def test_functional_diffgeom_ch3():
x0, y0 = symbols('x0, y0', extended_real=True)
x, y, t = symbols('x, y, t', extended_real=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
s_field = f(R2.x, R2.y)
v_field = b1(R2.x) * R2.e_x + b2(R2.y) * R2.e_y
assert v_field.rcall(s_field).rcall(p_r).doit() == b1(x0) * Derivative(
f(x0, y0), x0) + b2(y0) * Derivative(f(x0, y0), y0)
assert R2.e_x(R2.r**2).rcall(p_r) == 2 * x0
v = R2.e_x + 2 * R2.e_y
s = R2.r**2 + 3 * R2.x
assert v.rcall(s).rcall(p_r).doit() == 2 * x0 + 4 * y0 + 3
circ = -R2.y * R2.e_x + R2.x * R2.e_y
series = intcurve_series(circ, t, R2_r.point([1, 0]), coeffs=True)
series_x, series_y = zip(*series)
assert all(term == cos(t).taylor_term(i, t)
for i, term in enumerate(series_x))
assert all(term == sin(t).taylor_term(i, t)
for i, term in enumerate(series_y))
def test_functional_diffgeom_ch3():
x0, y0 = symbols('x0, y0', extended_real=True)
x, y, t = symbols('x, y, t', extended_real=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
s_field = f(R2.x, R2.y)
v_field = b1(R2.x)*R2.e_x + b2(R2.y)*R2.e_y
assert v_field.rcall(s_field).rcall(p_r).doit() == b1(
x0)*Derivative(f(x0, y0), x0) + b2(y0)*Derivative(f(x0, y0), y0)
assert R2.e_x(R2.r**2).rcall(p_r) == 2*x0
v = R2.e_x + 2*R2.e_y
s = R2.r**2 + 3*R2.x
assert v.rcall(s).rcall(p_r).doit() == 2*x0 + 4*y0 + 3
circ = -R2.y*R2.e_x + R2.x*R2.e_y
series = intcurve_series(circ, t, R2_r.point([1, 0]), coeffs=True)
series_x, series_y = zip(*series)
assert all(term == cos(t).taylor_term(i, t)
for i, term in enumerate(series_x))
assert all(term == sin(t).taylor_term(i, t)
for i, term in enumerate(series_y))
def test_Indexed_subs():
i, j, k = symbols('i j k', integer=True)
a, b = symbols('a b')
A = IndexedBase(a)
B = IndexedBase(b)
assert A[i, j] == B[i, j].subs(b, a)
assert A[i, j] == A[i, k].subs(k, j)
def test_functional_diffgeom_ch6():
u0, u1, u2, v0, v1, v2, w0, w1, w2 = symbols('u0:3, v0:3, w0:3', extended_real=True)
u = u0*R2.e_x + u1*R2.e_y
v = v0*R2.e_x + v1*R2.e_y
wp = WedgeProduct(R2.dx, R2.dy)
assert wp(u, v) == u0*v1 - u1*v0
u = u0*R3_r.e_x + u1*R3_r.e_y + u2*R3_r.e_z
v = v0*R3_r.e_x + v1*R3_r.e_y + v2*R3_r.e_z
w = w0*R3_r.e_x + w1*R3_r.e_y + w2*R3_r.e_z
wp = WedgeProduct(R3_r.dx, R3_r.dy, R3_r.dz)
assert wp(
u, v, w) == Matrix(3, 3, [u0, u1, u2, v0, v1, v2, w0, w1, w2]).det()
a, b, c = symbols('a, b, c', cls=Function)
a_f = a(R3_r.x, R3_r.y, R3_r.z)
b_f = b(R3_r.x, R3_r.y, R3_r.z)
c_f = c(R3_r.x, R3_r.y, R3_r.z)
theta = a_f*R3_r.dx + b_f*R3_r.dy + c_f*R3_r.dz
dtheta = Differential(theta)
da = Differential(a_f)
db = Differential(b_f)
dc = Differential(c_f)
expr = dtheta - WedgeProduct(
da, R3_r.dx) - WedgeProduct(db, R3_r.dy) - WedgeProduct(dc, R3_r.dz)
assert expr.rcall(R3_r.e_x, R3_r.e_y) == 0
def test_functional_diffgeom_ch6():
u0, u1, u2, v0, v1, v2, w0, w1, w2 = symbols('u0:3, v0:3, w0:3',
extended_real=True)
u = u0 * R2.e_x + u1 * R2.e_y
v = v0 * R2.e_x + v1 * R2.e_y
wp = WedgeProduct(R2.dx, R2.dy)
assert wp(u, v) == u0 * v1 - u1 * v0
u = u0 * R3_r.e_x + u1 * R3_r.e_y + u2 * R3_r.e_z
v = v0 * R3_r.e_x + v1 * R3_r.e_y + v2 * R3_r.e_z
w = w0 * R3_r.e_x + w1 * R3_r.e_y + w2 * R3_r.e_z
wp = WedgeProduct(R3_r.dx, R3_r.dy, R3_r.dz)
assert wp(u, v, w) == Matrix(3, 3,
[u0, u1, u2, v0, v1, v2, w0, w1, w2]).det()
a, b, c = symbols('a, b, c', cls=Function)
a_f = a(R3_r.x, R3_r.y, R3_r.z)
b_f = b(R3_r.x, R3_r.y, R3_r.z)
c_f = c(R3_r.x, R3_r.y, R3_r.z)
theta = a_f * R3_r.dx + b_f * R3_r.dy + c_f * R3_r.dz
dtheta = Differential(theta)
da = Differential(a_f)
db = Differential(b_f)
dc = Differential(c_f)
expr = dtheta - WedgeProduct(da, R3_r.dx) - WedgeProduct(
db, R3_r.dy) - WedgeProduct(dc, R3_r.dz)
assert expr.rcall(R3_r.e_x, R3_r.e_y) == 0
def test_functional_diffgeom_ch2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', extended_real=True)
x, y = symbols('x, y', extended_real=True)
f = Function('f')
assert (R2_p.point_to_coords(R2_r.point([x0, y0])) == Matrix(
[sqrt(x0**2 + y0**2), atan2(y0, x0)]))
assert (R2_r.point_to_coords(R2_p.point([r0, theta0])) == Matrix(
[r0 * cos(theta0), r0 * sin(theta0)]))
assert R2_p.jacobian(R2_r, [r0, theta0]) == Matrix(
[[cos(theta0), -r0 * sin(theta0)], [sin(theta0), r0 * cos(theta0)]])
field = f(R2.x, R2.y)
p1_in_rect = R2_r.point([x0, y0])
p1_in_polar = R2_p.point([sqrt(x0**2 + y0**2), atan2(y0, x0)])
assert field.rcall(p1_in_rect) == f(x0, y0)
assert field.rcall(p1_in_polar) == f(x0, y0)
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
assert R2.x(p_r) == x0
assert R2.x(p_p) == r0 * cos(theta0)
assert R2.r(p_p) == r0
assert R2.r(p_r) == sqrt(x0**2 + y0**2)
assert R2.theta(p_r) == atan2(y0, x0)
h = R2.x * R2.r**2 + R2.y**3
assert h.rcall(p_r) == x0 * (x0**2 + y0**2) + y0**3
assert h.rcall(p_p) == r0**3 * sin(theta0)**3 + r0**3 * cos(theta0)
def test_functional_diffgeom_ch2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', extended_real=True)
x, y = symbols('x, y', extended_real=True)
f = Function('f')
assert (R2_p.point_to_coords(R2_r.point([x0, y0])) ==
Matrix([sqrt(x0**2 + y0**2), atan2(y0, x0)]))
assert (R2_r.point_to_coords(R2_p.point([r0, theta0])) ==
Matrix([r0*cos(theta0), r0*sin(theta0)]))
assert R2_p.jacobian(R2_r, [r0, theta0]) == Matrix(
[[cos(theta0), -r0*sin(theta0)], [sin(theta0), r0*cos(theta0)]])
field = f(R2.x, R2.y)
p1_in_rect = R2_r.point([x0, y0])
p1_in_polar = R2_p.point([sqrt(x0**2 + y0**2), atan2(y0, x0)])
assert field.rcall(p1_in_rect) == f(x0, y0)
assert field.rcall(p1_in_polar) == f(x0, y0)
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
assert R2.x(p_r) == x0
assert R2.x(p_p) == r0*cos(theta0)
assert R2.r(p_p) == r0
assert R2.r(p_r) == sqrt(x0**2 + y0**2)
assert R2.theta(p_r) == atan2(y0, x0)
h = R2.x*R2.r**2 + R2.y**3
assert h.rcall(p_r) == x0*(x0**2 + y0**2) + y0**3
assert h.rcall(p_p) == r0**3*sin(theta0)**3 + r0**3*cos(theta0)
def test_Assignment():
x, y = symbols("x, y")
A = MatrixSymbol('A', 3, 1)
mat = Matrix([1, 2, 3])
B = IndexedBase('B')
n = symbols("n", integer=True)
i = Idx("i", n)
# Here we just do things to show they don't error
Assignment(x, y)
Assignment(x, 0)
Assignment(A, mat)
Assignment(A[1, 0], 0)
Assignment(A[1, 0], x)
Assignment(B[i], x)
Assignment(B[i], 0)
# Here we test things to show that they error
# Matrix to scalar
pytest.raises(ValueError, lambda: Assignment(B[i], A))
pytest.raises(ValueError, lambda: Assignment(B[i], mat))
pytest.raises(ValueError, lambda: Assignment(x, mat))
pytest.raises(ValueError, lambda: Assignment(x, A))
pytest.raises(ValueError, lambda: Assignment(A[1, 0], mat))
# Scalar to matrix
pytest.raises(ValueError, lambda: Assignment(A, x))
pytest.raises(ValueError, lambda: Assignment(A, 0))
# Non-atomic lhs
pytest.raises(TypeError, lambda: Assignment(mat, A))
pytest.raises(TypeError, lambda: Assignment(0, x))
pytest.raises(TypeError, lambda: Assignment(x * x, 1))
pytest.raises(TypeError, lambda: Assignment(A + A, mat))
pytest.raises(TypeError, lambda: Assignment(B, 0))
def test_Assignment():
x, y = symbols("x, y")
A = MatrixSymbol('A', 3, 1)
mat = Matrix([1, 2, 3])
B = IndexedBase('B')
n = symbols("n", integer=True)
i = Idx("i", n)
# Here we just do things to show they don't error
Assignment(x, y)
Assignment(x, 0)
Assignment(A, mat)
Assignment(A[1, 0], 0)
Assignment(A[1, 0], x)
Assignment(B[i], x)
Assignment(B[i], 0)
# Here we test things to show that they error
# Matrix to scalar
pytest.raises(ValueError, lambda: Assignment(B[i], A))
pytest.raises(ValueError, lambda: Assignment(B[i], mat))
pytest.raises(ValueError, lambda: Assignment(x, mat))
pytest.raises(ValueError, lambda: Assignment(x, A))
pytest.raises(ValueError, lambda: Assignment(A[1, 0], mat))
# Scalar to matrix
pytest.raises(ValueError, lambda: Assignment(A, x))
pytest.raises(ValueError, lambda: Assignment(A, 0))
# Non-atomic lhs
pytest.raises(TypeError, lambda: Assignment(mat, A))
pytest.raises(TypeError, lambda: Assignment(0, x))
pytest.raises(TypeError, lambda: Assignment(x*x, 1))
pytest.raises(TypeError, lambda: Assignment(A + A, mat))
pytest.raises(TypeError, lambda: Assignment(B, 0))
def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == cbrt((1 + I)**(12*I*f))
assert cbrt((1 + I)**(I*(1 + 7*f))).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, Rational(5, 2))) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**Rational(2, 3)
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == Rational(1, 2)
assert sqrt(r**Rational(4, 3)) != r**Rational(2, 3)
assert sqrt((p + I)**Rational(4, 3)) == (p + I)**Rational(2, 3)
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**Rational(5, 2)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def interpolating_poly(n, x, X='x', Y='y'):
"""Construct Lagrange interpolating polynomial for ``n`` data points. """
if isinstance(X, str):
X = symbols("%s:%s" % (X, n))
if isinstance(Y, str):
Y = symbols("%s:%s" % (Y, n))
coeffs = []
for i in range(0, n):
numer = []
denom = []
for j in range(0, n):
if i == j:
continue
numer.append(x - X[j])
denom.append(X[i] - X[j])
numer = Mul(*numer)
denom = Mul(*denom)
coeffs.append(numer / denom)
return Add(*[coeff * y for coeff, y in zip(coeffs, Y)])
def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**Rational(1, 3)
assert (((1 + I)**(I*(1 + 7*f)))**Rational(1, 3)).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**(5/Integer(4))
p = symbols('p', positive=True)
assert cbrt(p**2) == p**(2/Integer(3))
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**-S.Half == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == S.Half
assert sqrt(r**(4/Integer(3))) != r**(2/Integer(3))
assert sqrt((p + I)**(4/Integer(3))) == (p + I)**(2/Integer(3))
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_IndexedBase_subs():
i, j, k = symbols('i j k', integer=True)
a, b, c = symbols('a b c')
A = IndexedBase(a)
B = IndexedBase(b)
C = IndexedBase(c)
assert A[i] == B[i].subs(b, a)
assert isinstance(C[1].subs(C, Dict({1: 2})), type(A[1]))
def test_IndexedBase_sugar():
i, j = symbols('i j', integer=True)
a = symbols('a')
A1 = Indexed(a, i, j)
A2 = IndexedBase(a)
assert A1 == A2[i, j]
assert A1 == A2[(i, j)]
assert A1 == A2[[i, j]]
assert A1 == A2[Tuple(i, j)]
assert all(a.is_Integer for a in A2[1, 0].args[1:])
def test_ccode_user_functions():
x = symbols('x', integer=False)
n = symbols('n', integer=True)
custom_functions = {
"ceiling": "ceil",
"Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")],
}
assert ccode(ceiling(x), user_functions=custom_functions) == "ceil(x)"
assert ccode(Abs(x), user_functions=custom_functions) == "fabs(x)"
assert ccode(Abs(n), user_functions=custom_functions) == "abs(n)"
def test_Indexed_shape_precedence():
i, j = symbols('i j', integer=True)
o, p = symbols('o p', integer=True)
n, m = symbols('n m', integer=True)
a = IndexedBase('a', shape=(o, p))
assert a.shape == Tuple(o, p)
assert Indexed(
a, Idx(i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)]
assert Indexed(a, Idx(i, m), Idx(j, n)).shape == Tuple(o, p)
assert Indexed(
a, Idx(i, m), Idx(j)).ranges == [Tuple(0, m - 1), Tuple(None, None)]
assert Indexed(a, Idx(i, m), Idx(j)).shape == Tuple(o, p)
def test_ccode_user_functions():
x = symbols('x', integer=False)
n = symbols('n', integer=True)
custom_functions = {
"ceiling":
"ceil",
"Abs": [(lambda x: not x.is_integer, "fabs"),
(lambda x: x.is_integer, "abs")],
}
assert ccode(ceiling(x), user_functions=custom_functions) == "ceil(x)"
assert ccode(Abs(x), user_functions=custom_functions) == "fabs(x)"
assert ccode(Abs(n), user_functions=custom_functions) == "abs(n)"
def test_Indexed_shape_precedence():
i, j = symbols('i j', integer=True)
o, p = symbols('o p', integer=True)
n, m = symbols('n m', integer=True)
a = IndexedBase('a', shape=(o, p))
assert a.shape == Tuple(o, p)
assert Indexed(
a, Idx(i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)]
assert Indexed(a, Idx(i, m), Idx(j, n)).shape == Tuple(o, p)
assert Indexed(
a, Idx(i, m), Idx(j)).ranges == [Tuple(0, m - 1), Tuple(None, None)]
assert Indexed(a, Idx(i, m), Idx(j)).shape == Tuple(o, p)
def test_Indexed_properties():
i, j = symbols('i j', integer=True)
A = Indexed('A', i, j)
assert A.rank == 2
assert A.indices == (i, j)
assert A.base == IndexedBase('A')
assert A.ranges == [None, None]
pytest.raises(IndexException, lambda: A.shape)
n, m = symbols('n m', integer=True)
assert Indexed('A', Idx(
i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)]
assert Indexed('A', Idx(i, m), Idx(j, n)).shape == Tuple(m, n)
pytest.raises(IndexException, lambda: Indexed("A", Idx(i, m), Idx(j)).shape)
def test_jscode_inline_function():
x = symbols('x')
g = implemented_function('g', Lambda(x, 2*x))
assert jscode(g(x)) == "2*x"
g = implemented_function('g', Lambda(x, 2*x/Catalan))
assert jscode(g(x)) == "var Catalan = %s;\n2*x/Catalan" % Catalan.n()
A = IndexedBase('A')
i = Idx('i', symbols('n', integer=True))
g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x)))
assert jscode(g(A[i]), assign_to=A[i]) == (
"for (var i=0; i<n; i++){\n"
" A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n"
"}"
)
def test_Indexed_properties():
i, j = symbols('i j', integer=True)
A = Indexed('A', i, j)
assert A.rank == 2
assert A.indices == (i, j)
assert A.base == IndexedBase('A')
assert A.ranges == [None, None]
pytest.raises(IndexException, lambda: A.shape)
n, m = symbols('n m', integer=True)
assert Indexed('A', Idx(
i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)]
assert Indexed('A', Idx(i, m), Idx(j, n)).shape == Tuple(m, n)
pytest.raises(IndexException, lambda: Indexed("A", Idx(i, m), Idx(j)).shape)
def test_Idx_construction():
i, a, b = symbols('i a b', integer=True)
assert Idx(i) != Idx(i, 1)
assert Idx(i, a) == Idx(i, (0, a - 1))
assert Idx(i, oo) == Idx(i, (0, oo))
x = symbols('x')
pytest.raises(TypeError, lambda: Idx(x))
pytest.raises(TypeError, lambda: Idx(0.5))
pytest.raises(TypeError, lambda: Idx(i, x))
pytest.raises(TypeError, lambda: Idx(i, 0.5))
pytest.raises(TypeError, lambda: Idx(i, (x, 5)))
pytest.raises(TypeError, lambda: Idx(i, (2, x)))
pytest.raises(TypeError, lambda: Idx(i, (2, 3.5)))
def test_ccode_inline_function():
x = symbols('x')
g = implemented_function('g', Lambda(x, 2*x))
assert ccode(g(x)) == "2*x"
g = implemented_function('g', Lambda(x, 2*x/Catalan))
assert ccode(
g(x)) == "double const Catalan = %s;\n2*x/Catalan" % Catalan.evalf()
A = IndexedBase('A')
i = Idx('i', symbols('n', integer=True))
g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x)))
assert ccode(g(A[i]), assign_to=A[i]) == (
"for (int i=0; i<n; i++){\n"
" A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n"
"}"
)
def test_IndexedBase_subs():
i, j, k = symbols('i j k', integer=True)
A = IndexedBase(a)
B = IndexedBase(b)
C = IndexedBase(c)
assert A[i] == B[i].subs({b: a})
assert isinstance(C[1].subs({C: Dict({1: 2})}), type(A[1]))
def test_Indexed_constructor():
i, j = symbols('i j', integer=True)
A = Indexed('A', i, j)
assert A == Indexed(Symbol('A'), i, j)
assert A == Indexed(IndexedBase('A'), i, j)
pytest.raises(TypeError, lambda: Indexed(A, i, j))
pytest.raises(IndexException, lambda: Indexed("A"))
def test_R2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', extended_real=True)
point_r = R2_r.point([x0, y0])
point_p = R2_p.point([r0, theta0])
# r**2 = x**2 + y**2
assert (R2.r**2 - R2.x**2 - R2.y**2).rcall(point_r) == 0
assert trigsimp((R2.r**2 - R2.x**2 - R2.y**2).rcall(point_p)) == 0
assert trigsimp(R2.e_r(R2.x**2 + R2.y**2).rcall(point_p).doit()) == 2 * r0
# polar->rect->polar == Id
a, b = symbols('a b', positive=True)
m = Matrix([[a], [b]])
# TODO assert m == R2_r.coord_tuple_transform_to(R2_p, R2_p.coord_tuple_transform_to(R2_r, [a, b])).applyfunc(simplify)
assert m == R2_p.coord_tuple_transform_to(
R2_r, R2_r.coord_tuple_transform_to(R2_p, m)).applyfunc(simplify)
def test_ccode_loops_multiple_contractions():
from diofant.tensor import IndexedBase, Idx
from diofant import symbols
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = ('for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = y[i] + b[%s]*a[%s];\n' %
(j * o * p + k * p + l, i * n * o * p + j * o * p + k * p + l) +
' }\n'
' }\n'
' }\n'
'}')
c = ccode(b[j, k, l] * a[i, j, k, l], assign_to=y[i])
assert c == s
def test_FracElement___add__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f + g == g + f == (x + y)/(x*y)
z = symbols('z')
pytest.raises(TypeError, lambda: x + z)
assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2*x
F, x, y = field("x,y", ZZ)
assert x + 3 == 3 + x
assert x + QQ(3, 7) == QQ(3, 7) + x == (7*x + 3)/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u*v + x)/(y + u*v)
assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u*v + x)/(y + u*v)
assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
def test_MatrixSymbol():
n, m, t = symbols('n,m,t')
X = MatrixSymbol('X', n, m)
assert X.shape == (n, m)
pytest.raises(TypeError, lambda: MatrixSymbol('X', n, m)
(t)) # issue sympy/sympy#5855
assert X.doit() == X
def test_PolyElement_from_expr():
x, y, z = symbols("x,y,z")
R, X, Y, Z = ring((x, y, z), ZZ)
f = R.convert(1)
assert f == 1 and isinstance(f, R.dtype)
f = R.convert(x)
assert f == X and isinstance(f, R.dtype)
f = R.convert(x*y*z)
assert f == X*Y*Z and isinstance(f, R.dtype)
f = R.convert(x*y*z + x*y + x)
assert f == X*Y*Z + X*Y + X and isinstance(f, R.dtype)
f = R.convert(x**3*y*z + x**2*y**7 + 1)
assert f == X**3*Y*Z + X**2*Y**7 + 1 and isinstance(f, R.dtype)
pytest.raises(CoercionFailed, lambda: R.convert(1/x))
pytest.raises(CoercionFailed, lambda: R.convert(2**x))
pytest.raises(CoercionFailed, lambda: R.convert(7*x + sqrt(2)))
R, X, Y = ring((2**x, y), ZZ)
f = R.convert(2**(2*x) + 1)
assert f == X**2 + 1
def test_ccode_loops_multiple_contractions():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = y[i] + b[%s]*a[%s];\n' % (j*o*p + k*p + l, i*n*o*p + j*o*p + k*p + l) +
' }\n'
' }\n'
' }\n'
'}'
)
c = ccode(b[j, k, l]*a[i, j, k, l], assign_to=y[i])
assert c == s
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
pytest.raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
A / Trace(A) # Make sure this is possible
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(2 * A * B) == 2 * Trace(A * B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
pytest.raises(TypeError, lambda: Trace(S.One))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
def test_FracElement___add__():
F, x, y = field("x,y", QQ)
f, g = 1 / x, 1 / y
assert f + g == g + f == (x + y) / (x * y)
z = symbols('z')
pytest.raises(TypeError, lambda: x + z)
assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2 * x
F, x, y = field("x,y", ZZ)
assert x + 3 == 3 + x
assert x + QQ(3, 7) == QQ(3, 7) + x == (7 * x + 3) / 7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u * v + x) / (y + u * v)
assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u * v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u * v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u * v + x) / (y + u * v)
assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u * v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u * v}
def test_ccode_loops_matrix_vector():
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = x[j]*A[%s] + y[i];\n' % (i*n + j) +
' }\n'
'}'
)
c = ccode(A[i, j]*x[j], assign_to=y[i])
assert c == s
pytest.raises(ValueError, lambda: ccode(A[i, j]*x[j], assign_to=x[j]))
s2 = ('for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int i=0; i<m; i++){\n'
' y[i] = y[i] + A[m*i + i];\n'
' }\n}')
c = ccode(A[i, i], assign_to=y[i])
assert c == s2
def test_ccode_loops_multiple_terms():
from diofant.tensor import IndexedBase, Idx
from diofant import symbols
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
s0 = ('for (int i=0; i<m; i++){\n' ' y[i] = 0;\n' '}\n')
s1 = ('for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' y[i] = b[j]*b[k]*c[%s] + y[i];\n' %
(i * n * o + j * o + k) + ' }\n'
' }\n'
'}\n')
s2 = ('for (int i=0; i<m; i++){\n'
' for (int k=0; k<o; k++){\n'
' y[i] = b[k]*a[%s] + y[i];\n' % (i * o + k) + ' }\n'
'}\n')
s3 = ('for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = b[j]*a[%s] + y[i];\n' % (i * n + j) + ' }\n'
'}\n')
c = ccode(b[j] * a[i, j] + b[k] * a[i, k] + b[j] * b[k] * c[i, j, k],
assign_to=y[i])
assert (c == s0 + s1 + s2 + s3[:-1] or c == s0 + s1 + s3 + s2[:-1]
or c == s0 + s2 + s1 + s3[:-1] or c == s0 + s2 + s3 + s1[:-1]
or c == s0 + s3 + s1 + s2[:-1] or c == s0 + s3 + s2 + s1[:-1])
def test_ccode_loops_addfactor():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = (a[%s] + b[%s])*c[%s] + y[i];\n' % (i*n*o*p + j*o*p + k*p + l, i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +
' }\n'
' }\n'
' }\n'
'}'
)
c = ccode((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i])
assert c == s
def test_FractionField_methods():
F = ZZ.frac_field("x")
assert F.domain_new(2) == ZZ(2)
x = symbols("x")
assert F.field_new(x**2 + x) == F.x**2 + F.x
def test_Mul():
A, B, C, D = symbols('A B C D', commutative=False)
assert mcode(x*y*z) == "x*y*z"
assert mcode(x*y*A) == "x*y*A"
assert mcode(x*y*A*B) == "x*y*A**B"
assert mcode(x*y*A*B*C) == "x*y*A**B**C"
assert mcode(x*A*B*(C + D)*A*y) == "x*y*A**B**(C + D)**A"
def test_FractionField_methods():
F = ZZ.frac_field("x")
assert F.domain_new(2) == ZZ(2)
x = symbols("x")
assert F.field_new(x**2 + x) == F.x**2 + F.x
def test_PolyElement_from_expr():
x, y, z = symbols("x,y,z")
R, X, Y, Z = ring((x, y, z), ZZ)
f = R.from_expr(1)
assert f == 1 and isinstance(f, R.dtype)
f = R.from_expr(x)
assert f == X and isinstance(f, R.dtype)
f = R.from_expr(x * y * z)
assert f == X * Y * Z and isinstance(f, R.dtype)
f = R.from_expr(x * y * z + x * y + x)
assert f == X * Y * Z + X * Y + X and isinstance(f, R.dtype)
f = R.from_expr(x**3 * y * z + x**2 * y**7 + 1)
assert f == X**3 * Y * Z + X**2 * Y**7 + 1 and isinstance(f, R.dtype)
pytest.raises(ValueError, lambda: R.from_expr(1 / x))
pytest.raises(ValueError, lambda: R.from_expr(2**x))
pytest.raises(ValueError, lambda: R.from_expr(7 * x + sqrt(2)))
R, X, Y = ring((2**x, y), ZZ)
f = R.from_expr(2**(2 * x) + 1)
assert f == X**2 + 1
def test_Mul():
A, B, C, D = symbols('A B C D', commutative=False)
assert mcode(x*y*z) == "x*y*z"
assert mcode(x*y*A) == "x*y*A"
assert mcode(x*y*A*B) == "x*y*A**B"
assert mcode(x*y*A*B*C) == "x*y*A**B**C"
assert mcode(x*A*B*(C + D)*A*y) == "x*y*A**B**(C + D)**A"
def test_m_tensor_loops_multiple_contractions():
# see comments in previous test about vectorizing
n, m, o, p = symbols('n m o p', integer=True)
A = IndexedBase('A')
B = IndexedBase('B')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
result, = codegen(('tensorthing', Eq(y[i], B[j, k, l]*A[i, j, k, l])),
"Octave", header=False, empty=False)
source = result[1]
expected = (
'function y = tensorthing(A, B, m, n, o, p)\n'
' for i = 1:m\n'
' y(i) = 0;\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' for k = 1:o\n'
' for l = 1:p\n'
' y(i) = y(i) + B(j, k, l).*A(i, j, k, l);\n'
' end\n'
' end\n'
' end\n'
' end\n'
'end\n'
)
assert source == expected
def test_m_loops():
# Note: an Octave programmer would probably vectorize this across one or
# more dimensions. Also, size(A) would be used rather than passing in m
# and n. Perhaps users would expect us to vectorize automatically here?
# Or is it possible to represent such things using IndexedBase?
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
result, = codegen(('mat_vec_mult', Eq(y[i], A[i, j]*x[j])), "Octave",
header=False, empty=False)
source = result[1]
expected = (
'function y = mat_vec_mult(A, m, n, x)\n'
' for i = 1:m\n'
' y(i) = 0;\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' y(i) = %(rhs)s + y(i);\n'
' end\n'
' end\n'
'end\n'
)
assert (source == expected % {'rhs': 'A(%s, %s).*x(j)' % (i, j)} or
source == expected % {'rhs': 'x(j).*A(%s, %s)' % (i, j)})
def test_IndexedBase_subs():
i, j, k = symbols('i j k', integer=True)
A = IndexedBase(a)
B = IndexedBase(b)
C = IndexedBase(c)
assert A[i] == B[i].subs({b: a})
assert isinstance(C[1].subs({C: Dict({1: 2})}), type(A[1]))
def test_attrs():
a, b = symbols('a, b', cls=Dummy)
f = Hyper_Function([2, a], [b])
assert f.ap == Tuple(2, a)
assert f.bq == Tuple(b)
assert f.args == (Tuple(2, a), Tuple(b))
assert f.sizes == (2, 1)
def test_jscode_loops_addfactor():
from diofant.tensor import IndexedBase, Idx
from diofant import symbols
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (var i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (var i=0; i<m; i++){\n'
' for (var j=0; j<n; j++){\n'
' for (var k=0; k<o; k++){\n'
' for (var l=0; l<p; l++){\n'
' y[i] = (a[%s] + b[%s])*c[%s] + y[i];\n' % (i*n*o*p + j*o*p + k*p + l, i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +
' }\n'
' }\n'
' }\n'
'}'
)
c = jscode((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i])
assert c == s
def test_m_tensor_loops_multiple_contractions():
# see comments in previous test about vectorizing
from diofant.tensor import IndexedBase, Idx
from diofant import symbols
n, m, o, p = symbols('n m o p', integer=True)
A = IndexedBase('A')
B = IndexedBase('B')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
result, = codegen(('tensorthing', Eq(y[i], B[j, k, l] * A[i, j, k, l])),
"Octave",
header=False,
empty=False)
source = result[1]
expected = ('function y = tensorthing(A, B, m, n, o, p)\n'
' for i = 1:m\n'
' y(i) = 0;\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' for k = 1:o\n'
' for l = 1:p\n'
' y(i) = y(i) + B(j, k, l).*A(i, j, k, l);\n'
' end\n'
' end\n'
' end\n'
' end\n'
'end\n')
assert source == expected
def test_R2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', extended_real=True)
point_r = R2_r.point([x0, y0])
point_p = R2_p.point([r0, theta0])
# r**2 = x**2 + y**2
assert (R2.r**2 - R2.x**2 - R2.y**2).rcall(point_r) == 0
assert trigsimp( (R2.r**2 - R2.x**2 - R2.y**2).rcall(point_p) ) == 0
assert trigsimp(R2.e_r(R2.x**2 + R2.y**2).rcall(point_p).doit()) == 2*r0
# polar->rect->polar == Id
a, b = symbols('a b', positive=True)
m = Matrix([[a], [b]])
# TODO assert m == R2_r.coord_tuple_transform_to(R2_p, R2_p.coord_tuple_transform_to(R2_r, [a, b])).applyfunc(simplify)
assert m == R2_p.coord_tuple_transform_to(
R2_r, R2_r.coord_tuple_transform_to(R2_p, m)).applyfunc(simplify)
def test_global_vars_octave():
x, y, z, t = symbols("x y z t")
result = codegen(('f', x * y),
"Octave",
header=False,
empty=False,
global_vars=(y, ))
source = result[0][1]
expected = ("function out1 = f(x)\n"
" global y\n"
" out1 = x.*y;\n"
"end\n")
assert source == expected
result = codegen(('f', x * y + z),
"Octave",
header=False,
empty=False,
argument_sequence=(x, y),
global_vars=(z, t))
source = result[0][1]
expected = ("function out1 = f(x, y)\n"
" global t z\n"
" out1 = x.*y + z;\n"
"end\n")
assert source == expected
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
pytest.raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
A / Trace(A) # Make sure this is possible
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(2*A*B) == 2*Trace(A*B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
pytest.raises(TypeError, lambda: Trace(1))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
def test_Indexed_constructor():
i, j = symbols('i j', integer=True)
A = Indexed('A', i, j)
assert A == Indexed(Symbol('A'), i, j)
assert A == Indexed(IndexedBase('A'), i, j)
pytest.raises(TypeError, lambda: Indexed(A, i, j))
pytest.raises(IndexException, lambda: Indexed("A"))
def test_m_loops():
# Note: an Octave programmer would probably vectorize this across one or
# more dimensions. Also, size(A) would be used rather than passing in m
# and n. Perhaps users would expect us to vectorize automatically here?
# Or is it possible to represent such things using IndexedBase?
from diofant.tensor import IndexedBase, Idx
from diofant import symbols
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
result, = codegen(('mat_vec_mult', Eq(y[i], A[i, j] * x[j])),
"Octave",
header=False,
empty=False)
source = result[1]
expected = ('function y = mat_vec_mult(A, m, n, x)\n'
' for i = 1:m\n'
' y(i) = 0;\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' y(i) = %(rhs)s + y(i);\n'
' end\n'
' end\n'
'end\n')
assert (source == expected % {
'rhs': 'A(%s, %s).*x(j)' % (i, j)
} or source == expected % {
'rhs': 'x(j).*A(%s, %s)' % (i, j)
})
def test_coords():
r, theta = symbols('r, theta')
m = Manifold('M', 2)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
polar.coord_tuple_transform_to(rect, [0, 2]) == Matrix([[0], [0]])
def test_Idx_func_args():
i, a, b = symbols('i a b', integer=True)
ii = Idx(i)
assert ii.func(*ii.args) == ii
ii = Idx(i, a)
assert ii.func(*ii.args) == ii
ii = Idx(i, (a, b))
assert ii.func(*ii.args) == ii
def test_Idx_subs():
i, a, b = symbols('i a b', integer=True)
assert Idx(i, a).subs({a: b}) == Idx(i, b)
assert Idx(i, a).subs({i: b}) == Idx(b, a)
assert Idx(i).subs({i: 2}) == Idx(2)
assert Idx(i, a).subs({a: 2}) == Idx(i, 2)
assert Idx(i, (a, b)).subs({i: 2}) == Idx(2, (a, b))
def test_IndexedBase_sugar():
i, j = symbols('i j', integer=True)
A1 = Indexed(a, i, j)
A2 = IndexedBase(a)
assert A1 == A2[i, j]
assert A1 == A2[(i, j)]
assert A1 == A2[[i, j]]
assert A1 == A2[Tuple(i, j)]
assert all(a.is_Integer for a in A2[1, 0].args[1:])
def test_PolyElement_almosteq():
R, x, y = ring('x, y', RR)
z = symbols('z')
assert x.almosteq(y) is False
assert x.almosteq(1) is False
assert (x + 2*y).almosteq(2) is False
assert (x + 2*y).almosteq(2*x + y) is False
assert R.one.almosteq(2) is False
assert R.one.almosteq(z) is False
def test_minimize_linear():
assert minimize([-2*x - 3*y - 2*z, 2*x + y + z <= 4,
x + 2*y + z <= 7, z <= 5, x >= 0,
y >= 0, z >= 0], x, y, z) == (-11, {x: 0, y: 3, z: 1})
assert minimize([-2*x - 3*y - 2*z, 2*x + y + z <= 4,
x + 2*y + z <= 7, z <= 5, x >= 0,
y >= 0, z >= 0], x, y, z) == (-11, {x: 0, y: 3, z: 1})
assert minimize([-2*x - 3*y - 2*z, 2*x + y + z <= 4,
x + 2*y + z < 7, z <= 5, x >= 0,
y >= 0, z >= 0], x, y, z) is None
assert minimize([-2*x - 3*y - 4*z, 3*x + 2*y + z <= 10,
2*x + 5*y + 3*z <= 15, x >= 0,
y >= 0, z >= 0], x, y, z) == (-20, {x: 0, y: 0, z: 5})
assert maximize([12*x + 40*y, x + y <= 15, x + 3*y <= 36,
x <= 10, x >= 0, y >= 0], x, y) == (480, {x: 0, y: 12})
assert minimize([-2*x - 3*y - 4*z, Eq(3*x + 2*y + z, 10),
Eq(2*x + 5*y + 3*z, 15), x >= 0, y >= 0, z >= 0],
x, y, z) == (Rational(-130, 7),
{x: Rational(15, 7), y: 0,
z: Rational(25, 7)})
assert maximize([2*x + y, y - x <= 1, x - 2*y <= 2,
x >= 0, y >= 0], x, y) == (oo, {x: nan, y: nan})
assert minimize([2*x + 3*y - z, 1 <= x + y + z,
x + y + z <= 2, 1 <= x - y + z,
x - y + z <= 2, Eq(x - y - z, 3)],
x, y, z) == (3, {x: 2, y: Rational(-1, 2), z: Rational(-1, 2)})
x1, x2, x3, x4, x5 = symbols('x1:6')
assert minimize([-2*x1 + 4*x2 + 7*x3 + x4 + 5*x5,
Eq(-x1 + x2 + 2*x3 + x4 + 2*x5, 7),
Eq(-x1 + 2*x2 + 3*x3 + x4 + x5, 6),
Eq(-x1 + x2 + x3 + 2*x4 + x5, 4),
x2 >= 0, x3 >= 0, x4 >= 0, x5 >= 0],
x1, x2, x3, x4, x5) == (19, {x1: -1, x2: 0,
x3: 1, x4: 0, x5: 2})
assert minimize([-x - y, x + 2*y <= 8, 3*x + 2*y <= 12,
x + 3*y >= 6, x >= 0, y >= 0],
x, y) == (-5, {x: 2, y: 3})
pytest.raises(InfeasibleProblem,
lambda: minimize([-x - y, x + 2*y <= 8, 3*x + 2*y <= 12,
x + 3*y >= 13, x >= 0, y >= 0],
x, y))
pytest.raises(InfeasibleProblem,
lambda: minimize([-x - y, x + 2*y <= 8, 3*x + 2*y <= 12,
x + 3*y >= 13, x >= 0, y >= 0], x, y))
assert minimize([-x - y, 2*x + y >= 4,
Eq(x + 2*y, 6), x >= 0, y >= 0],
x, y) == (-6, {x: 6, y: 0})
assert minimize([6*x + 3*y, x + y >= 1, 2*x - y >= 1,
3*y <= 2, x >= 0,
y >= 0], x, y) == (5, {x: Rational(2, 3),
y: Rational(1, 3)})
def test_check_case_false_positive():
# The upper case/lower case exception should not be triggered by Diofant
# objects that differ only because of assumptions. (It may be useful to
# have a check for that as well, but here we only want to test against
# false positives with respect to case checking.)
x2 = symbols('x', my_assumption=True)
try:
codegen(('test', x*x2), 'f95', 'prefix')
except CodeGenError as e:
if e.args[0].startswith("Fortran ignores case."):
raise AssertionError("This exception should not be raised!")
def test_ccode_functions():
assert ccode(sin(x) ** cos(x)) == "pow(sin(x), cos(x))"
assert ccode(elliptic_e(x)) == ("// Not supported in C:\n"
"// elliptic_e\nelliptic_e(x)")
n = symbols('n', integer=True)
assert ccode(Abs(n)) == '// Not supported in C:\n// Abs\nAbs(n)'
assert ccode(Abs(x)) == 'fabs(x)'
pytest.raises(TypeError, lambda: ccode(sin(x), assign_to=1))
