def solve(self, rhs, autoTranspose=False):
"""
If autoTranspose == False, solves the system of equations
L^Tx = rhs, otherwise solves the system Lx = rhs. Here L denotes the
discrete Laplacian matrix with the boundary row and column vectors
extracted.
"""
# Both UMFPACK and SuperLU internally use CSC format, while L uses CSR
# format. Thus, passing L's data `as is' into those routines would give
# us the solution to L^Tx = rhs. When autoTranspose is set to True, we
# first have to convert L to CSR and then pass it on. We will use _left
# to label the structures for solving L^Tx = rhs (since x multiplies L
# on the left) and right for those that enable solving Lx = rhs (since
# x in this case multiplies L on the right). Thus, autoTranspose set
# really means that `left' structures are used.
if not autoTranspose:
if self.solve_left is None:
A = self.L.transpose() # a CSC matrix - no copy
self.solve_left = dsolve.factorized(A)
solve = self.solve_left
else:
if self.solve_right is None:
A = self.L.tocsc() # requires a copy
self.solve_right = dsolve.factorized(A)
solve = self.solve_right
x = solve(rhs)
return x
def geodesic_distance(pts, polys, verts):
"""
Computes the geodesic distance using the heat method (Crane et al, 2012)
:param pts: arrays of vertices
:param polys: arrays of faces
:param verts: array of
:return:
"""
num_pt = len(pts)
C, M = laplace_operator(pts, polys)
# Step 1: Solve the heat values u
# time of heat evolution
h = avg_edge_length(pts, polys)
t = h**2
# use backward Euler step
lfac = M - t * C
# Exclude rows with zero weight (these break the sparse LU)
goodrows = np.nonzero(~np.array(lfac.sum(0) == 0).ravel())[0]
# Prefactor matrices
_rlfac_solvers = factorized(lfac[goodrows][:, goodrows])
_nLC_solvers = factorized(C[goodrows][:, goodrows])
# Compute u
u0 = np.zeros((num_pt)) # initial heat values
u0[verts] = 1.0
goodu = _rlfac_solvers(u0[goodrows])
u = np.zeros((num_pt, ))
u[goodrows] = goodu
# Step 2: Compute X (normalized gradient)
# Compute gradients at each face
gradu = surface_gradient(pts, polys, u)
# Compute X
graduT = gradu.T
gusum = ne.evaluate("sum(gradu ** 2, 1)")
X = np.nan_to_num(ne.evaluate("-graduT / sqrt(gusum)").T)
# Step 3: Solve the poisson equation
# Compute integrated divergence of X at each vertex
divx = integrated_divergence(pts, polys, X)
# Compute phi (distance)
goodphi = _nLC_solvers(divx[goodrows])
phi = np.zeros((num_pt, ))
phi[goodrows] = goodphi - goodphi.min()
# Ensure that distance is zero for selected verts
phi[verts] = 0.0
return phi
def test_call_with_cast_to_complex_without_umfpack(self):
use_solver(useUmfpack=False)
solve = factorized(self.A)
b = random.rand(4)
for t in [np.complex64, np.complex128]:
with assert_raises(TypeError, match="Cannot cast array data"):
solve(b.astype(t))
def test_call_with_cast_to_complex_without_umfpack(self):
use_solver(useUmfpack=False)
solve = factorized(self.A)
b = random.rand(4)
for t in [np.complex64, np.complex128]:
with assert_raises(TypeError, message="Cannot cast array data"):
solve(b.astype(t))
def test_bug_8278(self):
check_free_memory(8000)
use_solver(useUmfpack=True)
A, b = setup_bug_8278()
A = A.tocsc()
f = factorized(A)
x = f(b)
assert_array_almost_equal(A @ x, b)
def _check_non_singular(self):
# Make a diagonal dominant, to make sure it is not singular
n = 5
a = csc_matrix(random.rand(n, n))
b = ones(n)
expected = splu(a).solve(b)
assert_array_almost_equal(factorized(a)(b), expected)
def _check_non_singular(self):
# Make a diagonal dominant, to make sure it is not singular
n = 5
a = csc_matrix(random.rand(n, n))
b = ones(n)
expected = splu(a).solve(b)
assert_array_almost_equal(factorized(a)(b), expected)
def test_call_with_incorrectly_sized_matrix_without_umfpack(self):
use_solver(useUmfpack=False)
solve = factorized(self.A)
b = random.rand(4)
B = random.rand(4, 3)
BB = random.rand(self.n, 3, 9)
assert_raises_regex(ValueError, "is of incompatible size", solve, b)
assert_raises_regex(ValueError, "is of incompatible size", solve, B)
assert_raises_regex(ValueError, "object too deep for desired array", solve, BB)
def test_assume_sorted_indices_flag(self):
# a sparse matrix with unsorted indices
unsorted_inds = np.array([2, 0, 1, 0])
data = np.array([10, 16, 5, 0.4])
indptr = np.array([0, 1, 2, 4])
A = csc_matrix((data, unsorted_inds, indptr), (3, 3))
b = ones(3)
# should raise when incorrectly assuming indices are sorted
use_solver(useUmfpack=True, assumeSortedIndices=True)
with assert_raises(RuntimeError, match="UMFPACK_ERROR_invalid_matrix"):
factorized(A)
# should sort indices and succeed when not assuming indices are sorted
use_solver(useUmfpack=True, assumeSortedIndices=False)
expected = splu(A.copy()).solve(b)
assert_equal(A.has_sorted_indices, 0)
assert_array_almost_equal(factorized(A)(b), expected)
def test_call_with_incorrectly_sized_matrix_with_umfpack(self):
use_solver(useUmfpack=True)
solve = factorized(self.A)
b = random.rand(4)
B = random.rand(4, 3)
BB = random.rand(self.n, 3, 9)
# does not raise
solve(b)
assert_raises_regex(ValueError, "object too deep for desired array", solve, B)
assert_raises_regex(ValueError, "object too deep for desired array", solve, BB)
def uzawa(A, B, f, g):
Asolve = ssld.factorized(A)
p = np.zeros(B.shape[0])
r = 0.0001
while True:
u = Asolve(f.flatten() - B.transpose() * p)
pd = B * u - g.flatten()
p += r * pd
pdl2 = math.sqrt(np.sum(pd * pd))
print pdl2
if pdl2 < 1E-6: break
return u, p
def test_assume_sorted_indices_flag(self):
# a sparse matrix with unsorted indices
unsorted_inds = np.array([2, 0, 1, 0])
data = np.array([10, 16, 5, 0.4])
indptr = np.array([0, 1, 2, 4])
A = csc_matrix((data, unsorted_inds, indptr), (3, 3))
b = ones(3)
# should raise when incorrectly assuming indices are sorted
use_solver(useUmfpack=True, assumeSortedIndices=True)
with assert_raises(RuntimeError,
message="UMFPACK_ERROR_invalid_matrix"):
factorized(A)
# should sort indices and succeed when not assuming indices are sorted
use_solver(useUmfpack=True, assumeSortedIndices=False)
expected = splu(A.copy()).solve(b)
assert_equal(A.has_sorted_indices, 0)
assert_array_almost_equal(factorized(A)(b), expected)
assert_equal(A.has_sorted_indices, 1)
def mixedcg(A, B, f, g):
Asolve = ssld.factorized(A)
Aif = Asolve(f.flatten())
BAf = B * Aif
def BAiBt(p):
return B * Asolve(B.transpose() * p)
p, info = ssl.bicgstab(ssl.LinearOperator((B.shape[0],)*2, matvec = BAiBt, dtype=float), BAf - g.flatten())
print "bicgstab result ",info
u = Aif - Asolve(B.transpose() * p)
return u,p
def test_call_with_incorrectly_sized_matrix_with_umfpack(self):
use_solver(useUmfpack=True)
solve = factorized(self.A)
b = random.rand(4)
B = random.rand(4, 3)
BB = random.rand(self.n, 3, 9)
# does not raise
solve(b)
msg = "object too deep for desired array"
with assert_raises(ValueError, message=msg):
solve(B)
with assert_raises(ValueError, message=msg):
solve(BB)
def test_call_with_incorrectly_sized_matrix_without_umfpack(self):
use_solver(useUmfpack=False)
solve = factorized(self.A)
b = random.rand(4)
B = random.rand(4, 3)
BB = random.rand(self.n, 3, 9)
with assert_raises(ValueError, match="is of incompatible size"):
solve(b)
with assert_raises(ValueError, match="is of incompatible size"):
solve(B)
with assert_raises(ValueError,
match="object too deep for desired array"):
solve(BB)
def test_call_with_incorrectly_sized_matrix_with_umfpack(self):
use_solver(useUmfpack=True)
solve = factorized(self.A)
b = random.rand(4)
B = random.rand(4, 3)
BB = random.rand(self.n, 3, 9)
# does not raise
solve(b)
msg = "object too deep for desired array"
with assert_raises(ValueError, match=msg):
solve(B)
with assert_raises(ValueError, match=msg):
solve(BB)
def test_call_with_incorrectly_sized_matrix_without_umfpack(self):
use_solver(useUmfpack=False)
solve = factorized(self.A)
b = random.rand(4)
B = random.rand(4, 3)
BB = random.rand(self.n, 3, 9)
with assert_raises(ValueError, message="is of incompatible size"):
solve(b)
with assert_raises(ValueError, message="is of incompatible size"):
solve(B)
with assert_raises(ValueError,
message="object too deep for desired array"):
solve(BB)
def _check_singular(self):
A = csc_matrix((5,5), dtype='d')
b = ones(5)
assert_array_almost_equal(0. * b, factorized(A)(b))
def test_factorizes_nonsquare_matrix_with_umfpack(self):
use_solver(useUmfpack=True)
# does not raise
factorized(self.A[:, :4])
def test_cannot_factorize_nonsquare_matrix_without_umfpack(self):
use_solver(useUmfpack=False)
msg = "can only factor square matrices"
with assert_raises(ValueError, match=msg):
factorized(self.A[:, :4])
def fitGlobal(self):
"""
Perform a global B-spline surface fit to determine the
coefficients of each patch. This is only used with an plot3D
init type
"""
print("Global Fitting")
nCtl = self.topo.nGlobal
print(" -> Copying Topology")
origTopo = copy.deepcopy(self.topo)
print(" -> Creating global numbering")
sizes = []
for isurf in range(self.nSurf):
sizes.append([self.surfs[isurf].Nu, self.surfs[isurf].Nv])
# Get the Global number of the original data
origTopo.calcGlobalNumbering(sizes)
N = origTopo.nGlobal
print(" -> Creating global point list")
pts = np.zeros((N, 3))
for ii in range(N):
pts[ii] = self.surfs[origTopo.gIndex[ii][0][0]].X[origTopo.gIndex[ii][0][1], origTopo.gIndex[ii][0][2]]
# Get the maximum k (ku, kv for each surf)
kmax = 2
for isurf in range(self.nSurf):
if self.surfs[isurf].ku > kmax:
kmax = self.surfs[isurf].ku
if self.surfs[isurf].kv > kmax:
kmax = self.surfs[isurf].kv
nnz = N * kmax * kmax
vals = np.zeros(nnz)
rowPtr = [0]
colInd = np.zeros(nnz, "intc")
for ii in range(N):
isurf = origTopo.gIndex[ii][0][0]
i = origTopo.gIndex[ii][0][1]
j = origTopo.gIndex[ii][0][2]
u = self.surfs[isurf].U[i, j]
v = self.surfs[isurf].V[i, j]
vals, colInd = self.surfs[isurf].getBasisPt(u, v, vals, rowPtr[ii], colInd, self.topo.lIndex[isurf])
kinc = self.surfs[isurf].ku * self.surfs[isurf].kv
rowPtr.append(rowPtr[-1] + kinc)
# Now we can crop out any additional values in col_ptr and vals
vals = vals[: rowPtr[-1]]
colInd = colInd[: rowPtr[-1]]
# Now make a sparse matrix
NN = sparse.csr_matrix((vals, colInd, rowPtr))
print(" -> Multiplying N^T * N")
NNT = NN.T
NTN = NNT * NN
print(" -> Factorizing...")
solve = factorized(NTN)
print(" -> Back Solving...")
self.coef = np.zeros((nCtl, 3))
for idim in range(3):
self.coef[:, idim] = solve(NNT * pts[:, idim])
print(" -> Setting Surface Coefficients...")
self._updateSurfaceCoef()
def _check_singular(self):
A = csc_matrix((5, 5), dtype='d')
b = ones(5)
assert_array_almost_equal(0. * b, factorized(A)(b))
def test_call_with_cast_to_complex_with_umfpack(self):
use_solver(useUmfpack=True)
solve = factorized(self.A)
b = random.rand(4)
for t in [np.complex64, np.complex128]:
assert_warns(np.ComplexWarning, solve, b.astype(t))
def test_call_with_cast_to_complex_with_umfpack(self):
use_solver(useUmfpack=True)
solve = factorized(self.A)
b = random.rand(4)
for t in [np.complex64, np.complex128]:
assert_warns(np.ComplexWarning, solve, b.astype(t))
def test_factorizes_nonsquare_matrix_with_umfpack(self):
use_solver(useUmfpack=True)
# does not raise
factorized(self.A[:,:4])
def test_cannot_factorize_nonsquare_matrix_without_umfpack(self):
use_solver(useUmfpack=False)
msg = "can only factor square matrices"
with assert_raises(ValueError, message=msg):
factorized(self.A[:, :4])