def bicgstab(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M=None, callback=None):
A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)
n = len(b)
if maxiter is None:
maxiter = n*10
matvec = A.matvec
psolve = M.matvec
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'bicgstabrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros(7*n,dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, work, iter_, resid, info, ndx1, ndx2, ijob)
if callback is not None and iter_ > olditer:
callback(x)
slice1 = slice(ndx1-1, ndx1-1+n)
slice2 = slice(ndx2-1, ndx2-1+n)
if (ijob == -1):
if callback is not None:
callback(x)
break
elif (ijob == 1):
if matvec is None:
matvec = get_matvec(A)
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(work[slice1])
elif (ijob == 2):
if psolve is None:
psolve = get_psolve(A)
work[slice1] = psolve(work[slice2])
elif (ijob == 3):
if matvec is None:
matvec = get_matvec(A)
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
ijob = 2
if info > 0 and iter_ == maxiter and resid > tol:
#info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
def bicgstab(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M=None, callback=None):
A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)
n = len(b)
if maxiter is None:
maxiter = n*10
matvec = A.matvec
psolve = M.matvec
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'bicgstabrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros(7*n,dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, work, iter_, resid, info, ndx1, ndx2, ijob)
if callback is not None and iter_ > olditer:
callback(x)
slice1 = slice(ndx1-1, ndx1-1+n)
slice2 = slice(ndx2-1, ndx2-1+n)
if (ijob == -1):
if callback is not None:
callback(x)
break
elif (ijob == 1):
if matvec is None:
matvec = get_matvec(A)
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(work[slice1])
elif (ijob == 2):
if psolve is None:
psolve = get_psolve(A)
work[slice1] = psolve(work[slice2])
elif (ijob == 3):
if matvec is None:
matvec = get_matvec(A)
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
ijob = 2
if info > 0 and iter_ == maxiter and resid > tol:
#info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
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 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
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, dot, 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, dot, scal = get_blas_funcs(['axpy', 'dot', '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 = dot(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 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 qmr(A,
b,
x0=None,
tol=1e-5,
maxiter=None,
xtype=None,
M1=None,
M2=None,
callback=None):
"""Use Quasi-Minimal Residual iteration to solve A x = b
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).
Returns
-------
x : {array, 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
Other Parameters
----------------
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.
M1 : {sparse matrix, dense matrix, LinearOperator}
Left preconditioner for A.
M2 : {sparse matrix, dense matrix, LinearOperator}
Right preconditioner for A. Used together with the left
preconditioner M1. The matrix M1*A*M2 should have better
conditioned than A alone.
callback : function
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
xtype : {'f','d','F','D'}
This parameter is DEPRECATED -- avoid using it.
The type of the result. If None, then it will be determined from
A.dtype.char and b. If A does not have a typecode method then it
will compute A.matvec(x0) to get a typecode. To save the extra
computation when A does not have a typecode attribute use xtype=0
for the same type as b or use xtype='f','d','F',or 'D'.
This parameter has been superceeded by LinearOperator.
See Also
--------
LinearOperator
"""
A_ = A
A, M, x, b, postprocess = make_system(A, None, x0, b, xtype)
if M1 is None and M2 is None:
if hasattr(A_, 'psolve'):
def left_psolve(b):
return A_.psolve(b, 'left')
def right_psolve(b):
return A_.psolve(b, 'right')
def left_rpsolve(b):
return A_.rpsolve(b, 'left')
def right_rpsolve(b):
return A_.rpsolve(b, 'right')
M1 = LinearOperator(A.shape,
matvec=left_psolve,
rmatvec=left_rpsolve)
M2 = LinearOperator(A.shape,
matvec=right_psolve,
rmatvec=right_rpsolve)
else:
def id(b):
return b
M1 = LinearOperator(A.shape, matvec=id, rmatvec=id)
M2 = LinearOperator(A.shape, matvec=id, rmatvec=id)
n = len(b)
if maxiter is None:
maxiter = n * 10
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'qmrrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros(11 * n, x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, work, iter_, resid, info, ndx1, ndx2, ijob)
if callback is not None and iter_ > olditer:
callback(x)
slice1 = slice(ndx1 - 1, ndx1 - 1 + n)
slice2 = slice(ndx2 - 1, ndx2 - 1 + n)
if (ijob == -1):
if callback is not None:
callback(x)
break
elif (ijob == 1):
work[slice2] *= sclr2
work[slice2] += sclr1 * A.matvec(work[slice1])
elif (ijob == 2):
work[slice2] *= sclr2
work[slice2] += sclr1 * A.rmatvec(work[slice1])
elif (ijob == 3):
work[slice1] = M1.matvec(work[slice2])
elif (ijob == 4):
work[slice1] = M2.matvec(work[slice2])
elif (ijob == 5):
work[slice1] = M1.rmatvec(work[slice2])
elif (ijob == 6):
work[slice1] = M2.rmatvec(work[slice2])
elif (ijob == 7):
work[slice2] *= sclr2
work[slice2] += sclr1 * A.matvec(x)
elif (ijob == 8):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
ijob = 2
if info > 0 and iter_ == maxiter and resid > tol:
#info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
def gmres(A,
b,
x0=None,
tol=1e-5,
restart=None,
maxiter=None,
xtype=None,
M=None,
callback=None,
restrt=None):
"""
Use Generalized Minimal RESidual iteration to solve A x = b.
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).
Returns
-------
x : {array, 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
Other parameters
----------------
x0 : {array, matrix}
Starting guess for the solution (a vector of zeros by default).
tol : float
Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below `tol`.
restart : int, optional
Number of iterations between restarts. Larger values increase
iteration cost, but may be necessary for convergence.
Default is 20.
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}
Inverse of the preconditioner of A. M should approximate the
inverse of A and be easy to solve for (see Notes). Effective
preconditioning dramatically improves the rate of convergence,
which implies that fewer iterations are needed to reach a given
error tolerance. By default, no preconditioner is used.
callback : function
User-supplied function to call after each iteration. It is called
as callback(rk), where rk is the current residual vector.
See Also
--------
LinearOperator
Notes
-----
A preconditioner, P, is chosen such that P is close to A but easy to solve for.
The preconditioner parameter required by this routine is ``M = P^-1``.
The inverse should preferably not be calculated explicitly. Rather, use the
following template to produce M::
# Construct a linear operator that computes P^-1 * x.
import scipy.sparse.linalg as spla
M_x = lambda x: spla.spsolve(P, x)
M = spla.LinearOperator((n, n), M_x)
Deprecated Parameters
---------------------
xtype : {'f','d','F','D'}
This parameter is DEPRECATED --- avoid using it.
The type of the result. If None, then it will be determined from
A.dtype.char and b. If A does not have a typecode method then it
will compute A.matvec(x0) to get a typecode. To save the extra
computation when A does not have a typecode attribute use xtype=0
for the same type as b or use xtype='f','d','F',or 'D'.
This parameter has been superceeded by LinearOperator.
See Also
--------
LinearOperator
"""
# Change 'restrt' keyword to 'restart'
if restrt is None:
restrt = restart
elif restart is not None:
raise ValueError("Cannot specify both restart and restrt keywords. "
"Preferably use 'restart' only.")
A, M, x, b, postprocess = make_system(A, M, x0, b, xtype)
n = len(b)
if maxiter is None:
maxiter = n * 10
if restrt is None:
restrt = 20
restrt = min(restrt, n)
matvec = A.matvec
psolve = M.matvec
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'gmresrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros((6 + restrt) * n, dtype=x.dtype)
work2 = np.zeros((restrt + 1) * (2 * restrt + 2), dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
old_ijob = ijob
first_pass = True
resid_ready = False
iter_num = 1
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, restrt, work, work2, iter_, resid, info, ndx1, ndx2, ijob)
#if callback is not None and iter_ > olditer:
# callback(x)
slice1 = slice(ndx1 - 1, ndx1 - 1 + n)
slice2 = slice(ndx2 - 1, ndx2 - 1 + n)
if (ijob == -1): # gmres success, update last residual
if resid_ready and callback is not None:
callback(resid)
resid_ready = False
break
elif (ijob == 1):
work[slice2] *= sclr2
work[slice2] += sclr1 * matvec(x)
elif (ijob == 2):
work[slice1] = psolve(work[slice2])
if not first_pass and old_ijob == 3:
resid_ready = True
first_pass = False
elif (ijob == 3):
work[slice2] *= sclr2
work[slice2] += sclr1 * matvec(work[slice1])
if resid_ready and callback is not None:
callback(resid)
resid_ready = False
iter_num = iter_num + 1
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
old_ijob = ijob
ijob = 2
if iter_num > maxiter:
break
if info >= 0 and resid > tol:
#info isn't set appropriately otherwise
info = maxiter
return postprocess(x), info
def qmr(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M1=None, M2=None, callback=None):
"""Use Quasi-Minimal Residual iteration to solve A x = b
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).
Returns
-------
x : {array, 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
Other Parameters
----------------
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.
M1 : {sparse matrix, dense matrix, LinearOperator}
Left preconditioner for A.
M2 : {sparse matrix, dense matrix, LinearOperator}
Right preconditioner for A. Used together with the left
preconditioner M1. The matrix M1*A*M2 should have better
conditioned than A alone.
callback : function
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
xtype : {'f','d','F','D'}
This parameter is DEPRECATED -- avoid using it.
The type of the result. If None, then it will be determined from
A.dtype.char and b. If A does not have a typecode method then it
will compute A.matvec(x0) to get a typecode. To save the extra
computation when A does not have a typecode attribute use xtype=0
for the same type as b or use xtype='f','d','F',or 'D'.
This parameter has been superceeded by LinearOperator.
See Also
--------
LinearOperator
"""
A_ = A
A,M,x,b,postprocess = make_system(A,None,x0,b,xtype)
if M1 is None and M2 is None:
if hasattr(A_,'psolve'):
def left_psolve(b):
return A_.psolve(b,'left')
def right_psolve(b):
return A_.psolve(b,'right')
def left_rpsolve(b):
return A_.rpsolve(b,'left')
def right_rpsolve(b):
return A_.rpsolve(b,'right')
M1 = LinearOperator(A.shape, matvec=left_psolve, rmatvec=left_rpsolve)
M2 = LinearOperator(A.shape, matvec=right_psolve, rmatvec=right_rpsolve)
else:
def id(b):
return b
M1 = LinearOperator(A.shape, matvec=id, rmatvec=id)
M2 = LinearOperator(A.shape, matvec=id, rmatvec=id)
n = len(b)
if maxiter is None:
maxiter = n*10
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'qmrrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros(11*n,x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, work, iter_, resid, info, ndx1, ndx2, ijob)
if callback is not None and iter_ > olditer:
callback(x)
slice1 = slice(ndx1-1, ndx1-1+n)
slice2 = slice(ndx2-1, ndx2-1+n)
if (ijob == -1):
if callback is not None:
callback(x)
break
elif (ijob == 1):
work[slice2] *= sclr2
work[slice2] += sclr1*A.matvec(work[slice1])
elif (ijob == 2):
work[slice2] *= sclr2
work[slice2] += sclr1*A.rmatvec(work[slice1])
elif (ijob == 3):
work[slice1] = M1.matvec(work[slice2])
elif (ijob == 4):
work[slice1] = M2.matvec(work[slice2])
elif (ijob == 5):
work[slice1] = M1.rmatvec(work[slice2])
elif (ijob == 6):
work[slice1] = M2.rmatvec(work[slice2])
elif (ijob == 7):
work[slice2] *= sclr2
work[slice2] += sclr1*A.matvec(x)
elif (ijob == 8):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
ijob = 2
if info > 0 and iter_ == maxiter and resid > tol:
#info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
def gmres(A, b, x0=None, tol=1e-5, restart=None, maxiter=None, xtype=None, M=None, callback=None, restrt=None):
"""Use Generalized Minimal RESidual iteration to solve A x = b
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).
Returns
-------
x : {array, 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
Other Parameters
----------------
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`.
restart : integer
Number of iterations between restarts. Larger values increase
iteration cost, but may be necessary for convergence.
(Default: 20)
maxiter : integer, 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}
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(rk), where rk is the current residual vector.
xtype : {'f','d','F','D'}
This parameter is DEPRECATED --- avoid using it.
The type of the result. If None, then it will be determined from
A.dtype.char and b. If A does not have a typecode method then it
will compute A.matvec(x0) to get a typecode. To save the extra
computation when A does not have a typecode attribute use xtype=0
for the same type as b or use xtype='f','d','F',or 'D'.
This parameter has been superceeded by LinearOperator.
See Also
--------
LinearOperator
"""
# Change 'restrt' keyword to 'restart'
if restrt is None:
restrt = restart
elif restart is not None:
raise ValueError("Cannot specify both restart and restrt keywords. "
"Preferably use 'restart' only.")
A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)
n = len(b)
if maxiter is None:
maxiter = n*10
if restrt is None:
restrt = 20
restrt = min(restrt, n)
matvec = A.matvec
psolve = M.matvec
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'gmresrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros((6+restrt)*n,dtype=x.dtype)
work2 = np.zeros((restrt+1)*(2*restrt+2),dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
old_ijob = ijob
first_pass = True
resid_ready = False
iter_num = 1
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, restrt, work, work2, iter_, resid, info, ndx1, ndx2, ijob)
#if callback is not None and iter_ > olditer:
# callback(x)
slice1 = slice(ndx1-1, ndx1-1+n)
slice2 = slice(ndx2-1, ndx2-1+n)
if (ijob == -1): # gmres success, update last residual
if resid_ready and callback is not None:
callback(resid)
resid_ready = False
break
elif (ijob == 1):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 2):
work[slice1] = psolve(work[slice2])
if not first_pass and old_ijob==3:
resid_ready = True
first_pass = False
elif (ijob == 3):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(work[slice1])
if resid_ready and callback is not None:
callback(resid)
resid_ready = False
iter_num = iter_num+1
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
old_ijob = ijob
ijob = 2
if iter_num > maxiter:
break
if info >= 0 and resid > tol:
#info isn't set appropriately otherwise
info = maxiter
return postprocess(x), info
def minres(A,
b,
x0=None,
shift=0.0,
tol=1e-5,
maxiter=None,
xtype=None,
M=None,
callback=None,
show=False,
check=False):
A, M, x, b, postprocess = make_system(A, M, x0, b, xtype)
matvec = A.matvec
psolve = M.matvec
first = 'Enter minres. '
last = 'Exit minres. '
n = A.shape[0]
if maxiter is None:
maxiter = 5 * n
msg = [
' beta2 = 0. If M = I, b and x are eigenvectors ', # -1
' beta1 = 0. The exact solution is x = 0 ', # 0
' A solution to Ax = b was found, given rtol ', # 1
' A least-squares solution was found, given rtol ', # 2
' Reasonable accuracy achieved, given eps ', # 3
' x has converged to an eigenvector ', # 4
' acond has exceeded 0.1/eps ', # 5
' The iteration limit was reached ', # 6
' A does not define a symmetric matrix ', # 7
' M does not define a symmetric matrix ', # 8
' M does not define a pos-def preconditioner '
] # 9
if show:
print first + 'Solution of symmetric Ax = b'
print first + 'n = %3g shift = %23.14e' % (n, shift)
print first + 'itnlim = %3g rtol = %11.2e' % (maxiter, tol)
print
istop = 0
itn = 0
Anorm = 0
Acond = 0
rnorm = 0
ynorm = 0
xtype = x.dtype
eps = finfo(xtype).eps
x = zeros(n, dtype=xtype)
# Set up y and v for the first Lanczos vector v1.
# y = beta1 P' v1, where P = C**(-1).
# v is really P' v1.
y = b
r1 = b
y = psolve(b)
beta1 = inner(b, y)
if beta1 < 0:
raise ValueError('indefinite preconditioner')
elif beta1 == 0:
return (postprocess(x), 0)
beta1 = sqrt(beta1)
if check:
# are these too strict?
# see if A is symmetric
w = matvec(y)
r2 = matvec(w)
s = inner(w, w)
t = inner(y, r2)
z = abs(s - t)
epsa = (s + eps) * eps**(1.0 / 3.0)
if z > epsa:
raise ValueError('non-symmetric matrix')
# see if M is symmetric
r2 = psolve(y)
s = inner(y, y)
t = inner(r1, r2)
z = abs(s - t)
epsa = (s + eps) * eps**(1.0 / 3.0)
if z > epsa:
raise ValueError('non-symmetric preconditioner')
# Initialize other quantities
oldb = 0
beta = beta1
dbar = 0
epsln = 0
qrnorm = beta1
phibar = beta1
rhs1 = beta1
rhs2 = 0
tnorm2 = 0
ynorm2 = 0
cs = -1
sn = 0
w = zeros(n, dtype=xtype)
w2 = zeros(n, dtype=xtype)
r2 = r1
if show:
print
print
print ' Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|'
while itn < maxiter:
itn += 1
s = 1.0 / beta
v = s * y
y = matvec(v)
y = y - shift * v
if itn >= 2:
y = y - (beta / oldb) * r1
alfa = inner(v, y)
y = y - (alfa / beta) * r2
r1 = r2
r2 = y
y = psolve(r2)
oldb = beta
beta = inner(r2, y)
if beta < 0:
raise ValueError('non-symmetric matrix')
beta = sqrt(beta)
tnorm2 += alfa**2 + oldb**2 + beta**2
if itn == 1:
if beta / beta1 <= 10 * eps:
istop = -1 # Terminate later
#tnorm2 = alfa**2 ??
gmax = abs(alfa)
gmin = gmax
# Apply previous rotation Qk-1 to get
# [deltak epslnk+1] = [cs sn][dbark 0 ]
# [gbar k dbar k+1] [sn -cs][alfak betak+1].
oldeps = epsln
delta = cs * dbar + sn * alfa # delta1 = 0 deltak
gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
epsln = sn * beta # epsln2 = 0 epslnk+1
dbar = -cs * beta # dbar 2 = beta2 dbar k+1
root = norm([gbar, dbar])
Arnorm = phibar * root
# Compute the next plane rotation Qk
gamma = norm([gbar, beta]) # gammak
gamma = max(gamma, eps)
cs = gbar / gamma # ck
sn = beta / gamma # sk
phi = cs * phibar # phik
phibar = sn * phibar # phibark+1
# Update x.
denom = 1.0 / gamma
w1 = w2
w2 = w
w = (v - oldeps * w1 - delta * w2) * denom
x = x + phi * w
# Go round again.
gmax = max(gmax, gamma)
gmin = min(gmin, gamma)
z = rhs1 / gamma
ynorm2 = z**2 + ynorm2
rhs1 = rhs2 - delta * z
rhs2 = -epsln * z
# Estimate various norms and test for convergence.
Anorm = sqrt(tnorm2)
ynorm = sqrt(ynorm2)
epsa = Anorm * eps
epsx = Anorm * ynorm * eps
epsr = Anorm * ynorm * tol
diag = gbar
if diag == 0: diag = epsa
qrnorm = phibar
rnorm = qrnorm
test1 = rnorm / (Anorm * ynorm) # ||r|| / (||A|| ||x||)
test2 = root / Anorm # ||Ar|| / (||A|| ||r||)
# Estimate cond(A).
# In this version we look at the diagonals of R in the
# factorization of the lower Hessenberg matrix, Q * H = R,
# where H is the tridiagonal matrix from Lanczos with one
# extra row, beta(k+1) e_k^T.
Acond = gmax / gmin
# See if any of the stopping criteria are satisfied.
# In rare cases, istop is already -1 from above (Abar = const*I).
if istop == 0:
t1 = 1 + test1 # These tests work if tol < eps
t2 = 1 + test2
if t2 <= 1: istop = 2
if t1 <= 1: istop = 1
if itn >= maxiter: istop = 6
if Acond >= 0.1 / eps: istop = 4
if epsx >= beta1: istop = 3
#if rnorm <= epsx : istop = 2
#if rnorm <= epsr : istop = 1
if test2 <= tol: istop = 2
if test1 <= tol: istop = 1
# See if it is time to print something.
prnt = False
if n <= 40: prnt = True
if itn <= 10: prnt = True
if itn >= maxiter - 10: prnt = True
if itn % 10 == 0: prnt = True
if qrnorm <= 10 * epsx: prnt = True
if qrnorm <= 10 * epsr: prnt = True
if Acond <= 1e-2 / eps: prnt = True
if istop != 0: prnt = True
if show and prnt:
str1 = '%6g %12.5e %10.3e' % (itn, x[0], test1)
str2 = ' %10.3e' % (test2, )
str3 = ' %8.1e %8.1e %8.1e' % (Anorm, Acond, gbar / Anorm)
print str1 + str2 + str3
if itn % 10 == 0: print
if callback is not None:
callback(x)
if istop != 0: break #TODO check this
if show:
print
print last + ' istop = %3g itn =%5g' % (istop, itn)
print last + ' Anorm = %12.4e Acond = %12.4e' % (Anorm, Acond)
print last + ' rnorm = %12.4e ynorm = %12.4e' % (rnorm, ynorm)
print last + ' Arnorm = %12.4e' % (Arnorm, )
print last + msg[istop + 1]
if istop == 6:
info = maxiter
else:
info = 0
return (postprocess(x), info)
def gmres(A, b, x0=None, tol=1e-5, restrt=20, maxiter=None, xtype=None, M=None, callback=None):
"""Use Generalized Minimal RESidual iteration to solve A x = b
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).
Optional Parameters
-------------------
x0 : {array, matrix}
Starting guess for the solution.
tol : float
Relative tolerance to achieve before terminating.
restrt : integer
Number of iterations between restarts. Larger values increase
iteration cost, but may be necessary for convergence.
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(rk), where rk is the current residual vector.
Outputs
-------
x : {array, 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
See Also
--------
LinearOperator
Deprecated Parameters
---------------------
xtype : {'f','d','F','D'}
The type of the result. If None, then it will be determined from
A.dtype.char and b. If A does not have a typecode method then it
will compute A.matvec(x0) to get a typecode. To save the extra
computation when A does not have a typecode attribute use xtype=0
for the same type as b or use xtype='f','d','F',or 'D'.
This parameter has been superceeded by LinearOperator.
"""
A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)
n = len(b)
if maxiter is None:
maxiter = n*10
restrt = min(restrt, n)
matvec = A.matvec
psolve = M.matvec
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'gmresrevcom')
stoptest = getattr(_iterative, ltr + 'stoptest2')
resid = tol
ndx1 = 1
ndx2 = -1
work = np.zeros((6+restrt)*n,dtype=x.dtype)
work2 = np.zeros((restrt+1)*(2*restrt+2),dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
bnrm2 = -1.0
iter_ = maxiter
old_ijob = ijob
first_pass = True
resid_ready = False
iter_num = 1
while True:
olditer = iter_
x, iter_, resid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, restrt, work, work2, iter_, resid, info, ndx1, ndx2, ijob)
#if callback is not None and iter_ > olditer:
# callback(x)
slice1 = slice(ndx1-1, ndx1-1+n)
slice2 = slice(ndx2-1, ndx2-1+n)
if (ijob == -1): # gmres success, update last residual
if resid_ready and callback is not None:
callback(resid)
resid_ready = False
break
elif (ijob == 1):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 2):
work[slice1] = psolve(work[slice2])
if not first_pass and old_ijob==3:
resid_ready = True
first_pass = False
elif (ijob == 3):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(work[slice1])
if resid_ready and callback is not None:
callback(resid)
resid_ready = False
iter_num = iter_num+1
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
bnrm2, resid, info = stoptest(work[slice1], b, bnrm2, tol, info)
old_ijob = ijob
ijob = 2
if iter_num > maxiter:
break
if info >= 0 and resid > tol:
#info isn't set appropriately otherwise
info = maxiter
return postprocess(x), info
def minres(A, b, x0=None, shift=0.0, tol=1e-5, maxiter=None, xtype=None,
M=None, callback=None, show=False, check=False):
A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)
matvec = A.matvec
psolve = M.matvec
first = 'Enter minres. '
last = 'Exit minres. '
n = A.shape[0]
if maxiter is None:
maxiter = 5 * n
msg =[' beta2 = 0. If M = I, b and x are eigenvectors ', # -1
' beta1 = 0. The exact solution is x = 0 ', # 0
' A solution to Ax = b was found, given rtol ', # 1
' A least-squares solution was found, given rtol ', # 2
' Reasonable accuracy achieved, given eps ', # 3
' x has converged to an eigenvector ', # 4
' acond has exceeded 0.1/eps ', # 5
' The iteration limit was reached ', # 6
' A does not define a symmetric matrix ', # 7
' M does not define a symmetric matrix ', # 8
' M does not define a pos-def preconditioner '] # 9
if show:
print first + 'Solution of symmetric Ax = b'
print first + 'n = %3g shift = %23.14e' % (n,shift)
print first + 'itnlim = %3g rtol = %11.2e' % (maxiter,tol)
print
istop = 0; itn = 0; Anorm = 0; Acond = 0;
rnorm = 0; ynorm = 0;
xtype = x.dtype
eps = finfo(xtype).eps
x = zeros( n, dtype=xtype )
# Set up y and v for the first Lanczos vector v1.
# y = beta1 P' v1, where P = C**(-1).
# v is really P' v1.
y = b
r1 = b
y = psolve(b)
beta1 = inner(b,y)
if beta1 < 0:
raise ValueError('indefinite preconditioner')
elif beta1 == 0:
return (postprocess(x), 0)
beta1 = sqrt( beta1 )
if check:
# are these too strict?
# see if A is symmetric
w = matvec(y)
r2 = matvec(w)
s = inner(w,w)
t = inner(y,r2)
z = abs( s - t )
epsa = (s + eps) * eps**(1.0/3.0)
if z > epsa:
raise ValueError('non-symmetric matrix')
# see if M is symmetric
r2 = psolve(y)
s = inner(y,y)
t = inner(r1,r2)
z = abs( s - t )
epsa = (s + eps) * eps**(1.0/3.0)
if z > epsa:
raise ValueError('non-symmetric preconditioner')
# Initialize other quantities
oldb = 0; beta = beta1; dbar = 0; epsln = 0;
qrnorm = beta1; phibar = beta1; rhs1 = beta1;
rhs2 = 0; tnorm2 = 0; ynorm2 = 0;
cs = -1; sn = 0;
w = zeros(n, dtype=xtype)
w2 = zeros(n, dtype=xtype)
r2 = r1
if show:
print
print
print ' Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|'
while itn < maxiter:
itn += 1
s = 1.0/beta
v = s*y
y = matvec(v)
y = y - shift * v
if itn >= 2:
y = y - (beta/oldb)*r1
alfa = inner(v,y)
y = y - (alfa/beta)*r2
r1 = r2
r2 = y
y = psolve(r2)
oldb = beta
beta = inner(r2,y)
if beta < 0:
raise ValueError('non-symmetric matrix')
beta = sqrt(beta)
tnorm2 += alfa**2 + oldb**2 + beta**2
if itn == 1:
if beta/beta1 <= 10*eps:
istop = -1 # Terminate later
#tnorm2 = alfa**2 ??
gmax = abs(alfa)
gmin = gmax
# Apply previous rotation Qk-1 to get
# [deltak epslnk+1] = [cs sn][dbark 0 ]
# [gbar k dbar k+1] [sn -cs][alfak betak+1].
oldeps = epsln
delta = cs * dbar + sn * alfa # delta1 = 0 deltak
gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
epsln = sn * beta # epsln2 = 0 epslnk+1
dbar = - cs * beta # dbar 2 = beta2 dbar k+1
root = norm([gbar, dbar])
Arnorm = phibar * root
# Compute the next plane rotation Qk
gamma = norm([gbar, beta]) # gammak
gamma = max(gamma, eps)
cs = gbar / gamma # ck
sn = beta / gamma # sk
phi = cs * phibar # phik
phibar = sn * phibar # phibark+1
# Update x.
denom = 1.0/gamma
w1 = w2
w2 = w
w = (v - oldeps*w1 - delta*w2) * denom
x = x + phi*w
# Go round again.
gmax = max(gmax, gamma)
gmin = min(gmin, gamma)
z = rhs1 / gamma
ynorm2 = z**2 + ynorm2
rhs1 = rhs2 - delta*z
rhs2 = - epsln*z
# Estimate various norms and test for convergence.
Anorm = sqrt( tnorm2 )
ynorm = sqrt( ynorm2 )
epsa = Anorm * eps
epsx = Anorm * ynorm * eps
epsr = Anorm * ynorm * tol
diag = gbar
if diag == 0: diag = epsa
qrnorm = phibar
rnorm = qrnorm
test1 = rnorm / (Anorm*ynorm) # ||r|| / (||A|| ||x||)
test2 = root / Anorm # ||Ar|| / (||A|| ||r||)
# Estimate cond(A).
# In this version we look at the diagonals of R in the
# factorization of the lower Hessenberg matrix, Q * H = R,
# where H is the tridiagonal matrix from Lanczos with one
# extra row, beta(k+1) e_k^T.
Acond = gmax/gmin
# See if any of the stopping criteria are satisfied.
# In rare cases, istop is already -1 from above (Abar = const*I).
if istop == 0:
t1 = 1 + test1 # These tests work if tol < eps
t2 = 1 + test2
if t2 <= 1 : istop = 2
if t1 <= 1 : istop = 1
if itn >= maxiter : istop = 6
if Acond >= 0.1/eps : istop = 4
if epsx >= beta1 : istop = 3
#if rnorm <= epsx : istop = 2
#if rnorm <= epsr : istop = 1
if test2 <= tol : istop = 2
if test1 <= tol : istop = 1
# See if it is time to print something.
prnt = False
if n <= 40 : prnt = True
if itn <= 10 : prnt = True
if itn >= maxiter-10 : prnt = True
if itn % 10 == 0 : prnt = True
if qrnorm <= 10*epsx : prnt = True
if qrnorm <= 10*epsr : prnt = True
if Acond <= 1e-2/eps : prnt = True
if istop != 0 : prnt = True
if show and prnt:
str1 = '%6g %12.5e %10.3e' % (itn, x[0], test1)
str2 = ' %10.3e' % (test2,)
str3 = ' %8.1e %8.1e %8.1e' % (Anorm, Acond, gbar/Anorm)
print str1 + str2 + str3
if itn % 10 == 0: print
if callback is not None:
callback(x)
if istop != 0: break #TODO check this
if show:
print
print last + ' istop = %3g itn =%5g' % (istop,itn)
print last + ' Anorm = %12.4e Acond = %12.4e' % (Anorm,Acond)
print last + ' rnorm = %12.4e ynorm = %12.4e' % (rnorm,ynorm)
print last + ' Arnorm = %12.4e' % (Arnorm,)
print last + msg[istop+1]
if istop == 6:
info = maxiter
else:
info = 0
return (postprocess(x),info)
