def _solve(v, alpha, cs, ds):
"""Evaluate w = M^-1 v"""
if len(cs) == 0:
return v/alpha
# (B + C D^H)^-1 = B^-1 - B^-1 C (I + D^H B^-1 C)^-1 D^H B^-1
axpy, dotc = blas.get_blas_funcs(['axpy', 'dotc'], cs[:1] + [v])
c0 = cs[0]
A = alpha * np.identity(len(cs), dtype=c0.dtype)
for i, d in enumerate(ds):
for j, c in enumerate(cs):
A[i,j] += dotc(d, c)
q = np.zeros(len(cs), dtype=c0.dtype)
for j, d in enumerate(ds):
q[j] = dotc(d, v)
q /= alpha
q = solve(A, q)
w = v/alpha
for c, qc in zip(cs, q):
w = axpy(c, w, w.size, -qc)
return w
def _matvec(v, alpha, cs, ds):
axpy, scal, dotc = blas.get_blas_funcs(["axpy", "scal", "dotc"], cs[:1] + [v])
w = alpha * v
for c, d in zip(cs, ds):
a = dotc(d, v)
w = axpy(c, w, w.size, a)
return w
def _solve(v, alpha, cs, ds):
"""Evaluate w = M^-1 v"""
if len(cs) == 0:
return v / alpha
# (B + C D^H)^-1 = B^-1 - B^-1 C (I + D^H B^-1 C)^-1 D^H B^-1
axpy, dotc = blas.get_blas_funcs(['axpy', 'dotc'], cs[:1] + [v])
c0 = cs[0]
A = alpha * np.identity(len(cs), dtype=c0.dtype)
for i, d in enumerate(ds):
for j, c in enumerate(cs):
A[i, j] += dotc(d, c)
q = np.zeros(len(cs), dtype=c0.dtype)
for j, d in enumerate(ds):
q[j] = dotc(d, v)
q /= alpha
q = solve(A, q)
w = v / alpha
for c, qc in zip(cs, q):
w = axpy(c, w, w.size, -qc)
return w
def axpy(x,y,a=1.0):
"""
Quick level-1 call to blas::
y = a*x+y
Parameters
----------
x : array_like
nx1 real or complex vector
y : array_like
nx1 real or complex vector
a : float
real or complex scalar
Returns
-------
y : array_like
Input variable y is rewritten
Notes
-----
The call to get_blas_funcs automatically determines the prefix for the blas
call.
"""
from scipy.lib.blas import get_blas_funcs
fn = get_blas_funcs(['axpy'], [x,y])[0]
fn(x,y,a)
def _matvec(v, alpha, cs, ds):
axpy, scal, dotc = blas.get_blas_funcs(['axpy', 'scal', 'dotc'],
cs[:1] + [v])
w = alpha * v
for c, d in zip(cs, ds):
a = dotc(d, v)
w = axpy(c, w, w.size, a)
return w
def _matvec(v, alpha, cs, ds):
axpy, scal, dotc = blas.get_blas_funcs(['axpy', 'scal', 'dotc'],
cs[:1] + [v])
w = alpha * v
for c, d in zip(cs, ds):
a = dotc(d, v)
w = axpy(c, w, w.size, a)
return w
def test_blas(self):
a = array([[1, 1, 1]])
b = array([[1], [1], [1]])
# get_blas_funcs is deprecated, silence the warning
with warnings.catch_warnings():
warnings.simplefilter('ignore', DeprecationWarning)
gemm, = get_blas_funcs(('gemm', ), (a, b))
assert_array_almost_equal(gemm(1, a, b), [[3]], 15)
def test_blas(self):
a = array([[1, 1, 1]])
b = array([[1], [1], [1]])
# get_blas_funcs is deprecated, silence the warning
with warnings.catch_warnings():
warnings.simplefilter("ignore", DeprecationWarning)
gemm, = get_blas_funcs(("gemm",), (a, b))
assert_array_almost_equal(gemm(1, a, b), [[3]], 15)
def test_blas(self):
a = array([[1,1,1]])
b = array([[1],[1],[1]])
# get_blas_funcs is deprecated, silence the warning
import warnings
warnings.simplefilter('ignore', DeprecationWarning)
gemm, = get_blas_funcs(('gemm',),(a,b))
# remove filter again
warnings.filters.pop(0)
assert_array_almost_equal(gemm(1,a,b),[[3]],15)
def test_blas(self):
a = array([[1, 1, 1]])
b = array([[1], [1], [1]])
# get_blas_funcs is deprecated, silence the warning
import warnings
warnings.simplefilter('ignore', DeprecationWarning)
gemm, = get_blas_funcs(('gemm', ), (a, b))
# remove filter again
warnings.filters.pop(0)
assert_array_almost_equal(gemm(1, a, b), [[3]], 15)
def test_blas(self):
a = array([[1,1,1]])
b = array([[1],[1],[1]])
# get_blas_funcs is deprecated, silence the warning
warn_ctx = WarningManager()
warn_ctx.__enter__()
try:
warnings.simplefilter('ignore', DeprecationWarning)
gemm, = get_blas_funcs(('gemm',),(a,b))
finally:
warn_ctx.__exit__()
assert_array_almost_equal(gemm(1,a,b),[[3]],15)
def test_blas(self):
a = array([[1, 1, 1]])
b = array([[1], [1], [1]])
# get_blas_funcs is deprecated, silence the warning
warn_ctx = WarningManager()
warn_ctx.__enter__()
try:
warnings.simplefilter('ignore', DeprecationWarning)
gemm, = get_blas_funcs(('gemm', ), (a, b))
finally:
warn_ctx.__exit__()
assert_array_almost_equal(gemm(1, a, b), [[3]], 15)
def test_blas(self):
a = array([[1, 1, 1]])
b = array([[1], [1], [1]])
gemm, = get_blas_funcs(("gemm",), (a, b))
assert_array_almost_equal(gemm(1, a, b), [[3]], 15)
def lgmres(A,
b,
x0=None,
tol=1e-5,
maxiter=1000,
M=None,
callback=None,
inner_m=30,
outer_k=3,
outer_v=None,
store_outer_Av=True):
"""
Solve a matrix equation using the LGMRES algorithm.
The LGMRES algorithm [BJM]_ [BPh]_ is designed to avoid some problems
in the convergence in restarted GMRES, and often converges in fewer
iterations.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : {array, matrix}
Starting guess for the solution.
tol : float
Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below `tol`.
maxiter : integer
Maximum number of iterations. Iteration will stop after maxiter
steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}
Preconditioner for A. The preconditioner should approximate the
inverse of A. Effective preconditioning dramatically improves the
rate of convergence, which implies that fewer iterations are needed
to reach a given error tolerance.
callback : function
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
Additional parameters
---------------------
inner_m : int, optional
Number of inner GMRES iterations per each outer iteration.
outer_k : int, optional
Number of vectors to carry between inner GMRES iterations.
According to [BJM]_, good values are in the range of 1...3.
However, note that if you want to use the additional vectors to
accelerate solving multiple similar problems, larger values may
be beneficial.
outer_v : list of tuples, optional
List containing tuples ``(v, Av)`` of vectors and corresponding
matrix-vector products, used to augment the Krylov subspace, and
carried between inner GMRES iterations. The element ``Av`` can
be `None` if the matrix-vector product should be re-evaluated.
This parameter is modified in-place by `lgmres`, and can be used
to pass "guess" vectors in and out of the algorithm when solving
similar problems.
store_outer_Av : bool, optional
Whether LGMRES should store also A*v in addition to vectors `v`
in the `outer_v` list. Default is True.
Returns
-------
x : array or matrix
The converged solution.
info : integer
Provides convergence information:
0 : successful exit
>0 : convergence to tolerance not achieved, number of iterations
<0 : illegal input or breakdown
Notes
-----
The LGMRES algorithm [BJM]_ [BPh]_ is designed to avoid the
slowing of convergence in restarted GMRES, due to alternating
residual vectors. Typically, it often outperforms GMRES(m) of
comparable memory requirements by some measure, or at least is not
much worse.
Another advantage in this algorithm is that you can supply it with
'guess' vectors in the `outer_v` argument that augment the Krylov
subspace. If the solution lies close to the span of these vectors,
the algorithm converges faster. This can be useful if several very
similar matrices need to be inverted one after another, such as in
Newton-Krylov iteration where the Jacobian matrix often changes
little in the nonlinear steps.
References
----------
.. [BJM] A.H. Baker and E.R. Jessup and T. Manteuffel,
SIAM J. Matrix Anal. Appl. 26, 962 (2005).
.. [BPh] A.H. Baker, PhD thesis, University of Colorado (2003).
http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps
"""
from scipy.linalg.basic import lstsq
A, M, x, b, postprocess = make_system(A, M, x0, b)
if not np.isfinite(b).all():
raise ValueError("RHS must contain only finite numbers")
matvec = A.matvec
psolve = M.matvec
if outer_v is None:
outer_v = []
axpy, dotc, scal = None, None, None
b_norm = norm2(b)
if b_norm == 0:
b_norm = 1
for k_outer in xrange(maxiter):
r_outer = matvec(x) - b
# -- callback
if callback is not None:
callback(x)
# -- determine input type routines
if axpy is None:
if np.iscomplexobj(r_outer) and not np.iscomplexobj(x):
x = x.astype(r_outer.dtype)
axpy, dotc, scal = blas.get_blas_funcs(['axpy', 'dotc', 'scal'],
(x, r_outer))
# -- check stopping condition
r_norm = norm2(r_outer)
if r_norm < tol * b_norm or r_norm < tol:
break
# -- inner LGMRES iteration
vs0 = -psolve(r_outer)
inner_res_0 = norm2(vs0)
if inner_res_0 == 0:
rnorm = norm2(r_outer)
raise RuntimeError("Preconditioner returned a zero vector; "
"|v| ~ %.1g, |M v| = 0" % rnorm)
vs0 = scal(1.0 / inner_res_0, vs0)
hs = []
vs = [vs0]
ws = []
y = None
for j in xrange(1, 1 + inner_m + len(outer_v)):
# -- Arnoldi process:
#
# Build an orthonormal basis V and matrices W and H such that
# A W = V H
# Columns of W, V, and H are stored in `ws`, `vs` and `hs`.
#
# The first column of V is always the residual vector, `vs0`;
# V has *one more column* than the other of the three matrices.
#
# The other columns in V are built by feeding in, one
# by one, some vectors `z` and orthonormalizing them
# against the basis so far. The trick here is to
# feed in first some augmentation vectors, before
# starting to construct the Krylov basis on `v0`.
#
# It was shown in [BJM]_ that a good choice (the LGMRES choice)
# for these augmentation vectors are the `dx` vectors obtained
# from a couple of the previous restart cycles.
#
# Note especially that while `vs0` is always the first
# column in V, there is no reason why it should also be
# the first column in W. (In fact, below `vs0` comes in
# W only after the augmentation vectors.)
#
# The rest of the algorithm then goes as in GMRES, one
# solves a minimization problem in the smaller subspace
# spanned by W (range) and V (image).
#
# XXX: Below, I'm lazy and use `lstsq` to solve the
# small least squares problem. Performance-wise, this
# is in practice acceptable, but it could be nice to do
# it on the fly with Givens etc.
#
# ++ evaluate
v_new = None
if j < len(outer_v) + 1:
z, v_new = outer_v[j - 1]
elif j == len(outer_v) + 1:
z = vs0
else:
z = vs[-1]
if v_new is None:
v_new = psolve(matvec(z))
else:
# Note: v_new is modified in-place below. Must make a
# copy to ensure that the outer_v vectors are not
# clobbered.
v_new = v_new.copy()
# ++ orthogonalize
hcur = []
for v in vs:
alpha = dotc(v, v_new)
hcur.append(alpha)
v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v
hcur.append(norm2(v_new))
if hcur[-1] == 0:
# Exact solution found; bail out.
# Zero basis vector (v_new) in the least-squares problem
# does no harm, so we can just use the same code as usually;
# it will give zero (inner) residual as a result.
bailout = True
else:
bailout = False
v_new = scal(1.0 / hcur[-1], v_new)
vs.append(v_new)
hs.append(hcur)
ws.append(z)
# XXX: Ugly: should implement the GMRES iteration properly,
# with Givens rotations and not using lstsq. Instead, we
# spare some work by solving the LSQ problem only every 5
# iterations.
if not bailout and j % 5 != 1 and j < inner_m + len(outer_v) - 1:
continue
# -- GMRES optimization problem
hess = np.zeros((j + 1, j), x.dtype)
e1 = np.zeros((j + 1, ), x.dtype)
e1[0] = inner_res_0
for q in xrange(j):
hess[:(q + 2), q] = hs[q]
y, resids, rank, s = lstsq(hess, e1)
inner_res = norm2(np.dot(hess, y) - e1)
# -- check for termination
if inner_res < tol * inner_res_0:
break
# -- GMRES terminated: eval solution
dx = ws[0] * y[0]
for w, yc in zip(ws[1:], y[1:]):
dx = axpy(w, dx, dx.shape[0], yc) # dx += w*yc
# -- Store LGMRES augmentation vectors
nx = norm2(dx)
if store_outer_Av:
q = np.dot(hess, y)
ax = vs[0] * q[0]
for v, qc in zip(vs[1:], q[1:]):
ax = axpy(v, ax, ax.shape[0], qc)
outer_v.append((dx / nx, ax / nx))
else:
outer_v.append((dx / nx, None))
# -- Retain only a finite number of augmentation vectors
while len(outer_v) > outer_k:
del outer_v[0]
# -- Apply step
x += dx
else:
# didn't converge ...
return postprocess(x), maxiter
return postprocess(x), 0
def norm2(q):
q = np.asarray(q)
nrm2, = blas.get_blas_funcs(['nrm2'], [q])
return nrm2(q)
def test_blas(self):
a = array([[1, 1, 1]])
b = array([[1], [1], [1]])
gemm, = get_blas_funcs(('gemm', ), (a, b))
assert_array_almost_equal(gemm(1, a, b), [[3]], 15)
def norm2(q):
q = np.asarray(q)
nrm2, = blas.get_blas_funcs(['nrm2'], [q])
return nrm2(q)
def lgmres(A, b, x0=None, tol=1e-5, maxiter=1000, M=None, callback=None,
inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True):
"""
Solve a matrix equation using the LGMRES algorithm.
The LGMRES algorithm [BJM]_ [BPh]_ is designed to avoid some problems
in the convergence in restarted GMRES, and often converges in fewer
iterations.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : {array, matrix}
Starting guess for the solution.
tol : float
Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below `tol`.
maxiter : integer
Maximum number of iterations. Iteration will stop after maxiter
steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}
Preconditioner for A. The preconditioner should approximate the
inverse of A. Effective preconditioning dramatically improves the
rate of convergence, which implies that fewer iterations are needed
to reach a given error tolerance.
callback : function
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
Additional parameters
---------------------
inner_m : int, optional
Number of inner GMRES iterations per each outer iteration.
outer_k : int, optional
Number of vectors to carry between inner GMRES iterations.
According to [BJM]_, good values are in the range of 1...3.
However, note that if you want to use the additional vectors to
accelerate solving multiple similar problems, larger values may
be beneficial.
outer_v : list of tuples, optional
List containing tuples ``(v, Av)`` of vectors and corresponding
matrix-vector products, used to augment the Krylov subspace, and
carried between inner GMRES iterations. The element ``Av`` can
be `None` if the matrix-vector product should be re-evaluated.
This parameter is modified in-place by `lgmres`, and can be used
to pass "guess" vectors in and out of the algorithm when solving
similar problems.
store_outer_Av : bool, optional
Whether LGMRES should store also A*v in addition to vectors `v`
in the `outer_v` list. Default is True.
Returns
-------
x : array or matrix
The converged solution.
info : integer
Provides convergence information:
0 : successful exit
>0 : convergence to tolerance not achieved, number of iterations
<0 : illegal input or breakdown
Notes
-----
The LGMRES algorithm [BJM]_ [BPh]_ is designed to avoid the
slowing of convergence in restarted GMRES, due to alternating
residual vectors. Typically, it often outperforms GMRES(m) of
comparable memory requirements by some measure, or at least is not
much worse.
Another advantage in this algorithm is that you can supply it with
'guess' vectors in the `outer_v` argument that augment the Krylov
subspace. If the solution lies close to the span of these vectors,
the algorithm converges faster. This can be useful if several very
similar matrices need to be inverted one after another, such as in
Newton-Krylov iteration where the Jacobian matrix often changes
little in the nonlinear steps.
References
----------
.. [BJM] A.H. Baker and E.R. Jessup and T. Manteuffel,
SIAM J. Matrix Anal. Appl. 26, 962 (2005).
.. [BPh] A.H. Baker, PhD thesis, University of Colorado (2003).
http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps
"""
from scipy.linalg.basic import lstsq
A,M,x,b,postprocess = make_system(A,M,x0,b)
if not np.isfinite(b).all():
raise ValueError("RHS must contain only finite numbers")
matvec = A.matvec
psolve = M.matvec
if outer_v is None:
outer_v = []
axpy, dotc, scal = None, None, None
b_norm = norm2(b)
if b_norm == 0:
b_norm = 1
for k_outer in xrange(maxiter):
r_outer = matvec(x) - b
# -- callback
if callback is not None:
callback(x)
# -- determine input type routines
if axpy is None:
if np.iscomplexobj(r_outer) and not np.iscomplexobj(x):
x = x.astype(r_outer.dtype)
axpy, dotc, scal = blas.get_blas_funcs(['axpy', 'dotc', 'scal'],
(x, r_outer))
# -- check stopping condition
r_norm = norm2(r_outer)
if r_norm < tol * b_norm or r_norm < tol:
break
# -- inner LGMRES iteration
vs0 = -psolve(r_outer)
inner_res_0 = norm2(vs0)
if inner_res_0 == 0:
rnorm = norm2(r_outer)
raise RuntimeError("Preconditioner returned a zero vector; "
"|v| ~ %.1g, |M v| = 0" % rnorm)
vs0 = scal(1.0/inner_res_0, vs0)
hs = []
vs = [vs0]
ws = []
y = None
for j in xrange(1, 1 + inner_m + len(outer_v)):
# -- Arnoldi process:
#
# Build an orthonormal basis V and matrices W and H such that
# A W = V H
# Columns of W, V, and H are stored in `ws`, `vs` and `hs`.
#
# The first column of V is always the residual vector, `vs0`;
# V has *one more column* than the other of the three matrices.
#
# The other columns in V are built by feeding in, one
# by one, some vectors `z` and orthonormalizing them
# against the basis so far. The trick here is to
# feed in first some augmentation vectors, before
# starting to construct the Krylov basis on `v0`.
#
# It was shown in [BJM]_ that a good choice (the LGMRES choice)
# for these augmentation vectors are the `dx` vectors obtained
# from a couple of the previous restart cycles.
#
# Note especially that while `vs0` is always the first
# column in V, there is no reason why it should also be
# the first column in W. (In fact, below `vs0` comes in
# W only after the augmentation vectors.)
#
# The rest of the algorithm then goes as in GMRES, one
# solves a minimization problem in the smaller subspace
# spanned by W (range) and V (image).
#
# XXX: Below, I'm lazy and use `lstsq` to solve the
# small least squares problem. Performance-wise, this
# is in practice acceptable, but it could be nice to do
# it on the fly with Givens etc.
#
# ++ evaluate
v_new = None
if j < len(outer_v) + 1:
z, v_new = outer_v[j-1]
elif j == len(outer_v) + 1:
z = vs0
else:
z = vs[-1]
if v_new is None:
v_new = psolve(matvec(z))
else:
# Note: v_new is modified in-place below. Must make a
# copy to ensure that the outer_v vectors are not
# clobbered.
v_new = v_new.copy()
# ++ orthogonalize
hcur = []
for v in vs:
alpha = dotc(v, v_new)
hcur.append(alpha)
v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v
hcur.append(norm2(v_new))
if hcur[-1] == 0:
# Exact solution found; bail out.
# Zero basis vector (v_new) in the least-squares problem
# does no harm, so we can just use the same code as usually;
# it will give zero (inner) residual as a result.
bailout = True
else:
bailout = False
v_new = scal(1.0/hcur[-1], v_new)
vs.append(v_new)
hs.append(hcur)
ws.append(z)
# XXX: Ugly: should implement the GMRES iteration properly,
# with Givens rotations and not using lstsq. Instead, we
# spare some work by solving the LSQ problem only every 5
# iterations.
if not bailout and j % 5 != 1 and j < inner_m + len(outer_v) - 1:
continue
# -- GMRES optimization problem
hess = np.zeros((j+1, j), x.dtype)
e1 = np.zeros((j+1,), x.dtype)
e1[0] = inner_res_0
for q in xrange(j):
hess[:(q+2),q] = hs[q]
y, resids, rank, s = lstsq(hess, e1)
inner_res = norm2(np.dot(hess, y) - e1)
# -- check for termination
if inner_res < tol * inner_res_0:
break
# -- GMRES terminated: eval solution
dx = ws[0]*y[0]
for w, yc in zip(ws[1:], y[1:]):
dx = axpy(w, dx, dx.shape[0], yc) # dx += w*yc
# -- Store LGMRES augmentation vectors
nx = norm2(dx)
if store_outer_Av:
q = np.dot(hess, y)
ax = vs[0]*q[0]
for v, qc in zip(vs[1:], q[1:]):
ax = axpy(v, ax, ax.shape[0], qc)
outer_v.append((dx/nx, ax/nx))
else:
outer_v.append((dx/nx, None))
# -- Retain only a finite number of augmentation vectors
while len(outer_v) > outer_k:
del outer_v[0]
# -- Apply step
x += dx
else:
# didn't converge ...
return postprocess(x), maxiter
return postprocess(x), 0