def test_der_abs():
x = np.linspace(-0.98, 0.98, 5)
h = 1e-8
der1 = abs(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.assert_allclose(der1, np.where(x < 0, -1, 1))
der2 = abs(bicomplex(x + h * 1j, h)).imag12 / h**2
np.testing.assert_allclose(der2, 0, atol=1e-6)
def test_division():
z1 = bicomplex(1, 2)
z2 = bicomplex(3, 4)
z3 = z1 / z2
z4 = z1 * (z2**-1)
np.testing.assert_allclose(z3.z1, z4.z1)
np.testing.assert_allclose(z3.z2, z4.z2)
def test_division():
z1 = bicomplex(1, 2)
z2 = bicomplex(3, 4)
z3 = z1 / z2
z4 = z1 * (z2**-1)
np.testing.assert_allclose(z3.z1, z4.z1)
np.testing.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
np.testing.assert_allclose(der1, 1. / x)
der2 = np.log(bicomplex(x + h * 1j, h)).imag12 / h**2
np.testing.assert_allclose(der2, -1. / x**2)
def test_der_abs():
x = np.linspace(-0.98, 0.98, 5)
h = 1e-8
der1 = abs(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.assert_allclose(der1, np.where(x < 0, -1, 1))
der2 = abs(bicomplex(x + h * 1j, h)).imag12 / h**2
np.testing.assert_allclose(der2, 0, atol=1e-6)
def test_der_log():
x = np.linspace(0.001, 5, 6)
h = 1e-15
der1 = np.log(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.assert_allclose(der1, 1./x)
der2 = np.log(bicomplex(x + h * 1j, h)).imag12 / h**2
np.testing.assert_allclose(der2, -1./x**2)
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_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_arctan():
x = np.linspace(0, 2, 5)
h = 1e-8
der1 = np.arctan(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.assert_allclose(der1, 1. / (1 + x**2))
der2 = bicomplex(x + h * 1j, h).arctan().imag12 / h**2
np.testing.assert_allclose(der2, -2 * x / (1 + x**2)**2)
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
np.testing.assert_array_equal(z2.z1, z0.z1 - z1.z1)
np.testing.assert_array_equal(z2.z2, z0.z2 - z1.z2)
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
np.testing.assert_allclose(der1, -np.sin(x))
h *= 100
der2 = np.cos(bicomplex(x + h * 1j, h)).imag12 / h**2
np.testing.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
np.testing.assert_array_equal(z2.z1, z0.z1 - z1.z1)
np.testing.assert_array_equal(z2.z2, z0.z2 - z1.z2)
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
np.testing.assert_allclose(der1, - np.sin(x))
h *= 100
der2 = np.cos(bicomplex(x + h * 1j, h)).imag12 / h**2
np.testing.assert_allclose(der2, -np.cos(x))
def test_shape(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = bicomplex(t, 2 * t)
self.assertEqual(z.shape, shape)
z = bicomplex(1, 2)
self.assertEqual(z.shape, ())
def test_der_arctan():
x = np.linspace(0, 2, 5)
h = 1e-8
der1 = np.arctan(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.assert_allclose(der1, 1. / (1 + x**2))
der2 = bicomplex(x+h*1j, h).arctan().imag12/h**2
np.testing.assert_allclose(der2, -2*x/(1+x**2)**2)
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_shape(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = bicomplex(t, 2 * t)
self.assertEqual(z.shape, shape)
z = bicomplex(1, 2)
self.assertEqual(z.shape, ())
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_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_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_gt():
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, False, True],
[ True, True, True],
[ True, True, True]], dtype=bool)
np.testing.assert_array_equal(val, truth)
def test_der_arccosh():
x = np.linspace(1.2, 5, 5)
h = 1e-8
der1 = np.arccosh(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.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)
np.testing.assert_allclose(der2, true_der2, atol=1e-5)
def test_der_arccosh():
x = np.linspace(1.2, 5, 5)
h = 1e-8
der1 = np.arccosh(bicomplex(x + h * 1j, 0)).imag1 / h
np.testing.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)
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)
np.testing.assert_array_equal(val, truth)
def test_der_arccos(self):
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_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
np.testing.assert_array_equal(z2.z1, z0.z1 + z1.z1)
np.testing.assert_array_equal(z2.z2, z0.z2 + z1.z2)
z3 = z0 + 1
np.testing.assert_array_equal(z3.z1, z0.z1 + 1)
np.testing.assert_array_equal(z3.z2, z0.z2)
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
np.testing.assert_array_equal(z2.z1, z0.z1 + z1.z1)
np.testing.assert_array_equal(z2.z2, z0.z2 + z1.z2)
z3 = z0 + 1
np.testing.assert_array_equal(z3.z1, z0.z1 + 1)
np.testing.assert_array_equal(z3.z2, z0.z2)
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)
np.testing.assert_allclose(der, der_true, err_msg=('{0!s}'.format(name)))
def test_pow():
z1 = bicomplex(1, 2)
z2 = z1 ** 2
z3 = z1 * z1
np.testing.assert_allclose(z2.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.assert_allclose(z2.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
np.testing.assert_allclose(z3.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.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
np.testing.assert_allclose(z2.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.assert_allclose(z2.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
np.testing.assert_allclose(z3.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.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)
np.testing.assert_allclose(der, der_true, err_msg=('{0!s}'.format(name)))
def test_pow():
z1 = bicomplex(1, 2)
z2 = z1**2
z3 = z1 * z1
np.testing.assert_allclose(z2.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.assert_allclose(z2.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
np.testing.assert_allclose(z3.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.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
np.testing.assert_allclose(z2.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.assert_allclose(z2.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
np.testing.assert_allclose(z3.z1, z1.z1 * z1.z1 - z1.z2 * z1.z2)
np.testing.assert_allclose(z3.z2, z1.z1 * z1.z2 + z1.z2 * z1.z1)
def _test_second_derivative(name):
x = np.linspace(0.01, 0.98, 5)
h = _default_base_step(x, scale=2.5)
# h = 1e-8
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 test_assign():
shape = (3, 3)
z = bicomplex(np.ones(shape), 2 * np.ones(shape))
z0 = z[0]
np.testing.assert_array_equal(z0.z1, z.z1[0])
np.testing.assert_array_equal(z0.z2, z.z2[0])
z1 = z[:]
np.testing.assert_array_equal(z1.z1, z.z1[:])
np.testing.assert_array_equal(z1.z2, z.z2[:])
def test_assign():
shape = (3, 3)
z = bicomplex(np.ones(shape), 2 * np.ones(shape))
z0 = z[0]
np.testing.assert_array_equal(z0.z1, z.z1[0])
np.testing.assert_array_equal(z0.z2, z.z2[0])
z1 = z[:]
np.testing.assert_array_equal(z1.z1, z.z1[:])
np.testing.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)
# h = 1e-8
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, *args, **kwargs):
"""Calculate Hessian with bicomplex-step derivative approximation"""
n = len(x)
ee = np.diag(h)
hess = np.outer(h, h)
for i in range(n):
for j in range(i, n):
zph = bicomplex(x + 1j * ee[i, :], ee[j, :])
hess[i, j] = (f(zph, *args, **kwargs)).imag12 / hess[j, i]
hess[j, i] = hess[i, j]
return hess
def test_subsref(self):
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = bicomplex(t, 2 * t)
z0 = z[0]
np.testing.assert_array_equal(z0.z1, z.z1[0])
np.testing.assert_array_equal(z0.z2, z.z2[0])
z1 = z[:]
np.testing.assert_array_equal(z1.z1, z.z1[:])
np.testing.assert_array_equal(z1.z2, z.z2[:])
z1 = z[1:3, 1:3]
np.testing.assert_array_equal(z1.z1, z.z1[1:3, 1:3])
np.testing.assert_array_equal(z1.z2, z.z2[1:3, 1:3])
def test_subsref():
shape = (3, 3)
t = np.arange(9).reshape(shape)
z = bicomplex(t, 2 * t)
z0 = z[0]
np.testing.assert_array_equal(z0.z1, z.z1[0])
np.testing.assert_array_equal(z0.z2, z.z2[0])
z1 = z[:]
np.testing.assert_array_equal(z1.z1, z.z1[:])
np.testing.assert_array_equal(z1.z2, z.z2[:])
z1 = z[1:3, 1:3]
np.testing.assert_array_equal(z1.z1, z.z1[1:3, 1:3])
np.testing.assert_array_equal(z1.z2, z.z2[1:3, 1:3])
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_rpow():
z2 = bicomplex(3, 4)
z3 = 2.**z2
z4 = np.exp(z2 * np.log(2))
np.testing.assert_allclose(z3.z1, z4.z1)
np.testing.assert_allclose(z3.z2, z4.z2)
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 test_multiplication():
z1 = bicomplex(1, 2)
z2 = bicomplex(3, 4)
z3 = z1 * z2
np.testing.assert_array_equal(z3.z1, z1.z1 * z2.z1 - z1.z2 * z2.z2)
np.testing.assert_array_equal(z3.z2, z1.z1 * z2.z2 + z1.z2 * z2.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_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_conjugate(self):
z = bicomplex(1, 2)
z2 = bicomplex(1, -2)
self.assertTrue(z.conjugate()==z2)
def test_rsub():
z1 = bicomplex(2, 1)
a = 1 + 1j
z2 = a - z1
np.testing.assert_array_equal(z2.z1, a - z1.z1)
np.testing.assert_array_equal(z2.z2, -z1.z2)
def test_init(self):
z = bicomplex(1, 2)
self.assertEqual(z.z1, 1)
self.assertEqual(z.z2, 2)
def test_multiplication():
z1 = bicomplex(1, 2)
z2 = bicomplex(3, 4)
z3 = z1 * z2
np.testing.assert_array_equal(z3.z1, z1.z1 * z2.z1 - z1.z2 * z2.z2)
np.testing.assert_array_equal(z3.z2, z1.z1 * z2.z2 + z1.z2 * z2.z1)
def test_init(self):
z = bicomplex(1, 2)
self.assertEqual(z.z1, 1)
self.assertEqual(z.z2, 2)
def test_neg(self):
z = bicomplex(1, 2)
z2 = -z
self.assertEqual(z2.z1, -z.z1)
self.assertEqual(z2.z2, -z.z2)
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 test_rsub():
z1 = bicomplex(2, 1)
a = 1 + 1j
z2 = a - z1
np.testing.assert_array_equal(z2.z1, a - z1.z1)
np.testing.assert_array_equal(z2.z2, -z1.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
self.assertEqual(z2.z1, -z.z1)
self.assertEqual(z2.z2, -z.z2)
def test_rpow():
z2 = bicomplex(3, 4)
z3 = 2. ** z2
z4 = np.exp(z2*np.log(2))
np.testing.assert_allclose(z3.z1, z4.z1)
np.testing.assert_allclose(z3.z2, z4.z2)
def test_repr(self):
z = bicomplex(1, 2)
txt = repr(z)
self.assertEqual(txt, "bicomplex(z1=(1+0j), z2=(2+0j))")
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_repr(self):
z = bicomplex(1, 2)
txt = repr(z)
self.assertEqual(txt, "bicomplex(z1=(1+0j), z2=(2+0j))")
