def expm_higham_2005(A):
"""
Compute the matrix exponential using the method of Higham 2005.
N. J. Higham,
"The Scaling and Squaring Method for the Matrix Exponential Revisited",
SIAM. J. Matrix Anal. & Appl. 26, 1179 (2005).
"""
n_squarings = 0
# FIXME: is there an algopy norm implementation?
A_L1 = numpy.linalg.norm(A, 1)
ident = numpy.eye(A.shape[0])
if A_L1 < 1.495585217958292e-002:
U,V = _expm_pade3(A, ident)
elif A_L1 < 2.539398330063230e-001:
U,V = _expm_pade5(A, ident)
elif A_L1 < 9.504178996162932e-001:
U,V = _expm_pade7(A, ident)
elif A_L1 < 2.097847961257068e+000:
U,V = _expm_pade9(A, ident)
else:
maxnorm = 5.371920351148152
# FIXME: this should probably use algopy log,
# and algopy max and ceil if they exist.
n_squarings = max(0, int(math.ceil(math.log(A_L1 / maxnorm, 2))))
A /= 2**n_squarings
U, V = _expm_pade13(A, ident)
R = solve(-U + V, U + V)
for i in range(n_squarings):
R = dot(R, R)
return R
def expm_higham_2005(A):
"""
Compute the matrix exponential using the method of Higham 2005.
N. J. Higham,
"The Scaling and Squaring Method for the Matrix Exponential Revisited",
SIAM. J. Matrix Anal. & Appl. 26, 1179 (2005).
"""
n_squarings = 0
# FIXME: is there an algopy norm implementation?
A_L1 = numpy.linalg.norm(A, 1)
ident = numpy.eye(A.shape[0])
if A_L1 < 1.495585217958292e-002:
U,V = _expm_pade3(A, ident)
elif A_L1 < 2.539398330063230e-001:
U,V = _expm_pade5(A, ident)
elif A_L1 < 9.504178996162932e-001:
U,V = _expm_pade7(A, ident)
elif A_L1 < 2.097847961257068e+000:
U,V = _expm_pade9(A, ident)
else:
maxnorm = 5.371920351148152
# FIXME: this should probably use algopy log,
# and algopy max and ceil if they exist.
n_squarings = max(0, int(math.ceil(math.log(A_L1 / maxnorm, 2))))
A /= 2**n_squarings
U, V = _expm_pade13(A, ident)
R = solve(-U + V, U + V)
for i in range(n_squarings):
R = dot(R, R)
return R
def expm_pade(A, q):
"""
Compute the matrix exponential using a fixed-order Pade approximation.
"""
q_to_pade = {
3 : _expm_pade3,
5 : _expm_pade5,
7 : _expm_pade7,
9 : _expm_pade9,
13 : _expm_pade13,
}
pade = q_to_pade[q]
ident = numpy.eye(A.shape[0])
U, V = pade(A, ident)
return solve(-U + V, U + V)
def expm_pade(A, q):
"""
Compute the matrix exponential using a fixed-order Pade approximation.
"""
q_to_pade = {
3 : _expm_pade3,
5 : _expm_pade5,
7 : _expm_pade7,
9 : _expm_pade9,
13 : _expm_pade13,
}
pade = q_to_pade[q]
ident = numpy.eye(A.shape[0])
U, V = pade(A, ident)
return solve(-U + V, U + V)
