1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
|
"""Test functions for the sparse.linalg._expm_multiply module
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import assert_allclose, assert_, assert_equal
from scipy._lib._numpy_compat import suppress_warnings
from scipy.sparse import SparseEfficiencyWarning
import scipy.linalg
from scipy.sparse.linalg._expm_multiply import (_theta, _compute_p_max,
_onenormest_matrix_power, expm_multiply, _expm_multiply_simple,
_expm_multiply_interval)
def less_than_or_close(a, b):
return np.allclose(a, b) or (a < b)
class TestExpmActionSimple(object):
"""
These tests do not consider the case of multiple time steps in one call.
"""
def test_theta_monotonicity(self):
pairs = sorted(_theta.items())
for (m_a, theta_a), (m_b, theta_b) in zip(pairs[:-1], pairs[1:]):
assert_(theta_a < theta_b)
def test_p_max_default(self):
m_max = 55
expected_p_max = 8
observed_p_max = _compute_p_max(m_max)
assert_equal(observed_p_max, expected_p_max)
def test_p_max_range(self):
for m_max in range(1, 55+1):
p_max = _compute_p_max(m_max)
assert_(p_max*(p_max - 1) <= m_max + 1)
p_too_big = p_max + 1
assert_(p_too_big*(p_too_big - 1) > m_max + 1)
def test_onenormest_matrix_power(self):
np.random.seed(1234)
n = 40
nsamples = 10
for i in range(nsamples):
A = scipy.linalg.inv(np.random.randn(n, n))
for p in range(4):
if not p:
M = np.identity(n)
else:
M = np.dot(M, A)
estimated = _onenormest_matrix_power(A, p)
exact = np.linalg.norm(M, 1)
assert_(less_than_or_close(estimated, exact))
assert_(less_than_or_close(exact, 3*estimated))
def test_expm_multiply(self):
np.random.seed(1234)
n = 40
k = 3
nsamples = 10
for i in range(nsamples):
A = scipy.linalg.inv(np.random.randn(n, n))
B = np.random.randn(n, k)
observed = expm_multiply(A, B)
expected = np.dot(scipy.linalg.expm(A), B)
assert_allclose(observed, expected)
def test_matrix_vector_multiply(self):
np.random.seed(1234)
n = 40
nsamples = 10
for i in range(nsamples):
A = scipy.linalg.inv(np.random.randn(n, n))
v = np.random.randn(n)
observed = expm_multiply(A, v)
expected = np.dot(scipy.linalg.expm(A), v)
assert_allclose(observed, expected)
def test_scaled_expm_multiply(self):
np.random.seed(1234)
n = 40
k = 3
nsamples = 10
for i in range(nsamples):
for t in (0.2, 1.0, 1.5):
with np.errstate(invalid='ignore'):
A = scipy.linalg.inv(np.random.randn(n, n))
B = np.random.randn(n, k)
observed = _expm_multiply_simple(A, B, t=t)
expected = np.dot(scipy.linalg.expm(t*A), B)
assert_allclose(observed, expected)
def test_scaled_expm_multiply_single_timepoint(self):
np.random.seed(1234)
t = 0.1
n = 5
k = 2
A = np.random.randn(n, n)
B = np.random.randn(n, k)
observed = _expm_multiply_simple(A, B, t=t)
expected = scipy.linalg.expm(t*A).dot(B)
assert_allclose(observed, expected)
def test_sparse_expm_multiply(self):
np.random.seed(1234)
n = 40
k = 3
nsamples = 10
for i in range(nsamples):
A = scipy.sparse.rand(n, n, density=0.05)
B = np.random.randn(n, k)
observed = expm_multiply(A, B)
with suppress_warnings() as sup:
sup.filter(SparseEfficiencyWarning,
"splu requires CSC matrix format")
sup.filter(SparseEfficiencyWarning,
"spsolve is more efficient when sparse b is in the CSC matrix format")
expected = scipy.linalg.expm(A).dot(B)
assert_allclose(observed, expected)
def test_complex(self):
A = np.array([
[1j, 1j],
[0, 1j]], dtype=complex)
B = np.array([1j, 1j])
observed = expm_multiply(A, B)
expected = np.array([
1j * np.exp(1j) + 1j * (1j*np.cos(1) - np.sin(1)),
1j * np.exp(1j)], dtype=complex)
assert_allclose(observed, expected)
class TestExpmActionInterval(object):
def test_sparse_expm_multiply_interval(self):
np.random.seed(1234)
start = 0.1
stop = 3.2
n = 40
k = 3
endpoint = True
for num in (14, 13, 2):
A = scipy.sparse.rand(n, n, density=0.05)
B = np.random.randn(n, k)
v = np.random.randn(n)
for target in (B, v):
X = expm_multiply(A, target,
start=start, stop=stop, num=num, endpoint=endpoint)
samples = np.linspace(start=start, stop=stop,
num=num, endpoint=endpoint)
with suppress_warnings() as sup:
sup.filter(SparseEfficiencyWarning,
"splu requires CSC matrix format")
sup.filter(SparseEfficiencyWarning,
"spsolve is more efficient when sparse b is in the CSC matrix format")
for solution, t in zip(X, samples):
assert_allclose(solution,
scipy.linalg.expm(t*A).dot(target))
def test_expm_multiply_interval_vector(self):
np.random.seed(1234)
start = 0.1
stop = 3.2
endpoint = True
for num in (14, 13, 2):
for n in (1, 2, 5, 20, 40):
A = scipy.linalg.inv(np.random.randn(n, n))
v = np.random.randn(n)
X = expm_multiply(A, v,
start=start, stop=stop, num=num, endpoint=endpoint)
samples = np.linspace(start=start, stop=stop,
num=num, endpoint=endpoint)
for solution, t in zip(X, samples):
assert_allclose(solution, scipy.linalg.expm(t*A).dot(v))
def test_expm_multiply_interval_matrix(self):
np.random.seed(1234)
start = 0.1
stop = 3.2
endpoint = True
for num in (14, 13, 2):
for n in (1, 2, 5, 20, 40):
for k in (1, 2):
A = scipy.linalg.inv(np.random.randn(n, n))
B = np.random.randn(n, k)
X = expm_multiply(A, B,
start=start, stop=stop, num=num, endpoint=endpoint)
samples = np.linspace(start=start, stop=stop,
num=num, endpoint=endpoint)
for solution, t in zip(X, samples):
assert_allclose(solution, scipy.linalg.expm(t*A).dot(B))
def test_sparse_expm_multiply_interval_dtypes(self):
# Test A & B int
A = scipy.sparse.diags(np.arange(5),format='csr', dtype=int)
B = np.ones(5, dtype=int)
Aexpm = scipy.sparse.diags(np.exp(np.arange(5)),format='csr')
assert_allclose(expm_multiply(A,B,0,1)[-1], Aexpm.dot(B))
# Test A complex, B int
A = scipy.sparse.diags(-1j*np.arange(5),format='csr', dtype=complex)
B = np.ones(5, dtype=int)
Aexpm = scipy.sparse.diags(np.exp(-1j*np.arange(5)),format='csr')
assert_allclose(expm_multiply(A,B,0,1)[-1], Aexpm.dot(B))
# Test A int, B complex
A = scipy.sparse.diags(np.arange(5),format='csr', dtype=int)
B = 1j*np.ones(5, dtype=complex)
Aexpm = scipy.sparse.diags(np.exp(np.arange(5)),format='csr')
assert_allclose(expm_multiply(A,B,0,1)[-1], Aexpm.dot(B))
def test_expm_multiply_interval_status_0(self):
self._help_test_specific_expm_interval_status(0)
def test_expm_multiply_interval_status_1(self):
self._help_test_specific_expm_interval_status(1)
def test_expm_multiply_interval_status_2(self):
self._help_test_specific_expm_interval_status(2)
def _help_test_specific_expm_interval_status(self, target_status):
np.random.seed(1234)
start = 0.1
stop = 3.2
num = 13
endpoint = True
n = 5
k = 2
nrepeats = 10
nsuccesses = 0
for num in [14, 13, 2] * nrepeats:
A = np.random.randn(n, n)
B = np.random.randn(n, k)
status = _expm_multiply_interval(A, B,
start=start, stop=stop, num=num, endpoint=endpoint,
status_only=True)
if status == target_status:
X, status = _expm_multiply_interval(A, B,
start=start, stop=stop, num=num, endpoint=endpoint,
status_only=False)
assert_equal(X.shape, (num, n, k))
samples = np.linspace(start=start, stop=stop,
num=num, endpoint=endpoint)
for solution, t in zip(X, samples):
assert_allclose(solution, scipy.linalg.expm(t*A).dot(B))
nsuccesses += 1
if not nsuccesses:
msg = 'failed to find a status-' + str(target_status) + ' interval'
raise Exception(msg)
|
