def _solve_P_Q(U, V, structure=None):
"""
A helper function for expm_2009.
Parameters
----------
U : ndarray
Pade numerator.
V : ndarray
Pade denominator.
structure : str, optional
A string describing the structure of both matrices `U` and `V`.
Only `upper_triangular` is currently supported.
Notes
-----
The `structure` argument is inspired by similar args
for theano and cvxopt functions.
"""
P = U + V
Q = -U + V
if isspmatrix(U):
return spsolve(Q, P)
elif structure is None:
return solve(Q, P)
elif structure == UPPER_TRIANGULAR:
return solve_triangular(Q, P)
else:
raise ValueError('unsupported matrix structure: ' + str(structure))
def _solve_P_Q(U, V, structure=None):
"""
A helper function for expm_2009.
Parameters
----------
U : ndarray
Pade numerator.
V : ndarray
Pade denominator.
structure : str, optional
A string describing the structure of both matrices `U` and `V`.
Only `upper_triangular` is currently supported.
Notes
-----
The `structure` argument is inspired by similar args
for theano and cvxopt functions.
"""
P = U + V
Q = -U + V
if isspmatrix(U):
return spsolve(Q, P)
elif structure is None:
return solve(Q, P)
elif structure == UPPER_TRIANGULAR:
return solve_triangular(Q, P)
else:
raise ValueError('unsupported matrix structure: ' + str(structure))
def _solve_P_Q(U, V):
"""
A helper function for expm_2009.
"""
P = U + V
Q = -U + V
if isspmatrix(U):
R = spsolve(Q, P)
else:
R = solve(Q, P)
return R
def _solve_P_Q(U, V):
"""
A helper function for expm_2009.
"""
P = U + V
Q = -U + V
if isspmatrix(U):
R = spsolve(Q, P)
else:
R = solve(Q, P)
return R
def generate(src, dst):
"""
Function to generate the distortion parameters between the source and
the destination. The name of the variables are the same as in the
original article.
"""
n = src.shape[0]
# Variables as defined by Bookstein
K = U(cdist(src, src, metric="euclidean"))
P = np.hstack((np.ones((n, 1)), src))
L = np.vstack((np.hstack((K, P)), np.hstack((P.T, np.zeros((3, 3))))))
V = np.hstack((dst.T, np.zeros((2, 3))))
# Matrix system solving
Wa = solve(L, V.T)
W = Wa[:-3, :]
a = Wa[-3:, :]
WK = np.dot(W.T, K)
WKW = np.dot(WK, W)
be = max(0.5 * np.trace(WKW), 0)
# Implementation of other variables not present in the original publication
surfaceratio = (a[1, 0] * a[2, 1]) - (a[2, 0] * a[1, 1])
scale = np.sqrt(np.abs(surfaceratio))
mirror = surfaceratio < 0
shearing = angle_between(Wa[-2:, 0], Wa[-2:, 1], True)
# Prepare the return dictionary
src = src.tolist()
dst = dst.tolist()
W = W.tolist()
a = a.tolist()
return {
'src': src,
'dst': dst,
'linear': a,
'scale': scale,
'mirror': mirror,
'shearing': shearing,
'weights': W,
'be': be
}
def expm(A):
"""
Compute the matrix exponential using Pade approximation.
.. versionadded:: 0.12.0
Parameters
----------
A : (M,M) array or sparse matrix
2D Array or Matrix (sparse or dense) to be exponentiated
Returns
-------
expA : (M,M) ndarray
Matrix exponential of `A`
References
----------
N. J. Higham,
"The Scaling and Squaring Method for the Matrix Exponential Revisited",
SIAM. J. Matrix Anal. & Appl. 26, 1179 (2005).
"""
n_squarings = 0
Aissparse = isspmatrix(A)
if Aissparse:
A_L1 = max(abs(A).sum(axis=0).flat)
ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
else:
A = asarray(A)
A_L1 = norm(A, 1)
ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
if A.dtype == 'float64' or A.dtype == 'complex128':
if A_L1 < 1.495585217958292e-002:
U, V = _pade3(A, ident)
elif A_L1 < 2.539398330063230e-001:
U, V = _pade5(A, ident)
elif A_L1 < 9.504178996162932e-001:
U, V = _pade7(A, ident)
elif A_L1 < 2.097847961257068e+000:
U, V = _pade9(A, ident)
else:
maxnorm = 5.371920351148152
n_squarings = max(0, int(ceil(log2(A_L1 / maxnorm))))
A = A / 2**n_squarings
U, V = _pade13(A, ident)
elif A.dtype == 'float32' or A.dtype == 'complex64':
if A_L1 < 4.258730016922831e-001:
U, V = _pade3(A, ident)
elif A_L1 < 1.880152677804762e+000:
U, V = _pade5(A, ident)
else:
maxnorm = 3.925724783138660
n_squarings = max(0, int(ceil(log2(A_L1 / maxnorm))))
A = A / 2**n_squarings
U, V = _pade7(A, ident)
else:
raise ValueError("invalid type: " + str(A.dtype))
P = U + V # p_m(A) : numerator
Q = -U + V # q_m(A) : denominator
if Aissparse:
from scipy.sparse.linalg import spsolve
R = spsolve(Q, P)
else:
R = solve(Q, P)
# squaring step to undo scaling
for i in range(n_squarings):
R = R.dot(R)
return R
def expm(A):
"""Compute the matrix exponential using Pade approximation.
.. versionadded:: 0.12.0
Parameters
----------
A : array or sparse matrix, shape(M,M)
2D Array or Matrix (sparse or dense) to be exponentiated
Returns
-------
expA : array, shape(M,M)
Matrix exponential of A
References
----------
N. J. Higham,
"The Scaling and Squaring Method for the Matrix Exponential Revisited",
SIAM. J. Matrix Anal. & Appl. 26, 1179 (2005).
"""
n_squarings = 0
Aissparse = isspmatrix(A)
if Aissparse:
A_L1 = max(abs(A).sum(axis=0).flat)
ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
else:
A = asarray(A)
A_L1 = norm(A,1)
ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
if A.dtype == 'float64' or A.dtype == 'complex128':
if A_L1 < 1.495585217958292e-002:
U,V = _pade3(A, ident)
elif A_L1 < 2.539398330063230e-001:
U,V = _pade5(A, ident)
elif A_L1 < 9.504178996162932e-001:
U,V = _pade7(A, ident)
elif A_L1 < 2.097847961257068e+000:
U,V = _pade9(A, ident)
else:
maxnorm = 5.371920351148152
n_squarings = max(0, int(ceil(log2(A_L1 / maxnorm))))
A = A / 2**n_squarings
U,V = _pade13(A, ident)
elif A.dtype == 'float32' or A.dtype == 'complex64':
if A_L1 < 4.258730016922831e-001:
U,V = _pade3(A, ident)
elif A_L1 < 1.880152677804762e+000:
U,V = _pade5(A, ident)
else:
maxnorm = 3.925724783138660
n_squarings = max(0, int(ceil(log2(A_L1 / maxnorm))))
A = A / 2**n_squarings
U,V = _pade7(A, ident)
else:
raise ValueError("invalid type: "+str(A.dtype))
P = U + V # p_m(A) : numerator
Q = -U + V # q_m(A) : denominator
if Aissparse:
from scipy.sparse.linalg import spsolve
R = spsolve(Q, P)
else:
R = solve(Q,P)
# squaring step to undo scaling
for i in range(n_squarings):
R = R.dot(R)
return R