doit(self, **hints):
f = self.function
for index, limit in enumerate(self.limits):
i, a, b = limit
dif = b - a
if dif.is_Integer and dif < 0:
a, b = b + 1, a - 1
f = 1 / f
g = self._eval_product(f, (i, a, b))
if g in (None, S.NaN):
return self.func(powsimp(f), *self.limits[index:])
else:
f = g
if hints.get('deep', True):
return f.doit(**hints)
else:
return powsimp(f)
def _eval_adjoint(self):
if self.is_commutative:
return self.func(self.function.adjoint(), *self.limits)
return None
def _eval_conjugate(self):
return self.func(self.function.conjugate(), *self.limits)
def _eval_product(self, term, limits):
from sympy.concrete.delta import deltaproduct, _has_simple_delta
from sympy.concrete.summations import summation
from sympy.functions import KroneckerDelta, RisingFactorial
(k, a, n) = limits
if k not in term.free_symbols:
if (term - 1).is_zero:
return S.One
return term**(n - a + 1)
if a == n:
return term.subs(k, a)
if term.has(KroneckerDelta) and _has_simple_delta(term, limits[0]):
return deltaproduct(term, limits)
dif = n - a
if dif.is_Integer:
return Mul(*[term.subs(k, a + i) for i in range(dif + 1)])
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
all_roots = roots(poly)
M = 0
for r, m in all_roots.items():
M += m
A *= RisingFactorial(a - r, n - a + 1)**m
Q *= (n - r)**m
if M < poly.degree():
arg = quo(poly, Q.as_poly(k))
B = self.func(arg, (k, a, n)).doit()
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
factored = factor_terms(term, fraction=True)
if factored.is_Mul:
return self._eval_product(factored, (k, a, n))
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(t, (k, a, n))
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
arg = term._new_rawargs(*include)
A = Mul(*exclude)
B = self.func(arg, (k, a, n)).doit()
return A * B
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(term.base, (k, a, n))
if p is not None:
return p**term.exp
elif isinstance(term, Product):
evaluated = term.doit()
f = self._eval_product(evaluated, limits)
if f is None:
return self.func(evaluated, limits)
else:
return f
def _eval_simplify(self, ratio, measure, rational, inverse):
from sympy.simplify.simplify import product_simplify
return product_simplify(self)
def _eval_transpose(self):
if self.is_commutative:
return self.func(self.function.transpose(), *self.limits)
return None
[docs] def is_convergent(self):
r"""
See docs of Sum.is_convergent() for explanation of convergence
in SymPy.
The infinite product:
.. math::
\prod_{1 \leq i < \infty} f(i)
is defined by the sequence of partial products:
.. math::
\prod_{i=1}^{n} f(i) = f(1) f(2) \cdots f(n)
as n increases without bound. The product converges to a non-zero
value if and only if the sum:
.. math::
\sum_{1 \leq i < \infty} \log{f(n)}
converges.
Examples
========
>>> from sympy import Interval, S, Product, Symbol, cos, pi, exp, oo
>>> n = Symbol('n', integer=True)
>>> Product(n/(n + 1), (n, 1, oo)).is_convergent()
False
>>> Product(1/n**2, (n, 1, oo)).is_convergent()
False
>>> Product(cos(pi/n), (n, 1, oo)).is_convergent()
True
>>> Product(exp(-n**2), (n, 1, oo)).is_convergent()
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Infinite_product
"""
from sympy.concrete.summations import Sum
sequence_term = self.function
log_sum = log(sequence_term)
lim = self.limits
try:
is_conv = Sum(log_sum, *lim).is_convergent()
except NotImplementedError:
if Sum(sequence_term - 1, *lim).is_absolutely_convergent() is S.true:
return S.true
raise NotImplementedError("The algorithm to find the product convergence of %s "
"is not yet implemented" % (sequence_term))
return is_conv
[docs] def reverse_order(expr, *indices):
"""
Reverse the order of a limit in a Product.
Usage
=====
``reverse_order(expr, *indices)`` reverses some limits in the expression
``expr`` 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 Product, simplify, RisingFactorial, gamma, Sum
>>> from sympy.abc import x, y, a, b, c, d
>>> P = Product(x, (x, a, b))
>>> Pr = P.reverse_order(x)
>>> Pr
Product(1/x, (x, b + 1, a - 1))
>>> Pr = Pr.doit()
>>> Pr
1/RisingFactorial(b + 1, a - b - 1)
>>> simplify(Pr)
gamma(b + 1)/gamma(a)
>>> P = P.doit()
>>> P
RisingFactorial(a, -a + b + 1)
>>> simplify(P)
gamma(b + 1)/gamma(a)
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*y, (x, a, b), (y, c, d))
>>> S
Sum(x*y, (x, a, b), (y, c, d))
>>> S0 = S.reverse_order(0)
>>> S0
Sum(-x*y, (x, b + 1, a - 1), (y, c, d))
>>> S1 = S0.reverse_order(1)
>>> S1
Sum(x*y, (x, b + 1, a - 1), (y, 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
========
index, reorder_limit, reorder
References
==========
.. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM,
Volume 28 Issue 2, April 1981, Pages 305-350
http://dl.acm.org/citation.cfm?doid=322248.322255
"""
l_indices = list(indices)
for i, indx in enumerate(l_indices):
if not isinstance(indx, int):
l_indices[i] = expr.index(indx)
e = 1
limits = []
for i, limit in enumerate(expr.limits):
l = limit
if i in l_indices:
e = -e
l = (limit[0], limit[2] + 1, limit[1] - 1)
limits.append(l)
return Product(expr.function ** e, *limits)
[docs]def product(*args, **kwargs):
r"""
Compute the product.
The notation for symbols is similar to the notation used in Sum or
Integral. product(f, (i, a, b)) computes the product of f with
respect to i from a to b, i.e.,
::
b
_____
product(f(n), (i, a, b)) = | | f(n)
| |
i = a
If it cannot compute the product, it returns an unevaluated Product object.
Repeated products can be computed by introducing additional symbols tuples::
>>> from sympy import product, symbols
>>> i, n, m, k = symbols('i n m k', integer=True)
>>> product(i, (i, 1, k))
factorial(k)
>>> product(m, (i, 1, k))
m**k
>>> product(i, (i, 1, k), (k, 1, n))
Product(factorial(k), (k, 1, n))
"""
prod = Product(*args, **kwargs)
if isinstance(prod, Product):
return prod.doit(deep=False)
else:
return prod
