def test_all(self):
A = self.structure(self.n)
order = np.arange(self.n)
groups_1 = group_columns(A, order)
np.random.shuffle(order)
groups_2 = group_columns(A, order)
for method, groups, l, u in product(['2-point', '3-point', 'cs'],
[groups_1, groups_2],
[-np.inf, self.lb],
[np.inf, self.ub]):
J = approx_derivative(self.fun,
self.x0,
method=method,
bounds=(l, u),
sparsity=(A, groups))
assert_(isinstance(J, csr_matrix))
assert_allclose(J.toarray(), self.J_true, rtol=1e-6)
rel_step = np.full_like(self.x0, 1e-8)
rel_step[::2] *= -1
J = approx_derivative(self.fun,
self.x0,
method=method,
rel_step=rel_step,
sparsity=(A, groups))
assert_allclose(J.toarray(), self.J_true, rtol=1e-5)
def test_group_columns():
structure = [
[1, 1, 0, 0, 0, 0],
[1, 1, 1, 0, 0, 0],
[0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 0, 1, 1, 1],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0]
]
for transform in [np.asarray, csr_matrix, csc_matrix, lil_matrix]:
A = transform(structure)
order = np.arange(6)
groups_true = np.array([0, 1, 2, 0, 1, 2])
groups = group_columns(A, order)
assert_equal(groups, groups_true)
order = [1, 2, 4, 3, 5, 0]
groups_true = np.array([2, 0, 1, 2, 0, 1])
groups = group_columns(A, order)
assert_equal(groups, groups_true)
# Test repeatability.
groups_1 = group_columns(A)
groups_2 = group_columns(A)
assert_equal(groups_1, groups_2)
def test_group_columns():
structure = [
[1, 1, 0, 0, 0, 0],
[1, 1, 1, 0, 0, 0],
[0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 0, 1, 1, 1],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0]
]
for transform in [np.asarray, csr_matrix, csc_matrix, lil_matrix]:
A = transform(structure)
order = np.arange(6)
groups_true = np.array([0, 1, 2, 0, 1, 2])
groups = group_columns(A, order)
assert_equal(groups, groups_true)
order = [1, 2, 4, 3, 5, 0]
groups_true = np.array([2, 0, 1, 2, 0, 1])
groups = group_columns(A, order)
assert_equal(groups, groups_true)
# Test repeatability.
groups_1 = group_columns(A)
groups_2 = group_columns(A)
assert_equal(groups_1, groups_2)
def test_check_derivative(self):
def jac(x):
return csr_matrix(self.jac(x))
accuracy = check_derivative(self.fun,
jac,
self.x0,
bounds=(self.lb, self.ub),
sparse_diff=True)
assert_(accuracy < 1e-9)
A = self.structure(self.n)
groups = group_columns(A)
accuracy = check_derivative(self.fun,
jac,
self.x0,
bounds=(self.lb, self.ub),
sparse_diff=True,
sparsity=(A, groups))
assert_(accuracy < 1e-9)
accuracy = check_derivative(self.fun,
jac,
self.x0,
bounds=(self.lb, self.ub),
sparse_diff=False)
# Slightly worse accuracy because all elements are computed.
# Floating point issues make true 0 to some smal value, then
# it is divided by small step and as a result we have ~1e-9 element,
# which is actually should be zero.
assert_(accuracy < 1e-8)
def test_check_derivative(self):
def jac(x):
return csr_matrix(self.jac(x))
accuracy = check_derivative(
self.fun, jac, self.x0,
bounds=(self.lb, self.ub), sparse_diff=True)
assert_(accuracy < 1e-9)
A = self.structure(self.n)
groups = group_columns(A)
accuracy = check_derivative(
self.fun, jac, self.x0, bounds=(self.lb, self.ub),
sparse_diff=True, sparsity=(A, groups)
)
assert_(accuracy < 1e-9)
accuracy = check_derivative(
self.fun, jac, self.x0,
bounds=(self.lb, self.ub), sparse_diff=False)
# Slightly worse accuracy because all elements are computed.
# Floating point issues make true 0 to some smal value, then
# it is divided by small step and as a result we have ~1e-9 element,
# which is actually should be zero.
assert_(accuracy < 1e-8)
def _validate_jac(y0, fun_vectorized, atol=1e-8, jac=None, sparsity=None):
""" Taken from Scipy's Radau implementation. Returns a validated jacobian estimation function,
using if possible the sparsity pattern of the Jacobian to optimize the computation"""
n = y0.size
f = fun_vectorized(y0)
global jac_factor
if not (jac_factor is None):
if jac_factor.size != y0.size: # happens if we use twice the damped_newton on different size problems...
jac_factor = None
if jac is None:
if sparsity is not None:
if issparse(sparsity):
sparsity = csc_matrix(sparsity)
groups = group_columns(sparsity)
sparsity = (sparsity, groups)
def jac_wrapped(y):
f = fun_vectorized(y)
t = None # Scipy's numjac method assumes time is a separate variable
global jac_factor # TODO: object oriented implementation of the Newton solver ?
J, jac_factor = num_jac(fun=lambda t, y: fun_vectorized(y),
t=t,
y=y,
f=f,
threshold=atol,
factor=jac_factor,
sparsity=sparsity)
return J
J = jac_wrapped(y0)
elif callable(jac):
J = jac(y0)
if issparse(J):
J = csc_matrix(J)
def jac_wrapped(y):
return csc_matrix(jac(y), dtype=float)
else:
J = np.asarray(J, dtype=float)
def jac_wrapped(y):
return np.asarray(jac(y), dtype=float)
if J.shape != (n, n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}.".format((n, n), J.shape))
else:
if issparse(jac):
J = csc_matrix(jac)
else:
J = np.asarray(jac, dtype=float)
if J.shape != (n, n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}.".format((n, n), J.shape))
jac_wrapped = None
return jac_wrapped, J
def test_num_jac_sparse():
def fun(t, y):
e = y[1:]**3 - y[:-1]**2
z = np.zeros(y.shape[1])
return np.vstack((z, 3 * e)) + np.vstack((2 * e, z))
def structure(n):
A = np.zeros((n, n), dtype=int)
A[0, 0] = 1
A[0, 1] = 1
for i in range(1, n - 1):
A[i, i - 1:i + 2] = 1
A[-1, -1] = 1
A[-1, -2] = 1
return A
np.random.seed(0)
n = 20
y = np.random.randn(n)
A = structure(n)
groups = group_columns(A)
f = fun(0, y[:, None]).ravel()
# Compare dense and sparse results, assuming that dense implementation
# is correct (as it is straightforward).
J_num_sparse, factor_sparse = num_jac(fun,
0,
y.ravel(),
f,
1e-8,
None,
sparsity=(A, groups))
J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, None)
assert_allclose(J_num_dense,
J_num_sparse.toarray(),
rtol=1e-12,
atol=1e-14)
assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
# Take small factors to trigger their recomputing inside.
factor = np.random.uniform(0, 1e-12, size=n)
J_num_sparse, factor_sparse = num_jac(fun,
0,
y.ravel(),
f,
1e-8,
factor,
sparsity=(A, groups))
J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, factor)
assert_allclose(J_num_dense,
J_num_sparse.toarray(),
rtol=1e-12,
atol=1e-14)
assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
def test_all(self):
A = self.structure(self.n)
order = np.arange(self.n)
groups_1 = group_columns(A, order)
np.random.shuffle(order)
groups_2 = group_columns(A, order)
for method, groups, l, u in product(
['2-point', '3-point', 'cs'], [groups_1, groups_2],
[-np.inf, self.lb], [np.inf, self.ub]):
J = approx_derivative(self.fun, self.x0, method=method,
bounds=(l, u), sparsity=(A, groups))
assert_(isinstance(J, csr_matrix))
assert_allclose(J.toarray(), self.J_true, rtol=1e-6)
rel_step = 1e-8 * np.ones_like(self.x0)
rel_step[::2] *= -1
J = approx_derivative(self.fun, self.x0, method=method,
rel_step=rel_step, sparsity=(A, groups))
assert_allclose(J.toarray(), self.J_true, rtol=1e-5)
def check_jac_sparsity(jac_sparsity, m, n):
if jac_sparsity is None:
return None
if not issparse(jac_sparsity):
jac_sparsity = np.atleast_2d(jac_sparsity)
if jac_sparsity.shape != (m, n):
raise ValueError("`jac_sparsity` has wrong shape.")
return jac_sparsity, group_columns(jac_sparsity)
def check_jac_sparsity(jac_sparsity, m, n):
if jac_sparsity is None:
return None
if not issparse(jac_sparsity):
jac_sparsity = np.atleast_2d(jac_sparsity)
if jac_sparsity.shape != (m, n):
raise ValueError("`jac_sparsity` has wrong shape.")
return jac_sparsity, group_columns(jac_sparsity)
def _validate_jac(self, jac, sparsity):
t0 = self.t
y0 = self.y
if jac is None:
if sparsity is not None:
if issparse(sparsity):
sparsity = csc_matrix(sparsity)
groups = group_columns(sparsity)
sparsity = (sparsity, groups)
def jac_wrapped(t, y):
self.njev += 1
f = self.fun_single(t, y)
J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
self.atol, self.jac_factor,
sparsity)
return J
J = jac_wrapped(t0, y0)
elif callable(jac):
J = jac(t0, y0)
self.njev += 1
if issparse(J):
J = csc_matrix(J, dtype=y0.dtype)
def jac_wrapped(t, y):
self.njev += 1
return csc_matrix(jac(t, y), dtype=y0.dtype)
else:
J = np.asarray(J, dtype=y0.dtype)
def jac_wrapped(t, y):
self.njev += 1
return np.asarray(jac(t, y), dtype=y0.dtype)
if J.shape != (self.n, self.n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}.".format((self.n, self.n),
J.shape))
else:
if issparse(jac):
J = csc_matrix(jac, dtype=y0.dtype)
else:
J = np.asarray(jac, dtype=y0.dtype)
if J.shape != (self.n, self.n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}.".format((self.n, self.n),
J.shape))
jac_wrapped = None
return jac_wrapped, J
def _validate_jac(self, jac, sparsity):
t0 = self.t
y0 = self.y
if jac is None:
if sparsity is not None:
if issparse(sparsity):
sparsity = csc_matrix(sparsity)
groups = group_columns(sparsity)
sparsity = (sparsity, groups)
def jac_wrapped(t, y, f):
self.njev += 1
J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
self.atol, self.jac_factor,
sparsity)
return J
J = jac_wrapped(t0, y0, self.f)
elif callable(jac):
J = jac(t0, y0)
self.njev = 1
if issparse(J):
J = csc_matrix(J)
def jac_wrapped(t, y, _=None):
self.njev += 1
return csc_matrix(jac(t, y), dtype=float)
else:
J = np.asarray(J, dtype=float)
def jac_wrapped(t, y, _=None):
self.njev += 1
return np.asarray(jac(t, y), dtype=float)
if J.shape != (self.n, self.n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}."
.format((self.n, self.n), J.shape))
else:
if issparse(jac):
J = csc_matrix(jac)
else:
J = np.asarray(jac, dtype=float)
if J.shape != (self.n, self.n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}."
.format((self.n, self.n), J.shape))
jac_wrapped = None
return jac_wrapped, J
def check_jacobian(self):
accuracy = []
if self.sparsity is not None:
sparse_diff = True
groups = group_columns(self.sparsity)
sparsity = (self.sparsity, groups)
else:
sparse_diff = False
sparsity = None
for x0, bounds in self.specs:
x = x0 + np.random.randn(self.n)
acc = check_derivative(self.fun, self.jac, x, bounds=bounds,
sparse_diff=sparse_diff, sparsity=sparsity)
accuracy.append(acc)
return accuracy
def test_num_jac_sparse():
def fun(t, y):
e = y[1:]**3 - y[:-1]**2
z = np.zeros(y.shape[1])
return np.vstack((z, 3 * e)) + np.vstack((2 * e, z))
def structure(n):
A = np.zeros((n, n), dtype=int)
A[0, 0] = 1
A[0, 1] = 1
for i in range(1, n - 1):
A[i, i - 1: i + 2] = 1
A[-1, -1] = 1
A[-1, -2] = 1
return A
np.random.seed(0)
n = 20
y = np.random.randn(n)
A = structure(n)
groups = group_columns(A)
f = fun(0, y[:, None]).ravel()
# Compare dense and sparse results, assuming that dense implementation
# is correct (as it is straightforward).
J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, None,
sparsity=(A, groups))
J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, None)
assert_allclose(J_num_dense, J_num_sparse.toarray(),
rtol=1e-12, atol=1e-14)
assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
# Take small factors to trigger their recomputing inside.
factor = np.random.uniform(0, 1e-12, size=n)
J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, factor,
sparsity=(A, groups))
J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, factor)
assert_allclose(J_num_dense, J_num_sparse.toarray(),
rtol=1e-12, atol=1e-14)
assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
