def pade9_i(self, h):
U, V = self.pade9()
p = (
335221286400.0,
8821612800.0,
11762150400.0,
302702400.0,
90810720.0,
2162160.0,
205920.0,
3960.0,
110.0,
1.0,
)
q = (
335221286400.0,
-158789030400.0,
35286451200.0,
-4843238400.0,
454053600.0,
-30270240.0,
1441440.0,
-47520.0,
990.0,
-10.0,
)
P = h * (
mf._smart_matrix_product(
self.A,
p[9] * self.A8
+ p[7] * self.A6
+ p[5] * self.A4
+ p[3] * self.A2
+ p[1] * self.ident,
structure=self.structure,
)
+ p[8] * self.A8
+ p[6] * self.A6
+ p[4] * self.A4
+ p[2] * self.A2
+ p[0] * self.ident
)
Q = (
mf._smart_matrix_product(
self.A,
q[9] * self.A8
+ q[7] * self.A6
+ q[5] * self.A4
+ q[3] * self.A2
+ q[1] * self.ident,
structure=self.structure,
)
+ q[8] * self.A8
+ q[6] * self.A6
+ q[4] * self.A4
+ q[2] * self.A2
+ q[0] * self.ident
)
return U, V, P, Q
def pade7_i(self, h):
U, V = self.pade7()
p = (259459200.0, 8648640.0, 8648640.0, 277200.0, 55440.0, 1512.0, 72.0, 1.0)
q = (
259459200.0,
-121080960.0,
25945920.0,
-3326400.0,
277200.0,
-15120.0,
504.0,
-8.0,
)
P = h * (
mf._smart_matrix_product(
self.A,
p[7] * self.A6 + p[5] * self.A4 + p[3] * self.A2 + p[1] * self.ident,
structure=self.structure,
)
+ p[6] * self.A6
+ p[4] * self.A4
+ p[2] * self.A2
+ p[0] * self.ident
)
Q = (
mf._smart_matrix_product(
self.A,
q[7] * self.A6 + q[5] * self.A4 + q[3] * self.A2 + q[1] * self.ident,
structure=self.structure,
)
+ q[6] * self.A6
+ q[4] * self.A4
+ q[2] * self.A2
+ q[0] * self.ident
)
return U, V, P, Q
def A5(self):
if self._A5 is None:
self._A5 = mf._smart_matrix_product(
self.A, self.A4, structure=self.structure
)
return self._A5
def _geti2(H, E, I, h, pade):
"""
A helper routine for expmint; computes I2. See :func:`expmint`.
"""
if pade <= 3:
p = (450240.0, 104860.0, 15480.0, 1001.0)
q = (900480.0, -390600.0, 66240.0, -4540.0)
P = (h * h) * (p[3] * H.A3 + p[2] * H.A2 + p[1] * H.A + p[0] * H.ident)
Q = q[3] * H.A3 + q[2] * H.A2 + q[1] * H.A + q[0] * H.ident
return _solve_P_Q_2(P, Q, H.structure)
if pade <= 5:
p = (
69363423360.0,
14569133040.0,
2595781440.0,
239983884.0,
14154000.0,
398959.0,
)
q = (
138726846720.0,
-63346298400.0,
12740716800.0,
-1425725280.0,
89937120.0,
-2602278.0,
)
P = (h * h) * (
mf._smart_matrix_product(
H.A, p[5] * H.A4 + p[3] * H.A2 + p[1] * H.ident, structure=H.structure
)
+ p[4] * H.A4
+ p[2] * H.A2
+ p[0] * H.ident
)
Q = (
mf._smart_matrix_product(
H.A, q[5] * H.A4 + q[3] * H.A2 + q[1] * H.ident, structure=H.structure
)
+ q[4] * H.A4
+ q[2] * H.A2
+ q[0] * H.ident
)
return _solve_P_Q_2(P, Q, H.structure)
if pade <= 7:
p = (
41840477770291200.0,
8321436219096000.0,
1617627908856960.0,
159144893827920.0,
12344957190720.0,
614984330664.0,
19815037200.0,
312129649.0,
)
q = (
83680955540582400.0,
-39144431255529600.0,
8411304436254720.0,
-1081869058670400.0,
90503101180800.0,
-4959510549840.0,
166265789760.0,
-2658297528.0,
)
P = (h * h) * (
mf._smart_matrix_product(
H.A,
p[7] * H.A6 + p[5] * H.A4 + p[3] * H.A2 + p[1] * H.ident,
structure=H.structure,
)
+ p[6] * H.A6
+ p[4] * H.A4
+ p[2] * H.A2
+ p[0] * H.ident
)
Q = (
mf._smart_matrix_product(
H.A,
q[7] * H.A6 + q[5] * H.A4 + q[3] * H.A2 + q[1] * H.ident,
structure=H.structure,
)
+ q[6] * H.A6
+ q[4] * H.A4
+ q[2] * H.A2
+ q[0] * H.ident
)
return _solve_P_Q_2(P, Q, H.structure)
if pade <= 9:
p = (
69491589579005577984000.0,
13362425784392593708800.0,
2733673676598876211200.0,
274218651095712523200.0,
24016200124299102720.0,
1393079626219366800.0,
63177322465033920.0,
1972643393629480.0,
40190548856040.0,
403978495031.0,
)
q = (
138983179158011155968000.0,
-65930601203222249894400.0,
14675286699176463360000.0,
-2017982140321398451200.0,
189583633133409715200.0,
-12669377873982429600.0,
604988181888330240.0,
-20009929231749600.0,
418494695659920.0,
-4247085597370.0,
)
P = (h * h) * (
mf._smart_matrix_product(
H.A,
p[9] * H.A8 + p[7] * H.A6 + p[5] * H.A4 + p[3] * H.A2 + p[1] * H.ident,
structure=H.structure,
)
+ p[8] * H.A8
+ p[6] * H.A6
+ p[4] * H.A4
+ p[2] * H.A2
+ p[0] * H.ident
)
Q = (
mf._smart_matrix_product(
H.A,
q[9] * H.A8 + q[7] * H.A6 + q[5] * H.A4 + q[3] * H.A2 + q[1] * H.ident,
structure=H.structure,
)
+ q[8] * H.A8
+ q[6] * H.A6
+ q[4] * H.A4
+ q[2] * H.A2
+ q[0] * H.ident
)
return _solve_P_Q_2(P, Q, H.structure)
# second, try direct solution:
n = H.A.shape[0]
with warnings.catch_warnings():
warnings.simplefilter("error", RuntimeWarning)
try:
lup = la.lu_factor(H.A)
# decomp maybe worked ... check it:
# I = A\(expm(A*h) - Ident)
I_test = la.lu_solve(lup, h * (E - np.eye(n)))
if np.allclose(I_test, I):
# I2 = A\(expm(A*h)*h - A\(expm(A*h) - Ident))
return la.lu_solve(lup, h * (E * h - I_test))
except RuntimeWarning:
pass
# finally, use taylor series:
msg = (
"Using power series expansion directly for `I2` (see"
" `expmint`).\nRecommendation: use a finer time step."
)
warnings.warn(msg, RuntimeWarning)
I2 = np.eye(n) / 2
term = H.A
j = 1.0
tol = 1e-15
maxloops = 200
while abs(term).max() > tol * abs(E).max() and j < maxloops:
j += 1.0
I2 += term / (j + 1)
term = term.dot(H.A) / j
if j >= maxloops:
raise RuntimeError(
f"maximum loops ({maxloops}) exceeded for power series expansion"
)
return (h * h) * I2
def A3(self):
if self._A3 is None:
self._A3 = mf._smart_matrix_product(
self.A, self.A2, structure=self.structure
)
return self._A3
def pade13_scaled_i(self, s, h):
b = (
64764752532480000.0,
32382376266240000.0,
7771770303897600.0,
1187353796428800.0,
129060195264000.0,
10559470521600.0,
670442572800.0,
33522128640.0,
1323241920.0,
40840800.0,
960960.0,
16380.0,
182.0,
1.0,
)
B = self.A * 2 ** -s
h = h * 2 ** -s
B2 = self.A2 * 2 ** (-2 * s)
B4 = self.A4 * 2 ** (-4 * s)
B6 = self.A6 * 2 ** (-6 * s)
U2 = mf._smart_matrix_product(
B6, b[13] * B6 + b[11] * B4 + b[9] * B2, structure=self.structure
)
U = mf._smart_matrix_product(
B,
(U2 + b[7] * B6 + b[5] * B4 + b[3] * B2 + b[1] * self.ident),
structure=self.structure,
)
V2 = mf._smart_matrix_product(
B6, b[12] * B6 + b[10] * B4 + b[8] * B2, structure=self.structure
)
V = V2 + b[6] * B6 + b[4] * B4 + b[2] * B2 + b[0] * self.ident
p = (
1748648318376960000.0,
32382376266240000.0,
64764752532480000.0,
1187353796428800.0,
593676898214400.0,
10559470521600.0,
2011327718400.0,
33522128640.0,
2793510720.0,
40840800.0,
1485120.0,
16380.0,
210.0,
1.0,
)
q = (
1748648318376960000.0,
-841941782922240000.0,
194294257597440000.0,
-28496491114291200.0,
2968384491072000.0,
-232308351475200.0,
14079294028800.0,
-670442572800.0,
25141596480.0,
-735134400.0,
16336320.0,
-262080.0,
2730.0,
-14.0,
)
_P2 = mf._smart_matrix_product(
B6, p[13] * B6 + p[11] * B4 + p[9] * B2, structure=self.structure
)
P2 = mf._smart_matrix_product(
B,
_P2 + p[7] * B6 + p[5] * B4 + p[3] * B2 + p[1] * self.ident,
structure=self.structure,
)
P1 = mf._smart_matrix_product(
B6, p[12] * B6 + p[10] * B4 + p[8] * B2, structure=self.structure
)
P = h * (P2 + P1 + p[6] * B6 + p[4] * B4 + p[2] * B2 + p[0] * self.ident)
_Q2 = mf._smart_matrix_product(
B6, q[13] * B6 + q[11] * B4 + q[9] * B2, structure=self.structure
)
Q2 = mf._smart_matrix_product(
B,
_Q2 + q[7] * B6 + q[5] * B4 + q[3] * B2 + q[1] * self.ident,
structure=self.structure,
)
Q1 = mf._smart_matrix_product(
B6, q[12] * B6 + q[10] * B4 + q[8] * B2, structure=self.structure
)
Q = Q2 + Q1 + q[6] * B6 + q[4] * B4 + q[2] * B2 + q[0] * self.ident
return U, V, P, Q