def test_cornercase(self):
np.random.seed(1234)
# Rounding error may prevent convergence with tol=0 --- ensure
# that the return values in this case are correct, and no
# exceptions are raised
for n in [3, 5, 10, 100]:
A = 2*eye(n)
b = np.ones(n)
x, info = lgmres(A, b, maxiter=10)
assert_equal(info, 0)
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
x, info = lgmres(A, b, tol=0, maxiter=10)
if info == 0:
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
b = np.random.rand(n)
x, info = lgmres(A, b, maxiter=10)
assert_equal(info, 0)
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
x, info = lgmres(A, b, tol=0, maxiter=10)
if info == 0:
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
def test_breakdown_underdetermined(self):
# Should find LSQ solution in the Krylov span in one inner
# iteration, despite solver breakdown from nilpotent A.
A = np.array([[0, 1, 1, 1], [0, 0, 1, 1], [0, 0, 0, 1], [0, 0, 0, 0]],
dtype=float)
bs = [
np.array([1, 1, 1, 1]),
np.array([1, 1, 1, 0]),
np.array([1, 1, 0, 0]),
np.array([1, 0, 0, 0]),
]
for b in bs:
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b, maxiter=1)
resp = np.linalg.norm(A.dot(xp) - b)
K = np.c_[b, A.dot(b), A.dot(A.dot(b)), A.dot(A.dot(A.dot(b)))]
y, _, _, _ = np.linalg.lstsq(A.dot(K), b, rcond=-1)
x = K.dot(y)
res = np.linalg.norm(A.dot(x) - b)
assert_allclose(resp, res, err_msg=repr(b))
def test_arnoldi(self):
np.random.rand(1234)
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 = 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) > 4e-4)
assert_allclose(x0, x1)
def test_nans(self):
A = eye(3, format='lil')
A[1,1] = np.nan
b = np.ones(3)
x, info = lgmres(A, b, tol=0, maxiter=10)
assert_equal(info, 1)
def test_breakdown_underdetermined(self):
# Should find LSQ solution in the Krylov span in one inner
# iteration, despite solver breakdown from nilpotent A.
A = np.array([[0, 1, 1, 1],
[0, 0, 1, 1],
[0, 0, 0, 1],
[0, 0, 0, 0]], dtype=float)
bs = [
np.array([1, 1, 1, 1]),
np.array([1, 1, 1, 0]),
np.array([1, 1, 0, 0]),
np.array([1, 0, 0, 0]),
]
for b in bs:
xp, info = lgmres(A, b, maxiter=1)
resp = np.linalg.norm(A.dot(xp) - b)
K = np.c_[b, A.dot(b), A.dot(A.dot(b)), A.dot(A.dot(A.dot(b)))]
y, _, _, _ = np.linalg.lstsq(A.dot(K), b)
x = K.dot(y)
res = np.linalg.norm(A.dot(x) - b)
assert_allclose(resp, res, err_msg=repr(b))
def test_breakdown_underdetermined(self):
# Should find LSQ solution in the Krylov span in one inner
# iteration, despite solver breakdown from nilpotent A.
A = np.array([[0, 1, 1, 1],
[0, 0, 1, 1],
[0, 0, 0, 1],
[0, 0, 0, 0]], dtype=float)
bs = [
np.array([1, 1, 1, 1]),
np.array([1, 1, 1, 0]),
np.array([1, 1, 0, 0]),
np.array([1, 0, 0, 0]),
]
for b in bs:
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b, maxiter=1)
resp = np.linalg.norm(A.dot(xp) - b)
K = np.c_[b, A.dot(b), A.dot(A.dot(b)), A.dot(A.dot(A.dot(b)))]
y, _, _, _ = np.linalg.lstsq(A.dot(K), b, rcond=-1)
x = K.dot(y)
res = np.linalg.norm(A.dot(x) - b)
assert_allclose(resp, res, err_msg=repr(b))
def do_solve(**kw):
count[0] = 0
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x0, flag = lgmres(A, b, x0=zeros(A.shape[0]),
inner_m=6, tol=1e-14, **kw)
count_0 = count[0]
assert_(allclose(A@x0, b, rtol=1e-12, atol=1e-12), norm(A@x0-b))
return x0, count_0
def do_solve(**kw):
count[0] = 0
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x0, flag = lgmres(A, b, x0=zeros(A.shape[0]),
inner_m=6, tol=1e-14, **kw)
count_0 = count[0]
assert_(allclose(A*x0, b, rtol=1e-12, atol=1e-12), norm(A*x0-b))
return x0, count_0
def test_nans(self):
A = eye(3, format='lil')
A[1, 1] = np.nan
b = np.ones(3)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x, info = lgmres(A, b, tol=0, maxiter=10)
assert_equal(info, 1)
def test_nans(self):
A = eye(3, format='lil')
A[1, 1] = np.nan
b = np.ones(3)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x, info = lgmres(A, b, tol=0, maxiter=10)
assert_equal(info, 1)
def test_breakdown_with_outer_v(self):
A = np.array([[1, 2], [3, 4]], dtype=float)
b = np.array([1, 2])
x = np.linalg.solve(A, b)
v0 = np.array([1, 0])
# The inner iteration should converge to the correct solution,
# since it's in the outer vector list
xp, info = lgmres(A, b, outer_v=[(v0, None), (x, None)], maxiter=1)
assert_allclose(xp, x, atol=1e-12)
def test_denormals(self):
# Check that no warnings are emitted if the matrix contains
# numbers for which 1/x has no float representation, and that
# the solver behaves properly.
A = np.array([[1, 2], [3, 4]], dtype=float)
A *= 100 * np.nextafter(0, 1)
b = np.array([1, 1])
xp, info = lgmres(A, b)
if info == 0:
assert_allclose(A.dot(xp), b)
def test_breakdown_with_outer_v(self):
A = np.array([[1, 2], [3, 4]], dtype=float)
b = np.array([1, 2])
x = np.linalg.solve(A, b)
v0 = np.array([1, 0])
# The inner iteration should converge to the correct solution,
# since it's in the outer vector list
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b, outer_v=[(v0, None), (x, None)], maxiter=1)
assert_allclose(xp, x, atol=1e-12)
def fd_solve(self):
"""
w = fd_solve()
where coeff is the sparse coefficient matrix output from function
coeff_matrix and qs is the array of loads (stresses)
Sparse solver for one-dimensional flexure of an elastic plate
"""
if self.Debug:
print("qs", self.qs.shape)
print("Te", self.Te.shape)
self.calc_max_flexural_wavelength()
print(
"maxFlexuralWavelength_ncells', self.maxFlexuralWavelength_ncells"
)
if self.Solver == "iterative" or self.Solver == "Iterative":
if self.Debug:
print(
"Using generalized minimal residual method for iterative solution"
)
if self.Verbose:
print("Converging to a tolerance of",
self.iterative_ConvergenceTolerance,
"m between iterations")
# qs negative so bends down with positive load, bends up with neative load
# (i.e. material removed)
w = isolve.lgmres(self.coeff_matrix,
-self.qs,
tol=self.iterative_ConvergenceTolerance)
self.w = w[0] # Reach into tuple to get my array back
else:
if self.Solver == 'direct' or self.Solver == 'Direct':
if self.Debug:
print("Using direct solution with UMFpack")
else:
print("Solution type not understood:")
print("Defaulting to direct solution with UMFpack")
# UMFpack is now the default, but setting true just to be sure in case
# anything changes
# qs negative so bends down with positive load, bends up with neative load
# (i.e. material removed)
self.w = spsolve(self.coeff_matrix, -self.qs, use_umfpack=True)
if self.Debug:
print("w.shape:")
print(self.w.shape)
print("w:")
print(self.w)
def test_breakdown_with_outer_v(self):
A = np.array([[1, 2], [3, 4]], dtype=float)
b = np.array([1, 2])
x = np.linalg.solve(A, b)
v0 = np.array([1, 0])
# The inner iteration should converge to the correct solution,
# since it's in the outer vector list
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b, outer_v=[(v0, None), (x, None)], maxiter=1)
assert_allclose(xp, x, atol=1e-12)
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_denormals(self):
# Check that no warnings are emitted if the matrix contains
# numbers for which 1/x has no float representation, and that
# the solver behaves properly.
A = np.array([[1, 2], [3, 4]], dtype=float)
A *= 100 * np.nextafter(0, 1)
b = np.array([1, 1])
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b)
if info == 0:
assert_allclose(A.dot(xp), b)
def test_denormals(self):
# Check that no warnings are emitted if the matrix contains
# numbers for which 1/x has no float representation, and that
# the solver behaves properly.
A = np.array([[1, 2], [3, 4]], dtype=float)
A *= 100 * np.nextafter(0, 1)
b = np.array([1, 1])
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b)
if info == 0:
assert_allclose(A.dot(xp), b)
def test_arnoldi(self):
np.random.rand(1234)
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 = 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) > 4e-4)
assert_allclose(x0, x1)
def fd_solve(self):
"""
w = fd_solve()
where coeff is the sparse coefficient matrix output from function
coeff_matrix and qs is the array of loads (stresses)
Sparse solver for one-dimensional flexure of an elastic plate
"""
if self.Debug:
print('qs', self.qs.shape)
print('Te', self.Te.shape)
self.calc_max_flexural_wavelength()
print('maxFlexuralWavelength_ncells', self.maxFlexuralWavelength_ncells)
if self.Solver == "iterative" or self.Solver == "Iterative":
if self.Debug:
print("Using generalized minimal residual method for iterative solution")
if self.Verbose:
print("Converging to a tolerance of", self.iterative_ConvergenceTolerance, "m between iterations")
# qs negative so bends down with positive load, bends up with neative load
# (i.e. material removed)
w = isolve.lgmres(self.coeff_matrix, -self.qs, tol=self.iterative_ConvergenceTolerance)
self.w = w[0] # Reach into tuple to get my array back
else:
if self.Solver == "direct" or self.Solver == "Direct":
if self.Debug:
print("Using direct solution with UMFpack")
else:
print("Solution type not understood:")
print("Defaulting to direct solution with UMFpack")
# UMFpack is now the default, but setting true just to be sure in case
# anything changes
# qs negative so bends down with positive load, bends up with neative load
# (i.e. material removed)
self.w = spsolve(self.coeff_matrix, -self.qs, use_umfpack=True)
if self.Debug:
print("w.shape:")
print(self.w.shape)
print("w:")
print(self.w)
def do_solve(**kw):
count[0] = 0
x0, flag = lgmres(A, b, x0=zeros(A.shape[0]), inner_m=6, tol=1e-14, **kw)
count_0 = count[0]
assert_(allclose(A*x0, b, rtol=1e-12, atol=1e-12), norm(A*x0-b))
return x0, count_0