def test_cgf_overlap(self):
"""
Test kinetic integrals for contracted Gaussians
Sij = <cgf_i | cgf_j>
"""
# construct integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule("H2")
mol.add_atom('H', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.0, 0.0, 1.4)
cgfs, nuclei = mol.build_basis('sto3g')
S = np.zeros((2, 2))
S[0, 0] = integrator.overlap(cgfs[0], cgfs[0])
S[0, 1] = S[1, 0] = integrator.overlap(cgfs[0], cgfs[1])
S[1, 1] = integrator.overlap(cgfs[1], cgfs[1])
S11 = 1.0
S12 = 0.65931845
np.testing.assert_almost_equal(S[0, 0], S11, 4)
np.testing.assert_almost_equal(S[1, 1], S11, 4)
np.testing.assert_almost_equal(S[0, 1], S12, 4)
def test_cgf_kinetic(self):
"""
Test kinetic integrals for contracted Gaussians
Tij = <cgf_i | -1/2 nabla^{2} | cgf_j>
"""
# construct integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule("H2")
mol.add_atom('H', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.0, 0.0, 1.4)
cgfs, nuclei = mol.build_basis('sto3g')
T = np.zeros((2, 2))
T[0, 0] = integrator.kinetic(cgfs[0], cgfs[0])
T[0, 1] = T[1, 0] = integrator.kinetic(cgfs[0], cgfs[1])
T[1, 1] = integrator.kinetic(cgfs[1], cgfs[1])
T11 = 0.7600315809249878
T12 = 0.2364544570446014
np.testing.assert_almost_equal(T[0, 0], T11, 4)
np.testing.assert_almost_equal(T[1, 1], T11, 4)
np.testing.assert_almost_equal(T[0, 1], T12, 4)
def test_cgf_nuclear(self):
"""
Test nuclear attraction integrals for contracted Gaussians
V^{(c)}_ij = <cgf_i | -Zc / |r-Rc| | cgf_j>
"""
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule("H2")
mol.add_atom('H', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.0, 0.0, 1.4)
cgfs, nuclei = mol.build_basis('sto3g')
V1 = np.zeros((2,2))
V1[0,0] = integrator.nuclear(cgfs[0], cgfs[0], cgfs[0].p, 1)
V1[0,1] = V1[1,0] = integrator.nuclear(cgfs[0], cgfs[1], cgfs[0].p, 1)
V1[1,1] = integrator.nuclear(cgfs[1], cgfs[1], cgfs[0].p, 1)
V2 = np.zeros((2,2))
V2[0,0] = integrator.nuclear(cgfs[0], cgfs[0], cgfs[1].p, 1)
V2[0,1] = V2[1,0] = integrator.nuclear(cgfs[0], cgfs[1], cgfs[1].p, 1)
V2[1,1] = integrator.nuclear(cgfs[1], cgfs[1], cgfs[1].p, 1)
V11 = -1.2266135215759277
V12 = -0.5974172949790955
V22 = -0.6538270711898804
np.testing.assert_almost_equal(V1[0,0], V11, 4)
np.testing.assert_almost_equal(V1[1,1], V22, 4)
np.testing.assert_almost_equal(V1[0,1], V12, 4)
np.testing.assert_almost_equal(V2[0,0], V22, 4)
np.testing.assert_almost_equal(V2[1,1], V11, 4)
np.testing.assert_almost_equal(V2[0,1], V12, 4)
def testFunctionsCGF(self):
"""
Test getting amplitude from CGF
"""
# construct integrator object
integrator = PyQInt()
# get compile info
compile_info = integrator.get_compile_info()
# build hydrogen molecule
mol = Molecule()
mol.add_atom('H', 0.0, 0.0, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
# test values at these coordinates
coords = []
for x in np.linspace(0, 10, 10):
coords.append([x, x, x])
# results
ans = [
6.2825e-01, 7.1229e-02, 6.8672e-03, 3.0000e-04, 3.7662e-06,
1.3536e-08, 1.3927e-11, 4.1021e-15, 3.4591e-19, 8.3505e-24
]
# test for each coord
amps = []
for i, coord in enumerate(coords):
amp = cgfs[0].get_amp(coord)
amps.append(amp)
np.testing.assert_almost_equal(amps, ans, 4)
def test_cgf_repulsion(self):
"""
Test two-electron integrals for contracted Gaussians
(ij|kl) = <cgf_i cgf_j | r_ij | cgf_k cgf_l>
"""
# construct integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule("H2")
mol.add_atom('H', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.0, 0.0, 1.4)
cgfs, nuclei = mol.build_basis('sto3g')
T1111 = integrator.repulsion(cgfs[0], cgfs[0], cgfs[0], cgfs[0])
T1122 = integrator.repulsion(cgfs[0], cgfs[0], cgfs[1], cgfs[1])
T1112 = integrator.repulsion(cgfs[0], cgfs[0], cgfs[0], cgfs[1])
T2121 = integrator.repulsion(cgfs[1], cgfs[0], cgfs[1], cgfs[0])
T1222 = integrator.repulsion(cgfs[0], cgfs[1], cgfs[1], cgfs[1])
T2211 = integrator.repulsion(cgfs[1], cgfs[1], cgfs[0], cgfs[0])
np.testing.assert_almost_equal(T1111, 0.7746056914329529, 4)
np.testing.assert_almost_equal(T1122, 0.5696758031845093, 4)
np.testing.assert_almost_equal(T1112, 0.4441076656879812, 4)
np.testing.assert_almost_equal(T2121, 0.2970285713672638, 4)
# test similarity between two-electron integrals
np.testing.assert_almost_equal(T1222, T1112, 4)
np.testing.assert_almost_equal(T1122, T2211, 4)
def testDerivH2(self):
"""
Test Derivatives of dihydrogen
"""
# build integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule('H2')
mol.add_atom('H', -0.5, 0.0, 0.0)
mol.add_atom('H', 0.5, 0.0, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
fx1 = integrator.nuclear_deriv(cgfs[0], cgfs[0], nuclei[0][0],
nuclei[0][1], nuclei[0][0], 0)
fx2 = integrator.nuclear_deriv(cgfs[1], cgfs[1], nuclei[0][0],
nuclei[0][1], nuclei[0][0], 0)
ans1 = calculate_force_finite_difference(cgfs[0], cgfs[0],
nuclei[0][0], nuclei[0][1], 0)
ans2 = calculate_force_finite_difference(cgfs[1], cgfs[1],
nuclei[0][0], nuclei[0][1], 0)
np.testing.assert_almost_equal(fx1, ans1, 4)
np.testing.assert_almost_equal(fx2, ans2, 4)
# assert that the nuclear gradient in the x direction is zero because the
# basis functions spawn from this atom
self.assertTrue(np.abs(fx1) < 1e-8)
# assert that the nuclear gradient has a meaningful number
self.assertTrue(np.abs(fx2) > 1e-1)
fx3 = integrator.nuclear_deriv(cgfs[0], cgfs[0], nuclei[1][0],
nuclei[1][1], nuclei[1][0], 0)
fx4 = integrator.nuclear_deriv(cgfs[1], cgfs[1], nuclei[1][0],
nuclei[1][1], nuclei[1][0], 0)
ans3 = calculate_force_finite_difference(cgfs[0], cgfs[0],
nuclei[1][0], nuclei[1][1], 0)
ans4 = calculate_force_finite_difference(cgfs[1], cgfs[1],
nuclei[1][0], nuclei[1][1], 0)
np.testing.assert_almost_equal(fx3, ans3, 4)
np.testing.assert_almost_equal(fx4, ans4, 4)
fy1 = integrator.nuclear_deriv(cgfs[0], cgfs[0], nuclei[0][0],
nuclei[0][1], nuclei[0][0], 1)
fy2 = integrator.nuclear_deriv(cgfs[1], cgfs[1], nuclei[0][0],
nuclei[0][1], nuclei[0][0], 1)
ans5 = calculate_force_finite_difference(cgfs[0], cgfs[0],
nuclei[0][0], nuclei[0][1], 1)
ans6 = calculate_force_finite_difference(cgfs[1], cgfs[1],
nuclei[0][0], nuclei[0][1], 1)
np.testing.assert_almost_equal(fy1, ans5, 4)
np.testing.assert_almost_equal(fy2, ans6, 4)
def test_integrals_h2(self):
"""
Test automatic integral evaluation for hydrogen molecule
"""
# construct integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule()
mol.add_atom('H', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.0, 0.0, 1.4)
cgfs, nuclei = mol.build_basis('sto3g')
# evaluate all integrals
ncpu = multiprocessing.cpu_count()
S, T, V, teint = integrator.build_integrals(cgfs,
nuclei,
npar=ncpu,
verbose=False)
# overlap integrals
S_result = np.matrix([[1.0, 0.65931845], [0.65931845, 1.0]])
# kinetic integrals
T_result = np.matrix([[0.7600315809249878, 0.2364544570446014],
[0.2364544570446014, 0.7600315809249878]])
# nuclear attraction integrals
V1_result = np.matrix([[-1.2266135215759277, -0.5974172949790955],
[-0.5974172949790955, -0.6538270711898804]])
V2_result = np.matrix([[-0.6538270711898804, -0.5974172949790955],
[-0.5974172949790955, -1.2266135215759277]])
V_result = V1_result + V2_result
# two-electron integrals
T1111 = 0.7746056914329529
T1122 = 0.5696758031845093
T1112 = 0.4441076219081878
T1212 = 0.2970285713672638
# perform tests
np.testing.assert_almost_equal(S, S_result, 4)
np.testing.assert_almost_equal(T, T_result, 4)
np.testing.assert_almost_equal(V, V_result, 4)
np.testing.assert_almost_equal(teint[integrator.teindex(0, 0, 0, 0)],
T1111, 4)
np.testing.assert_almost_equal(teint[integrator.teindex(0, 0, 1, 1)],
T1122, 4)
np.testing.assert_almost_equal(teint[integrator.teindex(0, 0, 0, 1)],
T1112, 4)
np.testing.assert_almost_equal(teint[integrator.teindex(0, 1, 0, 1)],
T1212, 4)
def testDerivH2(self):
"""
Test Derivatives of dihydrogen
"""
# build integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule('H2')
mol.add_atom('H', -0.5, 0.0, 0.0)
mol.add_atom('H', 0.5, 0.0, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
# calculate derivative towards H1 in the x-direction
fx1 = integrator.repulsion_deriv(cgfs[0], cgfs[0], cgfs[0], cgfs[0],
nuclei[0][0], 0)
fx2 = integrator.repulsion_deriv(cgfs[1], cgfs[1], cgfs[1], cgfs[1],
nuclei[0][0], 0)
# mol | nuc_id | cgf_id1 | cgf_id2 | cgf_id3 | cgf_id4 | coord
ans1 = calculate_force_finite_difference(mol, 0, 0, 0, 0, 0, 0)
ans2 = calculate_force_finite_difference(mol, 0, 1, 1, 1, 1, 0)
# assert that the repulsion of two CGFs that spawn from
# the same nucleus will not change in energy due to a
# change of the nucleus coordinates
np.testing.assert_almost_equal(fx1, ans1, 4)
np.testing.assert_almost_equal(fx1, 0.0, 4)
np.testing.assert_almost_equal(fx2, ans2, 4)
np.testing.assert_almost_equal(fx2, 0.0, 4)
# assert that the cross-terms will change
fx3 = integrator.repulsion_deriv(cgfs[0], cgfs[0], cgfs[1], cgfs[1],
nuclei[0][0], 0)
fx4 = integrator.repulsion_deriv(cgfs[0], cgfs[0], cgfs[1], cgfs[1],
nuclei[1][0], 0)
fx5 = integrator.repulsion_deriv(cgfs[1], cgfs[0], cgfs[1], cgfs[0],
nuclei[0][0], 0)
fx6 = integrator.repulsion_deriv(cgfs[1], cgfs[0], cgfs[1], cgfs[0],
nuclei[1][0], 0)
# mol | nuc_id | cgf_id1 | cgf_id2 | cgf_id3 | cgf_id4 | coord
ans3 = calculate_force_finite_difference(mol, 0, 0, 0, 1, 1, 0)
ans4 = calculate_force_finite_difference(mol, 1, 0, 0, 1, 1, 0)
ans5 = calculate_force_finite_difference(mol, 0, 1, 0, 1, 0, 0)
ans6 = calculate_force_finite_difference(mol, 1, 1, 0, 1, 0, 0)
np.testing.assert_almost_equal(fx3, ans3, 4)
np.testing.assert_almost_equal(fx4, ans4, 4)
np.testing.assert_almost_equal(fx5, ans5, 4)
np.testing.assert_almost_equal(fx6, ans6, 4)
def test_integrals_h2o_openmp_sto3g(self):
"""
Test OpenMP integral evaluation for H2O using sto-3g basis set
"""
# construct integrator object
integrator = PyQInt()
# build water molecule
mol = Molecule("H2O")
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
mol.add_atom('O', 0.0, 0.0, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
# evaluate all integrals
ncpu = multiprocessing.cpu_count()
S, T, V, teint = integrator.build_integrals_openmp(cgfs, nuclei)
# load results from npy files
S_result = np.load(os.path.join(os.path.dirname(__file__), 'results', 'h2o_overlap_sto3g.npy'))
T_result = np.load(os.path.join(os.path.dirname(__file__), 'results', 'h2o_kinetic_sto3g.npy'))
V_result = np.load(os.path.join(os.path.dirname(__file__), 'results', 'h2o_nuclear_sto3g.npy'))
teint_result = np.load(os.path.join(os.path.dirname(__file__), 'results', 'h2o_teint_sto3g.npy'))
# assessment
self.assertEqual(len(cgfs), 7)
self.assertEqual(S.shape[0], S.shape[1])
self.assertEqual(S.shape[0], len(cgfs))
np.testing.assert_almost_equal(S, S_result, 4)
np.testing.assert_almost_equal(T, T_result, 4)
np.testing.assert_almost_equal(V, V_result, 4)
np.testing.assert_almost_equal(teint, teint_result, 4)
def testPlotGrid(self):
"""
Test plotting of 1b2 molecular orbital of H2O
"""
# coefficients
coeff = [
8.37612e-17, -2.73592e-16, -0.713011, -1.8627e-17, 9.53496e-17,
-0.379323, 0.379323
]
# construct integrator object
integrator = PyQInt()
# build water molecule
mol = Molecule('H2O')
mol.add_atom('O', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
# build grid
x = np.linspace(-2, 2, 50)
y = np.linspace(-2, 2, 50)
xx, yy = np.meshgrid(x, y)
zz = np.zeros(len(x) * len(y))
grid = np.vstack([xx.flatten(), yy.flatten(), zz]).reshape(3, -1).T
res = integrator.plot_wavefunction(grid, coeff, cgfs).reshape(
(len(y), len(x)))
ans = np.load(
os.path.join(os.path.dirname(__file__), 'results',
'h2o_orb_1b2.npy'))
np.testing.assert_almost_equal(res, ans, 8)
def testHartreeFockForcesCore(self):
"""
Test Hartree-Fock calculation on water using STO-3G basis set
"""
mol = Molecule()
mol.add_atom('O', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
basis = 'sto3g'
# calculate forces using analytical derivatives
solver = HF()
res = solver.rhf(mol, basis, calc_forces=False)
# calculate forces via routines
forces = np.zeros((len(mol.atoms), 3))
for i in range(0, len(mol.atoms)): # loop over nuclei
for j in range(0, 3): # loop over directions
forces[i, j] = np.sum(res['density'] *
solver.rhf_force_core(mol, basis, i, j))
# calculate forces using finite difference
forces_ans = calculate_forces_core_finite_difference(mol)
np.testing.assert_almost_equal(forces, forces_ans)
def test_strings(self):
"""
Test automatic integral evaluation for hydrogen molecule
"""
# construct integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule('H2')
mol.add_atom('H', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.0, 0.0, 1.4)
cgfs, nuclei = mol.build_basis('sto3g')
ans = "CGF; R=(0.000000,0.000000,0.000000)\n"
ans += " 01 | GTO : c=0.154329, alpha=3.425251, l=0, m=0, n=0, R=(0.000000,0.000000,0.000000)\n"
ans += " 02 | GTO : c=0.535328, alpha=0.623914, l=0, m=0, n=0, R=(0.000000,0.000000,0.000000)\n"
ans += " 03 | GTO : c=0.444635, alpha=0.168855, l=0, m=0, n=0, R=(0.000000,0.000000,0.000000)\n"
self.assertEqual(str(cgfs[0]), ans)
ans = "Molecule: H2\n"
ans += " H (0.000000,0.000000,0.000000)\n"
ans += " H (0.000000,0.000000,1.400000)\n"
self.assertEqual(str(mol), ans)
def testHartreeFockWater(self):
"""
Test Hartree-Fock calculation on water using STO-3G basis set
"""
mol = Molecule()
mol.add_atom('O', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
energy = perform_hf(mol)
np.testing.assert_almost_equal(energy, -73.21447132, 4)
def testNuclearRepulsionDerivatives(self):
"""
Test Hartree-Fock calculation on water using STO-3G basis set
"""
mol = Molecule()
mol.add_atom('O', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
forces_fd = calculate_deriv_nuclear_repulsion_finite_difference(mol)
forces = np.zeros(forces_fd.shape)
for i in range(0, len(mol.atoms)): # loop over nuclei
for j in range(0, 3): # loop over directions
forces[i, j] = deriv_nuclear_repulsion(nuclei, i, j)
np.testing.assert_almost_equal(forces, forces_fd, 4)
def testDerivH2O(self):
"""
Test Derivatives of dihydrogen
"""
# build integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule()
mol.add_atom('O', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
# calculate derivative towards H1 in the x-direction
fx1 = integrator.overlap_deriv(cgfs[2], cgfs[2], nuclei[1][0], 0) # px
fx2 = integrator.overlap_deriv(cgfs[2], cgfs[3], nuclei[1][0], 0) # py
ans1 = calculate_force_finite_difference(mol, 1, 2, 2, 0)
ans2 = calculate_force_finite_difference(mol, 1, 3, 3, 0)
# assert that the overlap of two CGFs that spawn from
# the same nucleus will not change in energy due to a
# change of the nucleus coordinates
np.testing.assert_almost_equal(fx1, ans1, 4)
np.testing.assert_almost_equal(fx2, ans2, 4)
# assert that the cross-terms will change
fx3 = integrator.overlap_deriv(cgfs[2], cgfs[5], nuclei[1][0], 0)
fx4 = integrator.overlap_deriv(cgfs[2], cgfs[5], nuclei[1][0], 0)
ans3 = calculate_force_finite_difference(mol, 1, 2, 5, 0)
ans4 = calculate_force_finite_difference(mol, 1, 2, 5, 0)
np.testing.assert_almost_equal(fx3, ans3, 4)
self.assertFalse(fx3 == 0.0)
np.testing.assert_almost_equal(fx4, ans4, 4)
self.assertFalse(fx4 == 0.0)
def testDerivH2O(self):
"""
Test Derivatives of water
"""
# build integrator object
integrator = PyQInt()
# build hydrogen molecule
mol = Molecule()
mol.add_atom('O', 0.0, 0.0, 0.0)
mol.add_atom('H', 0.7570, 0.5860, 0.0)
mol.add_atom('H', -0.7570, 0.5860, 0.0)
cgfs, nuclei = mol.build_basis('sto3g')
# calculate derivative of 2s AO on oxygen towards H1 in the x-direction
fx1 = integrator.repulsion_deriv(cgfs[2], cgfs[2], cgfs[2], cgfs[2],
nuclei[1][0], 0) # px
fx2 = integrator.repulsion_deriv(cgfs[2], cgfs[3], cgfs[3], cgfs[3],
nuclei[1][0], 0) # py
ans1 = calculate_force_finite_difference(mol, 1, 2, 2, 2, 2, 0)
ans2 = calculate_force_finite_difference(mol, 1, 3, 3, 3, 3, 0)
# assert that the repulsion of two CGFs that spawn from
# the same nucleus will not change in energy due to a
# change of the nucleus coordinates
np.testing.assert_almost_equal(fx1, ans1, 4)
np.testing.assert_almost_equal(fx2, ans2, 4)
# assert that the cross-terms will change
fx3 = integrator.repulsion_deriv(cgfs[3], cgfs[3], cgfs[5], cgfs[5],
nuclei[0][0], 0)
fx4 = integrator.repulsion_deriv(cgfs[3], cgfs[3], cgfs[5], cgfs[5],
nuclei[1][0], 0)
fx5 = integrator.repulsion_deriv(cgfs[5], cgfs[3], cgfs[5], cgfs[3],
nuclei[0][0], 0)
fx6 = integrator.repulsion_deriv(cgfs[3], cgfs[5], cgfs[3], cgfs[5],
nuclei[1][0], 0)
# mol | nuc_id | cgf_id1 | cgf_id2 | cgf_id3 | cgf_id4 | coord
ans3 = calculate_force_finite_difference(mol, 0, 3, 3, 5, 5, 0)
ans4 = calculate_force_finite_difference(mol, 1, 3, 3, 5, 5, 0)
ans5 = calculate_force_finite_difference(mol, 0, 5, 3, 5, 3, 0)
ans6 = calculate_force_finite_difference(mol, 1, 3, 5, 3, 5, 0)
# assert that these are non-trivial tests
self.assertFalse(ans3 == 0.0)
self.assertFalse(ans4 == 0.0)
self.assertFalse(ans5 == 0.0)
self.assertFalse(ans6 == 0.0)
np.testing.assert_almost_equal(fx3, ans3, 10)
np.testing.assert_almost_equal(fx4, ans4, 10)
np.testing.assert_almost_equal(fx5, ans5, 10)
np.testing.assert_almost_equal(fx6, ans6, 10)
# assert that the cross-terms will change
fx7 = integrator.repulsion_deriv(cgfs[2], cgfs[3], cgfs[5], cgfs[6],
nuclei[0][0], 0)
fx8 = integrator.repulsion_deriv(cgfs[2], cgfs[3], cgfs[5], cgfs[6],
nuclei[1][0], 1)
fx9 = integrator.repulsion_deriv(cgfs[2], cgfs[3], cgfs[5], cgfs[6],
nuclei[2][0], 1)
# get answers
ans7 = calculate_force_finite_difference(mol, 0, 2, 3, 5, 6, 0)
ans8 = calculate_force_finite_difference(mol, 1, 2, 3, 5, 6, 1)
ans9 = calculate_force_finite_difference(mol, 2, 2, 3, 5, 6, 1)
# assert that these are non-trivial tests
self.assertFalse(ans7 == 0.0)
self.assertFalse(ans8 == 0.0)
self.assertFalse(ans9 == 0.0)
np.testing.assert_almost_equal(fx7, ans7, 10)
np.testing.assert_almost_equal(fx8, ans8, 10)
np.testing.assert_almost_equal(fx9, ans9, 10)