def test_left_mul_by_2(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# matrix independent of s
M2 = sp.Matrix([[1, 0], [-C, 1]])
# 1-forms
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
# vector 1-form
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
t = w.left_mul_by(M2, additional_symbols=[C])
# object to compare with:
t2 = -C * w1 + w2
self.assertEqual(t2.coeff, t.coeff.row(1).T)
def test_left_mul_by_1(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# s-dependent matrix
M1 = sp.Matrix([[1, 0], [-C * s, 1]])
# 1-forms
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
# vector 1-form
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
t = w.left_mul_by(M1, s, [C])
t2 = -C * w1.dot() + w2
self.assertEqual(t2.coeff, t.coeff.row(1).T)
def test_mul(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
W = pc.VectorDifferentialForm(1, xx, coeff=Q)
W1 = s * W
W2 = W * C
self.assertEqual(W1.coeff, nct.nc_mul(s, W.coeff))
self.assertNotEqual(W1.coeff, nct.nc_mul(W.coeff, s))
self.assertEqual(W2.coeff, nct.nc_mul(W.coeff, C))
self.assertNotEqual(W2.coeff, nct.nc_mul(C, W.coeff))
alpha = pc.DifferentialForm(1, xx)
with self.assertRaises(TypeError) as cm:
alpha * W1
with self.assertRaises(TypeError) as cm:
W1 * alpha
M = sp.eye(2)
with self.assertRaises(sp.SympifyError) as cm:
M * W1
with self.assertRaises(TypeError) as cm:
W1 * M
def test_vector_form_append(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
s = sp.Symbol('s', commutative=False)
C = sp.Symbol('C', commutative=False)
# vector 1-form
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
# 1-forms
Q2 = sp.Matrix([[x1, x2, x3]])
Q2_ = st.col_stack(Q2, sp.zeros(1, 6))
w2 = pc.DifferentialForm(1, XX, coeff=Q2_[0, :])
w.append(w2)
# vector form to compare with:
B = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3], [x1, x2, x3]])
B_ = st.col_stack(B, sp.zeros(3, 6))
b = pc.VectorDifferentialForm(1, XX, coeff=B_)
self.assertEqual(w.coeff, b.coeff)
def test_wedge_product1(self):
dx1, dx2, dx3, dx4, dx5 = self.dx
self.assertEqual(dx1 ^ dx2, dx1.wp(dx2))
self.assertEqual(dx5 ^ dx2, -dx2 ^ dx5)
self.assertEqual((dx5 ^ dx2 ^ dx1).degree, 3)
zero4_form = pc.DifferentialForm(4, self.xx)
self.assertEqual((dx5 ^ dx2 ^ dx1 ^ dx5), zero4_form)
self.assertFalse(any((dx4 ^ dx4).coeff))
def test_stack_to_vector_form(self):
x1, x2, x3 = xx = st.symb_vector('x1:4', commutative=False)
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
# 1-forms
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
w_stacked = pc.stack_to_vector_form(w1, w2)
# vector 1-form
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
self.assertEqual(w.coeff, w_stacked.coeff)
def test_vector_k_form(self):
x1, x2, x3 = xx = st.symb_vector('x1:4')
xdot1, xdot2, xdot3 = xxdot = st.time_deriv(xx, xx)
xddot1, xddot2, xddot3 = xxddot = st.time_deriv(xxdot, xxdot)
XX = st.row_stack(xx, xxdot, xxddot)
Q = sp.Matrix([[x3 / sin(x1), 1, 0], [-tan(x1), 0, x3]])
Q_ = st.col_stack(Q, sp.zeros(2, 6))
w1 = pc.DifferentialForm(1, XX, coeff=Q_[0, :])
w2 = pc.DifferentialForm(1, XX, coeff=Q_[1, :])
w = pc.VectorDifferentialForm(1, XX, coeff=Q_)
w1_tilde = w.get_differential_form(0)
self.assertEqual(w1.coeff, w1_tilde.coeff)
w2_tilde = w.get_differential_form(1)
self.assertEqual(w2.coeff, w2_tilde.coeff)
def test_wedge_product2(self):
# here we test the `*` operator which was extended to DifferentialForms
# after `^`
x1, x2, x3, x4, x5 = self.xx
dx1, dx2, dx3, dx4, dx5 = self.dx
self.assertEqual(dx1 * dx2, dx1.wp(dx2))
self.assertEqual(dx5 * dx2, -dx2 ^ dx5)
self.assertEqual((dx5 * dx2 * dx1).degree, 3)
# commutativity with scalar functions
self.assertEqual(dx5 * x2 * x3 * 10 * dx2 * dx1 * x1,
x2 * x3 * 10 * x1 * dx5 ^ dx2 ^ dx1)
zero4_form = pc.DifferentialForm(4, self.xx)
self.assertEqual((dx5 * dx2 * dx1 * dx5), zero4_form)
self.assertFalse(any((dx4 * dx4).coeff))
def test_dot(self):
x1, x2, x3 = xx = sp.Matrix(sp.symbols("x1, x2, x3"))
xdot1, xdot2, xdot3 = xxd = pc.st.time_deriv(xx, xx)
xx_tmp, ddx = pc.setup_objects(xx)
dx1, dx2, dx3 = ddx
full_basis = list(xx) + list(xxd)
dxdot2 = pc.DifferentialForm(1, full_basis, [0, 0, 0, 0, 1, 0])
w1 = x3 * dx2
with self.assertRaises(ValueError) as cm:
w1.dot()
w1.jet_extend_basis()
dx2.jet_extend_basis()
wdot1 = w1.dot()
self.assertEqual(wdot1, xdot3 * dx2 + x3 * dxdot2)
def test_calculation_a(self):
xx = st.symb_vector('x1:6')
dx1 = pc.DifferentialForm(1, xx, coeff=[1, 0, 0, 0, 0])
