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 [1]_ [2]_ 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 real or complex 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, optional
Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below `tol`.
maxiter : int, optional
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}, optional
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, optional
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
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 [1]_, 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 : int
Provides convergence information:
- 0 : successful exit
- >0 : convergence to tolerance not achieved, number of iterations
- <0 : illegal input or breakdown
Notes
-----
The LGMRES algorithm [1]_ [2]_ 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
----------
.. [1] A.H. Baker and E.R. Jessup and T. Manteuffel,
SIAM J. Matrix Anal. Appl. 26, 962 (2005).
.. [2] A.H. Baker, PhD thesis, University of Colorado (2003).
http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps
"""
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, dot, scal = None, None, None
nrm2 = get_blas_funcs('nrm2', [b])
b_norm = nrm2(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, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'],
(x, r_outer))
trtrs = get_lapack_funcs('trtrs', (x, r_outer))
# -- check stopping condition
r_norm = nrm2(r_outer)
if r_norm <= tol * b_norm or r_norm <= tol:
break
# -- inner LGMRES iteration
vs0 = -psolve(r_outer)
inner_res_0 = nrm2(vs0)
if inner_res_0 == 0:
rnorm = nrm2(r_outer)
raise RuntimeError("Preconditioner returned a zero vector; "
"|v| ~ %.1g, |M v| = 0" % rnorm)
vs0 = scal(1.0/inner_res_0, vs0)
vs = [vs0]
ws = []
y = None
# H is stored in QR factorized form
Q = np.ones((1, 1), dtype=vs0.dtype)
R = np.zeros((1, 0), dtype=vs0.dtype)
eps = np.finfo(vs0.dtype).eps
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).
#
# ++ 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 = np.zeros(j+1, dtype=Q.dtype)
for i, v in enumerate(vs):
alpha = dot(v, v_new)
hcur[i] = alpha
v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v
hcur[-1] = nrm2(v_new)
with np.errstate(over='ignore', divide='ignore'):
# Careful with denormals
alpha = 1/hcur[-1]
if np.isfinite(alpha):
v_new = scal(alpha, v_new)
else:
# v_new either zero (solution in span of previous
# vectors) or we have nans. If we already have
# previous vectors in R, we can discard the current
# vector and bail out.
if j > 1:
j -= 1
break
vs.append(v_new)
ws.append(z)
# -- GMRES optimization problem
# Add new column to H=Q*R, padding other columns with zeros
Q2 = np.zeros((j+1, j+1), dtype=Q.dtype, order='F')
Q2[:j,:j] = Q
Q2[j,j] = 1
R2 = np.zeros((j+1, j-1), dtype=R.dtype, order='F')
R2[:j,:] = R
Q, R = qr_insert(Q2, R2, hcur, j-1, which='col',
overwrite_qru=True, check_finite=False)
# Transformed least squares problem
# || Q R y - inner_res_0 * e_1 ||_2 = min!
# Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]
# Residual is immediately known
inner_res = abs(Q[0,-1]) * inner_res_0
# -- check for termination
if inner_res <= tol * inner_res_0:
break
# -- Get the LSQ problem solution
y, info = trtrs(R[:j,:j], Q[0,:j].conj())
if info != 0:
# Zero diagonal -> exact solution, but we handled that above
raise RuntimeError("QR solution failed")
y *= inner_res_0
if not np.isfinite(y).all():
# Floating point over/underflow, non-finite result from
# matmul etc. -- report failure.
return postprocess(x), k_outer + 1
# -- 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 = nrm2(dx)
if nx > 0:
if store_outer_Av:
q = Q.dot(R.dot(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 _fgmres(matvec,
v0,
m,
atol,
lpsolve=None,
rpsolve=None,
cs=(),
outer_v=(),
prepend_outer_v=False):
"""
FGMRES Arnoldi process, with optional projection or augmentation
Parameters
----------
matvec : callable
Operation A*x
v0 : ndarray
Initial vector, normalized to nrm2(v0) == 1
m : int
Number of GMRES rounds
atol : float
Absolute tolerance for early exit
lpsolve : callable
Left preconditioner L
rpsolve : callable
Right preconditioner R
CU : list of (ndarray, ndarray)
Columns of matrices C and U in GCROT
outer_v : list of ndarrays
Augmentation vectors in LGMRES
prepend_outer_v : bool, optional
Whether augmentation vectors come before or after
Krylov iterates
Raises
------
LinAlgError
If nans encountered
Returns
-------
Q, R : ndarray
QR decomposition of the upper Hessenberg H=QR
B : ndarray
Projections corresponding to matrix C
vs : list of ndarray
Columns of matrix V
zs : list of ndarray
Columns of matrix Z
y : ndarray
Solution to ||H y - e_1||_2 = min!
res : float
The final (preconditioned) residual norm
"""
if lpsolve is None:
lpsolve = lambda x: x
if rpsolve is None:
rpsolve = lambda x: x
axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'],
(v0, ))
vs = [v0]
zs = []
y = None
res = np.nan
m = m + len(outer_v)
# Orthogonal projection coefficients
B = np.zeros((len(cs), m), dtype=v0.dtype)
# H is stored in QR factorized form
Q = np.ones((1, 1), dtype=v0.dtype)
R = np.zeros((1, 0), dtype=v0.dtype)
eps = np.finfo(v0.dtype).eps
breakdown = False
# FGMRES Arnoldi process
for j in range(m):
# L A Z = C B + V H
if prepend_outer_v and j < len(outer_v):
z, w = outer_v[j]
elif prepend_outer_v and j == len(outer_v):
z = rpsolve(v0)
w = None
elif not prepend_outer_v and j >= m - len(outer_v):
z, w = outer_v[j - (m - len(outer_v))]
else:
z = rpsolve(vs[-1])
w = None
if w is None:
w = lpsolve(matvec(z))
else:
# w is clobbered below
w = w.copy()
w_norm = nrm2(w)
# GCROT projection: L A -> (1 - C C^H) L A
# i.e. orthogonalize against C
for i, c in enumerate(cs):
alpha = dot(c, w)
B[i, j] = alpha
w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c
# Orthogonalize against V
hcur = np.zeros(j + 2, dtype=Q.dtype)
for i, v in enumerate(vs):
alpha = dot(v, w)
hcur[i] = alpha
w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v
hcur[i + 1] = nrm2(w)
with np.errstate(over='ignore', divide='ignore'):
# Careful with denormals
alpha = 1 / hcur[-1]
if np.isfinite(alpha):
w = scal(alpha, w)
if not (hcur[-1] > eps * w_norm):
# w essentially in the span of previous vectors,
# or we have nans. Bail out after updating the QR
# solution.
breakdown = True
vs.append(w)
zs.append(z)
# Arnoldi LSQ problem
# Add new column to H=Q@R, padding other columns with zeros
Q2 = np.zeros((j + 2, j + 2), dtype=Q.dtype, order='F')
Q2[:j + 1, :j + 1] = Q
Q2[j + 1, j + 1] = 1
R2 = np.zeros((j + 2, j), dtype=R.dtype, order='F')
R2[:j + 1, :] = R
Q, R = qr_insert(Q2,
R2,
hcur,
j,
which='col',
overwrite_qru=True,
check_finite=False)
# Transformed least squares problem
# || Q R y - inner_res_0 * e_1 ||_2 = min!
# Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]
# Residual is immediately known
res = abs(Q[0, -1])
# Check for termination
if res < atol or breakdown:
break
if not np.isfinite(R[j, j]):
# nans encountered, bail out
raise LinAlgError()
# -- Get the LSQ problem solution
# The problem is triangular, but the condition number may be
# bad (or in case of breakdown the last diagonal entry may be
# zero), so use lstsq instead of trtrs.
y, _, _, _, = lstsq(R[:j + 1, :j + 1], Q[0, :j + 1].conj())
B = B[:, :j + 1]
return Q, R, B, vs, zs, y, res
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 [1]_ [2]_ 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 real or complex 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, optional
Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below `tol`.
maxiter : int, optional
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}, optional
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, optional
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
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 [1]_, 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 : int
Provides convergence information:
- 0 : successful exit
- >0 : convergence to tolerance not achieved, number of iterations
- <0 : illegal input or breakdown
Notes
-----
The LGMRES algorithm [1]_ [2]_ 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
----------
.. [1] A.H. Baker and E.R. Jessup and T. Manteuffel,
SIAM J. Matrix Anal. Appl. 26, 962 (2005).
.. [2] A.H. Baker, PhD thesis, University of Colorado (2003).
http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps
"""
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, dot, scal = None, None, None
nrm2 = get_blas_funcs('nrm2', [b])
b_norm = nrm2(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, dot, scal, nrm2 = get_blas_funcs(
['axpy', 'dot', 'scal', 'nrm2'], (x, r_outer))
trtrs = get_lapack_funcs('trtrs', (x, r_outer))
# -- check stopping condition
r_norm = nrm2(r_outer)
if r_norm <= tol * b_norm or r_norm <= tol:
break
# -- inner LGMRES iteration
vs0 = -psolve(r_outer)
inner_res_0 = nrm2(vs0)
if inner_res_0 == 0:
rnorm = nrm2(r_outer)
raise RuntimeError("Preconditioner returned a zero vector; "
"|v| ~ %.1g, |M v| = 0" % rnorm)
vs0 = scal(1.0 / inner_res_0, vs0)
vs = [vs0]
ws = []
y = None
# H is stored in QR factorized form
Q = np.ones((1, 1), dtype=vs0.dtype)
R = np.zeros((1, 0), dtype=vs0.dtype)
eps = np.finfo(vs0.dtype).eps
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).
#
# ++ 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 = np.zeros(j + 1, dtype=Q.dtype)
for i, v in enumerate(vs):
alpha = dot(v, v_new)
hcur[i] = alpha
v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v
hcur[-1] = nrm2(v_new)
with np.errstate(over='ignore', divide='ignore'):
# Careful with denormals
alpha = 1 / hcur[-1]
if np.isfinite(alpha):
v_new = scal(alpha, v_new)
else:
# v_new either zero (solution in span of previous
# vectors) or we have nans. If we already have
# previous vectors in R, we can discard the current
# vector and bail out.
if j > 1:
j -= 1
break
vs.append(v_new)
ws.append(z)
# -- GMRES optimization problem
# Add new column to H=Q*R, padding other columns with zeros
Q2 = np.zeros((j + 1, j + 1), dtype=Q.dtype, order='F')
Q2[:j, :j] = Q
Q2[j, j] = 1
R2 = np.zeros((j + 1, j - 1), dtype=R.dtype, order='F')
R2[:j, :] = R
Q, R = qr_insert(Q2,
R2,
hcur,
j - 1,
which='col',
overwrite_qru=True,
check_finite=False)
# Transformed least squares problem
# || Q R y - inner_res_0 * e_1 ||_2 = min!
# Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]
# Residual is immediately known
inner_res = abs(Q[0, -1]) * inner_res_0
# -- check for termination
if inner_res <= tol * inner_res_0:
break
# -- Get the LSQ problem solution
y, info = trtrs(R[:j, :j], Q[0, :j].conj())
if info != 0:
# Zero diagonal -> exact solution, but we handled that above
raise RuntimeError("QR solution failed")
y *= inner_res_0
if not np.isfinite(y).all():
# Floating point over/underflow, non-finite result from
# matmul etc. -- report failure.
return postprocess(x), k_outer + 1
# -- 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 = nrm2(dx)
if nx > 0:
if store_outer_Av:
q = Q.dot(R.dot(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 _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(),
prepend_outer_v=False):
"""
FGMRES Arnoldi process, with optional projection or augmentation
Parameters
----------
matvec : callable
Operation A*x
v0 : ndarray
Initial vector, normalized to nrm2(v0) == 1
m : int
Number of GMRES rounds
atol : float
Absolute tolerance for early exit
lpsolve : callable
Left preconditioner L
rpsolve : callable
Right preconditioner R
CU : list of (ndarray, ndarray)
Columns of matrices C and U in GCROT
outer_v : list of ndarrays
Augmentation vectors in LGMRES
prepend_outer_v : bool, optional
Whether augmentation vectors come before or after
Krylov iterates
Raises
------
LinAlgError
If nans encountered
Returns
-------
Q, R : ndarray
QR decomposition of the upper Hessenberg H=QR
B : ndarray
Projections corresponding to matrix C
vs : list of ndarray
Columns of matrix V
zs : list of ndarray
Columns of matrix Z
y : ndarray
Solution to ||H y - e_1||_2 = min!
res : float
The final (preconditioned) residual norm
"""
if lpsolve is None:
lpsolve = lambda x: x
if rpsolve is None:
rpsolve = lambda x: x
axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (v0,))
vs = [v0]
zs = []
y = None
res = np.nan
m = m + len(outer_v)
# Orthogonal projection coefficients
B = np.zeros((len(cs), m), dtype=v0.dtype)
# H is stored in QR factorized form
Q = np.ones((1, 1), dtype=v0.dtype)
R = np.zeros((1, 0), dtype=v0.dtype)
eps = np.finfo(v0.dtype).eps
breakdown = False
# FGMRES Arnoldi process
for j in xrange(m):
# L A Z = C B + V H
if prepend_outer_v and j < len(outer_v):
z, w = outer_v[j]
elif prepend_outer_v and j == len(outer_v):
z = rpsolve(v0)
w = None
elif not prepend_outer_v and j >= m - len(outer_v):
z, w = outer_v[j - (m - len(outer_v))]
else:
z = rpsolve(vs[-1])
w = None
if w is None:
w = lpsolve(matvec(z))
else:
# w is clobbered below
w = w.copy()
w_norm = nrm2(w)
# GCROT projection: L A -> (1 - C C^H) L A
# i.e. orthogonalize against C
for i, c in enumerate(cs):
alpha = dot(c, w)
B[i,j] = alpha
w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c
# Orthogonalize against V
hcur = np.zeros(j+2, dtype=Q.dtype)
for i, v in enumerate(vs):
alpha = dot(v, w)
hcur[i] = alpha
w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v
hcur[i+1] = nrm2(w)
with np.errstate(over='ignore', divide='ignore'):
# Careful with denormals
alpha = 1/hcur[-1]
if np.isfinite(alpha):
w = scal(alpha, w)
if not (hcur[-1] > eps * w_norm):
# w essentially in the span of previous vectors,
# or we have nans. Bail out after updating the QR
# solution.
breakdown = True
vs.append(w)
zs.append(z)
# Arnoldi LSQ problem
# Add new column to H=Q*R, padding other columns with zeros
Q2 = np.zeros((j+2, j+2), dtype=Q.dtype, order='F')
Q2[:j+1,:j+1] = Q
Q2[j+1,j+1] = 1
R2 = np.zeros((j+2, j), dtype=R.dtype, order='F')
R2[:j+1,:] = R
Q, R = qr_insert(Q2, R2, hcur, j, which='col',
overwrite_qru=True, check_finite=False)
# Transformed least squares problem
# || Q R y - inner_res_0 * e_1 ||_2 = min!
# Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]
# Residual is immediately known
res = abs(Q[0,-1])
# Check for termination
if res < atol or breakdown:
break
if not np.isfinite(R[j,j]):
# nans encountered, bail out
raise LinAlgError()
# -- Get the LSQ problem solution
# The problem is triangular, but the condition number may be
# bad (or in case of breakdown the last diagonal entry may be
# zero), so use lstsq instead of trtrs.
y, _, _, _, = lstsq(R[:j+1,:j+1], Q[0,:j+1].conj())
B = B[:,:j+1]
return Q, R, B, vs, zs, y, res
def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(),
prepend_outer_v=False):
if lpsolve is None:
def lpsolve(x): return x
if rpsolve is None:
def rpsolve(x): return x
axpy, dot, scal, nrm2 = get_blas_funcs(
['axpy', 'dot', 'scal', 'nrm2'], (v0,))
vs = [v0]
zs = []
y = None
m = m + len(outer_v)
B = np.zeros((len(cs), m), dtype=v0.dtype)
Q = np.ones((1, 1), dtype=v0.dtype)
R = np.zeros((1, 0), dtype=v0.dtype)
eps = np.finfo(v0.dtype).eps
breakdown = False
for j in xrange(m):
if prepend_outer_v and j < len(outer_v):
z, w = outer_v[j]
elif prepend_outer_v and j == len(outer_v):
z = rpsolve(v0)
w = None
elif not prepend_outer_v and j >= m - len(outer_v):
z, w = outer_v[j - (m - len(outer_v))]
else:
z = rpsolve(vs[-1])
w = None
if w is None:
w = lpsolve(matvec(z))
else:
w = w.copy()
w_norm = nrm2(w)
for i, c in enumerate(cs):
alpha = dot(c, w)
B[i, j] = alpha
w = axpy(c, w, c.shape[0], -alpha)
hcur = np.zeros(j + 2, dtype=Q.dtype)
for i, v in enumerate(vs):
alpha = dot(v, w)
hcur[i] = alpha
w = axpy(v, w, v.shape[0], -alpha)
hcur[i + 1] = nrm2(w)
with np.errstate(over='ignore', divide='ignore'):
alpha = 1 / hcur[-1]
if np.isfinite(alpha):
w = scal(alpha, w)
if not (hcur[-1] > eps * w_norm):
breakdown = True
vs.append(w)
zs.append(z)
Q2 = np.zeros((j + 2, j + 2), dtype=Q.dtype, order='F')
Q2[:j + 1, :j + 1] = Q
Q2[j + 1, j + 1] = 1
R2 = np.zeros((j + 2, j), dtype=R.dtype, order='F')
R2[:j + 1, :] = R
Q, R = qr_insert(Q2, R2, hcur, j, which='col',
overwrite_qru=True, check_finite=False)
res = abs(Q[0, -1])
if res < atol or breakdown:
break
if not np.isfinite(R[j, j]):
raise LinAlgError()
y, _, _, _, = lstsq(R[:j + 1, :j + 1], Q[0, :j + 1].conj())
B = B[:, :j + 1]
return Q, R, B, vs, zs, y
def quadprog_solve(quadr_coeff_G, linear_coeff_a, n_ineq, ineq_coef_C,
ineq_vec_d, m_eq, eq_coef_A, eq_vec_b):
DONE = False
FULL_STEP = False
if eq_coef_A is not None:
ADDING_EQ_CONSTRAINTS = True
else:
ADDING_EQ_CONSTRAINTS = False
first_pass = True
###-------------------------------------###
## Solution iterate
sol = (-1) * np.linalg.inv(quadr_coeff_G) * linear_coeff_a
## We only need to keep ahold of L^{-1} for the implementation.
## L is the triangular result of
L = np.linalg.cholesky(quadr_coeff_G)
Linv = np.linalg.inv(L)
## Need to adopt the conventions used for directly computing
## the values of ``z`` and ``r``, we need to hold onto
## the values of the factorization QR = B = L^{-1} N.
Q = None
R = None
J1 = None
J2 = None
## indices of constraints being considered
active_set = np.array([], dtype=(np.dtype(int)))
## normal vector of a given constraint
n_p = None
## index of n_p in the choice of equality or inequality constraint
p = None
## number of considered constraints in active set
q = 0
u = None
lagr = None
# index out of all constraints which was dropped
k_dropped = None
# index of active constraint which was dropped
j_dropped = None
z = None ## Step direction in `primal' space
r = 0 ## Step direction in `dual' space
###-------------------------------------###
###~~~~~~~~ Step 1 ~~~~~~~~###
while not DONE:
ineq = np.ravel((ineq_coef_C.T * sol) - ineq_vec_d)
if ADDING_EQ_CONSTRAINTS:
eq_prb = np.ravel((eq_coef_A.T * sol) - eq_vec_b)
if (len(active_set) == m_eq):
ADDING_EQ_CONSTRAINTS = False
if (np.any(ineq < 0) or ADDING_EQ_CONSTRAINTS):
if (ADDING_EQ_CONSTRAINTS):
p = len(active_set)
n_p = eq_coef_A[:, p]
else:
## Choose a violated constraint not in active set.
## This is the most naive way, can be improved.
#violated_constraints = np.ravel(np.where(ineq < 0))
#v = [x for x in violated_constraints if x not in active_set]
## Pick the first violated constraint.
#p = v[0]
## normal vector for each constraint, vector normal to the plane.
#n_p = ineq_coef_C[:,p]
## Instead, just loop through inequality constraints
for blah in range(0, n_ineq):
if ineq[blah] < 0:
p = blah
burn_flag = False
for jack in range(m_eq + 1, q):
if active_set[jack] == p:
burn_flag = True
break
if burn_flag:
continue
else:
break
n_p = ineq_coef_C[:, p]
if q == 0:
u = 0
lagr = np.hstack((u, 0))
###~~~~~~~~ Step 2 ~~~~~~~~###
FULL_STEP = False
while not FULL_STEP:
# algo as writ will cycle back here after taking a step
# in dual space, update inequality portion
ineq = np.ravel((ineq_coef_C.T * sol) - ineq_vec_d)
if ADDING_EQ_CONSTRAINTS:
eq_prb = np.ravel((eq_coef_A.T * sol) - eq_vec_b)
###~~~~~~~~ Step 2(a) ~~~~~~~~###
## Calculate step directions
if first_pass:
z = np.linalg.inv(quadr_coeff_G) * n_p
first_pass = False
else:
# step direction in the primal space
z = J2 * J2.T * n_p
if (q > 0):
# negative of step direction in the dual space
# r will have num_rows = len(active_set)
r = np.linalg.inv(R[0:q, 0:q]) * J1.T * n_p
###~~~~~~~~ Step 2(b) ~~~~~~~~###
# partial step length t1 - max step in dual space
if ((q == 0) or (np.any(r <= 0)) or ADDING_EQ_CONSTRAINTS):
t1 = np.inf
else:
t1 = np.inf
k_dropped = None
for j in range(m_eq, len(active_set)):
k = active_set[j]
if (r[j] > 0) and (lagr[j] / r[j]) < t1:
t1 = lagr[j] / r[j]
k_dropped = k
j_dropped = j
t1 = np.ravel(t1)[0]
# full step length t2 - min step in primal space
if (np.all(z == 0)):
# If no step in primal space
t2 = np.inf
else:
if ADDING_EQ_CONSTRAINTS:
t2 = (-1) * eq_prb[p] / (z.T * n_p)
else:
t2 = (-1) * ineq[p] / (z.T * n_p)
t2 = np.ravel(t2)[0]
# current step length
t = np.min([t1, t2])
###~~~~~~~~ Step 2(c) ~~~~~~~~###
if (t == np.inf):
print("infeasible! Stop here!")
FULL_STEP = True
DONE = True
return None
#break
# If t2 is infinite, then we took a partial step in the dual space.
if (t2 == np.inf):
#print("t2 infinite")
# Update lagrangian
lagr = lagr + t * np.hstack((np.ravel(-1 * r), 1))
# Drop the constraint which minimized the step we took at that
# point.
active_set = np.delete(active_set, j_dropped)
q = q - 1
Q, R = qr_delete(Q, R, j_dropped, 1, 'col')
J1 = Linv.T * Q[:, [x for x in range(0, q)]]
J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]]
# go back to step 2(a)
continue
# Update iterate for x, and the lagrangian
sol = sol + t * z
lagr = lagr + t * np.hstack((np.ravel(-1 * r), 1))
# if we took a full step
if (t == t2):
#print("full step")
## Add the constraint to the active set
active_set = np.hstack((active_set, p))
q = q + 1
u = lagr[-q:]
if Q is None:
Q, R = np.linalg.qr(Linv * n_p, mode="complete")
J1 = Linv.T * Q[:, [x for x in range(0, q)]]
J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]]
else:
Q, R = qr_insert(Q, R, (Linv * n_p),
len(active_set) - 1, 'col')
J1 = Linv.T * Q[:, [x for x in range(0, q)]]
J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]]
# Exit current loop for Step 2, go back to Step 1
FULL_STEP = True
break
# if we took a partial step
if (t == t1):
#print("partial step")
# Drop constraint k
active_set = np.delete(active_set, j_dropped)
q = q - 1
Q, R = qr_delete(Q, R, j_dropped, 1, 'col')
J1 = Linv.T * Q[:, [x for x in range(0, q)]]
J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]]
# Go back to step 2(a)
continue
else:
DONE = True
#print(ineq)
#print(sol)
return sol