def d8_loose(self):
if self.use_exact_onenorm:
return self.d8_tight
if self._d8_exact is not None:
return self._d8_exact
else:
if self._d8_approx is None:
self._d8_approx = matfuncs._onenormest_matrix_power(
self.A4, 2, structure=self.structure)**(1 / 8.)
return self._d8_approx
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