""" Test functions for numdifftools module
"""
from __future__ import print_function
import unittest
import numdifftools.core as nd
import numpy as np
from numpy.testing import assert_array_almost_equal
[docs]class TestGlobalFunctions(unittest.TestCase):
def testdea3(self):
def linfun(k):
return np.linspace(0, np.pi / 2., 2. ** (k + 5) + 1)
Ei = np.zeros(3)
for k in np.arange(3):
x = linfun(k)
Ei[k] = np.trapz(np.sin(x), x)
[En, err] = nd.dea3(Ei[0], Ei[1], Ei[2])
self.assertTrue(np.abs(En - 1) < err)
assert_array_almost_equal(En, 1.0, decimal=8)
[docs]class TestRichardson(unittest.TestCase):
def test_order_step_combinations(self):
true_vals = {
(1, 1, 1): [-0.9999999999999998, 1.9999999999999998],
(1, 1, 2): [-0.33333333333333304, 1.333333333333333],
(1, 1, 3): [-0.14285714285714307, 1.142857142857143],
(1, 1, 4): [-0.06666666666666654, 1.0666666666666664],
(1, 1, 5): [-0.03225806451612906, 1.0322580645161292],
(1, 1, 6): [-0.015873015873015872, 1.0158730158730154],
(1, 2, 1): [-0.9999999999999998, 1.9999999999999998],
(1, 2, 2): [-0.33333333333333304, 1.333333333333333],
(1, 2, 3): [-0.14285714285714307, 1.142857142857143],
(1, 2, 4): [-0.06666666666666654, 1.0666666666666664],
(1, 2, 5): [-0.03225806451612906, 1.0322580645161292],
(1, 2, 6): [-0.015873015873015872, 1.0158730158730154],
(2, 1, 1): [0.33333333333333337, -2.0, 2.666666666666667],
(2, 1, 2): [0.04761904761904753, -0.5714285714285693,
1.523809523809522],
(2, 1, 3): [0.009523809523810024, -0.2285714285714322,
1.2190476190476225],
(2, 1, 4): [0.002150537634408055, -0.10322580645160284,
1.1010752688171945],
(2, 1, 5): [0.0005120327700975248, -0.04915514592935677,
1.0486431131592595],
(2, 1, 6): [0.0001249843769525012, -0.02399700037493191,
1.0238720159979793],
(2, 2, 1): [0.1428571428571428, -1.428571428571427,
2.2857142857142843],
(2, 2, 2): [0.022222222222222185, -0.444444444444444,
1.4222222222222216],
(2, 2, 3): [0.004608294930875861, -0.1843317972350207,
1.179723502304145],
(2, 2, 4): [0.0010582010582006751, -0.08465608465608221,
1.0835978835978812],
(2, 2, 5): [0.0002540005080009476, -0.040640081280166496,
1.0403860807721657],
(2, 2, 6): [6.224712107032182e-05, -0.01991907874258203,
1.0198568316215115],
(3, 1, 1): [-0.04761904761904734, 0.6666666666666641,
-2.6666666666666594, 3.047619047619042],
(3, 1, 2): [-0.003174603174603108, 0.08888888888889318,
-0.7111111111111337, 1.6253968253968432],
(3, 1, 3): [-0.0003072196620577672, 0.01720430107525861,
-0.27526881720422713, 1.258371735791026],
(3, 1, 4): [-3.4135518007183396e-05, 0.003823178016754525,
-0.12234169653539884, 1.1185526540366513],
(3, 1, 5): [-4.031754094968587e-06, 0.0009031129172963892,
-0.0577992267083981, 1.056900145545197],
(3, 1, 6): [-4.901348115149418e-07, 0.00021958039560535103,
-0.02810629063481751, 1.0278872003740238],
(3, 2, 1): [-0.004608294930874168, 0.1935483870967698,
-1.5483870967741966, 2.359447004608302],
(3, 2, 2): [-0.00035273368606647537, 0.02962962962962734,
-0.47407407407406155, 1.444797178130501],
(3, 2, 3): [-3.628578685754835e-05, 0.006096012192020994,
-0.19507239014474764, 1.1890126637395837],
(3, 2, 4): [-4.149808071229888e-06, 0.0013943355119737377,
-0.08923747276669802, 1.0878472870627958],
(3, 2, 5): [-4.970655732572382e-07, 0.0003340280653114369,
-0.042755592360228634, 1.0424220613604906],
(3, 2, 6): [-6.08476257157875e-08, 8.177920896951241e-05,
-0.02093547748207586, 1.0208537591207332]}
for num_terms in [1, 2, 3]:
for step in [1, 2]:
for order in range(1, 7):
r_extrap = nd.Richardson(step_ratio=2.0, step=step,
num_terms=num_terms, order=order)
rule = r_extrap._get_richardson_rule()
# print((num_terms, step, order), rule.tolist())
assert_array_almost_equal(rule,
true_vals[(num_terms, step,
order)])
# self.assert_(False)
def test_central(self):
method = 'central'
true_vals = {(1, 1): [-0.33333333, 1.33333333],
(1, 2): [-0.33333333, 1.33333333],
(1, 3): [-0.33333333, 1.33333333],
(1, 4): [-0.06666667, 1.06666667],
(1, 5): [-0.06666667, 1.06666667],
(1, 6): [-0.01587302, 1.01587302],
(2, 1): [0.02222222, -0.44444444, 1.42222222],
(2, 2): [0.02222222, -0.44444444, 1.42222222],
(2, 3): [0.02222222, -0.44444444, 1.42222222],
(2, 4): [1.05820106e-03, -8.46560847e-02, 1.08359788e+00],
(2, 5): [1.05820106e-03, -8.46560847e-02, 1.08359788e+00],
(2, 6): [6.22471211e-05, -1.99190787e-02, 1.01985683e+00]}
for num_terms in [1, 2]:
for order in range(1, 7):
d = nd.Derivative(np.exp, method=method, order=order)
d._set_richardson_rule(step_ratio=2.0, num_terms=num_terms)
rule = d._richardson_extrapolate._get_richardson_rule()
assert_array_almost_equal(rule,
true_vals[(num_terms, order)])
def test_complex(self):
truth = {
(1, 2, 8): [9.576480164718605e-07, -0.004167684167715291,
1.004166726519699],
(4, 2, 2): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(1, 2, 4): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(4, 1, 8): [-0.0039215686274510775, 1.0039215686274505],
(2, 2, 4): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(4, 2, 8): [9.576480164718605e-07, -0.004167684167715291,
1.004166726519699],
(3, 1, 8): [-0.0039215686274510775, 1.0039215686274505],
(4, 1, 2): [-0.06666666666666654, 1.0666666666666664],
(3, 1, 6): [-0.06666666666666654, 1.0666666666666664],
(1, 1, 8): [-0.0039215686274510775, 1.0039215686274505],
(2, 1, 8): [-0.0039215686274510775, 1.0039215686274505],
(4, 1, 4): [-0.06666666666666654, 1.0666666666666664],
(3, 1, 4): [-0.06666666666666654, 1.0666666666666664],
(2, 1, 4): [-0.06666666666666654, 1.0666666666666664],
(3, 2, 2): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(2, 2, 8): [9.576480164718605e-07, -0.004167684167715291,
1.004166726519699],
(2, 1, 6): [-0.06666666666666654, 1.0666666666666664],
(3, 1, 2): [-0.06666666666666654, 1.0666666666666664],
(4, 1, 6): [-0.06666666666666654, 1.0666666666666664],
(1, 1, 6): [-0.06666666666666654, 1.0666666666666664],
(1, 2, 2): [0.022222222222222185, -0.444444444444444,
1.4222222222222216],
(3, 2, 6): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(1, 1, 4): [-0.06666666666666654, 1.0666666666666664],
(2, 1, 2): [-0.06666666666666654, 1.0666666666666664],
(4, 2, 4): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(3, 2, 4): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(2, 2, 2): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(1, 2, 6): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(4, 2, 6): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616],
(1, 1, 2): [-0.33333333333333304, 1.333333333333333],
(3, 2, 8): [9.576480164718605e-07, -0.004167684167715291,
1.004166726519699],
(2, 2, 6): [0.0002614379084968331, -0.07111111111111235,
1.070849673202616]}
# t = dict()
for n in [1, 2, 3, 4]:
for num_terms in [1, 2]:
for order in range(2, 9, 2):
d = nd.Derivative(np.exp, n=n, method='complex',
order=order)
d._set_richardson_rule(step_ratio=2.0, num_terms=num_terms)
rule = d._richardson_extrapolate._get_richardson_rule()
# t[(n, num_terms, order)] = rule.tolist()
assert_array_almost_equal(rule,
truth[(n, num_terms, order)])
# print(t)
# self.assert_(False)
def test_forward_backward(self):
truth = {(1, 1): [-1., 2.],
(1, 2): [-0.33333333, 1.33333333],
(1, 3): [-0.14285714, 1.14285714],
(1, 4): [-0.06666667, 1.06666667],
(1, 5): [-0.03225806, 1.03225806],
(1, 6): [-0.01587302, 1.01587302],
(2, 1): [0.33333333, -2., 2.66666667],
(2, 2): [0.04761905, -0.57142857, 1.52380952],
(2, 3): [0.00952381, -0.22857143, 1.21904762],
(2, 4): [0.00215054, -0.10322581, 1.10107527],
(2, 5): [5.12032770e-04, -4.91551459e-02, 1.04864311e+00],
(2, 6): [1.24984377e-04, -2.39970004e-02, 1.02387202e+00]}
for method in ['forward', 'backward']:
for num_terms in [1, 2]:
for order in range(1, 7):
d = nd.Derivative(np.exp, method=method, order=order)
d._set_richardson_rule(step_ratio=2.0, num_terms=num_terms)
rule = d._richardson_extrapolate._get_richardson_rule()
assert_array_almost_equal(rule,
truth[(num_terms, order)])
def _example_(self):
def f(x, h):
return (np.exp(x + h) - np.exp(x - h)) / (2.)
# f = lambda x, h: (np.exp(x+h)-np.exp(x))
steps = [h for h in 2.0**-np.arange(10)]
df = [f(1, h) for h in steps]
print([dfi / hi for dfi, hi in zip(df, steps)])
step = nd.MaxStepGenerator(step_ratio=2.0)
for method in ['central']:
d = nd.Derivative(np.exp, step=step, method=method)
for order in [2, 6]:
d.order = order
r_extrap = nd.Richardson(step_ratio=2.0, method=method,
num_terms=2, order=order)
fd_rule = d._get_finite_difference_rule(step_ratio=2.0)
print(fd_rule)
df1, stepsi, _shape = d._apply_fd_rule(fd_rule, df, steps)
rule = r_extrap._get_richardson_rule()
df2, error, hi = r_extrap(df1, stepsi)
print(rule)
print(np.hstack((df2, error)))
self.assert_(False)
[docs]class TestStepGenerator(unittest.TestCase):
def test_default_generator(self):
step_gen = nd.MinStepGenerator(base_step=None, num_steps=10,
step_ratio=4, offset=-1)
h = np.array([h for h in step_gen(0)])
desired = np.array([3.58968236e-02, 8.97420590e-03, 2.24355147e-03,
5.60887869e-04, 1.40221967e-04, 3.50554918e-05,
8.76387295e-06, 2.19096824e-06, 5.47742059e-07,
1.36935515e-07])
assert_array_almost_equal((h - desired) / desired, 0)
def test_default_base_step(self):
step_gen = nd.MinStepGenerator(num_steps=1, offset=0)
h = [h for h in step_gen(0)]
desired = nd.EPS ** (1. / 2.5)
assert_array_almost_equal((h[0] - desired) / desired, 0)
def test_fixed_base_step(self):
desired = 0.1
step_gen = nd.MinStepGenerator(base_step=desired, num_steps=1, scale=2,
offset=0)
h = [h for h in step_gen(0)]
assert_array_almost_equal((h[0] - desired) / desired, 0)
[docs]class TestFornbergWeights(unittest.TestCase):
def test_weights(self):
x = np.r_[-1, 0, 1]
xbar = 0
k = 1
weights = nd.fornberg_weights(x, xbar, k)
np.testing.assert_allclose(weights, [-.5, 0, .5])
[docs]class TestDerivative(unittest.TestCase):
def test_infinite_functions(self):
def finf(x):
return np.inf
df = nd.Derivative(finf)
val = df(0)
self.assert_(np.isnan(val))
def _example_fd_mat(self):
fdmat = nd.Derivative._fd_matrix(step_ratio=2.0, parity=1, nterms=3)
_fd_rules = np.linalg.pinv(fdmat)
self.assert_(False)
def test_high_order_derivative_cos(self):
true_vals = (-1.0, 0.0, 1.0, 0.0, -1.0, 0.0)
methods = ['complex', 'multicomplex', 'central',
'forward', 'backward']
for method in methods:
n_max = dict(multicomplex=2, central=6).get(method, 5)
for n in range(1, n_max + 1):
true_val = true_vals[n - 1]
for order in range(2, 9, 2):
d3cos = nd.Derivative(np.cos, n=n, order=order,
method=method, full_output=True)
y, info = d3cos(np.pi / 2.0)
error = np.abs(y - true_val)
small = error <= info.error_estimate
if not small:
small = error < 10**(-12 + n)
if not small:
print('method=%s, n=%d, order=%d' % (method, n, order))
print(error, info.error_estimate)
# self.assertTrue(small)
assert_array_almost_equal(y, true_val, decimal=4)
# self.assert_(False)
def test_derivative_of_cos_x(self):
x = np.r_[0, np.pi / 6.0, np.pi / 2.0]
true_vals = (-np.sin(x), -np.cos(x), np.sin(x), np.cos(x), -np.sin(x),
-np.cos(x))
for method in ['complex', 'central', 'forward', 'backward']:
n_max = dict(complex=2, central=6).get(method, 5)
for n in range(1, n_max + 1):
true_val = true_vals[n - 1]
start, stop, step = dict(central=(2, 7, 2),
complex=(2, 3, 1)).get(method,
(1, 5, 1))
for order in range(start, stop, step):
d3cos = nd.Derivative(np.cos, n=n, order=order,
method=method, full_output=True)
y, info = d3cos(x)
error = np.abs(y - true_val)
small = error <= info.error_estimate
if not small.all():
small = np.where(small, small, error <= 10**(-11 + n))
if not small.all():
print('method=%s, n=%d, order=%d' % (method, n, order))
print(error, info.error_estimate)
assert_array_almost_equal(y, true_val, decimal=4)
# self.assertTrue(small.all())
# assert_allclose(y, true_val)
# self.assert_(False)
def test_default_scale(self):
for method, scale in zip(['complex', 'central', 'forward', 'backward',
'multicomplex'],
[1.35, 2.5, 2.5, 2.5, 1.35]):
np.testing.assert_allclose(scale, nd.default_scale(method, n=1))
def test_fun_with_additional_parameters(self):
'''Test for issue #9'''
def func(x, a, b=1):
return b * a * x * x * x
methods = ['forward', 'backward', 'central', 'complex', 'multicomplex']
dfuns = [nd.Gradient, nd.Derivative, nd.Jacobian, nd.Hessdiag,
nd.Hessian]
for dfun in dfuns:
for method in methods:
df = dfun(func, method=method)
val = df(0.0, 1.0, b=2)
assert_array_almost_equal(val, 0)
def test_derivative_cube(self):
'''Test for Issue 7'''
def cube(x):
return x * x * x
dcube = nd.Derivative(cube)
shape = (3, 2)
x = np.ones(shape) * 2
dx = dcube(x)
assert_array_almost_equal(list(dx.shape), list(shape),
decimal=8,
err_msg='Shape mismatch')
txt = 'First differing element %d\n value = %g,\n true value = %g'
for i, (val, tval) in enumerate(zip(dx.ravel(), (3 * x**2).ravel())):
assert_array_almost_equal(val, tval, decimal=8,
err_msg=txt % (i, val, tval))
def test_derivative_exp(self):
# derivative of exp(x), at x == 0
dexp = nd.Derivative(np.exp)
assert_array_almost_equal(dexp(0), np.exp(0), decimal=8)
def test_derivative_sin(self):
# Evaluate the indicated (default = first)
# derivative at multiple points
dsin = nd.Derivative(np.sin)
x = np.linspace(0, 2. * np.pi, 13)
y = dsin(x)
np.testing.assert_almost_equal(y, np.cos(x), decimal=8)
def test_backward_derivative_on_sinh(self):
# Compute the derivative of a function using a backward difference
# scheme. A backward scheme will only look below x0.
dsinh = nd.Derivative(np.sinh, method='backward')
self.assertAlmostEqual(dsinh(0.0), np.cosh(0.0))
def test_central_and_forward_derivative_on_log(self):
# Although a central rule may put some samples in the wrong places, it
# may still succeed
epsilon = nd.MinStepGenerator(num_steps=15, offset=0, step_ratio=2)
dlog = nd.Derivative(np.log, method='central', step=epsilon)
x = 0.001
self.assertAlmostEqual(dlog(x), 1.0 / x)
# But forcing the use of a one-sided rule may be smart anyway
dlog = nd.Derivative(np.log, method='forward', step=epsilon)
self.assertAlmostEqual(dlog(x), 1 / x)
[docs]class TestJacobian(unittest.TestCase):
def testjacobian(self):
xdata = np.reshape(np.arange(0, 1, 0.1), (-1, 1))
ydata = 1 + 2 * np.exp(0.75 * xdata)
def fun(c):
return (c[0] + c[1] * np.exp(c[2] * xdata) - ydata) ** 2
for method in ['complex', 'central', 'forward', 'backward']:
for order in [2, 4]:
Jfun = nd.Jacobian(fun, method=method, order=order)
J = Jfun([1, 2, 0.75]) # should be numerically zero
assert_array_almost_equal(J, np.zeros(J.shape))
[docs]class TestGradient(unittest.TestCase):
def testgradient(self):
def fun(x):
return np.sum(x ** 2)
dtrue = [2., 4., 6.]
for method in ['complex', 'central', 'backward', 'forward']:
for order in [2, 4]:
dfun = nd.Gradient(fun, method=method, order=order)
d = dfun([1, 2, 3])
assert_array_almost_equal(d, dtrue)
# self.assert_(False)
[docs]class TestHessdiag(unittest.TestCase):
def test_complex(self):
def fun(x):
return x[0] + x[1] ** 2 + x[2] ** 3
htrue = np.array([0., 2., 18.])
method = 'complex'
for num_steps in range(3, 7, 1):
steps = nd.MinStepGenerator(num_steps=num_steps,
use_exact_steps=True,
step_ratio=2.0, offset=4)
Hfun = nd.Hessdiag(fun, step=steps, method=method,
full_output=True)
hd, _info = Hfun([1, 2, 3])
_error = hd - htrue
assert_array_almost_equal(hd, htrue)
def test_fixed_step(self):
def fun(x):
return x[0] + x[1] ** 2 + x[2] ** 3
htrue = np.array([0., 2., 18.])
methods = ['multicomplex', 'complex', 'central', 'forward', 'backward']
for order in range(2, 7, 2):
steps = nd.MinStepGenerator(num_steps=order + 1,
use_exact_steps=True,
step_ratio=3., offset=0)
for method in methods:
Hfun = nd.Hessdiag(fun, step=steps, method=method, order=order,
full_output=True)
hd, _info = Hfun([1, 2, 3])
_error = hd - htrue
assert_array_almost_equal(hd, htrue)
def test_default_step(self):
def fun(x):
return x[0] + x[1] ** 2 + x[2] ** 3
htrue = np.array([0., 2., 18.])
methods = ['central2', 'central', 'multicomplex', 'complex', 'forward',
'backward']
for order in range(2, 7, 2):
for method in methods:
Hfun = nd.Hessdiag(fun, method=method, order=order,
full_output=True)
hd, _info = Hfun([1, 2, 3])
_error = hd - htrue
assert_array_almost_equal(hd, htrue)
[docs]class TestHessian(unittest.TestCase):
def test_hessian_cosIx_yI_at_I0_0I(self):
# cos(x-y), at (0,0)
def fun(xy):
return np.cos(xy[0] - xy[1])
htrue = [[-1., 1.], [1., -1.]]
methods = ['multicomplex', 'complex', 'central', 'central2', 'forward',
'backward']
for num_steps in [10, 1]:
step = nd.MinStepGenerator(num_steps=num_steps)
for method in methods:
Hfun2 = nd.Hessian(fun, method=method, step=step,
full_output=True)
h2, _info = Hfun2([0, 0])
# print(method, (h2-np.array(htrue)))
assert_array_almost_equal(h2, htrue)
if __name__ == '__main__':
unittest.main()