def d2psi(x, c):
"""Second derivative of psi"""
from numpy.polynomial.hermite import hermval, hermder
yp = hermval(x, hermder(hermder(c))) * np.exp(-x**2 / 2)
yp += -x * hermval(x, hermder(c)) * np.exp(-x**2 / 2)
yp += -psi(x, c) - x * dpsi(x, c)
return yp
def test_hermder_axis(self):
# get_inds that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([herm.hermder(c) for c in c2d.T]).T
res = herm.hermder(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([herm.hermder(c) for c in c2d])
res = herm.hermder(c2d, axis=1)
assert_almost_equal(res, tgt)
def test_hermder_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([herm.hermder(c) for c in c2d.T]).T
res = herm.hermder(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([herm.hermder(c) for c in c2d])
res = herm.hermder(c2d, axis=1)
assert_almost_equal(res, tgt)
def evaluate_basis_derivative_all(self, x=None, k=0):
if x is None:
x = self.mesh(False, False)
V = self.vandermonde(x)
M = V.shape[1]
X = x[:, np.newaxis]
if k == 1:
D = np.zeros((M, M))
D[:-1, :] = hermite.hermder(np.eye(M), 1)
W = np.dot(V, D)
W -= V*X
V = W*np.exp(-X**2/2)
V *= self.factor(np.arange(M))[np.newaxis, :]
elif k == 2:
W = (X**2 - 1 - 2*np.arange(M)[np.newaxis, :])*V
V = W*self.factor(np.arange(M))[np.newaxis, :]*np.exp(-X**2/2)
#D = np.zeros((M, M))
#D[:-1, :] = hermite.hermder(np.eye(M), 1)
#W = np.dot(V, D)
#W = -2*X*W
#D[-2:] = 0
#D[:-2, :] = hermite.hermder(np.eye(M), 2)
#W += np.dot(V, D)
#W += (X**2-1)*V
#W *= np.exp(-X**2/2)*self.factor(np.arange(M))[np.newaxis, :]
#V[:] = W
elif k == 0:
V *= np.exp(-X**2/2)*self.factor(np.arange(M))[np.newaxis, :]
else:
raise NotImplementedError
return V
def test_hermder(self):
# get_inds exceptions
assert_raises(TypeError, herm.hermder, [0], .5)
assert_raises(ValueError, herm.hermder, [0], -1)
# get_inds that zeroth derivative does nothing
for i in range(5):
tgt = [0] * i + [1]
res = herm.hermder(tgt, m=0)
assert_equal(trim(res), trim(tgt))
# get_inds that derivation is the inverse of integration
for i in range(5):
for j in range(2, 5):
tgt = [0] * i + [1]
res = herm.hermder(herm.hermint(tgt, m=j), m=j)
assert_almost_equal(trim(res), trim(tgt))
# get_inds derivation with scaling
for i in range(5):
for j in range(2, 5):
tgt = [0] * i + [1]
res = herm.hermder(herm.hermint(tgt, m=j, scl=2), m=j, scl=.5)
assert_almost_equal(trim(res), trim(tgt))
def test_hermder(self) :
# check exceptions
assert_raises(ValueError, herm.hermder, [0], .5)
assert_raises(ValueError, herm.hermder, [0], -1)
# check that zeroth deriviative does nothing
for i in range(5) :
tgt = [0]*i + [1]
res = herm.hermder(tgt, m=0)
assert_equal(trim(res), trim(tgt))
# check that derivation is the inverse of integration
for i in range(5) :
for j in range(2,5) :
tgt = [0]*i + [1]
res = herm.hermder(herm.hermint(tgt, m=j), m=j)
assert_almost_equal(trim(res), trim(tgt))
# check derivation with scaling
for i in range(5) :
for j in range(2,5) :
tgt = [0]*i + [1]
res = herm.hermder(herm.hermint(tgt, m=j, scl=2), m=j, scl=.5)
assert_almost_equal(trim(res), trim(tgt))
def evaluate_basis_derivative(self, x=None, i=0, k=0, output_array=None):
if x is None:
x = self.mesh(False, False)
x = np.atleast_1d(x)
v = eval_hermite(i, x, out=output_array)
if k == 1:
D = np.zeros((self.N, self.N))
D[:-1, :] = hermite.hermder(np.eye(self.N), 1)
V = np.dot(v, D)
V -= v*x[:, np.newaxis]
V *= np.exp(-x**2/2)[:, np.newaxis]*self.factor(i)
v[:] = V
elif k == 2:
W = (x[:, np.newaxis]**2 - 1 - 2*i)*V
v[:] = W*self.factor(i)*np.exp(-x**2/2)[:, np.newaxis]
elif k == 0:
v *= np.exp(-x**2/2)[:, np.newaxis]*self.factor(i)
else:
raise NotImplementedError
return v
def dpsi(x, c):
from numpy.polynomial.hermite import hermval, hermder
yp = hermval(x, hermder(c)) * np.exp(-x**2 / 2) - x * psi(x, c)
return yp