/

# Source code for sympy.concrete.products

from sympy.core.containers import Tuple
from sympy.core.core import C
from sympy.core.expr import Expr
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.core.sympify import sympify
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.polys import quo, roots
from sympy.simplify import powsimp

[docs]class Product(Expr):
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()
nan

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(-pi/2 + 1, 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(-pi/2 + 1)*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

========

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] http://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation
.. [3] http://en.wikipedia.org/wiki/Empty_product
"""

__slots__ = ['is_commutative']

def __new__(cls, function, *symbols, **assumptions):
from sympy.integrals.integrals import _process_limits

# Any embedded piecewise functions need to be brought out to the
# top level so that integration can go into piecewise mode at the
# earliest possible moment.
function = piecewise_fold(sympify(function))

if function is S.NaN:
return S.NaN

if not symbols:
raise ValueError("Product variables must be given")

limits, sign = _process_limits(*symbols)

# Only limits with lower and upper bounds are supported; the indefinite
# Product is not supported
if any(len(l) != 3 or None in l for l in limits):
raise ValueError(
'Product requires values for lower and upper bounds.')

obj = Expr.__new__(cls, **assumptions)
arglist = [sign*function]
arglist.extend(limits)
obj._args = tuple(arglist)
obj.is_commutative = function.is_commutative  # limits already checked

return obj

@property
def term(self):
return self._args[0]
function = term

@property
def limits(self):
return self._args[1:]

@property
[docs]    def variables(self):
"""Return a list of the product variables

>>> from sympy import Product
>>> from sympy.abc import x, i
>>> Product(x**i, (i, 1, 3)).variables
[i]
"""
return [l[0] for l in self.limits]

@property
[docs]    def free_symbols(self):
"""
This method returns the symbols that will affect the value of
the Product when evaluated. This is useful if one is trying to
determine whether a product depends on a certain symbol or not.

>>> from sympy import Product
>>> from sympy.abc import x, y
>>> Product(x, (x, y, 1)).free_symbols
set([y])
"""
from sympy.integrals.integrals import _free_symbols
if self.function.is_zero or self.function == 1:
return set()
return _free_symbols(self)

@property
[docs]    def is_zero(self):
"""A Product is zero only if its term is zero.
"""
return self.term.is_zero

@property
[docs]    def is_number(self):
"""
Return True if the Product will result in a number, else False.

Examples
========

>>> from sympy import log, Product
>>> from sympy.abc import x, y, z
>>> log(2).is_number
True
>>> Product(x, (x, 1, 2)).is_number
True
>>> Product(y, (x, 1, 2)).is_number
False
>>> Product(1, (x, y, z)).is_number
True
>>> Product(2, (x, y, z)).is_number
False
"""

return self.function.is_zero or self.function == 1 or not self.free_symbols

def as_dummy(self):
from sympy.integrals.integrals import _as_dummy
return _as_dummy(self)

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 is None:
return Product(powsimp(f), *self.limits[index:])
else:
f = g

if hints.get('deep', True):
return f.doit(**hints)
else:
return powsimp(f)

if self.is_commutative:
return None

def _eval_conjugate(self):
return Product(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

(k, a, n) = limits

if k not in term.free_symbols:
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 xrange(dif + 1)])

elif term.is_polynomial(k):
poly = term.as_poly(k)

A = B = Q = S.One

all_roots = roots(poly, multiple=True)

for r in all_roots:
A *= C.RisingFactorial(a - r, n - a + 1)
Q *= n - r

if len(all_roots) < poly.degree():
arg = quo(poly, Q.as_poly(k))
B = Product(arg, (k, a, n)).doit()

return poly.LC()**(n - a + 1) * A * B

p, q = term.as_numer_denom()

p = self._eval_product(p, (k, a, n))
q = self._eval_product(q, (k, a, n))

return p / q

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 = Product(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 Product(evaluated, limits)
else:
return f

def _eval_transpose(self):
if self.is_commutative:
return Product(self.function.transpose(), *self.limits)
return None

def _eval_subs(self, old, new):
from sympy.integrals.integrals import _eval_subs
return _eval_subs(self, old, new)

[docs]def product(*args, **kwargs):
r"""
Compute the product.

The notation for symbols is similiar 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