def test_Sum():
assert mcode(Sum(sin(x), (x, 0, 10))) == "Hold[Sum[Sin[x], {x, 0, 10}]]"
assert mcode(Sum(exp(-x**2 - y**2),
(x, -oo, oo),
(y, -oo, oo))) == \
"Hold[Sum[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, " \
"{y, -Infinity, Infinity}]]"
def test_log_product():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
from sympy.concrete import Product, Sum
f, g = Function('f'), Function('g')
assert simplify(log(Product(x**i, (i, 1, n)))) == Sum(log(x**i), (i, 1, n))
assert simplify(log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))) == \
Sum(log(x**i*y**j), (i, 1, n), (j, 1, m))
expr = log(Product(-2, (n, 0, 4)))
assert Eq(simplify(expr), expr)
def _eval_expand_log(self, deep=True, **hints):
from sympy import unpolarify
from sympy.concrete import Sum, Product
force = hints.get('force', False)
arg = self.args[0]
if arg.is_Mul:
expr = []
nonpos = []
for x in arg.args:
if force or x.is_positive or x.is_polar:
a = self.func(x)
if isinstance(a, log):
expr.append(self.func(x)._eval_expand_log(**hints))
else:
expr.append(a)
else:
nonpos.append(x)
return Add(*expr) + log(Mul(*nonpos))
elif arg.is_Pow:
if force or (arg.exp.is_real and arg.base.is_positive) or \
arg.base.is_polar:
b = arg.base
e = arg.exp
a = self.func(b)
if isinstance(a, log):
return unpolarify(e) * a._eval_expand_log(**hints)
else:
return unpolarify(e) * a
elif isinstance(arg, Product):
if arg.function.is_positive:
return Sum(log(arg.function), *arg.limits)
return self.func(arg)
def infinite(self):
"""Returns an infinite representation of the series"""
from sympy.concrete import Sum
ak, xk = self.ak, self.xk
k = ak.variables[0]
inf_sum = Sum(ak.formula * xk.formula, (k, ak.start, ak.stop))
return self.ind + inf_sum
def as_Add(self):
if self.arg.is_Mul:
return Add(*(self.func(arg).simplify() for arg in self.arg.args))
from sympy import Sum
if self.arg.is_Product:
product = self.arg
return Sum(self.func(product.function), *product.limits)
return self
def _eval_expand_log(self, deep=True, **hints):
from sympy import unpolarify, expand_log
from sympy.concrete import Sum, Product
force = hints.get('force', False)
if (len(self.args) == 2):
return expand_log(self.func(*self.args), deep=deep, force=force)
arg = self.args[0]
if arg.is_Integer:
# remove perfect powers
p = perfect_power(int(arg))
if p is not False:
return p[1] * self.func(p[0])
elif arg.is_Rational:
return log(arg.p) - log(arg.q)
elif arg.is_Mul:
expr = []
nonpos = []
for x in arg.args:
if force or x.is_positive or x.is_polar:
a = self.func(x)
if isinstance(a, log):
expr.append(self.func(x)._eval_expand_log(**hints))
else:
expr.append(a)
elif x.is_negative:
a = self.func(-x)
expr.append(a)
nonpos.append(S.NegativeOne)
else:
nonpos.append(x)
return Add(*expr) + log(Mul(*nonpos))
elif arg.is_Pow or isinstance(arg, exp):
if force or (
arg.exp.is_extended_real and
(arg.base.is_positive or
((arg.exp + 1).is_positive and
(arg.exp - 1).is_nonpositive))) or arg.base.is_polar:
b = arg.base
e = arg.exp
a = self.func(b)
if isinstance(a, log):
return unpolarify(e) * a._eval_expand_log(**hints)
else:
return unpolarify(e) * a
elif isinstance(arg, Product):
if arg.function.is_positive:
return Sum(log(arg.function), *arg.limits)
return self.func(arg)
def test_deltasummation_basic_numerical():
n = symbols('n', integer=True, nonzero=True)
assert ds(KD(n, 0), (n, 1, 3)) == 0
# return unevaluated, until it gets implemented
assert ds(KD(i**2, j**2), (j, -oo, oo)) == \
Sum(KD(i**2, j**2), (j, -oo, oo))
assert Piecewise((KD(i, k), And(S(1) <= i, i <= 3)), (0, True)) == \
ds(KD(i, j)*KD(j, k), (j, 1, 3)) == \
ds(KD(j, k)*KD(i, j), (j, 1, 3))
assert ds(KD(i, k), (k, -oo, oo)) == 1
assert ds(KD(i, k), (k, 0, oo)) == Piecewise((1, S(0) <= i), (0, True))
assert ds(KD(i, k), (k, 1, 3)) == \
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True))
assert ds(k * KD(i, j) * KD(j, k), (k, -oo, oo)) == j * KD(i, j)
assert ds(j * KD(i, j), (j, -oo, oo)) == i
assert ds(i * KD(i, j), (i, -oo, oo)) == j
assert ds(x, (i, 1, 3)) == 3 * x
assert ds((i + j) * KD(i, j), (j, -oo, oo)) == 2 * i
def change_index(expr, var, trafo, newvar=None):
"""
Change index of a Sum or Product.
Perform a linear transformation `x \mapsto a x + b` on the index variable
`x`. For `a` the only values allowed are `\pm 1`. A new variable to be used
after the change of index can also be specified.
Usage
=====
``change_index(expr, var, trafo, newvar=None)`` where ``var`` specifies the
index variable `x` to transform. The transformation ``trafo`` must be linear
and given in terms of ``var``. If the optional argument ``newvar`` is
provided then ``var`` gets replaced by ``newvar`` in the final expression.
Examples
========
>>> from sympy.concrete.simplification import change_index
>>> from sympy import Sum, Product, simplify
>>> from sympy.abc import x, y, a, b, c, d, u, v, i, j, k, l
>>> S = Sum(x, (x, a, b))
>>> S.doit()
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = change_index(S, x, x + 1, y)
>>> Sn
Sum(y - 1, (y, a + 1, b + 1))
>>> Sn.doit()
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = change_index(S, x, -x, y)
>>> Sn
Sum(-y, (y, -b, -a))
>>> Sn.doit()
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = change_index(S, x, x+u)
>>> Sn
Sum(-u + x, (x, a + u, b + u))
>>> Sn.doit()
-a**2/2 - a*u + a/2 + b**2/2 + b*u + b/2 - u*(-a + b + 1) + u
>>> simplify(Sn.doit())
-a**2/2 + a/2 + b**2/2 + b/2
>>> Sn = change_index(S, x, -x - u, y)
>>> Sn
Sum(-u - y, (y, -b - u, -a - u))
>>> Sn.doit()
-a**2/2 - a*u + a/2 + b**2/2 + b*u + b/2 - u*(-a + b + 1) + u
>>> simplify(Sn.doit())
-a**2/2 + a/2 + b**2/2 + b/2
>>> P = Product(i*j**2, (i, a, b), (j, c, d))
>>> P
Product(i*j**2, (i, a, b), (j, c, d))
>>> P2 = change_index(P, i, i+3, k)
>>> P2
Product(j**2*(k - 3), (k, a + 3, b + 3), (j, c, d))
>>> P3 = change_index(P2, j, -j, l)
>>> P3
Product(l**2*(k - 3), (k, a + 3, b + 3), (l, -d, -c))
When dealing with symbols only, we can make a
general linear transformation:
>>> Sn = change_index(S, x, u*x+v, y)
>>> Sn
Sum((-v + y)/u, (y, b*u + v, a*u + v))
>>> Sn.doit()
-v*(a*u - b*u + 1)/u + (a**2*u**2/2 + a*u*v + a*u/2 - b**2*u**2/2 - b*u*v + b*u/2 + v)/u
>>> simplify(Sn.doit())
a**2*u/2 + a/2 - b**2*u/2 + b/2
However, the last result can be inconsistent with usual
summation where the index increment is always 1. This is
obvious as we get back the original value only for ``u``
equal +1 or -1.
See Also
========
sympy.concrete.simplification.index,
sympy.concrete.simplification.reorder_limit,
sympy.concrete.simplification.reorder,
sympy.concrete.simplification.reverse_order
"""
if newvar is None:
newvar = var
limits = []
for limit in expr.limits:
if limit[0] == var:
p = trafo.as_poly(var)
if p.degree() != 1:
raise ValueError("Index transformation is not linear")
alpha = p.coeff_monomial(var)
beta = p.coeff_monomial(S.One)
if alpha.is_number:
if alpha == S.One:
limits.append((newvar, alpha * limit[1] + beta,
alpha * limit[2] + beta))
elif alpha == S.NegativeOne:
limits.append((newvar, alpha * limit[2] + beta,
alpha * limit[1] + beta))
else:
raise ValueError(
"Linear transformation results in non-linear summation stepsize"
)
else:
# Note that the case of alpha being symbolic can give issues if alpha < 0.
limits.append(
(newvar, alpha * limit[2] + beta, alpha * limit[1] + beta))
else:
limits.append(limit)
function = expr.function.subs(var, (var - beta) / alpha)
function = function.subs(var, newvar)
if isinstance(expr, Sum):
return Sum(function, *tuple(limits))
elif isinstance(expr, Product):
return Product(function, *tuple(limits))
else:
raise NotImplementedError(
expr, "change_index only implemented for Sum and Product")
def reverse_order(expr, *indices):
"""
Reverse the order of a limit in a Sum or 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.concrete.simplification import reverse_order
>>> from sympy import Sum
>>> from sympy.abc import x, y, a, b, c, d
>>> reverse_order(Sum(x, (x, 0, 3)), x)
Sum(-x, (x, 4, -1))
>>> reverse_order(Sum(x*y, (x, 1, 5), (y, 0, 6)), x, y)
Sum(x*y, (x, 6, 0), (y, 7, -1))
>>> reverse_order(Sum(x, (x, a, b)), x)
Sum(-x, (x, b + 1, a - 1))
>>> reverse_order(Sum(x, (x, a, b)), 0)
Sum(-x, (x, b + 1, a - 1))
>>> from sympy import Product, simplify, RisingFactorial, gamma
>>> P = Product(x, (x, a, b))
>>> Pr = reverse_order(P, 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**2, (x, a, b), (x, c, d))
>>> S
Sum(x**2, (x, a, b), (x, c, d))
>>> S0 = reverse_order(S, 0)
>>> S0
Sum(-x**2, (x, b + 1, a - 1), (x, c, d))
>>> S1 = reverse_order(S0, 1)
>>> S1
Sum(x**2, (x, b + 1, a - 1), (x, d + 1, c - 1))
Of course we can mix both notations:
>>> reverse_order(Sum(x*y, (x, a, b), (y, 2, 5)), x, 1)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
>>> reverse_order(Sum(x*y, (x, a, b), (y, 2, 5)), y, x)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
See Also
========
sympy.concrete.simplification.index,
sympy.concrete.simplification.change_index,
sympy.concrete.simplification.reorder_limit,
sympy.concrete.simplification.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] = index(expr, indx)
if isinstance(expr, Sum) or isinstance(expr, Product):
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)
if isinstance(expr, Sum):
return Sum(e * expr.function, *limits)
elif isinstance(expr, Product):
return Product(expr.function**e, *limits)
else:
return expr
def reorder_limit(expr, x, y):
"""
Interchange two limit tuples of a Sum or Product expression.
Usage
=====
``reorder_limit(expr, x, y)`` interchanges two limit tuples. The
arguments ``x`` and ``y`` are integers corresponding to the index
variables of the two limits which are to be interchanged. The
expression ``expr`` has to be either a Sum or a Product.
Examples
========
>>> from sympy.concrete.simplification import reorder_limit
>>> from sympy.abc import x, y, z, a, b, c, d, e, f
>>> from sympy import Sum, Product
>>> reorder_limit(Sum(x*y*z, (x, a, b), (y, c, d), (z, e, f)), 0, 2)
Sum(x*y*z, (z, e, f), (y, c, d), (x, a, b))
>>> reorder_limit(Sum(x**2, (x, a, b), (x, c, d)), 1, 0)
Sum(x**2, (x, c, d), (x, a, b))
>>> reorder_limit(Product(x*y*z, (x, a, b), (y, c, d), (z, e, f)), 0, 2)
Product(x*y*z, (z, e, f), (y, c, d), (x, a, b))
See Also
========
sympy.concrete.simplification.index,
sympy.concrete.simplification.change_index,
sympy.concrete.simplification.reorder,
sympy.concrete.simplification.reverse_order
"""
var = set([limit[0] for limit in expr.limits])
limit_x = expr.limits[x]
limit_y = expr.limits[y]
if (len(set(limit_x[1].free_symbols).intersection(var)) == 0
and len(set(limit_x[2].free_symbols).intersection(var)) == 0
and len(set(limit_y[1].free_symbols).intersection(var)) == 0
and len(set(limit_y[2].free_symbols).intersection(var)) == 0):
limits = []
for i, limit in enumerate(expr.limits):
if i == x:
limits.append(limit_y)
elif i == y:
limits.append(limit_x)
else:
limits.append(limit)
if isinstance(expr, Sum):
return Sum(expr.function, *limits)
elif isinstance(expr, Product):
return Product(expr.function, *limits)
else:
raise NotImplementedError(
expr, "reorder only implemented for Sum and Product")
else:
raise ReorderError(expr,
"could not interchange the two limits specified")
def test_trace_rewrite():
assert trace(A).rewrite(Sum) == Sum(A[i, i], (i, 0, n - 1))
assert trace(eye(3)).rewrite(Sum) == 3
def test_deltasummation():
ds = deltasummation
assert ds(x, (j, 1, 0)) == 0
assert ds(x, (j, 1, 3)) == 3*x
assert ds(x + y, (j, 1, 3)) == 3*(x + y)
assert ds(x*y, (j, 1, 3)) == 3*x*y
assert ds(KD(i, j), (k, 1, 3)) == 3*KD(i, j)
assert ds(x*KD(i, j), (k, 1, 3)) == 3*x*KD(i, j)
assert ds(x*y*KD(i, j), (k, 1, 3)) == 3*x*y*KD(i, j)
n = symbols('n', integer=True, nonzero=True)
assert ds(KD(n, 0), (n, 1, 3)) == 0
# return unevaluated, until it gets implemented
assert ds(KD(i**2, j**2), (j, -oo, oo)) == \
Sum(KD(i**2, j**2), (j, -oo, oo))
assert Piecewise((KD(i, k), And(S(1) <= i, i <= 3)), (0, True)) == \
ds(KD(i, j)*KD(j, k), (j, 1, 3)) == \
ds(KD(j, k)*KD(i, j), (j, 1, 3))
assert ds(KD(i, k), (k, -oo, oo)) == 1
assert ds(KD(i, k), (k, 0, oo)) == Piecewise((1, i >= 0), (0, True))
assert ds(KD(i, k), (k, 1, 3)) == \
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True))
assert ds(k*KD(i, j)*KD(j, k), (k, -oo, oo)) == j*KD(i, j)
assert ds(j*KD(i, j), (j, -oo, oo)) == i
assert ds(i*KD(i, j), (i, -oo, oo)) == j
assert ds(x, (i, 1, 3)) == 3*x
assert ds((i + j)*KD(i, j), (j, -oo, oo)) == 2*i
assert ds(KD(i, j), (j, 1, 3)) == \
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True))
assert ds(KD(i, j), (j, 1, 1)) == Piecewise((1, Eq(i, 1)), (0, True))
assert ds(KD(i, j), (j, 2, 2)) == Piecewise((1, Eq(i, 2)), (0, True))
assert ds(KD(i, j), (j, 3, 3)) == Piecewise((1, Eq(i, 3)), (0, True))
assert ds(KD(i, j), (j, 1, k)) == \
Piecewise((1, And(S(1) <= i, i <= k)), (0, True))
assert ds(KD(i, j), (j, k, 3)) == \
Piecewise((1, And(k <= i, i <= 3)), (0, True))
assert ds(KD(i, j), (j, k, l)) == \
Piecewise((1, And(k <= i, i <= l)), (0, True))
assert ds(x*KD(i, j), (j, 1, 3)) == \
Piecewise((x, And(S(1) <= i, i <= 3)), (0, True))
assert ds(x*KD(i, j), (j, 1, 1)) == Piecewise((x, Eq(i, 1)), (0, True))
assert ds(x*KD(i, j), (j, 2, 2)) == Piecewise((x, Eq(i, 2)), (0, True))
assert ds(x*KD(i, j), (j, 3, 3)) == Piecewise((x, Eq(i, 3)), (0, True))
assert ds(x*KD(i, j), (j, 1, k)) == \
Piecewise((x, And(S(1) <= i, i <= k)), (0, True))
assert ds(x*KD(i, j), (j, k, 3)) == \
Piecewise((x, And(k <= i, i <= 3)), (0, True))
assert ds(x*KD(i, j), (j, k, l)) == \
Piecewise((x, And(k <= i, i <= l)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, 3)) == \
Piecewise((x + y, And(S(1) <= i, i <= 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, 1)) == \
Piecewise((x + y, Eq(i, 1)), (0, True))
assert ds((x + y)*KD(i, j), (j, 2, 2)) == \
Piecewise((x + y, Eq(i, 2)), (0, True))
assert ds((x + y)*KD(i, j), (j, 3, 3)) == \
Piecewise((x + y, Eq(i, 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, k)) == \
Piecewise((x + y, And(S(1) <= i, i <= k)), (0, True))
assert ds((x + y)*KD(i, j), (j, k, 3)) == \
Piecewise((x + y, And(k <= i, i <= 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, k, l)) == \
Piecewise((x + y, And(k <= i, i <= l)), (0, True))
assert ds(KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
Piecewise((1, Eq(i, 1)), (0, True)) +
Piecewise((1, Eq(j, 1)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
Piecewise((1, Eq(i, 2)), (0, True)) +
Piecewise((1, Eq(j, 2)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
Piecewise((1, Eq(i, 3)), (0, True)) +
Piecewise((1, Eq(j, 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
Piecewise((1, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= l)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
Piecewise((1, And(l <= i, i <= 3)), (0, True)) +
Piecewise((1, And(l <= j, j <= 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
Piecewise((1, And(l <= i, i <= m)), (0, True)) +
Piecewise((1, And(l <= j, j <= m)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
Piecewise((x, Eq(i, 1)), (0, True)) +
Piecewise((1, Eq(j, 1)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
Piecewise((x, Eq(i, 2)), (0, True)) +
Piecewise((1, Eq(j, 2)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
Piecewise((x, Eq(i, 3)), (0, True)) +
Piecewise((1, Eq(j, 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= l)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
Piecewise((1, And(l <= j, j <= 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= m)), (0, True)) +
Piecewise((1, And(l <= j, j <= m)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((x, And(S(1) <= j, j <= 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x, Eq(i, 1)), (0, True)) +
Piecewise((x, Eq(j, 1)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x, Eq(i, 2)), (0, True)) +
Piecewise((x, Eq(j, 2)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x, Eq(i, 3)), (0, True)) +
Piecewise((x, Eq(j, 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((x, And(S(1) <= j, j <= l)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x, And(l <= j, j <= 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= m)), (0, True)) +
Piecewise((x, And(l <= j, j <= m)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x + y, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(S(1) <= j, j <= 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x + y, Eq(i, 1)), (0, True)) +
Piecewise((x + y, Eq(j, 1)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x + y, Eq(i, 2)), (0, True)) +
Piecewise((x + y, Eq(j, 2)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x + y, Eq(i, 3)), (0, True)) +
Piecewise((x + y, Eq(j, 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x + y, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((x + y, And(S(1) <= j, j <= l)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= m)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= m)), (0, True)))
assert ds(x*y + x*KD(i, j), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(S(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*y + x*KD(i, j), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 1, k)) == \
Piecewise((k*x*y + x, And(S(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*y + x*KD(i, j), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*y + x*KD(i, j), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(S(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + x, And(S(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + KD(i, j)), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*(y + KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + 2*x, And(S(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, 1)) == \
Piecewise((x*y + 2*x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 2, 2)) == \
Piecewise((x*y + 2*x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 3, 3)) == \
Piecewise((x*y + 2*x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + 2*x, And(S(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, k, 3)) == Piecewise(
((4 - k)*x*y + 2*x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + 2*x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, 3)) == Piecewise(
(3*(x + y)*y + x + y, And(S(1) <= i, i <= 3)), (3*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, 1)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 1)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 2, 2)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 2)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 3, 3)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 3)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, k)) == Piecewise(
(k*(x + y)*y + x + y, And(S(1) <= i, i <= k)), (k*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, k, 3)) == Piecewise(
((4 - k)*(x + y)*y + x + y, And(k <= i, i <= 3)),
((4 - k)*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*(x + y)*y + x + y, And(k <= i, i <= l)),
((l - k + 1)*(x + y)*y, True))
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(S(1) <= i, i <= 3)), (0, True)) +
3*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 1)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 2)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 3)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(S(1) <= i, i <= k)), (0, True)) +
k*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(k <= i, i <= 3)), (0, True)) +
(4 - k)*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(k <= i, i <= l)), (0, True)) +
(l - k + 1)*(KD(i, k) + x)*y)
def definition(self):
from sympy.concrete.summations import Sum
x = self.arg
return exp(x) / Sum(exp(x))