def test_auto_diff_square(self):
"""Test automatic differentiation using
dual numbers for the square function.
f(x) = x^2
f'(x) = d/dx f(x) = 2x
"""
ga = GeometricAlgebra(metric=dual_metric)
one = ga.from_scalar(1.0)
five = ga.from_scalar(5.0)
eps = ga.from_tensor_with_kind(tf.ones(1), kind="pseudoscalar")
x = one + eps
# f(1) = 1^2 = 1, f'(1) = 2
x_squared = ga.geom_prod(x, x)
self.assertTensorsEqual(ga.select_blades_with_name(x_squared, ""), 1.0)
self.assertTensorsEqual(ga.select_blades_with_name(x_squared, "0"),
2.0)
y = five + eps
# f(5) = 5^2 = 25, f'(5) = 10
y_squared = ga.geom_prod(y, y)
self.assertTensorsEqual(ga.select_blades_with_name(y_squared, ""),
25.0)
self.assertTensorsEqual(ga.select_blades_with_name(y_squared, "0"),
10.0)
def test_mul_inverse(self):
ga = GeometricAlgebra(metric=dual_metric)
# a = 2
a = ga.from_tensor_with_kind(tf.fill([1], 2.0), kind="scalar")
# b = 3 + 3e0
b = ga.from_tensor_with_kind(tf.fill([2], 3.0), kind="mv")
# a * b = 2 * (3 + 3e0) = 6 + 6e0
c = ga.geom_prod(a, b)
self.assertTensorsEqual(c, ga.from_scalar(6.0) + 6.0 * ga.e("0"))
# a^-1 = 1 / 2
a_inv = ga.inverse(a)
self.assertTensorsEqual(ga.select_blades_with_name(a_inv, ""), 0.5)
# c = a * b
# => a_inv * c = b
self.assertTensorsEqual(ga.geom_prod(a_inv, c), b)
# Since a is scalar, should commute too.
# => c * a_inv = b
self.assertTensorsEqual(ga.geom_prod(c, a_inv), b)
# b is not simply invertible (because it does not square to a scalar)
# and will throw an exception
self.assertRaises(Exception, ga.simple_inverse, b)
# b is invertible with the shirokov inverse
b_inv = ga.inverse(b)
self.assertTensorsEqual(ga.geom_prod(b, b_inv), 1 * ga.e(""))