class CanonicalChange(unittest.TestCase):
def setUp(self):
"""Set up an example from Hill's equations."""
x_2 = Powers((2, 0, 0, 0, 0, 0))
px2 = Powers((0, 2, 0, 0, 0, 0))
y_2 = Powers((0, 0, 2, 0, 0, 0))
py2 = Powers((0, 0, 0, 2, 0, 0))
z_2 = Powers((0, 0, 0, 0, 2, 0))
pz2 = Powers((0, 0, 0, 0, 0, 2))
xpy = Powers((1, 0, 0, 1, 0, 0))
ypx = Powers((0, 1, 1, 0, 0, 0))
terms = {px2: 0.5, py2: 0.5, pz2: 0.5,
xpy: -1.0, ypx: 1.0,
x_2: -4.0, y_2: 2.0, z_2: 2.0}
assert len(terms) == 8
self.h_2 = Polynomial(6, terms=terms)
self.lie = LieAlgebra(3)
self.diag = Diagonalizer(self.lie)
self.eq_type = 'scc'
e = []
e.append(+sqrt(2.0*sqrt(7.0)+1.0))
e.append(-e[-1])
e.append(complex(0.0, +sqrt(2.0*sqrt(7.0)-1.0)))
e.append(-e[-1])
e.append(complex(0.0, +2.0))
e.append(-e[-1])
self.eig_vals = e
def test_basic_matrix(self):
"""This test is rather monolithic, but at least it implements
a concrete example that we can compare with our earlier
computations. It also tests the mutual-inverse character of
the equi-to-diag and diag-to-equi transformations."""
tolerance = 5.0e-15
eig = self.diag.compute_eigen_system(self.h_2, tolerance)
self.diag.compute_diagonal_change()
eq_type = eig.get_equilibrium_type()
self.assertEquals(eq_type, self.eq_type)
eigs = [pair.val for pair in eig.get_raw_eigen_value_vector_pairs()]
for actual, expected in zip(eigs, self.eig_vals):
self.assert_(abs(actual-expected) < tolerance, (actual, expected))
mat = self.diag.get_matrix_diag_to_equi()
assert self.diag.matrix_is_symplectic(mat)
sub_diag_into_equi = self.diag.matrix_as_vector_of_row_polynomials(mat)
mat_inv = LinearAlgebra.inverse(MLab.array(mat))
sub_equi_into_diag = self.diag.matrix_as_vector_of_row_polynomials(mat_inv)
h_diag_2 = self.h_2.substitute(sub_diag_into_equi)
h_2_inv = h_diag_2.substitute(sub_equi_into_diag)
self.assert_(h_2_inv) #non-zero
self.assert_(not h_2_inv.is_constant())
self.assert_(self.lie.is_isograde(h_2_inv, 2))
self.assert_((self.h_2-h_2_inv).l1_norm() < 1.0e-14)
comp = Complexifier(self.diag.get_lie_algebra(), eq_type)
sub_complex_into_real = comp.calc_sub_complex_into_real()
h_comp_2 = h_diag_2.substitute(sub_complex_into_real)
h_comp_2 = h_comp_2.with_small_coeffs_removed(tolerance)
self.assert_(self.lie.is_diagonal_polynomial(h_comp_2))
class CanonicalChange(unittest.TestCase):
def setUp(self):
"""Set up an example from Hill's equations."""
x_2 = Powers((2, 0, 0, 0, 0, 0))
px2 = Powers((0, 2, 0, 0, 0, 0))
y_2 = Powers((0, 0, 2, 0, 0, 0))
py2 = Powers((0, 0, 0, 2, 0, 0))
z_2 = Powers((0, 0, 0, 0, 2, 0))
pz2 = Powers((0, 0, 0, 0, 0, 2))
xpy = Powers((1, 0, 0, 1, 0, 0))
ypx = Powers((0, 1, 1, 0, 0, 0))
terms = {
px2: 0.5,
py2: 0.5,
pz2: 0.5,
xpy: -1.0,
ypx: 1.0,
x_2: -4.0,
y_2: 2.0,
z_2: 2.0
}
assert len(terms) == 8
self.h_2 = Polynomial(6, terms=terms)
self.lie = LieAlgebra(3)
self.diag = Diagonalizer(self.lie)
self.eq_type = 'scc'
e = []
e.append(+sqrt(2.0 * sqrt(7.0) + 1.0))
e.append(-e[-1])
e.append(complex(0.0, +sqrt(2.0 * sqrt(7.0) - 1.0)))
e.append(-e[-1])
e.append(complex(0.0, +2.0))
e.append(-e[-1])
self.eig_vals = e
def test_basic_matrix(self):
"""This test is rather monolithic, but at least it implements
a concrete example that we can compare with our earlier
computations. It also tests the mutual-inverse character of
the equi-to-diag and diag-to-equi transformations."""
tolerance = 5.0e-15
eig = self.diag.compute_eigen_system(self.h_2, tolerance)
self.diag.compute_diagonal_change()
eq_type = eig.get_equilibrium_type()
self.assertEquals(eq_type, self.eq_type)
eigs = [pair.val for pair in eig.get_raw_eigen_value_vector_pairs()]
for actual, expected in zip(eigs, self.eig_vals):
self.assert_(
abs(actual - expected) < tolerance, (actual, expected))
mat = self.diag.get_matrix_diag_to_equi()
assert self.diag.matrix_is_symplectic(mat)
sub_diag_into_equi = self.diag.matrix_as_vector_of_row_polynomials(mat)
mat_inv = LinearAlgebra.inverse(MLab.array(mat))
sub_equi_into_diag = self.diag.matrix_as_vector_of_row_polynomials(
mat_inv)
h_diag_2 = self.h_2.substitute(sub_diag_into_equi)
h_2_inv = h_diag_2.substitute(sub_equi_into_diag)
self.assert_(h_2_inv) #non-zero
self.assert_(not h_2_inv.is_constant())
self.assert_(self.lie.is_isograde(h_2_inv, 2))
self.assert_((self.h_2 - h_2_inv).l1_norm() < 1.0e-14)
comp = Complexifier(self.diag.get_lie_algebra(), eq_type)
sub_complex_into_real = comp.calc_sub_complex_into_real()
h_comp_2 = h_diag_2.substitute(sub_complex_into_real)
h_comp_2 = h_comp_2.with_small_coeffs_removed(tolerance)
self.assert_(self.lie.is_diagonal_polynomial(h_comp_2))
