def test_deltasummation_trivial():
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)
def test_deltasummation_trivial():
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)
def test_deltasummation_mul_x_kd():
assert ds(x*Kd(i, j), (j, 1, 3)) == \
Piecewise((x, And(Integer(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(Integer(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))
def test_deltasummation_mul_x_kd():
assert ds(x*KD(i, j), (j, 1, 3)) == \
Piecewise((x, And(Integer(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(Integer(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))
def test_deltasummation_basic_symbolic():
assert ds(KD(i, j), (j, 1, 3)) == \
Piecewise((1, And(Integer(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(Integer(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))
def test_deltasummation_mul_x_add_y_twokd():
assert ds(x*(y + 2*Kd(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + 2*x, And(Integer(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(Integer(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))
def test_deltasummation_mul_x_add_y_kd():
assert ds(x*(y + KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(Integer(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(Integer(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))
def test_deltasummation_mul_add_x_y_add_y_kd():
assert ds((x + y)*(y + Kd(i, j)), (j, 1, 3)) == Piecewise(
(3*(x + y)*y + x + y, And(Integer(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(Integer(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))
def test_deltasummation_mul_add_x_kd_add_y_kd():
assert ds((x + KD(i, k)) * (y + KD(i, j)), (j, 1, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(Integer(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(Integer(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 test_deltasummation_mul_add_x_kd_add_y_kd():
assert ds((x + Kd(i, k))*(y + Kd(i, j)), (j, 1, 3)) == piecewise_fold(
Piecewise((Kd(i, k) + x, And(Integer(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(Integer(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 test_deltasummation_mul_add_x_y_add_kd_kd():
assert ds((x + y)*(Kd(i, k) + Kd(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x + y, And(Integer(1) <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(Integer(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(Integer(1) <= i, i <= l)), (0, True)) +
Piecewise((x + y, And(Integer(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)))
def test_deltasummation_mul_add_x_y_add_kd_kd():
assert ds((x + y) * (KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x + y, And(Integer(1) <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(Integer(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(Integer(1) <= i, i <= l)), (0, True)) +
Piecewise((x + y, And(Integer(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)))
def test_deltasummation_add_mul_x_kd_kd():
assert ds(x * Kd(i, k) + Kd(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(Integer(1) <= i, i <= 3)), (0, True)) +
Piecewise((1, And(Integer(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(Integer(1) <= i, i <= l)), (0, True)) +
Piecewise((1, And(Integer(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)))
def test_deltasummation_basic_symbolic():
assert ds(Kd(exp(i), 0), (i, 1, 3)) == 0
assert ds(Kd(exp(i), 0), (i, -1, 3)) == 0
assert ds(Kd(exp(i), 1), (i, 0, 3)) == 1
assert ds(Kd(exp(i), 1), (i, 1, 3)) == 0
assert ds(Kd(exp(i), 1), (i, -10, 3)) == 1
assert ds(Kd(i, j), (j, 1, 3)) == \
Piecewise((1, And(Integer(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(Integer(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))
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(Integer(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, Integer(0) <= i), (0, True))
assert ds(Kd(i, k), (k, 1, 3)) == \
Piecewise((1, And(Integer(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 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(Integer(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, Integer(0) <= i),
(0, True))
assert ds(KD(i, k), (k, 1, 3)) == \
Piecewise((1, And(Integer(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
