def get_R_der(x, y):
"""
Calculate the derivatives of the correlation matrix with respect
to the Cartesian coordinates.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
Returns
-------
numpy.ndarray
u, w, i, j :
First two dimensions are (n_atoms, 3), the variables being differentiated
Second two dimensions are (3, 3), the elements of the R-matrix derivatives with respect to atom u, dimension w
"""
x = x - np.mean(x, axis=0)
y = y - np.mean(y, axis=0)
# 3 x 3 x N_atoms x 3
ADiffR = np.zeros((x.shape[0], 3, 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
for i in range(3):
for j in range(3):
if i == w:
ADiffR[u, w, i, j] = y[u, j]
fdcheck = False
if fdcheck:
h = 1e-4
R0 = build_correlation(x, y)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
RPlus = build_correlation(x, y)
x[u, w] -= 2 * h
RMinus = build_correlation(x, y)
x[u, w] += h
FDiffR = (RPlus - RMinus) / (2 * h)
logger.info("%i %i %12.6f\n" %
(u, w, np.max(np.abs(ADiffR[u, w] - FDiffR))))
return ADiffR
def get_R_der(x, y):
"""
Calculate the derivatives of the correlation matrix with respect
to the Cartesian coordinates.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
Returns
-------
numpy.ndarray
u, w, i, j :
First two dimensions are (n_atoms, 3), the variables being differentiated
Second two dimensions are (3, 3), the elements of the R-matrix derivatives with respect to atom u, dimension w
"""
x = x - np.mean(x,axis=0)
y = y - np.mean(y,axis=0)
# 3 x 3 x N_atoms x 3
ADiffR = np.zeros((x.shape[0], 3, 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
for i in range(3):
for j in range(3):
if i == w:
ADiffR[u, w, i, j] = y[u, j]
fdcheck = False
if fdcheck:
h = 1e-4
R0 = build_correlation(x, y)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
RPlus = build_correlation(x, y)
x[u, w] -= 2*h
RMinus = build_correlation(x, y)
x[u, w] += h
FDiffR = (RPlus-RMinus)/(2*h)
logger.info("%i %i %12.6f\n" % (u, w, np.max(np.abs(ADiffR[u, w]-FDiffR))))
return ADiffR
def get_expmap_der(x, y, second=False, fdcheck=False, use_loops=False):
"""
Given trial coordinates x and target coordinates y,
return the derivatives of the exponential map that brings
x into maximal coincidence (minimum RMSD) with y, with
respect to the coordinates of x.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
second : bool
If true, return the second derivative matrix as well.
fdcheck : bool
If true, perform a finite difference check and return finite difference gradients
use_loops : bool
If true, use the slower reference implementation that uses for loops
Returns
-------
numpy.ndarray
u, w, i:
First two dimensions are (n_atoms, 3), the variables being differentiated
Third dimension is 3, the elements of the exponential map derivatives with respect to atom u, dimension w
numpy.ndarray (if second=True)
u, w, a, b, i:
First four dimensions are (n_atoms, 3, n_atoms, 3), the variables being differentiated
Fifth dimension is 3, the elements of the exponential map second derivatives with respect to atom u, dimension w, atom a, dimension b
"""
# t0 = time.time()
q = get_quat(x,y)
v = get_expmap(x,y)
if second:
fac, dfac, dfac2 = calc_fac_dfac(q[0], second=True)
else:
fac, dfac = calc_fac_dfac(q[0])
dvdq = np.zeros((4, 3), dtype=float)
dvdq[0, :] = dfac*q[1:]
for i in range(3):
dvdq[i+1, i] = fac
if second:
dvdq2 = np.zeros((4, 4, 3), dtype=float)
dvdq2[0, 0, :] = dfac2*q[1:]
for i in range(3):
dvdq2[0, i+1, i] = dfac
dvdq2[i+1, 0, i] = dfac
if fdcheck:
h = 1e-6
fac, _ = calc_fac_dfac(q[0])
V0 = fac*q[1:]
logger.info("-=# Now checking first derivatives of exponential map w/r.t. quaternion #=-\n")
for p in range(4):
# Do backwards difference only, because arccos of q[0] > 1 is undefined
q[p] -= h
fac, _ = calc_fac_dfac(q[0])
VMinus = fac*q[1:]
q[p] += h
FDiffV = (V0-VMinus)/h
maxerr = np.max(np.abs(dvdq[p]-FDiffV))
logger.info("q %3i : maxerr = %.3e %s\n" % (i, maxerr, 'X' if maxerr > 1e-6 else ''))
# logger.info(i, dvdq[i], FDiffV, np.max(np.abs(dvdq[i]-FDiffV)))
if second:
h = 1e-5
logger.info("-=# Now checking second derivatives of exponential map w/r.t. quaternion #=-\n")
def V_(q_):
fac_, _ = calc_fac_dfac(q_[0])
return fac_*q_[1:]
for p in range(4):
for r in range(4):
if p == r:
FDiffV2 = V0.copy()
q[p] -= h
FDiffV2 -= 2*V_(q)
q[p] -= h
FDiffV2 += V_(q)
q[p] += 2*h
else:
FDiffV2 = V0.copy()
q[p] -= h
q[r] -= h
FDiffV2 += V_(q)
q[r] += h
FDiffV2 -= V_(q)
q[r] -= h
q[p] += h
FDiffV2 -= V_(q)
q[r] += h
FDiffV2 /= h**2
maxerr = np.max(np.abs(dvdq2[p, r]-FDiffV2))
if maxerr > 1e-7:
logger.info("q %3i %3i : maxerr = %.3e %s\n" % (p, r, maxerr, 'X' if maxerr > 1e-5 else ''))
# logger.info("q %3i %3i : analytic %s numerical %s maxerr = %.3e %s" % (i, j, str(dvdq2[i, j]), str(FDiffV2), maxerr, 'X' if maxerr > 1e-4 else ''))
# Dimensionality: Number of atoms, number of dimensions (3), number of elements in q (4)
if second:
dqdx, dqdx2 = get_q_der(x, y, second=True)
else:
dqdx = get_q_der(x, y)
# Dimensionality: Number of atoms, number of dimensions (3), number of elements in v (3)
dvdx = np.zeros((x.shape[0], 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
dqdx_uw = dqdx[u, w]
for p in range(4):
dvdx[u, w, :] += dvdq[p, :] * dqdx[u, w, p]
if second:
if use_loops:
# Reference implementation using for loops
dvdx2 = np.zeros((x.shape[0], 3, x.shape[0], 3, 3), dtype=float)
dvdqx = np.zeros((4, x.shape[0], 3, 3), dtype=float)
for p in range(4):
for u in range(x.shape[0]):
for w in range(3):
for i in range(3):
for r in range(4):
dvdqx[p, u, w, i] += dvdq2[p, r, i] * dqdx[u, w, r]
for u in range(x.shape[0]):
for w in range(3):
for a in range(x.shape[0]):
for b in range(3):
for i in range(3):
for p in range(4):
dvdx2[u, w, a, b, i] += dvdqx[p, a, b, i] * dqdx[u, w, p]
dvdx2[u, w, a, b, i] += dvdq[p, i] * dqdx2[u, w, a, b, p]
else:
dvdqx = np.einsum('pri,uwr->puwi', dvdq2, dqdx, optimize=True)
dvdx2 = np.einsum('pabi,uwp->uwabi', dvdqx, dqdx, optimize=True)
dvdx2 += np.einsum('pi,uwabp->uwabi', dvdq, dqdx2, optimize=True)
# LPW 2019-03-09: Using einsum gives 166x speedup over for loops for trp-cage (200 atoms); 0.06 vs. 10.1 s
# print(time.time()-t0)
if fdcheck:
h = 1e-3
logger.info("-=# Now checking first derivatives of exponential map w/r.t. Cartesians #=-\n")
FDiffV = np.zeros((x.shape[0], 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
VPlus = get_expmap(x, y)
x[u, w] -= 2*h
VMinus = get_expmap(x, y)
x[u, w] += h
FDiffV[u, w] = (VPlus-VMinus)/(2*h)
maxerr = np.max(np.abs(dvdx[u, w]-FDiffV[u, w]))
logger.info("atom %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], maxerr, 'X' if maxerr > 1e-6 else ''))
dvdx = FDiffV
if second:
logger.info("-=# Now checking second derivatives of exponential map w/r.t. Cartesians #=-\n")
FDiffV2 = np.zeros((x.shape[0], 3, y.shape[0], 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
for a in range(x.shape[0]):
for b in range(3):
if a == u and b == w:
x[u, w] += h
VPlus = get_expmap(x, y)
x[u, w] -= 2*h
VMinus = get_expmap(x, y)
x[u, w] += h
FDiffV2[u, w, a, b] = (VPlus+VMinus-2*V0)/h**2
else:
x[u, w] += h
x[a, b] += h # (+, +)
FDiffV2[u, w, a, b] = get_expmap(x, y)
x[u, w] -= 2*h # (-, +)
FDiffV2[u, w, a, b] -= get_expmap(x, y)
x[a, b] -= 2*h # (-, -)
FDiffV2[u, w, a, b] += get_expmap(x, y)
x[u, w] += 2*h # (+, -)
FDiffV2[u, w, a, b] -= get_expmap(x, y)
x[u, w] -= h
x[a, b] += h
FDiffV2[u, w, a, b] /= (4*h**2)
maxerr = np.max(np.abs(dvdx2[u, w, a, b]-FDiffV2[u, w, a, b]))
if maxerr > 1e-8:
logger.info("atom %3i %s, %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], a, 'xyz'[b], maxerr, 'X' if maxerr > 1e-6 else ''))
dvdx2 = FDiffV2
if second:
return dvdx, dvdx2
else:
return dvdx
def get_q_der(x, y, second=False, fdcheck=False, use_loops=False):
"""
Calculate the derivatives of the quaternion with respect
to the Cartesian coordinates.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
second : bool
If true, return the second derivative matrix as well.
fdcheck : bool
If true, perform a finite difference check and return finite difference gradients
use_loops : bool
If true, use the slower reference implementation that uses for loops (only in second derivs)
Returns
-------
numpy.ndarray
u, w, i:
First two dimensions are (n_atoms, 3), the variables being differentiated
Third dimension is 4, the elements of the quaternion derivatives with respect to atom u, dimension w
numpy.ndarray (if second=True)
u, w, a, b, i:
First four dimensions are (n_atoms, 3, n_atoms, 3), the variables being differentiated
Fifth dimension is 4, the elements of the quaternion second derivatives with respect to atom u, dimension w, atom a, dimension b
"""
x = x - np.mean(x,axis=0)
y = y - np.mean(y,axis=0)
q, l = get_quat(x, y, eig=True)
F = build_F(x, y)
dF = get_F_der(x, y)
mat = np.eye(4)*l - F
# pinv = np.matrix(np.linalg.pinv(np.eye(4)*l - F))
Minv = invert_svd(np.eye(4)*l - F, thresh=1e-6)
dq = np.zeros((x.shape[0], 3, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
# dquw = Minv*np.matrix(dF[u, w])*np.matrix(q).T
dquw = multi_dot([Minv,dF[u, w],q.T])
dq[u, w] = np.array(dquw).flatten()
if second:
if use_loops:
# Reference implementation using for loops
dl = np.zeros((x.shape[0], 3), dtype=float)
dM = np.zeros((x.shape[0], 3, 4, 4), dtype=float)
dq2 = np.zeros((x.shape[0], 3, x.shape[0], 3, 4), dtype=float)
dinv = np.zeros((x.shape[0], 3, 4, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
dl[u, w] = multi_dot([q, dF[u, w], q.T])
dM[u, w] = np.eye(4)*dl[u, w] - dF[u, w]
term1 = -multi_dot([Minv, dM[u, w], Minv])
term2 = multi_dot([Minv, Minv.T, dM[u, w].T, (np.eye(4)-np.dot(mat, Minv))])
term3 = multi_dot([(np.eye(4)-np.dot(Minv, mat)), dM[u, w].T, Minv.T, Minv])
dinv[u, w] = term1 + term2 + term3
for a in range(x.shape[0]):
for b in range(3):
dq2[u, w, a, b] += multi_dot([dinv[u, w], dF[a, b], q])
dq2[u, w, a, b] += multi_dot([Minv, dF[a, b], dq[u, w]])
else:
# t0 = time.time()
# LPW 2019-03-09: Setting optimize=True gives a 15x speedup for trp-cage (200 atoms, 0.02 vs. 0.3 s)
dl = np.einsum('p,uwpr,r->uw', q, dF, q, optimize=True)
dM = np.einsum('uw,pr->uwpr', dl, np.eye(4), optimize=True) - dF
dinv = -np.einsum('ps,uwst,tr->uwpr', Minv, dM, Minv, optimize=True)
dinv += np.einsum('ps,ts,uwvt,vr->uwpr', Minv, Minv, dM, np.eye(4)-np.dot(mat,Minv), optimize=True)
dinv += np.einsum('ps,uwts,vt,vr->uwpr', np.eye(4)-np.dot(Minv, mat), dM, Minv, Minv, optimize=True)
dq2 = np.einsum('uwpr,abrs,s->uwabp', dinv, dF, q, optimize=True)
dq2 += np.einsum('pr,abrs,uws->uwabp', Minv, dF, dq, optimize=True)
# print(time.time()-t0)
if fdcheck:
# If fdcheck = True, then return finite difference derivatives
h = 1e-6
logger.info("-=# Now checking first derivatives of superposition quaternion w/r.t. Cartesians #=-\n")
FDiffQ = np.zeros((x.shape[0], 3, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
QPlus = get_quat(x, y)
x[u, w] -= 2*h
QMinus = get_quat(x, y)
x[u, w] += h
FDiffQ[u, w] = (QPlus-QMinus)/(2*h)
maxerr = np.max(np.abs(dq[u, w]-FDiffQ[u, w]))
logger.info("atom %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], maxerr, 'X' if maxerr > 1e-6 else ''))
dq = FDiffQ
if second:
h = 1.0e-3
logger.info("-=# Now checking second derivatives of superposition quaternion w/r.t. Cartesians #=-\n")
Q0 = get_quat(x, y)
FDiffQ2 = np.zeros((x.shape[0], 3, y.shape[0], 3, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
for a in range(x.shape[0]):
for b in range(3):
if a == u and b == w:
x[u, w] += h
QPlus = get_quat(x, y)
x[u, w] -= 2*h
QMinus = get_quat(x, y)
x[u, w] += h
FDiffQ2[u, w, a, b] = (QPlus+QMinus-2*Q0)/h**2
else:
x[u, w] += h
x[a, b] += h # (+, +)
FDiffQ2[u, w, a, b] = get_quat(x, y)
x[u, w] -= 2*h # (-, +)
FDiffQ2[u, w, a, b] -= get_quat(x, y)
x[a, b] -= 2*h # (-, -)
FDiffQ2[u, w, a, b] += get_quat(x, y)
x[u, w] += 2*h # (+, -)
FDiffQ2[u, w, a, b] -= get_quat(x, y)
x[u, w] -= h
x[a, b] += h
FDiffQ2[u, w, a, b] /= (4*h**2)
maxerr = np.max(np.abs(dq2[u, w, a, b]-FDiffQ2[u, w, a, b]))
if maxerr > 1e-8:
logger.info("atom %3i %s, %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], a, 'xyz'[b], maxerr, 'X' if maxerr > 1e-6 else ''))
dq2 = FDiffQ2
if second:
return dq, dq2
else:
return dq
def get_F_der(x, y):
"""
Calculate the derivatives of the F-matrix with respect
to the Cartesian coordinates.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
Returns
-------
numpy.ndarray
u, w, i, j :
First two dimensions are (n_atoms, 3), the variables being differentiated
Second two dimensions are (4, 4), the elements of the R-matrix derivatives with respect to atom u, dimension w
"""
x = x - np.mean(x,axis=0)
y = y - np.mean(y,axis=0)
dR = get_R_der(x, y)
dF = np.zeros((x.shape[0], 3, 4, 4),dtype=float)
for u in range(x.shape[0]):
for w in range(3):
dR11 = dR[u,w,0,0]
dR12 = dR[u,w,0,1]
dR13 = dR[u,w,0,2]
dR21 = dR[u,w,1,0]
dR22 = dR[u,w,1,1]
dR23 = dR[u,w,1,2]
dR31 = dR[u,w,2,0]
dR32 = dR[u,w,2,1]
dR33 = dR[u,w,2,2]
dF[u,w,0,0] = dR11 + dR22 + dR33
dF[u,w,0,1] = dR23 - dR32
dF[u,w,0,2] = dR31 - dR13
dF[u,w,0,3] = dR12 - dR21
dF[u,w,1,0] = dR23 - dR32
dF[u,w,1,1] = dR11 - dR22 - dR33
dF[u,w,1,2] = dR12 + dR21
dF[u,w,1,3] = dR13 + dR31
dF[u,w,2,0] = dR31 - dR13
dF[u,w,2,1] = dR12 + dR21
dF[u,w,2,2] = dR22 - dR33 - dR11
dF[u,w,2,3] = dR23 + dR32
dF[u,w,3,0] = dR12 - dR21
dF[u,w,3,1] = dR13 + dR31
dF[u,w,3,2] = dR23 + dR32
dF[u,w,3,3] = dR33 - dR22 - dR11
fdcheck = False
if fdcheck:
h = 1e-4
F0 = build_F(x, y)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
FPlus = build_F(x, y)
x[u, w] -= 2*h
FMinus = build_F(x, y)
x[u, w] += h
FDiffF = (FPlus-FMinus)/(2*h)
logger.info("%i %i %12.6f\n" % (u, w, np.max(np.abs(dF[u, w]-FDiffF))))
return dF
def get_expmap_der(x, y, second=False, fdcheck=False, use_loops=False):
"""
Given trial coordinates x and target coordinates y,
return the derivatives of the exponential map that brings
x into maximal coincidence (minimum RMSD) with y, with
respect to the coordinates of x.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
second : bool
If true, return the second derivative matrix as well.
fdcheck : bool
If true, perform a finite difference check and return finite difference gradients
use_loops : bool
If true, use the slower reference implementation that uses for loops
Returns
-------
numpy.ndarray
u, w, i:
First two dimensions are (n_atoms, 3), the variables being differentiated
Third dimension is 3, the elements of the exponential map derivatives with respect to atom u, dimension w
numpy.ndarray (if second=True)
u, w, a, b, i:
First four dimensions are (n_atoms, 3, n_atoms, 3), the variables being differentiated
Fifth dimension is 3, the elements of the exponential map second derivatives with respect to atom u, dimension w, atom a, dimension b
"""
# t0 = time.time()
q = get_quat(x,y)
v = get_expmap(x,y)
if second:
fac, dfac, dfac2 = calc_fac_dfac(q[0], second=True)
else:
fac, dfac = calc_fac_dfac(q[0])
dvdq = np.zeros((4, 3), dtype=float)
dvdq[0, :] = dfac*q[1:]
for i in range(3):
dvdq[i+1, i] = fac
if second:
dvdq2 = np.zeros((4, 4, 3), dtype=float)
dvdq2[0, 0, :] = dfac2*q[1:]
for i in range(3):
dvdq2[0, i+1, i] = dfac
dvdq2[i+1, 0, i] = dfac
if fdcheck:
h = 1e-6
fac, _ = calc_fac_dfac(q[0])
V0 = fac*q[1:]
logger.info("-=# Now checking first derivatives of exponential map w/r.t. quaternion #=-\n")
for p in range(4):
# Do backwards difference only, because arccos of q[0] > 1 is undefined
q[p] -= h
fac, _ = calc_fac_dfac(q[0])
VMinus = fac*q[1:]
q[p] += h
FDiffV = (V0-VMinus)/h
maxerr = np.max(np.abs(dvdq[p]-FDiffV))
logger.info("q %3i : maxerr = %.3e %s\n" % (i, maxerr, 'X' if maxerr > 1e-6 else ''))
# logger.info(i, dvdq[i], FDiffV, np.max(np.abs(dvdq[i]-FDiffV)))
if second:
h = 1e-5
logger.info("-=# Now checking second derivatives of exponential map w/r.t. quaternion #=-\n")
def V_(q_):
fac_, _ = calc_fac_dfac(q_[0])
return fac_*q_[1:]
for p in range(4):
for r in range(4):
if p == r:
FDiffV2 = V0.copy()
q[p] -= h
FDiffV2 -= 2*V_(q)
q[p] -= h
FDiffV2 += V_(q)
q[p] += 2*h
else:
FDiffV2 = V0.copy()
q[p] -= h
q[r] -= h
FDiffV2 += V_(q)
q[r] += h
FDiffV2 -= V_(q)
q[r] -= h
q[p] += h
FDiffV2 -= V_(q)
q[r] += h
FDiffV2 /= h**2
maxerr = np.max(np.abs(dvdq2[p, r]-FDiffV2))
if maxerr > 1e-7:
logger.info("q %3i %3i : maxerr = %.3e %s\n" % (p, r, maxerr, 'X' if maxerr > 1e-5 else ''))
# logger.info("q %3i %3i : analytic %s numerical %s maxerr = %.3e %s" % (i, j, str(dvdq2[i, j]), str(FDiffV2), maxerr, 'X' if maxerr > 1e-4 else ''))
# Dimensionality: Number of atoms, number of dimensions (3), number of elements in q (4)
if second:
dqdx, dqdx2 = get_q_der(x, y, second=True)
else:
dqdx = get_q_der(x, y)
# Dimensionality: Number of atoms, number of dimensions (3), number of elements in v (3)
dvdx = np.zeros((x.shape[0], 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
dqdx_uw = dqdx[u, w]
for p in range(4):
dvdx[u, w, :] += dvdq[p, :] * dqdx[u, w, p]
if second:
if use_loops:
# Reference implementation using for loops
dvdx2 = np.zeros((x.shape[0], 3, x.shape[0], 3, 3), dtype=float)
dvdqx = np.zeros((4, x.shape[0], 3, 3), dtype=float)
for p in range(4):
for u in range(x.shape[0]):
for w in range(3):
for i in range(3):
for r in range(4):
dvdqx[p, u, w, i] += dvdq2[p, r, i] * dqdx[u, w, r]
for u in range(x.shape[0]):
for w in range(3):
for a in range(x.shape[0]):
for b in range(3):
for i in range(3):
for p in range(4):
dvdx2[u, w, a, b, i] += dvdqx[p, a, b, i] * dqdx[u, w, p]
dvdx2[u, w, a, b, i] += dvdq[p, i] * dqdx2[u, w, a, b, p]
else:
dvdqx = np.einsum('pri,uwr->puwi', dvdq2, dqdx, optimize=True)
dvdx2 = np.einsum('pabi,uwp->uwabi', dvdqx, dqdx, optimize=True)
dvdx2 += np.einsum('pi,uwabp->uwabi', dvdq, dqdx2, optimize=True)
# LPW 2019-03-09: Using einsum gives 166x speedup over for loops for trp-cage (200 atoms); 0.06 vs. 10.1 s
# print(time.time()-t0)
if fdcheck:
h = 1e-3
logger.info("-=# Now checking first derivatives of exponential map w/r.t. Cartesians #=-\n")
FDiffV = np.zeros((x.shape[0], 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
VPlus = get_expmap(x, y)
x[u, w] -= 2*h
VMinus = get_expmap(x, y)
x[u, w] += h
FDiffV[u, w] = (VPlus-VMinus)/(2*h)
maxerr = np.max(np.abs(dvdx[u, w]-FDiffV[u, w]))
logger.info("atom %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], maxerr, 'X' if maxerr > 1e-6 else ''))
dvdx = FDiffV
if second:
logger.info("-=# Now checking second derivatives of exponential map w/r.t. Cartesians #=-\n")
FDiffV2 = np.zeros((x.shape[0], 3, y.shape[0], 3, 3), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
for a in range(x.shape[0]):
for b in range(3):
if a == u and b == w:
x[u, w] += h
VPlus = get_expmap(x, y)
x[u, w] -= 2*h
VMinus = get_expmap(x, y)
x[u, w] += h
FDiffV2[u, w, a, b] = (VPlus+VMinus-2*V0)/h**2
else:
x[u, w] += h
x[a, b] += h # (+, +)
FDiffV2[u, w, a, b] = get_expmap(x, y)
x[u, w] -= 2*h # (-, +)
FDiffV2[u, w, a, b] -= get_expmap(x, y)
x[a, b] -= 2*h # (-, -)
FDiffV2[u, w, a, b] += get_expmap(x, y)
x[u, w] += 2*h # (+, -)
FDiffV2[u, w, a, b] -= get_expmap(x, y)
x[u, w] -= h
x[a, b] += h
FDiffV2[u, w, a, b] /= (4*h**2)
maxerr = np.max(np.abs(dvdx2[u, w, a, b]-FDiffV2[u, w, a, b]))
if maxerr > 1e-8:
logger.info("atom %3i %s, %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], a, 'xyz'[b], maxerr, 'X' if maxerr > 1e-6 else ''))
dvdx2 = FDiffV2
if second:
return dvdx, dvdx2
else:
return dvdx
def get_q_der(x, y, second=False, fdcheck=False, use_loops=False):
"""
Calculate the derivatives of the quaternion with respect
to the Cartesian coordinates.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
second : bool
If true, return the second derivative matrix as well.
fdcheck : bool
If true, perform a finite difference check and return finite difference gradients
use_loops : bool
If true, use the slower reference implementation that uses for loops (only in second derivs)
Returns
-------
numpy.ndarray
u, w, i:
First two dimensions are (n_atoms, 3), the variables being differentiated
Third dimension is 4, the elements of the quaternion derivatives with respect to atom u, dimension w
numpy.ndarray (if second=True)
u, w, a, b, i:
First four dimensions are (n_atoms, 3, n_atoms, 3), the variables being differentiated
Fifth dimension is 4, the elements of the quaternion second derivatives with respect to atom u, dimension w, atom a, dimension b
"""
x = x - np.mean(x,axis=0)
y = y - np.mean(y,axis=0)
q, l = get_quat(x, y, eig=True)
F = build_F(x, y)
dF = get_F_der(x, y)
mat = np.eye(4)*l - F
# pinv = np.matrix(np.linalg.pinv(np.eye(4)*l - F))
Minv = invert_svd(np.eye(4)*l - F, thresh=1e-6)
dq = np.zeros((x.shape[0], 3, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
# dquw = Minv*np.matrix(dF[u, w])*np.matrix(q).T
dquw = multi_dot([Minv,dF[u, w],q.T])
dq[u, w] = np.array(dquw).flatten()
if second:
if use_loops:
# Reference implementation using for loops
dl = np.zeros((x.shape[0], 3), dtype=float)
dM = np.zeros((x.shape[0], 3, 4, 4), dtype=float)
dq2 = np.zeros((x.shape[0], 3, x.shape[0], 3, 4), dtype=float)
dinv = np.zeros((x.shape[0], 3, 4, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
dl[u, w] = multi_dot([q, dF[u, w], q.T])
dM[u, w] = np.eye(4)*dl[u, w] - dF[u, w]
term1 = -multi_dot([Minv, dM[u, w], Minv])
term2 = multi_dot([Minv, Minv.T, dM[u, w].T, (np.eye(4)-np.dot(mat, Minv))])
term3 = multi_dot([(np.eye(4)-np.dot(Minv, mat)), dM[u, w].T, Minv.T, Minv])
dinv[u, w] = term1 + term2 + term3
for a in range(x.shape[0]):
for b in range(3):
dq2[u, w, a, b] += multi_dot([dinv[u, w], dF[a, b], q])
dq2[u, w, a, b] += multi_dot([Minv, dF[a, b], dq[u, w]])
else:
# t0 = time.time()
# LPW 2019-03-09: Setting optimize=True gives a 15x speedup for trp-cage (200 atoms, 0.02 vs. 0.3 s)
dl = np.einsum('p,uwpr,r->uw', q, dF, q, optimize=True)
dM = np.einsum('uw,pr->uwpr', dl, np.eye(4), optimize=True) - dF
dinv = -np.einsum('ps,uwst,tr->uwpr', Minv, dM, Minv, optimize=True)
dinv += np.einsum('ps,ts,uwvt,vr->uwpr', Minv, Minv, dM, np.eye(4)-np.dot(mat,Minv), optimize=True)
dinv += np.einsum('ps,uwts,vt,vr->uwpr', np.eye(4)-np.dot(Minv, mat), dM, Minv, Minv, optimize=True)
dq2 = np.einsum('uwpr,abrs,s->uwabp', dinv, dF, q, optimize=True)
dq2 += np.einsum('pr,abrs,uws->uwabp', Minv, dF, dq, optimize=True)
# print(time.time()-t0)
if fdcheck:
# If fdcheck = True, then return finite difference derivatives
h = 1e-6
logger.info("-=# Now checking first derivatives of superposition quaternion w/r.t. Cartesians #=-\n")
FDiffQ = np.zeros((x.shape[0], 3, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
QPlus = get_quat(x, y)
x[u, w] -= 2*h
QMinus = get_quat(x, y)
x[u, w] += h
FDiffQ[u, w] = (QPlus-QMinus)/(2*h)
maxerr = np.max(np.abs(dq[u, w]-FDiffQ[u, w]))
logger.info("atom %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], maxerr, 'X' if maxerr > 1e-6 else ''))
dq = FDiffQ
if second:
h = 1.0e-3
logger.info("-=# Now checking second derivatives of superposition quaternion w/r.t. Cartesians #=-\n")
Q0 = get_quat(x, y)
FDiffQ2 = np.zeros((x.shape[0], 3, y.shape[0], 3, 4), dtype=float)
for u in range(x.shape[0]):
for w in range(3):
for a in range(x.shape[0]):
for b in range(3):
if a == u and b == w:
x[u, w] += h
QPlus = get_quat(x, y)
x[u, w] -= 2*h
QMinus = get_quat(x, y)
x[u, w] += h
FDiffQ2[u, w, a, b] = (QPlus+QMinus-2*Q0)/h**2
else:
x[u, w] += h
x[a, b] += h # (+, +)
FDiffQ2[u, w, a, b] = get_quat(x, y)
x[u, w] -= 2*h # (-, +)
FDiffQ2[u, w, a, b] -= get_quat(x, y)
x[a, b] -= 2*h # (-, -)
FDiffQ2[u, w, a, b] += get_quat(x, y)
x[u, w] += 2*h # (+, -)
FDiffQ2[u, w, a, b] -= get_quat(x, y)
x[u, w] -= h
x[a, b] += h
FDiffQ2[u, w, a, b] /= (4*h**2)
maxerr = np.max(np.abs(dq2[u, w, a, b]-FDiffQ2[u, w, a, b]))
if maxerr > 1e-8:
logger.info("atom %3i %s, %3i %s : maxerr = %.3e %s\n" % (u, 'xyz'[w], a, 'xyz'[b], maxerr, 'X' if maxerr > 1e-6 else ''))
dq2 = FDiffQ2
if second:
return dq, dq2
else:
return dq
def get_F_der(x, y):
"""
Calculate the derivatives of the F-matrix with respect
to the Cartesian coordinates.
Parameters
----------
x : numpy.ndarray
Trial coordinates, dimensionality (number of atoms) x 3
y : numpy.ndarray
Target coordinates, dimensionalty must match trial coordinates
Returns
-------
numpy.ndarray
u, w, i, j :
First two dimensions are (n_atoms, 3), the variables being differentiated
Second two dimensions are (4, 4), the elements of the R-matrix derivatives with respect to atom u, dimension w
"""
x = x - np.mean(x,axis=0)
y = y - np.mean(y,axis=0)
dR = get_R_der(x, y)
dF = np.zeros((x.shape[0], 3, 4, 4),dtype=float)
for u in range(x.shape[0]):
for w in range(3):
dR11 = dR[u,w,0,0]
dR12 = dR[u,w,0,1]
dR13 = dR[u,w,0,2]
dR21 = dR[u,w,1,0]
dR22 = dR[u,w,1,1]
dR23 = dR[u,w,1,2]
dR31 = dR[u,w,2,0]
dR32 = dR[u,w,2,1]
dR33 = dR[u,w,2,2]
dF[u,w,0,0] = dR11 + dR22 + dR33
dF[u,w,0,1] = dR23 - dR32
dF[u,w,0,2] = dR31 - dR13
dF[u,w,0,3] = dR12 - dR21
dF[u,w,1,0] = dR23 - dR32
dF[u,w,1,1] = dR11 - dR22 - dR33
dF[u,w,1,2] = dR12 + dR21
dF[u,w,1,3] = dR13 + dR31
dF[u,w,2,0] = dR31 - dR13
dF[u,w,2,1] = dR12 + dR21
dF[u,w,2,2] = dR22 - dR33 - dR11
dF[u,w,2,3] = dR23 + dR32
dF[u,w,3,0] = dR12 - dR21
dF[u,w,3,1] = dR13 + dR31
dF[u,w,3,2] = dR23 + dR32
dF[u,w,3,3] = dR33 - dR22 - dR11
fdcheck = False
if fdcheck:
h = 1e-4
F0 = build_F(x, y)
for u in range(x.shape[0]):
for w in range(3):
x[u, w] += h
FPlus = build_F(x, y)
x[u, w] -= 2*h
FMinus = build_F(x, y)
x[u, w] += h
FDiffF = (FPlus-FMinus)/(2*h)
logger.info("%i %i %12.6f\n" % (u, w, np.max(np.abs(dF[u, w]-FDiffF))))
return dF
