def test_diff_point_3D():
x0 = Var("x0")
x1 = Var("x1")
x2 = Var("x2")
f5_x0 = Fun(x0 * x2, [x0, x1, x2], "f5_x0")
f5_x1 = Fun(x0 * 5, [x0, x1, x2], "f5_x1")
f5_x2 = Fun(x2 * 0.5, [x0, x1, x2], "f5_x2")
z0 = Point({"coordarray": [3.0, 2.0, 1.0], "coordnames": ["x0", "x1", "x2"]})
# could also have defined F directly from f5_x[i] definitions
F1 = Fun([f5_x0(x0, x1, x2), f5_x1(x0, x1, x2), f5_x2(x0, x1, x2)], [x0, x1, x2], "F")
F2 = [f5_x0(x0, x1, x2), f5_x1(x0, x1, x2), f5_x2(x0, x1, x2)]
assert Diff(F1, [x0, x1, x2]) == Diff(F2, [x0, x1, x2])
assert_array_almost_equal(
Diff(F1, [x0, x1, x2]).eval(z0).tonumeric(), array([[1.0, 0, 3.0], [5, 0, 0], [0, 0, 0.5]])
)
assert_array_almost_equal(
Diff(F2, [x0, x1, x2]).eval(z0).tonumeric(), array([[1.0, 0, 3.0], [5, 0, 0], [0, 0, 0.5]])
)
def F3(z):
return Point({"coorddict": {"x0": z("x0") * z("x2"), "x1": z("x0") * 5.0, "x2": z("x2") * 0.5}})
assert_array_almost_equal(
simplifyMatrixRepr(diff(F3, z0, axes=["x1", "x2"])), Diff(F1, [x0, x1, x2]).eval(z0).tonumeric()[1:]
)
# Comparing 1st order Taylor series for nearby point to actual value:
z1 = Point({"coordarray": array([3.1, 2.0, 0.94]), "coordnames": ["x0", "x1", "x2"]})
actual = F3(z1)
approx = F3(z0) + simplifyMatrixRepr(diff(F3, z0)).dot(z1 - z0)
assert all([err < 0.01 for err in abs(approx - actual)])
def test_diff_vector_3D_complex():
def f(x):
"""f: R^3->R^3"""
return array([
x[0] * x[2],
x[0] * 5,
x[2] * 0.5
])
def jac(x):
return array([
[x[2], 0.0, x[0]],
[5.0, 0.0, 0.0],
[0.0, 0.0, 0.5]
])
for i in range(10):
x = i * random(3)
assert_array_almost_equal(diff(f, x), jac(x))
assert_array_almost_equal(
simplifyMatrixRepr(diff(f, x, axes=[0])), jac(x)[0])
assert_array_almost_equal(
simplifyMatrixRepr(diff(f, x, axes=[1])), jac(x)[1])
assert_array_almost_equal(
simplifyMatrixRepr(diff(f, x, axes=[2])), jac(x)[2])
# Comparing 1st order Taylor series for nearby point to actual value:
x0 = zeros(3)
dx = array([0.2, 0., -.2])
actual = f(x0 + dx)
approx = f(x0) + simplifyMatrixRepr(diff(f, x0)).dot(dx)
assert allclose((actual - approx), array([-0.04, 0., 0.]))
def test_diff_point_3D():
x0 = Var('x0')
x1 = Var('x1')
x2 = Var('x2')
f5_x0 = Fun(x0 * x2, [x0, x1, x2], 'f5_x0')
f5_x1 = Fun(x0 * 5, [x0, x1, x2], 'f5_x1')
f5_x2 = Fun(x2 * 0.5, [x0, x1, x2], 'f5_x2')
z0 = Point({
'coordarray': [3., 2., 1.],
'coordnames': ['x0', 'x1', 'x2']
})
# could also have defined F directly from f5_x[i] definitions
F1 = Fun(
[
f5_x0(x0, x1, x2),
f5_x1(x0, x1, x2),
f5_x2(x0, x1, x2)
],
[x0, x1, x2],
'F'
)
F2 = [
f5_x0(x0, x1, x2),
f5_x1(x0, x1, x2),
f5_x2(x0, x1, x2)
]
assert Diff(F1, [x0, x1, x2]) == Diff(F2, [x0, x1, x2])
assert_array_almost_equal(
Diff(F1, [x0, x1, x2]).eval(z0).tonumeric(),
array([[1.0, 0, 3.0], [5, 0, 0], [0, 0, 0.5]])
)
assert_array_almost_equal(
Diff(F2, [x0, x1, x2]).eval(z0).tonumeric(),
array([[1.0, 0, 3.0], [5, 0, 0], [0, 0, 0.5]])
)
def F3(z):
return Point({
'coorddict': {
'x0': z('x0') * z('x2'),
'x1': z('x0') * 5.0,
'x2': z('x2') * 0.5
}
})
assert_array_almost_equal(
simplifyMatrixRepr(diff(F3, z0, axes=['x1', 'x2'])),
Diff(F1, [x0, x1, x2]).eval(z0).tonumeric()[1:]
)
# Comparing 1st order Taylor series for nearby point to actual value:
z1 = Point({
'coordarray': array([3.1, 2., .94]),
'coordnames': ['x0', 'x1', 'x2']
})
actual = F3(z1)
approx = F3(z0) + simplifyMatrixRepr(diff(F3, z0)).dot(z1 - z0)
assert all([err < 0.01 for err in abs(approx - actual)])
def test_diff_scalar_on_vector_2D():
def f(x):
"f: R^2 -> R"
return array([x[0] * x[1] + 2, x[1] * x[1]])
x0 = array([1.0, 3.0])
df = diff(f, x0, vars=[1])
assert simplifyMatrixRepr(diff(f, x0, vars=[1], axes=[0])) == 1.0
def test_diff_scalar_on_vector_2D():
def f(x):
"f: R^2 -> R"
return array([
x[0] * x[1] + 2,
x[1] * x[1]]
)
x0 = array([1., 3.])
df = diff(f, x0, vars=[1])
assert simplifyMatrixRepr(diff(f, x0, vars=[1], axes=[0])) == 1.0
def test_diff_vector_3D_complex():
def f(x):
"""f: R^3->R^3"""
return array([x[0] * x[2], x[0] * 5, x[2] * 0.5])
def jac(x):
return array([[x[2], 0.0, x[0]], [5.0, 0.0, 0.0], [0.0, 0.0, 0.5]])
for i in range(10):
x = i * random(3)
assert_array_almost_equal(diff(f, x), jac(x))
assert_array_almost_equal(simplifyMatrixRepr(diff(f, x, axes=[0])), jac(x)[0])
assert_array_almost_equal(simplifyMatrixRepr(diff(f, x, axes=[1])), jac(x)[1])
assert_array_almost_equal(simplifyMatrixRepr(diff(f, x, axes=[2])), jac(x)[2])
# Comparing 1st order Taylor series for nearby point to actual value:
x0 = zeros(3)
dx = array([0.2, 0.0, -0.2])
actual = f(x0 + dx)
approx = f(x0) + simplifyMatrixRepr(diff(f, x0)).dot(dx)
assert allclose((actual - approx), array([-0.04, 0.0, 0.0]))
def test_diff_scalar_on_vector_4D():
def f(x):
"""f: R^4->R"""
return sqrt(sum((x - array([1.0, 0.0, 1.0, 1.0])) ** 2))
# Compare the 1st order Taylor expansion of f2 at x0+dx = x0+[dx1,dx2,0,0]
# in the first argument
x0 = zeros(4)
dx1 = 0.25
dx2 = 0.25
actual = f(x0 + [dx1, dx2, 0.0, 0.0])
approx = simplifyMatrixRepr(f(x0) + diff(f, x0, [0]) * dx1)
assert actual - approx < 0.033
assert actual - approx > 0.0324
def test_diff_scalar_on_vector_4D():
def f(x):
"""f: R^4->R"""
return sqrt(sum((x - array([1., 0., 1., 1.])) ** 2))
# Compare the 1st order Taylor expansion of f2 at x0+dx = x0+[dx1,dx2,0,0]
# in the first argument
x0 = zeros(4)
dx1 = 0.25
dx2 = 0.25
actual = f(x0 + [dx1, dx2, 0., 0.])
approx = simplifyMatrixRepr(f(x0) + diff(f, x0, [0]) * dx1)
assert actual - approx < 0.033
assert actual - approx > 0.0324