def expm(A, use_exact_onenorm="auto", verbose=False):
from scipy.linalg import expm as expm_scipy
return expm_scipy(A)
# Core of expm, separated to allow testing exact and approximate
# algorithms.
# Hardcode a matrix order threshold for exact vs. estimated one-norms.
use_exact_onenorm = A.shape[0] < 200
h = _ExpmPadeHelper(A, use_exact_onenorm=use_exact_onenorm)
# Use Pade order 13.
eta_3 = max(h.d6_tight, h.d8_loose)
eta_4 = max(h.d8_loose, h.d10_loose)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
# Choose smallest s>=0 such that 2**(-s) eta_5 <= theta_13
if eta_5 == 0:
# Nilpotent special case
s = 0
else:
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + _ell(2**-s * h.A, 13)
U, V = h.pade13_scaled(s)
X = _solve_P_Q(U, V)
# X = r_13(A)^(2^s) by repeated squaring.
for i in range(s):
X = X.dot(X)
return X
def _expm_SS(A, ssA, order): # , use_exact_onenorm='auto'):
# Track functions of A to help compute the matrix exponential.
h = _ExpmPadeHelper_SS(A, ssA, order)
structure = None
# Try Pade order 3.
eta_1 = max(h.d4_loose, h.d6_loose)
if eta_1 < 1.495585217958292e-002 and mf._ell(h.A, 3) == 0:
U, V = h.pade3()
return mf._solve_P_Q(U, V, structure=structure)
# Try Pade order 5.
eta_2 = max(h.d4_tight, h.d6_loose)
if eta_2 < 2.539398330063230e-001 and mf._ell(h.A, 5) == 0:
U, V = h.pade5()
return mf._solve_P_Q(U, V, structure=structure)
# Try Pade orders 7 and 9.
eta_3 = max(h.d6_tight, h.d8_loose)
if eta_3 < 9.504178996162932e-001 and mf._ell(h.A, 7) == 0:
U, V = h.pade7()
return mf._solve_P_Q(U, V, structure=structure)
if eta_3 < 2.097847961257068e000 and mf._ell(h.A, 9) == 0:
U, V = h.pade9()
return mf._solve_P_Q(U, V, structure=structure)
# Use Pade order 13.
eta_4 = max(h.d8_loose, h.d10_loose)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + mf._ell(2 ** -s * h.A, 13)
U, V = h.pade13_scaled(s)
X = mf._solve_P_Q(U, V, structure=structure)
# X = r_13(A)^(2^s) by repeated squaring.
for _ in range(s):
X = X.dot(X)
return X
def Return(U, V, P, Q, geti2, pade):
E = mf._solve_P_Q(U, V, structure=structure)
I = _solve_P_Q_2(P, Q, structure=structure)
if geti2:
return E, I, _geti2(H, E, I, h, pade)
return E, I
def expmint(A, h, geti2=False):
"""
Compute the matrix exponential and its integral(s) using Pade
approximation.
Parameters
----------
A : (M, M) array_like or sparse matrix
2D Array or Matrix (sparse or dense) to be exponentiated
h : scalar
Time step
geti2 : bool
If True, also return the `I2` integral below. Useful for
first order holds.
Returns
-------
E : (M, M) ndarray
Matrix exponential of `A*h`: exp(A*h)
I : (M, M) ndarray
Integral of exp(A*t) dt from 0 to h
I2 : (M, M) ndarray; optional
Integral of exp(A*t)*t dt from 0 to h. Only returned if
`geti2` is True.
Notes
-----
This routine is modeled after and augments
:func:`scipy.linalg.expm`. The power series expansions for these
matrices are (I = identity):
.. code-block:: none
E = I + A*h + (A*h)**2/2! + (A*h)**3/3! + ...
I1 = h*(I + A*h/2 + (A*h)**2/3! + (A*h)**3/4! + ...)
I2 = h*h*(I/2 + A*h/3 + (A*h)**2/(4*2!) + (A*h)**3/(5*3!) + ...)
If `A` is non-singular, the exact solutions for `I1` and `I2` are::
E = exp(A*h)
I1 = inv(A)*(E-I)
I2 = inv(A)*(E*h-I1)
The Pade approximants for those power series are used for `E` and
`I1`. If necessary, the 'squaring and scaling' method will be used
such that a 13th order Pade approximation will be accurate. See
references [#exp1]_, [#exp2]_, and [#exp3]_ for more information.
For `I2`, a Pade approximation is used if a 3rd, 5th, 7th or 9th
order is accurate. If not, `A` is checked for singularity. If
non-singular, the exact solution show above is used. If A is
singular, a the power series is used directly until it converges
(and a warning message is printed about using a finer time step.)
References
----------
.. [#exp1] Awad H. Al-Mohy and Nicholas J. Higham (2009)
"A New Scaling and Squaring Algorithm for the Matrix
Exponential."
SIAM Journal on Matrix Analysis and Applications.
31 (3). pp. 970-989. ISSN 1095-7162
.. [#exp2] Nicholas J. Higham (2005)
"The Scaling and Squaring Method for the Matrix Exponential
Revisited."
SIAM Journal on Matrix Analysis and Applications.
Vol 26, No. 4, pp 1179-1193.
.. [#exp3] David Westreich (1990)
"A Practical Method for Computing the Exponential of a Matrix
and its Integral."
Communications in Applied Numerical Methods, Vol 6, 375-380.
Examples
--------
>>> from pyyeti import expmint
>>> import numpy as np
>>> import scipy.linalg as la
>>> np.set_printoptions(4)
>>> a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> e, i, i2 = expmint.expmint(a, .05, True)
>>> e
array([[ 1.0996, 0.1599, 0.2202],
[ 0.3099, 1.3849, 0.46 ],
[ 0.5202, 0.61 , 1.6998]])
>>> i
array([[ 0.052 , 0.0034, 0.0048],
[ 0.0067, 0.0583, 0.01 ],
[ 0.0114, 0.0133, 0.0651]])
>>> i2
array([[ 0.0013, 0.0001, 0.0002],
[ 0.0002, 0.0015, 0.0003],
[ 0.0004, 0.0005, 0.0018]])
"""
# Avoid indiscriminate asarray() to allow sparse or other strange
# arrays.
if isinstance(A, (list, tuple)):
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError("expected a square matrix")
# Detect upper triangularity.
if mf._is_upper_triangular(A):
structure = mf.UPPER_TRIANGULAR
else:
structure = None
# Hardcode a matrix order threshold for exact vs. estimated
# one-norms.
use_exact_onenorm = A.shape[0] < 200
# Track functions of A to help compute the matrix exponential.
H = _ExpmIntPadeHelper(
A * h, structure=structure, use_exact_onenorm=use_exact_onenorm
)
def Return(U, V, P, Q, geti2, pade):
E = mf._solve_P_Q(U, V, structure=structure)
I = _solve_P_Q_2(P, Q, structure=structure)
if geti2:
return E, I, _geti2(H, E, I, h, pade)
return E, I
# Try Pade order 3.
eta_1 = max(H.d4_loose, H.d6_loose)
if eta_1 < 1.495585217958292e-002 and mf._ell(H.A, 3) == 0:
U, V, P, Q = H.pade3_i(h)
return Return(U, V, P, Q, geti2, 3)
# Try Pade order 5.
eta_2 = max(H.d4_tight, H.d6_loose)
if eta_2 < 2.539398330063230e-001 and mf._ell(H.A, 5) == 0:
U, V, P, Q = H.pade5_i(h)
return Return(U, V, P, Q, geti2, 5)
# Try Pade orders 7 and 9.
eta_3 = max(H.d6_tight, H.d8_loose)
if eta_3 < 9.504178996162932e-001 and mf._ell(H.A, 7) == 0:
U, V, P, Q = H.pade7_i(h)
return Return(U, V, P, Q, geti2, 7)
if eta_3 < 2.097847961257068e000 and mf._ell(H.A, 9) == 0:
U, V, P, Q = H.pade9_i(h)
return Return(U, V, P, Q, geti2, 9)
# Use Pade order 13.
eta_4 = max(H.d8_loose, H.d10_loose)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + mf._ell(2 ** -s * H.A, 13)
U, V, P, Q = H.pade13_scaled_i(s, h)
E = mf._solve_P_Q(U, V, structure=structure)
I = _solve_P_Q_2(P, Q, structure=structure)
# E = r_13(A)^(2^s) by repeated squaring.
for _ in range(s):
I += I.dot(E)
E = E.dot(E)
if geti2:
return E, I, _geti2(H, E, I, h, 13)
return E, I
def expm(self, A, max_mat_mult=100, balance=True):
''' max_mat_mult can be used to reduce the computational complexity of the exponential
6: at most a Pade order 13 can be used but with no scaling
5: at most a Pade order 9 is used
4: at most a Pade order 7 is used
3: at most a Pade order 5 is used
2: at most a Pade order 3 is used
'''
if (np.any(np.isnan(A))):
print("Matrix A contains nan")
# Compute expm(A)*v
if balance:
A_bal, D = matrix_balance(A, permute=False)
Dinv = np.copy(D)
for i in range(D.shape[0]):
Dinv[i, i] = 1.0 / D[i, i]
# assert(np.max(np.abs(A_bal-(Dinv@A@D)))==0.0)
else:
A_bal = A
# Hardcode a matrix order threshold for exact vs. estimated one-norms.
use_exact_onenorm = A.shape[0] < 200
h = _ExpmPadeHelper(A_bal, use_exact_onenorm=use_exact_onenorm)
structure = None
# Compute the number of mat-mat multiplications needed in theory
self.mat_mult_in_theory = self.compute_mat_mult(A_bal)
self.mat_mult = min(self.mat_mult_in_theory, max_mat_mult)
self.mat_norm = np.linalg.norm(A_bal, 1)
if self.mat_mult <= 0:
U, V = self.pade1(A_bal)
X = _solve_P_Q(U, V, structure=structure)
if self.mat_mult == 1:
U, V = self.pade2(A_bal)
X = _solve_P_Q(U, V, structure=structure)
if self.mat_mult == 2:
U, V = h.pade3()
# U_, V_ = self.pade3(A_bal)
# assert(np.max(np.abs(U-U_))==0.0)
# assert(np.max(np.abs(V-V_))==0.0)
X = _solve_P_Q(U, V, structure=structure)
# Try Pade order 5.
if self.mat_mult == 3:
U, V = h.pade5()
X = _solve_P_Q(U, V, structure=structure)
# Try Pade orders 7 and 9.
if self.mat_mult == 4:
U, V = h.pade7()
X = _solve_P_Q(U, V, structure=structure)
if self.mat_mult == 5:
U, V = h.pade9()
X = _solve_P_Q(U, V, structure=structure)
if self.mat_mult > 5:
s = self.mat_mult - 6
U, V = h.pade13_scaled(s)
X = _solve_P_Q(U, V)
# X = r_13(A)^(2^s) by repeated squaring.
for i in range(s):
X = X.dot(X)
if (balance):
X = D @ X @ Dinv
# assert(np.max(np.abs(expm(A) - X)) == 0.0)
return X
def expm(A, A2=A2):
"""
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`
Notes
-----
This is algorithm (6.1) which is a simplification of algorithm (5.1).
References
----------
.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009)
"A New Scaling and Squaring Algorithm for the Matrix Exponential."
SIAM Journal on Matrix Analysis and Applications.
31 (3). pp. 970-989. ISSN 1095-7162
"""
# Use Pade order 3, no matter what.
multiply(A, A, out=A2)
U, V = _pade3(A, ident, A2)
return _solve_P_Q(U, V, structure=None)
# Detect upper triangularity.
# structure = UPPER_TRIANGULAR if _is_upper_triangular(A) else None
structure = None
# Define the identity matrix depending on sparsity.
# ident = _ident_like(A)
# Try Pade order 3.
A2 = A.dot(A)
d6 = _onenormest_matrix_power(A2, 3)**(1 / 6.)
eta_1 = max(_onenormest_matrix_power(A2, 2)**(1 / 4.), d6)
if eta_1 < 1.495585217958292e-002 and _ell(A, 3) == 0:
U, V = _pade3(A, ident, A2)
return _solve_P_Q(U, V, structure=structure)
# Try Pade order 5.
A4 = A2.dot(A2)
d4 = _exact_1_norm(A4)**(1 / 4.)
eta_2 = max(d4, d6)
if eta_2 < 2.539398330063230e-001 and _ell(A, 5) == 0:
U, V = _pade5(A, ident, A2, A4)
return _solve_P_Q(U, V, structure=structure)
# Try Pade orders 7 and 9.
A6 = A2.dot(A4)
d6 = _exact_1_norm(A6)**(1 / 6.)
d8 = _onenormest_matrix_power(A4, 2)**(1 / 8.)
eta_3 = max(d6, d8)
if eta_3 < 9.504178996162932e-001 and _ell(A, 7) == 0:
U, V = _pade7(A, ident, A2, A4, A6)
return _solve_P_Q(U, V, structure=structure)
if eta_3 < 2.097847961257068e+000 and _ell(A, 9) == 0:
U, V = _pade9(A, ident, A2, A4, A6)
return _solve_P_Q(U, V, structure=structure)
# Use Pade order 13.
d10 = _onenormest_product((A4, A6))**(1 / 10.)
eta_4 = max(d8, d10)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + _ell(2**-s * A, 13)
B = A * 2**-s
B2 = A2 * 2**(-2 * s)
B4 = A4 * 2**(-4 * s)
B6 = A6 * 2**(-6 * s)
U, V = _pade13(B, ident, B2, B4, B6)
X = _solve_P_Q(U, V, structure=structure)
if structure == UPPER_TRIANGULAR:
# Invoke Code Fragment 2.1.
X = _fragment_2_1(X, A, s)
else:
# X = r_13(A)^(2^s) by repeated squaring.
for i in range(s):
X = X.dot(X)
return X
def expm(A,A2 = A2):
"""
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`
Notes
-----
This is algorithm (6.1) which is a simplification of algorithm (5.1).
References
----------
.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009)
"A New Scaling and Squaring Algorithm for the Matrix Exponential."
SIAM Journal on Matrix Analysis and Applications.
31 (3). pp. 970-989. ISSN 1095-7162
"""
# Use Pade order 3, no matter what.
multiply(A,A,out=A2)
U, V = _pade3(A, ident, A2)
return _solve_P_Q(U, V, structure=None)
# Detect upper triangularity.
# structure = UPPER_TRIANGULAR if _is_upper_triangular(A) else None
structure = None
# Define the identity matrix depending on sparsity.
# ident = _ident_like(A)
# Try Pade order 3.
A2 = A.dot(A)
d6 = _onenormest_matrix_power(A2, 3)**(1/6.)
eta_1 = max(_onenormest_matrix_power(A2, 2)**(1/4.), d6)
if eta_1 < 1.495585217958292e-002 and _ell(A, 3) == 0:
U, V = _pade3(A, ident, A2)
return _solve_P_Q(U, V, structure=structure)
# Try Pade order 5.
A4 = A2.dot(A2)
d4 = _exact_1_norm(A4)**(1/4.)
eta_2 = max(d4, d6)
if eta_2 < 2.539398330063230e-001 and _ell(A, 5) == 0:
U, V = _pade5(A, ident, A2, A4)
return _solve_P_Q(U, V, structure=structure)
# Try Pade orders 7 and 9.
A6 = A2.dot(A4)
d6 = _exact_1_norm(A6)**(1/6.)
d8 = _onenormest_matrix_power(A4, 2)**(1/8.)
eta_3 = max(d6, d8)
if eta_3 < 9.504178996162932e-001 and _ell(A, 7) == 0:
U, V = _pade7(A, ident, A2, A4, A6)
return _solve_P_Q(U, V, structure=structure)
if eta_3 < 2.097847961257068e+000 and _ell(A, 9) == 0:
U, V = _pade9(A, ident, A2, A4, A6)
return _solve_P_Q(U, V, structure=structure)
# Use Pade order 13.
d10 = _onenormest_product((A4, A6))**(1/10.)
eta_4 = max(d8, d10)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + _ell(2**-s * A, 13)
B = A * 2**-s
B2 = A2 * 2**(-2*s)
B4 = A4 * 2**(-4*s)
B6 = A6 * 2**(-6*s)
U, V = _pade13(B, ident, B2, B4, B6)
X = _solve_P_Q(U, V, structure=structure)
if structure == UPPER_TRIANGULAR:
# Invoke Code Fragment 2.1.
X = _fragment_2_1(X, A, s)
else:
# X = r_13(A)^(2^s) by repeated squaring.
for i in range(s):
X = X.dot(X)
return X
def expm_times_v(A, v, scaling_reduction=0, verbose=False):
''' scaling_reduction can be used to reduce the computational complexity of the exponential
1: no scaling can be performed
2: at most a Pade order 9 is used
3: at most a Pade order 7 is used
4: at most a Pade order 5 is used
5: at most a Pade order 3 is used
'''
# Compute expm(A)*v
# Hardcode a matrix order threshold for exact vs. estimated one-norms.
use_exact_onenorm = A.shape[0] < 200
h = _ExpmPadeHelper(A, use_exact_onenorm=use_exact_onenorm)
structure = None
# Try Pade order 3.
eta_1 = max(h.d4_loose, h.d6_loose)
if scaling_reduction > 4 or (eta_1 < 1.495585217958292e-002
and _ell(h.A, 3) == 0):
U, V = h.pade3()
X = _solve_P_Q(U, V, structure=structure)
return X.dot(v)
# Try Pade order 5.
eta_2 = max(h.d4_tight, h.d6_loose)
if scaling_reduction > 3 or (eta_2 < 2.539398330063230e-001
and _ell(h.A, 5) == 0):
U, V = h.pade5()
X = _solve_P_Q(U, V, structure=structure)
return X.dot(v)
# Try Pade orders 7 and 9.
eta_3 = max(h.d6_tight, h.d8_loose)
if scaling_reduction > 2 or (eta_3 < 9.504178996162932e-001
and _ell(h.A, 7) == 0):
U, V = h.pade7()
X = _solve_P_Q(U, V, structure=structure)
return X.dot(v)
if scaling_reduction > 1 or (eta_3 < 2.097847961257068e+000
and _ell(h.A, 9) == 0):
U, V = h.pade9()
X = _solve_P_Q(U, V, structure=structure)
return X.dot(v)
# Use Pade order 13.
eta_3 = max(h.d6_tight, h.d8_loose)
eta_4 = max(h.d8_loose, h.d10_loose)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
# Choose smallest s>=0 such that 2**(-s) eta_5 <= theta_13
if eta_5 == 0:
# Nilpotent special case
s = 0
else:
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + _ell(2**-s * h.A, 13)
if (scaling_reduction > 0):
s = 0
U, V = h.pade13_scaled(s)
X = _solve_P_Q(U, V)
# X = r_13(A)^(2^s) by repeated squaring.
for i in range(s):
X = X.dot(X)
res = X.dot(v)
return res