def _help_product_norm_fast(self, A, B):
# for profiling
t = 2
itmax = 5
D = MatrixProductOperator(A, B)
est, v, w, nmults, nresamples = _onenormest_core(D, D.T, t, itmax)
return est
def _help_product_norm_fast(self, A, B):
# for profiling
t = 2
itmax = 5
D = MatrixProductOperator(A, B)
est, v, w, nmults, nresamples = _onenormest_core(D, D.T, t, itmax)
return est
def resolvent_onenormest(t, T, Z, eps_recip):
op = get_resolvent_operator(T, Z, eps_recip)
#return scipy.sparse.linalg.onenormest(op, t=t)
stuff = _onenormest_core(op, op.H, t=t, itmax=5)
#stuff = _onenormest_core(op, op.H, t=t, itmax=10)
est, v, w, nmults, nresamples = stuff
#print(nmults, nresamples)
#assert_equal(v.dtype, np.dtype(float))
#assert_equal(w.dtype, np.dtype(complex))
return est
def test_onenormest_table_6_t_1(self):
#TODO this test seems to give estimates that match the table,
#TODO even though no attempt has been made to deal with
#TODO complex numbers in the one-norm estimation.
# This will take multiple seconds if your computer is slow like mine.
# It is stochastic, so the tolerance could be too strict.
np.random.seed(1234)
t = 1
n = 100
itmax = 5
nsamples = 5000
observed = []
expected = []
nmult_list = []
nresample_list = []
for i in range(nsamples):
A_inv = np.random.rand(n, n) + 1j * np.random.rand(n, n)
A = scipy.linalg.inv(A_inv)
est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax)
observed.append(est)
expected.append(scipy.linalg.norm(A, 1))
nmult_list.append(nmults)
nresample_list.append(nresamples)
observed = np.array(observed, dtype=float)
expected = np.array(expected, dtype=float)
relative_errors = np.abs(observed - expected) / expected
# check the mean underestimation ratio
underestimation_ratio = observed / expected
underestimation_ratio_mean = np.mean(underestimation_ratio)
assert_(0.90 < underestimation_ratio_mean < 0.99)
# check the required column resamples
max_nresamples = np.max(nresample_list)
assert_equal(max_nresamples, 0)
# check the proportion of norms computed exactly correctly
nexact = np.count_nonzero(relative_errors < 1e-14)
proportion_exact = nexact / float(nsamples)
assert_(0.7 < proportion_exact < 0.8)
# check the average number of matrix*vector multiplications
mean_nmult = np.mean(nmult_list)
assert_(4 < mean_nmult < 5)
def test_onenormest_table_6_t_1(self):
#TODO this test seems to give estimates that match the table,
#TODO even though no attempt has been made to deal with
#TODO complex numbers in the one-norm estimation.
# This will take multiple seconds if your computer is slow like mine.
# It is stochastic, so the tolerance could be too strict.
np.random.seed(1234)
t = 1
n = 100
itmax = 5
nsamples = 5000
observed = []
expected = []
nmult_list = []
nresample_list = []
for i in range(nsamples):
A_inv = np.random.rand(n, n) + 1j * np.random.rand(n, n)
A = scipy.linalg.inv(A_inv)
est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax)
observed.append(est)
expected.append(scipy.linalg.norm(A, 1))
nmult_list.append(nmults)
nresample_list.append(nresamples)
observed = np.array(observed, dtype=float)
expected = np.array(expected, dtype=float)
relative_errors = np.abs(observed - expected) / expected
# check the mean underestimation ratio
underestimation_ratio = observed / expected
underestimation_ratio_mean = np.mean(underestimation_ratio)
assert_(0.90 < underestimation_ratio_mean < 0.99)
# check the required column resamples
max_nresamples = np.max(nresample_list)
assert_equal(max_nresamples, 0)
# check the proportion of norms computed exactly correctly
nexact = np.count_nonzero(relative_errors < 1e-14)
proportion_exact = nexact / float(nsamples)
assert_(0.7 < proportion_exact < 0.8)
# check the average number of matrix*vector multiplications
mean_nmult = np.mean(nmult_list)
assert_(4 < mean_nmult < 5)
def test_onenormest_table_5_t_1(self):
# "note that there is no randomness and hence only one estimate for t=1"
t = 1
n = 100
itmax = 5
alpha = 1 - 1e-6
A = -scipy.linalg.inv(np.identity(n) + alpha * np.eye(n, k=1))
first_col = np.array([1] + [0] * (n - 1))
first_row = np.array([(-alpha)**i for i in range(n)])
B = -scipy.linalg.toeplitz(first_col, first_row)
assert_allclose(A, B)
est, v, w, nmults, nresamples = _onenormest_core(B, B.T, t, itmax)
exact_value = scipy.linalg.norm(B, 1)
underest_ratio = est / exact_value
assert_allclose(underest_ratio, 0.05, rtol=1e-4)
assert_equal(nmults, 11)
assert_equal(nresamples, 0)
# check the non-underscored version of onenormest
est_plain = scipy.sparse.linalg.onenormest(B, t=t, itmax=itmax)
assert_allclose(est, est_plain)
def test_onenormest_table_5_t_1(self):
# "note that there is no randomness and hence only one estimate for t=1"
t = 1
n = 100
itmax = 5
alpha = 1 - 1e-6
A = -scipy.linalg.inv(np.identity(n) + alpha*np.eye(n, k=1))
first_col = np.array([1] + [0]*(n-1))
first_row = np.array([(-alpha)**i for i in range(n)])
B = -scipy.linalg.toeplitz(first_col, first_row)
assert_allclose(A, B)
est, v, w, nmults, nresamples = _onenormest_core(B, B.T, t, itmax)
exact_value = scipy.linalg.norm(B, 1)
underest_ratio = est / exact_value
assert_allclose(underest_ratio, 0.05, rtol=1e-4)
assert_equal(nmults, 11)
assert_equal(nresamples, 0)
# check the non-underscored version of onenormest
est_plain = scipy.sparse.linalg.onenormest(B, t=t, itmax=itmax)
assert_allclose(est, est_plain)
def test_onenormest_table_3_t_2(self):
# This will take multiple seconds if your computer is slow like mine.
# It is stochastic, so the tolerance could be too strict.
np.random.seed(1234)
t = 2
n = 100
itmax = 5
nsamples = 5000
observed = []
expected = []
nmult_list = []
nresample_list = []
for i in range(nsamples):
A = scipy.linalg.inv(np.random.randn(n, n))
est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax)
observed.append(est)
expected.append(scipy.linalg.norm(A, 1))
nmult_list.append(nmults)
nresample_list.append(nresamples)
observed = np.array(observed, dtype=float)
expected = np.array(expected, dtype=float)
relative_errors = np.abs(observed - expected) / expected
# check the mean underestimation ratio
underestimation_ratio = observed / expected
assert_(0.99 < np.mean(underestimation_ratio) < 1.0)
# check the max and mean required column resamples
assert_equal(np.max(nresample_list), 2)
assert_(0.05 < np.mean(nresample_list) < 0.2)
# check the proportion of norms computed exactly correctly
nexact = np.count_nonzero(relative_errors < 1e-14)
proportion_exact = nexact / float(nsamples)
assert_(0.9 < proportion_exact < 0.95)
# check the average number of matrix*vector multiplications
assert_(3.5 < np.mean(nmult_list) < 4.5)
def test_onenormest_table_3_t_2(self):
# This will take multiple seconds if your computer is slow like mine.
# It is stochastic, so the tolerance could be too strict.
np.random.seed(1234)
t = 2
n = 100
itmax = 5
nsamples = 5000
observed = []
expected = []
nmult_list = []
nresample_list = []
for i in range(nsamples):
A = scipy.linalg.inv(np.random.randn(n, n))
est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax)
observed.append(est)
expected.append(scipy.linalg.norm(A, 1))
nmult_list.append(nmults)
nresample_list.append(nresamples)
observed = np.array(observed, dtype=float)
expected = np.array(expected, dtype=float)
relative_errors = np.abs(observed - expected) / expected
# check the mean underestimation ratio
underestimation_ratio = observed / expected
assert_(0.99 < np.mean(underestimation_ratio) < 1.0)
# check the max and mean required column resamples
assert_equal(np.max(nresample_list), 2)
assert_(0.05 < np.mean(nresample_list) < 0.2)
# check the proportion of norms computed exactly correctly
nexact = np.count_nonzero(relative_errors < 1e-14)
proportion_exact = nexact / float(nsamples)
assert_(0.9 < proportion_exact < 0.95)
# check the average number of matrix*vector multiplications
assert_(3.5 < np.mean(nmult_list) < 4.5)
