def test_univariate(self):
np.random.seed(12345)
for x in np.linspace(-5, 5, num=11):
A = np.array([[x]])
assert_allclose(expm_cond(A), abs(x))
for x in np.logspace(-2, 2, num=11):
A = np.array([[x]])
assert_allclose(expm_cond(A), abs(x))
for i in range(10):
A = np.random.randn(1, 1)
assert_allclose(expm_cond(A), np.absolute(A)[0, 0])
def test_expm_cond_fuzz(self):
np.random.seed(12345)
eps = 1e-5
nsamples = 10
for i in range(nsamples):
n = np.random.randint(2, 5)
A = np.random.randn(n, n)
A_norm = scipy.linalg.norm(A)
X = expm(A)
X_norm = scipy.linalg.norm(X)
kappa = expm_cond(A)
# Look for the small perturbation that gives the greatest
# relative error.
f = functools.partial(_help_expm_cond_search, A, A_norm, X, X_norm, eps)
guess = np.ones(n * n)
out = minimize(f, guess, method="L-BFGS-B")
xopt = out.x
yopt = f(xopt)
p_best = eps * _normalized_like(np.reshape(xopt, A.shape), A)
p_best_relerr = _relative_error(expm, A, p_best)
assert_allclose(p_best_relerr, -yopt * eps)
# Check that the identified perturbation indeed gives greater
# relative error than random perturbations with similar norms.
for j in range(5):
p_rand = eps * _normalized_like(np.random.randn(*A.shape), A)
assert_allclose(norm(p_best), norm(p_rand))
p_rand_relerr = _relative_error(expm, A, p_rand)
assert_array_less(p_rand_relerr, p_best_relerr)
# The greatest relative error should not be much greater than
# eps times the condition number kappa.
# In the limit as eps approaches zero it should never be greater.
assert_array_less(p_best_relerr, (1 + 2 * eps) * eps * kappa)
def test_expm_bad_condition_number(self):
A = np.array([
[-1.128679820, 9.614183771e4, -4.524855739e9, 2.924969411e14],
[0, -1.201010529, 9.634696872e4, -4.681048289e9],
[0, 0, -1.132893222, 9.532491830e4],
[0, 0, 0, -1.179475332],
])
kappa = expm_cond(A)
assert_array_less(1e36, kappa)
def test_expm_bad_condition_number(self):
A = np.array([
[-1.128679820, 9.614183771e4, -4.524855739e9, 2.924969411e14],
[0, -1.201010529, 9.634696872e4, -4.681048289e9],
[0, 0, -1.132893222, 9.532491830e4],
[0, 0, 0, -1.179475332],
])
kappa = expm_cond(A)
assert_array_less(1e36, kappa)
def test_expm_cond_fuzz(self):
np.random.seed(12345)
eps = 1e-5
nsamples = 10
for i in range(nsamples):
n = np.random.randint(2, 5)
A = np.random.randn(n, n)
A_norm = scipy.linalg.norm(A)
X = expm(A)
X_norm = scipy.linalg.norm(X)
kappa = expm_cond(A)
# Look for the small perturbation that gives the greatest
# relative error.
f = functools.partial(_help_expm_cond_search, A, A_norm, X, X_norm,
eps)
guess = np.ones(n * n)
out = minimize(f, guess, method='L-BFGS-B')
xopt = out.x
yopt = f(xopt)
p_best = eps * _normalized_like(np.reshape(xopt, A.shape), A)
p_best_relerr = _relative_error(expm, A, p_best)
assert_allclose(p_best_relerr, -yopt * eps)
# Check that the identified perturbation indeed gives greater
# relative error than random perturbations with similar norms.
for j in range(5):
p_rand = eps * _normalized_like(np.random.randn(*A.shape), A)
assert_allclose(norm(p_best), norm(p_rand))
p_rand_relerr = _relative_error(expm, A, p_rand)
assert_array_less(p_rand_relerr, p_best_relerr)
# The greatest relative error should not be much greater than
# eps times the condition number kappa.
# In the limit as eps approaches zero it should never be greater.
assert_array_less(p_best_relerr, (1 + 2 * eps) * eps * kappa)
def test_expm_cond_smoke(self):
np.random.seed(1234)
for n in range(1, 4):
A = np.random.randn(n, n)
kappa = expm_cond(A)
assert_array_less(0, kappa)
def test_expm_cond_smoke(self):
np.random.seed(1234)
for n in range(1, 4):
A = np.random.randn(n, n)
kappa = expm_cond(A)
assert_array_less(0, kappa)
