def rmatrixu(u, theta):
"""Return a rotation matrix caused by a right hand rotation of theta
radians around vector u.
"""
if numpy.allclose(theta, 0.0) or numpy.allclose(numpy.dot(u, u), 0.0):
return numpy.identity(3, float)
x, y, z = normalize(u)
sa = math.sin(theta)
ca = math.cos(theta)
R = numpy.array([[
1.0 + (1.0 - ca) * (x * x - 1.0), -z * sa +
(1.0 - ca) * x * y, y * sa + (1.0 - ca) * x * z
],
[
z * sa + (1.0 - ca) * x * y, 1.0 + (1.0 - ca) *
(y * y - 1.0), -x * sa + (1.0 - ca) * y * z
],
[
-y * sa + (1.0 - ca) * x * z, x * sa +
(1.0 - ca) * y * z, 1.0 + (1.0 - ca) * (z * z - 1.0)
]], float)
try:
assert numpy.allclose(linalg.determinant(R), 1.0)
except AssertionError:
print "rmatrixu(%s, %f) determinant(R)=%f" % (u, theta,
linalg.determinant(R))
raise
return R
def rmatrixz(vec):
"""Return a rotation matrix which transforms the coordinate system
such that the vector vec is aligned along the z axis.
"""
u, v, w = normalize(vec)
d = math.sqrt(u*u + v*v)
if d != 0.0:
Rxz = numpy.array([ [ u/d, v/d, 0.0 ],
[ -v/d, u/d, 0.0 ],
[ 0.0, 0.0, 1.0 ] ], float)
else:
Rxz = numpy.identity(3, float)
Rxz2z = numpy.array([ [ w, 0.0, -d],
[ 0.0, 1.0, 0.0],
[ d, 0.0, w] ], float)
R = numpy.dot(Rxz2z, Rxz)
try:
assert numpy.allclose(linalg.determinant(R), 1.0)
except AssertionError:
print "rmatrixz(%s) determinant(R)=%f" % (vec, linalg.determinant(R))
raise
return R
def rmatrixz(vec):
"""Return a rotation matrix which transforms the coordinate system
such that the vector vec is aligned along the z axis.
"""
u, v, w = normalize(vec)
d = math.sqrt(u * u + v * v)
if d != 0.0:
Rxz = numpy.array(
[[u / d, v / d, 0.0], [-v / d, u / d, 0.0], [0.0, 0.0, 1.0]],
float)
else:
Rxz = numpy.identity(3, float)
Rxz2z = numpy.array([[w, 0.0, -d], [0.0, 1.0, 0.0], [d, 0.0, w]], float)
R = numpy.dot(Rxz2z, Rxz)
try:
assert numpy.allclose(linalg.determinant(R), 1.0)
except AssertionError:
print "rmatrixz(%s) determinant(R)=%f" % (vec, linalg.determinant(R))
raise
return R
def calc_inertia_tensor(atom_iter, origin):
"""Calculate a moment-of-inertia tensor at the given origin assuming all
atoms have the same mass.
"""
I = numpy.zeros((3, 3), float)
for atm in atom_iter:
x = atm.position - origin
I[0, 0] += x[1]**2 + x[2]**2
I[1, 1] += x[0]**2 + x[2]**2
I[2, 2] += x[0]**2 + x[1]**2
I[0, 1] += -x[0] * x[1]
I[1, 0] += -x[0] * x[1]
I[0, 2] += -x[0] * x[2]
I[2, 0] += -x[0] * x[2]
I[1, 2] += -x[1] * x[2]
I[2, 1] += -x[1] * x[2]
evals, evecs = linalg.eigenvectors(I)
## order the tensor such that the largest
## principal component is along the z-axis, and
## the second largest is along the y-axis
if evals[0] >= evals[1] and evals[0] >= evals[2]:
if evals[1] >= evals[2]:
R = numpy.array((evecs[2], evecs[1], evecs[0]), float)
else:
R = numpy.array((evecs[1], evecs[2], evecs[0]), float)
elif evals[1] >= evals[0] and evals[1] >= evals[2]:
if evals[0] >= evals[2]:
R = numpy.array((evecs[2], evecs[0], evecs[1]), float)
else:
R = numpy.array((evecs[0], evecs[2], evecs[1]), float)
elif evals[2] >= evals[0] and evals[2] >= evals[1]:
if evals[0] >= evals[1]:
R = numpy.array((evecs[1], evecs[0], evecs[2]), float)
else:
R = numpy.array((evecs[0], evecs[1], evecs[2]), float)
## make sure the tensor is right-handed
if numpy.allclose(linalg.determinant(R), -1.0):
I = numpy.identity(3, float)
I[0, 0] = -1.0
R = numpy.dot(I, R)
assert numpy.allclose(linalg.determinant(R), 1.0)
return R
def calc_CCuij(U, V):
"""Calculate the cooralation coefficent for anisotropic ADP tensors U
and V.
"""
invU = linalg.inverse(U)
invV = linalg.inverse(V)
det_invU = linalg.determinant(invU)
det_invV = linalg.determinant(invV)
return (math.sqrt(math.sqrt(det_invU * det_invV)) / math.sqrt(
(1.0 / 8.0) * linalg.determinant(invU + invV)))
def calc_inertia_tensor(atom_iter, origin):
"""Calculate a moment-of-inertia tensor at the given origin assuming all
atoms have the same mass.
"""
I = numpy.zeros((3,3), float)
for atm in atom_iter:
x = atm.position - origin
I[0,0] += x[1]**2 + x[2]**2
I[1,1] += x[0]**2 + x[2]**2
I[2,2] += x[0]**2 + x[1]**2
I[0,1] += - x[0]*x[1]
I[1,0] += - x[0]*x[1]
I[0,2] += - x[0]*x[2]
I[2,0] += - x[0]*x[2]
I[1,2] += - x[1]*x[2]
I[2,1] += - x[1]*x[2]
evals, evecs = linalg.eigenvectors(I)
## order the tensor such that the largest
## principal component is along the z-axis, and
## the second largest is along the y-axis
if evals[0] >= evals[1] and evals[0] >= evals[2]:
if evals[1] >= evals[2]:
R = numpy.array((evecs[2], evecs[1], evecs[0]), float)
else:
R = numpy.array((evecs[1], evecs[2], evecs[0]), float)
elif evals[1] >= evals[0] and evals[1] >= evals[2]:
if evals[0] >= evals[2]:
R = numpy.array((evecs[2], evecs[0], evecs[1]), float)
else:
R = numpy.array((evecs[0], evecs[2], evecs[1]), float)
elif evals[2] >= evals[0] and evals[2] >= evals[1]:
if evals[0] >= evals[1]:
R = numpy.array((evecs[1], evecs[0], evecs[2]), float)
else:
R = numpy.array((evecs[0], evecs[1], evecs[2]), float)
## make sure the tensor is right-handed
if numpy.allclose(linalg.determinant(R), -1.0):
I = numpy.identity(3, float)
I[0,0] = -1.0
R = numpy.dot(I, R)
assert numpy.allclose(linalg.determinant(R), 1.0)
return R
def calc_CCuij(U, V):
"""Calculate the correlation coefficient for anisotropic ADP tensors U
and V.
"""
## FIXME: Check for non-positive Uij's, 2009-08-19
invU = linalg.inverse(U)
invV = linalg.inverse(V)
#invU = internal_inv3x3(U)
#invV = internal_inv3x3(V)
det_invU = linalg.determinant(invU)
det_invV = linalg.determinant(invV)
return ( math.sqrt(math.sqrt(det_invU * det_invV)) /
math.sqrt((1.0/8.0) * linalg.determinant(invU + invV)) )
def calc_CCuij(U, V):
"""Calculate the correlation coefficient for anisotropic ADP tensors U
and V.
"""
## FIXME: Check for non-positive Uij's, 2009-08-19
invU = linalg.inverse(U)
invV = linalg.inverse(V)
#invU = internal_inv3x3(U)
#invV = internal_inv3x3(V)
det_invU = linalg.determinant(invU)
det_invV = linalg.determinant(invV)
return (math.sqrt(math.sqrt(det_invU * det_invV)) / math.sqrt(
(1.0 / 8.0) * linalg.determinant(invU + invV)))
def matrix_is_symplectic(self, m, tolerance=1.0e-12):
"""
Confirm that a given matrix $M$ is symplectic to within
numerical tolerance.
This is done by taking the 4 submatrices:
1. $A = M[0::2, 0::2]$, i.e., configuration coordinates only,
2. $B = M[0::2, 1::2]$, i.e., configuration rows, momenta cols,
3. $C = M[1::2, 0::2]$, i.e., momenta rows, configuration cols,
4. $D = M[1::2, 1::2]$, i.e., momenta only,
and verifying the following symplectic identities:-
1. $MJM^{T} = J$, the $2n\\times 2n$ sympletic matrix,
2. $AD^{T}-BC^{T} = I$, the $n\\times n$ identity,
3. $AB^{T}-BA^{T} = Z$, the $n\\times n$ zero,
4. $CD^{T}-DC^{T} = Z$.
Finally, we confirm that $\\det{M} = 1$.
"""
det = determinant(m)
j = self.skew_symmetric_matrix()
approx_j = matrixmultiply(m, matrixmultiply(j, transpose(m)))
a = m[0::2, 0::2] #even, even
b = m[0::2, 1::2] #even, odd
c = m[1::2, 0::2] #odd, even
d = m[1::2, 1::2] #odd, odd
i = self.identity_matrix(self.dof())
approx_i = matrixmultiply(a, transpose(d)) - matrixmultiply(b, transpose(c))
approx_z0 = matrixmultiply(a, transpose(b)) - matrixmultiply(b, transpose(a))
approx_z1 = matrixmultiply(c, transpose(d)) - matrixmultiply(d, transpose(c))
norm = self.matrix_norm
logger.info('Matrix from diagonal to equilibrium coordinates:')
logger.info('[output supressed]') #logger.info(m)
logger.info('error in determinant: %s', abs(det-1.0))
logger.info('error in symplectic identity #1: %s', norm(approx_j - j))
logger.info('error in symplectic identity #2: %s', norm(approx_i - i))
logger.info('error in symplectic identity #3: %s', norm(approx_z0))
logger.info('error in symplectic identity #4: %s', norm(approx_z1))
okay = True
if not (abs(det-1.0) < tolerance):
okay = False
if not (norm(approx_j - j) < tolerance):
okay = False
if not (norm(approx_i - i) < tolerance):
okay = False
if not (norm(approx_z0) < tolerance):
okay = False
if not (norm(approx_z1) < tolerance):
okay = False
return okay
def rmatrixquaternion(q):
"""Create a rotation matrix from q quaternion rotation.
Quaternions are typed as Numeric Python numpy.arrays of length 4.
"""
assert numpy.allclose(math.sqrt(numpy.dot(q, q)), 1.0)
x, y, z, w = q
xx = x * x
xy = x * y
xz = x * z
xw = x * w
yy = y * y
yz = y * z
yw = y * w
zz = z * z
zw = z * w
r00 = 1.0 - 2.0 * (yy + zz)
r01 = 2.0 * (xy - zw)
r02 = 2.0 * (xz + yw)
r10 = 2.0 * (xy + zw)
r11 = 1.0 - 2.0 * (xx + zz)
r12 = 2.0 * (yz - xw)
r20 = 2.0 * (xz - yw)
r21 = 2.0 * (yz + xw)
r22 = 1.0 - 2.0 * (xx + yy)
R = numpy.array([[r00, r01, r02], [r10, r11, r12], [r20, r21, r22]], float)
assert numpy.allclose(linalg.determinant(R), 1.0)
return R
def calc_DP2uij(U, V):
"""Calculate the square of the volumetric difference in the probability
density function of anisotropic ADP tensors U and V.
"""
invU = linalg.inverse(U)
invV = linalg.inverse(V)
det_invU = linalg.determinant(invU)
det_invV = linalg.determinant(invV)
Pu2 = math.sqrt( det_invU / (64.0 * Constants.PI3) )
Pv2 = math.sqrt( det_invV / (64.0 * Constants.PI3) )
Puv = math.sqrt(
(det_invU * det_invV) / (8.0*Constants.PI3 * linalg.determinant(invU + invV)))
dP2 = Pu2 + Pv2 - (2.0 * Puv)
return dP2
def calc_DP2uij(U, V):
"""Calculate the square of the volumetric difference in the probability
density function of anisotropic ADP tensors U and V.
"""
invU = linalg.inverse(U)
invV = linalg.inverse(V)
det_invU = linalg.determinant(invU)
det_invV = linalg.determinant(invV)
Pu2 = math.sqrt(det_invU / (64.0 * Constants.PI3))
Pv2 = math.sqrt(det_invV / (64.0 * Constants.PI3))
Puv = math.sqrt((det_invU * det_invV) /
(8.0 * Constants.PI3 * linalg.determinant(invU + invV)))
dP2 = Pu2 + Pv2 - (2.0 * Puv)
return dP2
def rmatrixu(u, theta):
"""Return a rotation matrix caused by a right hand rotation of theta
radians around vector u.
"""
if numpy.allclose(theta, 0.0) or numpy.allclose(numpy.dot(u,u), 0.0):
return numpy.identity(3, float)
x, y, z = normalize(u)
sa = math.sin(theta)
ca = math.cos(theta)
R = numpy.array(
[[1.0+(1.0-ca)*(x*x-1.0), -z*sa+(1.0-ca)*x*y, y*sa+(1.0-ca)*x*z],
[z*sa+(1.0-ca)*x*y, 1.0+(1.0-ca)*(y*y-1.0), -x*sa+(1.0-ca)*y*z],
[-y*sa+(1.0-ca)*x*z, x*sa+(1.0-ca)*y*z, 1.0+(1.0-ca)*(z*z-1.0)]], float)
try:
assert numpy.allclose(linalg.determinant(R), 1.0)
except AssertionError:
print "rmatrixu(%s, %f) determinant(R)=%f" % (
u, theta, linalg.determinant(R))
raise
return R
def _gaussian(self, mean, cvm, x):
m = len(mean)
assert cvm.shape == (m, m), \
'bad sized covariance matrix, %s' % str(cvm.shape)
try:
det = LinearAlgebra.determinant(cvm)
inv = LinearAlgebra.inverse(cvm)
a = det ** -0.5 * (2 * Numeric.pi) ** (-m / 2.0)
dx = x - mean
b = -0.5 * Numeric.matrixmultiply( \
Numeric.matrixmultiply(dx, inv), dx)
return a * Numeric.exp(b)
except OverflowError:
# happens when the exponent is negative infinity - i.e. b = 0
# i.e. the inverse of cvm is huge (cvm is almost zero)
return 0
def rmatrix(alpha, beta, gamma):
"""Return a rotation matrix based on the Euler angles alpha,
beta, and gamma in radians.
"""
cosA = math.cos(alpha)
cosB = math.cos(beta)
cosG = math.cos(gamma)
sinA = math.sin(alpha)
sinB = math.sin(beta)
sinG = math.sin(gamma)
R = numpy.array(
[[cosB*cosG, cosG*sinA*sinB-cosA*sinG, cosA*cosG*sinB+sinA*sinG],
[cosB*sinG, cosA*cosG+sinA*sinB*sinG, cosA*sinB*sinG-cosG*sinA ],
[-sinB, cosB*sinA, cosA*cosB ]], float)
assert numpy.allclose(linalg.determinant(R), 1.0)
return R
def quaternionrmatrix(R):
"""Return a quaternion calculated from the argument rotation matrix R.
"""
assert numpy.allclose(linalg.determinant(R), 1.0)
t = numpy.trace(R) + 1.0
if t > 1e-5:
w = math.sqrt(1.0 + numpy.trace(R)) / 2.0
w4 = 4.0 * w
x = (R[2, 1] - R[1, 2]) / w4
y = (R[0, 2] - R[2, 0]) / w4
z = (R[1, 0] - R[0, 1]) / w4
else:
if R[0, 0] > R[1, 1] and R[0, 0] > R[2, 2]:
S = math.sqrt(1.0 + R[0, 0] - R[1, 1] - R[2, 2]) * 2.0
x = 0.25 * S
y = (R[0, 1] + R[1, 0]) / S
z = (R[0, 2] + R[2, 0]) / S
w = (R[1, 2] - R[2, 1]) / S
elif R[1, 1] > R[2, 2]:
S = math.sqrt(1.0 + R[1, 1] - R[0, 0] - R[2, 2]) * 2.0
x = (R[0, 1] + R[1, 0]) / S
y = 0.25 * S
z = (R[1, 2] + R[2, 1]) / S
w = (R[0, 2] - R[2, 0]) / S
else:
S = math.sqrt(1.0 + R[2, 2] - R[0, 0] - R[1, 1]) * 2
x = (R[0, 2] + R[2, 0]) / S
y = (R[1, 2] + R[2, 1]) / S
z = 0.25 * S
w = (R[0, 1] - R[1, 0]) / S
q = numpy.array((x, y, z, w), float)
assert numpy.allclose(math.sqrt(numpy.dot(q, q)), 1.0)
return q
def quaternionrmatrix(R):
"""Return a quaternion calculated from the argument rotation matrix R.
"""
assert numpy.allclose(linalg.determinant(R), 1.0)
t = numpy.trace(R) + 1.0
if t>1e-5:
w = math.sqrt(1.0 + numpy.trace(R)) / 2.0
w4 = 4.0 * w
x = (R[2,1] - R[1,2]) / w4
y = (R[0,2] - R[2,0]) / w4
z = (R[1,0] - R[0,1]) / w4
else:
if R[0,0]>R[1,1] and R[0,0]>R[2,2]:
S = math.sqrt(1.0 + R[0,0] - R[1,1] - R[2,2]) * 2.0
x = 0.25 * S
y = (R[0,1] + R[1,0]) / S
z = (R[0,2] + R[2,0]) / S
w = (R[1,2] - R[2,1]) / S
elif R[1,1]>R[2,2]:
S = math.sqrt(1.0 + R[1,1] - R[0,0] - R[2,2]) * 2.0;
x = (R[0,1] + R[1,0]) / S;
y = 0.25 * S;
z = (R[1,2] + R[2,1]) / S;
w = (R[0,2] - R[2,0]) / S;
else:
S = math.sqrt(1.0 + R[2,2] - R[0,0] - R[1,1]) * 2;
x = (R[0,2] + R[2,0]) / S;
y = (R[1,2] + R[2,1]) / S;
z = 0.25 * S;
w = (R[0,1] - R[1,0] ) / S;
q = numpy.array((x, y, z, w), float)
assert numpy.allclose(math.sqrt(numpy.dot(q,q)), 1.0)
return q
def rmatrix(alpha, beta, gamma):
"""Return a rotation matrix based on the Euler angles alpha,
beta, and gamma in radians.
"""
cosA = math.cos(alpha)
cosB = math.cos(beta)
cosG = math.cos(gamma)
sinA = math.sin(alpha)
sinB = math.sin(beta)
sinG = math.sin(gamma)
R = numpy.array([[
cosB * cosG, cosG * sinA * sinB - cosA * sinG,
cosA * cosG * sinB + sinA * sinG
],
[
cosB * sinG, cosA * cosG + sinA * sinB * sinG,
cosA * sinB * sinG - cosG * sinA
], [-sinB, cosB * sinA, cosA * cosB]], float)
assert numpy.allclose(linalg.determinant(R), 1.0)
return R
def rmatrixquaternion(q):
"""Create a rotation matrix from q quaternion rotation.
Quaternions are typed as Numeric Python numpy.arrays of length 4.
"""
assert numpy.allclose(math.sqrt(numpy.dot(q,q)), 1.0)
x, y, z, w = q
xx = x*x
xy = x*y
xz = x*z
xw = x*w
yy = y*y
yz = y*z
yw = y*w
zz = z*z
zw = z*w
r00 = 1.0 - 2.0 * (yy + zz)
r01 = 2.0 * (xy - zw)
r02 = 2.0 * (xz + yw)
r10 = 2.0 * (xy + zw)
r11 = 1.0 - 2.0 * (xx + zz)
r12 = 2.0 * (yz - xw)
r20 = 2.0 * (xz - yw)
r21 = 2.0 * (yz + xw)
r22 = 1.0 - 2.0 * (xx + yy)
R = numpy.array([[r00, r01, r02],
[r10, r11, r12],
[r20, r21, r22]], float)
assert numpy.allclose(linalg.determinant(R), 1.0)
return R
def matrix_is_symplectic(self, m, tolerance=1.0e-12):
"""
Confirm that a given matrix $M$ is symplectic to within
numerical tolerance.
This is done by taking the 4 submatrices:
1. $A = M[0::2, 0::2]$, i.e., configuration coordinates only,
2. $B = M[0::2, 1::2]$, i.e., configuration rows, momenta cols,
3. $C = M[1::2, 0::2]$, i.e., momenta rows, configuration cols,
4. $D = M[1::2, 1::2]$, i.e., momenta only,
and verifying the following symplectic identities:-
1. $MJM^{T} = J$, the $2n\\times 2n$ sympletic matrix,
2. $AD^{T}-BC^{T} = I$, the $n\\times n$ identity,
3. $AB^{T}-BA^{T} = Z$, the $n\\times n$ zero,
4. $CD^{T}-DC^{T} = Z$.
Finally, we confirm that $\\det{M} = 1$.
"""
det = determinant(m)
j = self.skew_symmetric_matrix()
approx_j = matrixmultiply(m, matrixmultiply(j, transpose(m)))
a = m[0::2, 0::2] #even, even
b = m[0::2, 1::2] #even, odd
c = m[1::2, 0::2] #odd, even
d = m[1::2, 1::2] #odd, odd
i = self.identity_matrix(self.dof())
approx_i = matrixmultiply(a, transpose(d)) - matrixmultiply(
b, transpose(c))
approx_z0 = matrixmultiply(a, transpose(b)) - matrixmultiply(
b, transpose(a))
approx_z1 = matrixmultiply(c, transpose(d)) - matrixmultiply(
d, transpose(c))
norm = self.matrix_norm
logger.info('Matrix from diagonal to equilibrium coordinates:')
logger.info('[output supressed]') #logger.info(m)
logger.info('error in determinant: %s', abs(det - 1.0))
logger.info('error in symplectic identity #1: %s', norm(approx_j - j))
logger.info('error in symplectic identity #2: %s', norm(approx_i - i))
logger.info('error in symplectic identity #3: %s', norm(approx_z0))
logger.info('error in symplectic identity #4: %s', norm(approx_z1))
okay = True
if not (abs(det - 1.0) < tolerance):
okay = False
if not (norm(approx_j - j) < tolerance):
okay = False
if not (norm(approx_i - i) < tolerance):
okay = False
if not (norm(approx_z0) < tolerance):
okay = False
if not (norm(approx_z1) < tolerance):
okay = False
return okay