def test_atol_legacy(self):
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
# Check the strange legacy behavior: the tolerance is interpreted
# as atol, but only for the initial residual
A = eye(2)
b = 1e-6 * ones(2)
x, info = gmres(A, b, tol=1e-5)
assert_array_equal(x, np.zeros(2))
A = eye(2)
b = ones(2)
x, info = gmres(A, b, tol=1e-5)
assert_(np.linalg.norm(A.dot(x) - b) <= 1e-5*np.linalg.norm(b))
assert_allclose(x, b, atol=0, rtol=1e-8)
rndm = np.random.RandomState(12345)
A = rndm.rand(30, 30)
b = 1e-6 * ones(30)
x, info = gmres(A, b, tol=1e-7, restart=20)
assert_(np.linalg.norm(A.dot(x) - b) > 1e-7)
A = eye(2)
b = 1e-10 * ones(2)
x, info = gmres(A, b, tol=1e-8, atol=0)
assert_(np.linalg.norm(A.dot(x) - b) <= 1e-8*np.linalg.norm(b))
def test_callback(self):
def store_residual(r, rvec):
rvec[rvec.nonzero()[0].max()+1] = r
# Define, A,b
A = csr_matrix(array([[-2,1,0,0,0,0],[1,-2,1,0,0,0],[0,1,-2,1,0,0],[0,0,1,-2,1,0],[0,0,0,1,-2,1],[0,0,0,0,1,-2]]))
b = ones((A.shape[0],))
maxiter = 1
rvec = zeros(maxiter+1)
rvec[0] = 1.0
callback = lambda r:store_residual(r, rvec)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x,flag = gmres(A, b, x0=zeros(A.shape[0]), tol=1e-16, maxiter=maxiter, callback=callback)
# Expected output from Scipy 1.0.0
assert_allclose(rvec, array([1.0, 0.81649658092772603]), rtol=1e-10)
# Test preconditioned callback
M = 1e-3 * np.eye(A.shape[0])
rvec = zeros(maxiter+1)
rvec[0] = 1.0
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x, flag = gmres(A, b, M=M, tol=1e-16, maxiter=maxiter, callback=callback)
# Expected output from Scipy 1.0.0 (callback has preconditioned residual!)
assert_allclose(rvec, array([1.0, 1e-3 * 0.81649658092772603]), rtol=1e-10)
def test_abi(self):
# Check we don't segfault on gmres with complex argument
A = eye(2)
b = ones(2)
r_x, r_info = gmres(A, b)
r_x = r_x.astype(complex)
x, info = gmres(A.astype(complex), b.astype(complex))
assert_(iscomplexobj(x))
assert_allclose(r_x, x)
assert_(r_info == info)
def test_abi(self):
# Check we don't segfault on gmres with complex argument
A = eye(2)
b = ones(2)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
r_x, r_info = gmres(A, b)
r_x = r_x.astype(complex)
x, info = gmres(A.astype(complex), b.astype(complex))
assert_(iscomplexobj(x))
assert_allclose(r_x, x)
assert_(r_info == info)
def test_callback(self):
def store_residual(r, rvec):
rvec[rvec.nonzero()[0].max() + 1] = r
# Define, A,b
A = csr_matrix(
array(
[
[-2, 1, 0, 0, 0, 0],
[1, -2, 1, 0, 0, 0],
[0, 1, -2, 1, 0, 0],
[0, 0, 1, -2, 1, 0],
[0, 0, 0, 1, -2, 1],
[0, 0, 0, 0, 1, -2],
]
)
)
b = ones((A.shape[0],))
maxiter = 1
rvec = zeros(maxiter + 1)
rvec[0] = 1.0
callback = lambda r: store_residual(r, rvec)
x, flag = gmres(A, b, x0=zeros(A.shape[0]), tol=1e-16, maxiter=maxiter, callback=callback)
diff = max(abs((rvec - array([1.0, 0.81649658092772603]))))
assert_(diff < 1e-5)
def test_gmres_basic():
A = np.vander(np.arange(10) + 1)[:, ::-1]
b = np.zeros(10)
b[0] = 1
x = np.linalg.solve(A, b)
x_gm, err = gmres(A, b, restart=5, maxiter=1)
assert_allclose(x_gm[0], 0.359, rtol=1e-2)
def test_gmres_basic():
A = np.vander(np.arange(10) + 1)[:, ::-1]
b = np.zeros(10)
b[0] = 1
x = np.linalg.solve(A, b)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x_gm, err = gmres(A, b, restart=5, maxiter=1)
assert_allclose(x_gm[0], 0.359, rtol=1e-2)
def test_defective_matrix_breakdown(self):
# Breakdown due to defective matrix
A = np.array([[0, 1, 0], [1, 0, 0], [0, 0, 0]])
b = np.array([1, 0, 1])
x, info = gmres(A, b, tol=1e-8, atol=0)
# Should not return nans, nor terminate with false success
assert_(not np.isnan(x).any())
if info == 0:
assert_(np.linalg.norm(A.dot(x) - b) <= 1e-8*np.linalg.norm(b))
# The solution should be OK outside null space of A
assert_allclose(A.dot(A.dot(x)), A.dot(b))
def test_arnoldi(self):
np.random.rand(1234)
A = eye(10000) + rand(10000,10000,density=1e-4)
b = np.random.rand(10000)
# The inner arnoldi should be equivalent to gmres
x0, flag0 = lgmres(A, b, x0=zeros(A.shape[0]), inner_m=15, maxiter=1)
x1, flag1 = gmres(A, b, x0=zeros(A.shape[0]), restart=15, maxiter=1)
assert_equal(flag0, 1)
assert_equal(flag1, 1)
assert_(np.linalg.norm(A.dot(x0) - b) > 1e-3)
assert_allclose(x0, x1)
def test_arnoldi(self):
np.random.rand(1234)
A = eye(10000) + rand(10000,10000,density=1e-4)
b = np.random.rand(10000)
# The inner arnoldi should be equivalent to gmres
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x0, flag0 = gcrotmk(A, b, x0=zeros(A.shape[0]), m=15, k=0, maxiter=1)
x1, flag1 = gmres(A, b, x0=zeros(A.shape[0]), restart=15, maxiter=1)
assert_equal(flag0, 1)
assert_equal(flag1, 1)
assert_(np.linalg.norm(A.dot(x0) - b) > 1e-3)
assert_allclose(x0, x1)
def test_arnoldi(self):
np.random.seed(1)
A = eye(2000) + rand(2000, 2000, density=5e-4)
b = np.random.rand(2000)
# The inner arnoldi should be equivalent to gmres
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x0, flag0 = gcrotmk(A, b, x0=zeros(A.shape[0]), m=15, k=0, maxiter=1)
x1, flag1 = gmres(A, b, x0=zeros(A.shape[0]), restart=15, maxiter=1)
assert_equal(flag0, 1)
assert_equal(flag1, 1)
assert np.linalg.norm(A.dot(x0) - b) > 1e-3
assert_allclose(x0, x1)
def test_defective_precond_breakdown(self):
# Breakdown due to defective preconditioner
M = np.eye(3)
M[2,2] = 0
b = np.array([0, 1, 1])
x = np.array([1, 0, 0])
A = np.diag([2, 3, 4])
x, info = gmres(A, b, x0=x, M=M, tol=1e-15, atol=0)
# Should not return nans, nor terminate with false success
assert_(not np.isnan(x).any())
if info == 0:
assert_(np.linalg.norm(A.dot(x) - b) <= 1e-15*np.linalg.norm(b))
# The solution should be OK outside null space of M
assert_allclose(M.dot(A.dot(x)), M.dot(b))
def test_defective_precond_breakdown(self):
# Breakdown due to defective preconditioner
M = np.eye(3)
M[2,2] = 0
b = np.array([0, 1, 1])
x = np.array([1, 0, 0])
A = np.diag([2, 3, 4])
x, info = gmres(A, b, x0=x, M=M, tol=1e-15, atol=0)
# Should not return nans, nor terminate with false success
assert_(not np.isnan(x).any())
if info == 0:
assert_(np.linalg.norm(A.dot(x) - b) <= 1e-15*np.linalg.norm(b))
# The solution should be OK outside null space of M
assert_allclose(M.dot(A.dot(x)), M.dot(b))
def test_callback_x_monotonic(self):
# Check that callback_type='x' gives monotonic norm decrease
np.random.seed(1)
A = np.random.rand(20, 20) + np.eye(20)
b = np.random.rand(20)
prev_r = [np.inf]
count = [0]
def x_cb(x):
r = np.linalg.norm(A.dot(x) - b)
assert r <= prev_r[0]
prev_r[0] = r
count[0] += 1
x, info = gmres(A, b, tol=1e-6, atol=0, callback=x_cb, maxiter=20, restart=10,
callback_type='x')
assert info == 20
assert count[0] == 21
x_cb(x)
def test_callback(self):
def store_residual(r, rvec):
rvec[rvec.nonzero()[0].max() + 1] = r
# Define, A,b
A = csr_matrix(
array([[-2, 1, 0, 0, 0, 0], [1, -2, 1, 0, 0,
0], [0, 1, -2, 1, 0, 0],
[0, 0, 1, -2, 1, 0], [0, 0, 0, 1, -2, 1],
[0, 0, 0, 0, 1, -2]]))
b = ones((A.shape[0], ))
maxiter = 1
rvec = zeros(maxiter + 1)
rvec[0] = 1.0
callback = lambda r: store_residual(r, rvec)
x, flag = gmres(A,
b,
x0=zeros(A.shape[0]),
tol=1e-16,
maxiter=maxiter,
callback=callback)
diff = max(abs((rvec - array([1.0, 0.81649658092772603]))))
assert_(diff < 1e-5)
def test_callback_type(self):
# The legacy callback type changes meaning of 'maxiter'
np.random.seed(1)
A = np.random.rand(20, 20)
b = np.random.rand(20)
cb_count = [0]
def pr_norm_cb(r):
cb_count[0] += 1
assert_(isinstance(r, float))
def x_cb(x):
cb_count[0] += 1
assert_(isinstance(x, np.ndarray))
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
# 2 iterations is not enough to solve the problem
cb_count = [0]
x, info = gmres(A,
b,
tol=1e-6,
atol=0,
callback=pr_norm_cb,
maxiter=2,
restart=50)
assert info == 2
assert cb_count[0] == 2
# With `callback_type` specified, no warning should be raised
cb_count = [0]
x, info = gmres(A,
b,
tol=1e-6,
atol=0,
callback=pr_norm_cb,
maxiter=2,
restart=50,
callback_type='legacy')
assert info == 2
assert cb_count[0] == 2
# 2 restart cycles is enough to solve the problem
cb_count = [0]
x, info = gmres(A,
b,
tol=1e-6,
atol=0,
callback=pr_norm_cb,
maxiter=2,
restart=50,
callback_type='pr_norm')
assert info == 0
assert cb_count[0] > 2
# 2 restart cycles is enough to solve the problem
cb_count = [0]
x, info = gmres(A,
b,
tol=1e-6,
atol=0,
callback=x_cb,
maxiter=2,
restart=50,
callback_type='x')
assert info == 0
assert cb_count[0] == 2
