# Source code for diofant.concrete.products

from ..core import Integer, Mul, nan
from ..functions import exp, log
from ..polys import quo, roots
from ..simplify import powsimp
from .expr_with_intlimits import ExprWithIntLimits

[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.

For finite products (and products with symbolic limits assumed to be finite)
we follow the analogue of the summation convention described by
Karr :cite:Karr1981summation, 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 diofant.abc import i

>>> 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:

>>> 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):

>>> 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

========

diofant.concrete.summations.Sum
diofant.concrete.summations.summation
diofant.concrete.products.product

References
==========

* https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation
* https://en.wikipedia.org/wiki/Empty_product

"""

def __new__(cls, function, *symbols, **assumptions):
obj = ExprWithIntLimits.__new__(cls, function, *symbols, **assumptions)
return obj

def _eval_rewrite_as_Sum(self, *args):
from .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):
if self.term.is_zero:
return True

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, 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)

if self.is_commutative:
return

def _eval_conjugate(self):
return self.func(self.function.conjugate(), *self.limits)

def _eval_product(self, term, limits):
from .delta import deltaproduct, _has_simple_delta
from .summations import summation
from ..functions import KroneckerDelta, RisingFactorial

(k, a, n) = limits

if k not in term.free_symbols:
if (term - 1).is_zero:
return Integer(1)
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 = Integer(1)

all_roots = roots(poly)

M = 0
for r, m in all_roots.items():
M += m
A *= RisingFactorial(a - r, n - a + 1)**m
Q *= (k - r)**m

if M < poly.degree() and M > 0:
arg = quo(poly, Q.as_poly(k)).as_expr(k)
B = self.func(arg, (k, a, n)).doit()

if M > 0:
return poly.LC()**(n - a + 1) * A * B
else:
return

p, q = term.as_numer_denom()
q = self._eval_product(q, (k, a, n))
if q.is_Number:
# There is expression, which couldn't change by
# as_numer_denom(). E.g. n**(2/3) + 1 --> (n**(2/3) + 1, 1).
p = sum(self._eval_product(i, (k, a, n)) for i in p.as_coeff_Add())
else:
p = self._eval_product(p, (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
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

def _eval_simplify(self, ratio, measure):
from ..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

[docs]    def reverse_order(self, *indices):
r"""
Reverse the order of a limit in a Product.

Parameters
==========

\*indices : list
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
========

>>> 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))

========

diofant.concrete.expr_with_intlimits.ExprWithIntLimits.index,
diofant.concrete.expr_with_intlimits.ExprWithIntLimits.reorder_limit,
diofant.concrete.expr_with_intlimits.ExprWithIntLimits.reorder

References
==========

* :cite:Karr1981summation

"""
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 Product(self.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::

>>> i = symbols('i', 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