def test_dot():
z1 = Bicomplex(1, 2)
z2 = Bicomplex(3, 4)
z3 = z1.dot(z2)
z4 = z1 * z2
np.testing.assert_array_equal(z3.z1, z4.z1)
np.testing.assert_array_equal(z3.z2, z4.z2)
def test_der_abs():
x = np.linspace(-0.98, 0.98, 5)
h = 1e-8
der1 = abs(Bicomplex(x + h * 1j, 0)).imag1 / h
assert_allclose(der1, np.where(x < 0, -1, 1))
der2 = abs(Bicomplex(x + h * 1j, h)).imag12 / h ** 2
assert_allclose(der2, 0, atol=1e-6)
def test_dot():
z1 = Bicomplex(1, 2)
z2 = Bicomplex(3, 4)
z3 = z1.dot(z2)
z4 = z1 * z2
assert_array_equal(z3.z1, z4.z1)
assert_array_equal(z3.z2, z4.z2)
def test_division():
z1 = Bicomplex(1, 2)
z2 = Bicomplex(3, 4)
z3 = z1 / z2
z4 = z1 * (z2 ** -1)
assert_allclose(z3.z1, z4.z1)
assert_allclose(z3.z2, z4.z2)
def test_der_log():
x = np.linspace(0.001, 5, 6)
h = 1e-15
der1 = np.log(Bicomplex(x + h * 1j, 0)).imag1 / h
assert_allclose(der1, 1. / x)
der2 = np.log(Bicomplex(x + h * 1j, h)).imag12 / h ** 2
assert_allclose(der2, -1. / x ** 2)
def test_gt():
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
z2 = Bicomplex(1, 2)
val = z > z2
truth = [[False, False, True], [True, True, True], [True, True, True]]
np.testing.assert_array_equal(val, truth)
def test_arg_c():
z1 = Bicomplex(np.linspace(0, np.pi, 5), 0)
z2 = z1.arg_c()
np.testing.assert_array_equal(z2, np.arctan2(z1.z2.real, z1.z1.real))
z3 = Bicomplex(0.1, np.linspace(0, np.pi, 5))
z4 = z3.arg_c()
np.testing.assert_allclose(z4.real, np.arctan2(z3.z2.real, z3.z1.real))
def test_der_cos():
x = np.linspace(-0.99, 0.99, 5)
h = 1e-9
der1 = np.cos(Bicomplex(x + h * 1j, 0)).imag1 / h
assert_allclose(der1, -np.sin(x))
h *= 100
der2 = np.cos(Bicomplex(x + h * 1j, h)).imag12 / h ** 2
assert_allclose(der2, -np.cos(x))
def test_sub():
shape = (3, 3)
z0 = Bicomplex(np.ones(shape), 2 * np.ones(shape))
z1 = Bicomplex(3 * np.ones(shape), 4 * np.ones(shape))
z2 = z0 - z1
assert_array_equal(z2.z1, z0.z1 - z1.z1)
assert_array_equal(z2.z2, z0.z2 - z1.z2)
def test_norm(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
np.testing.assert_array_equal(z.norm(), np.sqrt(5*t**2))
z = Bicomplex(1, 2)
self.assertEqual(z.norm(), np.sqrt(5))
def test_der_arctan():
x = np.linspace(0, 2, 5)
h = 1e-8
der1 = np.arctan(Bicomplex(x + h * 1j, 0)).imag1 / h
assert_allclose(der1, 1. / (1 + x ** 2))
der2 = Bicomplex(x + h * 1j, h).arctan().imag12 / h ** 2
assert_allclose(der2, -2 * x / (1 + x ** 2) ** 2)
def test_shape(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
assert z.shape == shape
z = Bicomplex(1, 2)
assert z.shape == ()
def test_der_arccosh():
x = np.linspace(1.2, 5, 5)
h = 1e-8
der1 = np.arccosh(Bicomplex(x + h * 1j, 0)).imag1 / h
assert_allclose(der1, 1. / np.sqrt(x ** 2 - 1))
h = (_default_base_step(x, scale=2.5) + 1) - 1
der2 = np.arccosh(Bicomplex(x + h * 1j, h)).imag12 / h ** 2
true_der2 = -x / (x ** 2 - 1) ** (3. / 2)
assert_allclose(der2, true_der2, atol=1e-5)
def test_der_arccos():
x = np.linspace(-0.98, 0.98, 5)
h = 1e-8
der1 = np.arccos(Bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.assert_allclose(der1, -1. / np.sqrt(1 - x**2))
h = (_default_base_step(x, scale=2.5) + 1) - 1
der2 = np.arccos(Bicomplex(x + h * 1j, h)).imag12 / h**2
true_der2 = -x / (1 - x**2)**(3. / 2)
np.testing.assert_allclose(der2, true_der2, atol=1e-5)
def test_eq():
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
z2 = Bicomplex(1, 2)
val = z == z2
truth = np.array([[False, True, False],
[False, False, False],
[False, False, False]], dtype=bool)
assert_array_equal(val, truth)
def test_le():
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
z2 = Bicomplex(1, 2)
val = z <= z2
truth = [[True, True, False],
[False, False, False],
[False, False, False]]
assert_array_equal(val, truth)
def test_add():
shape = (3, 3)
z0 = Bicomplex(np.ones(shape), 2 * np.ones(shape))
z1 = Bicomplex(3 * np.ones(shape), 4 * np.ones(shape))
z2 = z0 + z1
assert_array_equal(z2.z1, z0.z1 + z1.z1)
assert_array_equal(z2.z2, z0.z2 + z1.z2)
z3 = z0 + 1
assert_array_equal(z3.z1, z0.z1 + 1)
assert_array_equal(z3.z2, z0.z2)
def _multicomplex2(f, fx, x, h):
n = len(x)
increments = np.identity(n) * h
cmplx_wrap = Bicomplex.__array_wrap__
partials = [cmplx_wrap(f(Bicomplex(x + 1j * hi, hi))).imag12
for hi in increments]
return np.array(partials)
def _multicomplex(f, fx, x, h):
n = len(x)
steps = _JacobianDifferenceFunctions.increments(n, h)
cmplx_wrap = Bicomplex.__array_wrap__
partials = [
cmplx_wrap(f(Bicomplex(x + 1j * hi, 0))).imag for hi in steps
]
return np.array(partials)
def _multicomplex(f, fx, x, h):
n = len(x)
cmplx_wrap = Bicomplex.__array_wrap__
partials = [
cmplx_wrap(f(Bicomplex(x + 1j * hi, 0))).imag
for hi in Jacobian._increments(n, h)
]
return np.array(partials)
def test_pow():
z1 = Bicomplex(1, 2)
z2 = z1 ** 2
z3 = z1 * z1
assert_allclose(z2.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
assert_allclose(z2.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
assert_allclose(z3.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
assert_allclose(z3.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
z1 = Bicomplex(z1=-1j, z2=-1 - 0j)
z2 = z1 * z1
z3 = z1 ** 2
assert_allclose(z2.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
assert_allclose(z2.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
assert_allclose(z3.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
assert_allclose(z3.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
def _test_first_derivative(name):
x = np.linspace(0.0001, 0.98, 5)
h = _default_base_step(x, scale=2)
f, df = get_function(name, n=1)
der = f(Bicomplex(x + h * 1j, 0)).imag1 / h
der_true = df(x)
assert_allclose(der, der_true, err_msg='{0!s}'.format(name))
def test_assign():
shape = (3, 3)
z = Bicomplex(np.ones(shape), 2 * np.ones(shape))
z0 = z[0]
assert_array_equal(z0.z1, z.z1[0])
assert_array_equal(z0.z2, z.z2[0])
z1 = z[:]
assert_array_equal(z1.z1, z.z1[:])
assert_array_equal(z1.z2, z.z2[:])
def _test_second_derivative(name):
x = np.linspace(0.01, 0.98, 5)
h = _default_base_step(x, scale=2.5)
f, df = get_function(name, n=2)
der = f(Bicomplex(x + h * 1j, h)).imag12 / h**2
der_true = df(x)
np.testing.assert_allclose(der, der_true, err_msg='{0!s}'.format(name))
def _multicomplex2(f, fx, x, h):
"""Calculate Hessian with Bicomplex-step derivative approximation"""
n = len(x)
ee = np.diag(h)
hess = np.outer(h, h)
cmplx_wrap = Bicomplex.__array_wrap__
for i in range(n):
for j in range(i, n):
zph = Bicomplex(x + 1j * ee[i, :], ee[j, :])
hess[i, j] = cmplx_wrap(f(zph)).imag12 / hess[j, i]
hess[j, i] = hess[i, j]
return hess
def test_mod_c():
z1 = Bicomplex(np.linspace(0, np.pi, 5), 0)
z2 = z1.mod_c()
assert_array_equal(z2, np.sqrt(z1.z1**2 + z1.z2**2))
z3 = Bicomplex(0.1, np.linspace(0, np.pi, 5))
z4 = z3.mod_c()
trueval = np.sqrt(z3*z3.conjugate())
assert_allclose(z4, np.sqrt(z3.z1**2 + z3.z2**2))
assert_allclose(z4, trueval.z1)
def test_arg_c():
z1 = Bicomplex(np.linspace(0, np.pi, 5), 0)
z2 = z1.arg_c()
assert_array_equal(z2, np.arctan2(z1.z2.real, z1.z1.real))
z3 = Bicomplex(0.1, np.linspace(0, np.pi, 5))
z4 = z3.arg_c()
assert_allclose(z4.real, np.arctan2(z3.z2.real, z3.z1.real))
def test_norm(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
assert_array_equal(z.norm(), np.sqrt(5 * t ** 2))
z = Bicomplex(1, 2)
assert z.norm() == np.sqrt(5)
def test_subsref():
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
z0 = z[0]
assert_array_equal(z0.z1, z.z1[0])
assert_array_equal(z0.z2, z.z2[0])
z1 = z[:]
assert_array_equal(z1.z1, z.z1[:])
assert_array_equal(z1.z2, z.z2[:])
z1 = z[1:3, 1:3]
assert_array_equal(z1.z1, z.z1[1:3, 1:3])
assert_array_equal(z1.z2, z.z2[1:3, 1:3])
def test_rdivision(self):
"""
Test issue # 39
"""
z2 = Bicomplex(3, 4)
z3 = 1 / z2
z4 = (z2**-1)
z5 = 1.0 / z2
assert_array_equal(z3.z1, z4.z1)
assert_array_equal(z3.z2, z4.z2)
assert_array_equal(z5.z1, z4.z1)
assert_array_equal(z5.z2, z4.z2)
def test_conjugate(self):
z = Bicomplex(1, 2)
z2 = Bicomplex(1, -2)
self.assertTrue(z.conjugate() == z2)
def test_neg(self):
z = Bicomplex(1, 2)
z2 = -z
assert z2.z1 == -z.z1
assert z2.z2 == -z.z2
def test_cos():
z1 = Bicomplex(np.linspace(0, np.pi, 5), 0)
z2 = z1.cos() # np.cos(z1)
np.testing.assert_array_equal(z2.z1, np.cos(z1.z1))
def test_cos():
z1 = Bicomplex(np.linspace(0, np.pi, 5), 0)
z2 = z1.cos() # np.cos(z1)
assert_array_equal(z2.z1, np.cos(z1.z1))
def test_arccos():
z1 = Bicomplex(np.linspace(-0.98, 0.98, 5), 0)
z2 = z1.arccos()
np.testing.assert_allclose(z2.real, np.arccos(z1.z1).real, atol=1e-15)
np.testing.assert_allclose(z2.imag1, np.arccos(z1.z1).imag, atol=1e-15)
def test_arccos():
z1 = Bicomplex(np.linspace(-0.98, 0.98, 5), 0)
z2 = z1.arccos()
assert_allclose(z2.real, np.arccos(z1.z1).real, atol=1e-15)
assert_allclose(z2.imag1, np.arccos(z1.z1).imag, atol=1e-15)
def test_flat(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = Bicomplex(t, 2 * t)
t = z.flat(1)
self.assertTrue(t == Bicomplex(1, 2))
def _multicomplex2(f, fx, x, h, *args, **kwds):
z = Bicomplex(x + 1j * h, h)
return Bicomplex.__array_wrap__(f(z, *args, **kwds)).imag12
