def test_poorly_conditioned(self):
""" Testing against a single poorly-conditioned matrix.
Generated as a random 6x6 array, and then divided the first row and column by 1e6
"""
mat = np.array([[
6.6683264e-07, 6.3376014e-07, 4.5754601e-07, 7.1252750e-07,
7.5942456e-07, 5.9028134e-07
],
[
5.7484930e-07, 6.5606222e-01, 6.5073522e-01,
2.4251825e-01, 1.0735555e-01, 2.6956707e-01
],
[
6.1098206e-07, 4.0445840e-01, 5.6152644e-01,
5.8278476e-01, 8.0418942e-01, 7.7306821e-01
],
[
8.5656473e-07, 4.3833014e-01, 5.7838875e-01,
2.4317287e-01, 1.6641302e-01, 3.9590950e-01
],
[
4.4858020e-07, 1.1731434e-01, 7.3598305e-01,
4.5670520e-01, 5.8185932e-01, 9.4438709e-01
],
[
4.1073805e-07, 9.6286350e-02, 7.7365790e-01,
1.3120009e-01, 9.3908360e-01, 1.4665616e-01
]])
phi = expm(mat)
out_scipy = expm_scipy(mat)
assert_arrays_equal(out_scipy, phi, decimal=8)
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 test_expm_nearly_identity(self):
""" nearly the identity matrix """
a = np.array(
[[1.0012, 0, 0.0003], [0.0075, 0.9876543, 0.0011], [0, 0, 0.9873]],
dtype=np.float64)
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_one_huge_element(self):
""" one value swamping the rest """
a = np.array(
[[999.875, 0.2, 0.3], [0.1, 0.002, 0.001], [0.01, 0, -0.003]],
dtype=np.float64)
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_one_large_element(self):
""" one value slightly swamping the rest """
a = np.array(
[[0.1, 0.02, 0.003], [1.875, 0.12, 0.11], [0.1234567, 0, 0.3]],
dtype=np.float64)
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_random_non_zero_floats(self):
""" arbitrarily large values, and some negatives """
a = np.array([[99.23452, 2.0000234523, 0.0003, 378.2362],
[1.00001, 8754.236, 1.1007, 33.333333],
[111, 0.00034, 3922.323, -999.333],
[-1234.5678, -0.00034, 333.65, 13]],
dtype=np.float64)
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_random_sparse(self):
""" Testing against a large set of random 2D matricies with defined sparsity
"""
np.random.seed(7)
for size in range(5, 40):
for d in [x / 100 for x in range(10, 90)]:
mat = sparse.random(size, size, density=d).toarray()
phi = expm(mat)
out_scipy = expm_scipy(mat)
assert_arrays_equal(out_scipy, phi, decimal=7)
def expm(M):
""" Computes the matrix exponential of M.
Args:
M (torch.tensor): Square matrix (N, N)
Returns:
M (torch.tensor): Matrix exponential (N, N)
"""
device = M.device
dtype = M.dtype
M = M.detach().cpu().numpy()
M = expm_scipy(M)
M = torch.from_numpy(M).type(dtype).to(device)
return M
def test_expm_random_integers(self):
""" a simple, arbitrary matrix """
a = np.array([[1, 2, 3], [1, 2, 1.1], [1, 0, 3]], dtype=np.float64)
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_all_ones(self):
""" Test against a matrix of all ones """
a = np.ones((33, 33), dtype='float64')
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_all_zeros(self):
""" Test against a matrix of all zeros """
a = np.zeros((20, 20), dtype='float32')
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)
def test_expm_smallish_floats(self):
""" This is just some arbitrary set of small-ish floats """
a = np.arange(1, 17, dtype='float64').reshape(4, 4) / 1e3
phi = expm(a)
out_scipy = expm_scipy(a)
assert_arrays_equal(out_scipy, phi, decimal=12)