# Copyright (C) 2015, Pauli Virtanen <pav@iki.fi>
# Distributed under the same license as SciPy.
import numpy as np
from numpy.linalg import LinAlgError
from scipy.linalg import (get_blas_funcs, qr, solve, svd, qr_insert, lstsq)
from .iterative import _get_atol_rtol
from scipy.sparse.linalg._isolve.utils import make_system
__all__ = ['gcrotmk']
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
cs : 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:
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
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
[文档]
def gcrotmk(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=1000, M=None, callback=None,
m=20, k=None, CU=None, discard_C=False, truncate='oldest'):
"""
Solve a matrix equation using flexible GCROT(m,k) algorithm.
Parameters
----------
A : {sparse matrix, ndarray, LinearOperator}
The real or complex N-by-N matrix of the linear system.
Alternatively, ``A`` can be a linear operator which can
produce ``Ax`` using, e.g.,
``scipy.sparse.linalg.LinearOperator``.
b : ndarray
Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : ndarray
Starting guess for the solution.
rtol, atol : float, optional
Parameters for the convergence test. For convergence,
``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied.
The default is ``rtol=1e-5``, the default for ``atol`` is ``0.0``.
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, ndarray, LinearOperator}, optional
Preconditioner for A. The preconditioner should approximate the
inverse of A. gcrotmk is a 'flexible' algorithm and the preconditioner
can vary from iteration to iteration. 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.
m : int, optional
Number of inner FGMRES iterations per each outer iteration.
Default: 20
k : int, optional
Number of vectors to carry between inner FGMRES iterations.
According to [2]_, good values are around m.
Default: m
CU : list of tuples, optional
List of tuples ``(c, u)`` which contain the columns of the matrices
C and U in the GCROT(m,k) algorithm. For details, see [2]_.
The list given and vectors contained in it are modified in-place.
If not given, start from empty matrices. The ``c`` elements in the
tuples can be ``None``, in which case the vectors are recomputed
via ``c = A u`` on start and orthogonalized as described in [3]_.
discard_C : bool, optional
Discard the C-vectors at the end. Useful if recycling Krylov subspaces
for different linear systems.
truncate : {'oldest', 'smallest'}, optional
Truncation scheme to use. Drop: oldest vectors, or vectors with
smallest singular values using the scheme discussed in [1,2].
See [2]_ for detailed comparison.
Default: 'oldest'
Returns
-------
x : ndarray
The solution found.
info : int
Provides convergence information:
* 0 : successful exit
* >0 : convergence to tolerance not achieved, number of iterations
Examples
--------
>>> import numpy as np
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import gcrotmk
>>> R = np.random.randn(5, 5)
>>> A = csc_matrix(R)
>>> b = np.random.randn(5)
>>> x, exit_code = gcrotmk(A, b, atol=1e-5)
>>> print(exit_code)
0
>>> np.allclose(A.dot(x), b)
True
References
----------
.. [1] E. de Sturler, ''Truncation strategies for optimal Krylov subspace
methods'', SIAM J. Numer. Anal. 36, 864 (1999).
.. [2] J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant
of GCROT for solving nonsymmetric linear systems'',
SIAM J. Sci. Comput. 32, 172 (2010).
.. [3] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti,
''Recycling Krylov subspaces for sequences of linear systems'',
SIAM J. Sci. Comput. 28, 1651 (2006).
"""
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")
if truncate not in ('oldest', 'smallest'):
raise ValueError(f"Invalid value for 'truncate': {truncate!r}")
matvec = A.matvec
psolve = M.matvec
if CU is None:
CU = []
if k is None:
k = m
axpy, dot, scal = None, None, None
if x0 is None:
r = b.copy()
else:
r = b - matvec(x)
axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (x, r))
b_norm = nrm2(b)
# we call this to get the right atol/rtol and raise errors as necessary
atol, rtol = _get_atol_rtol('gcrotmk', b_norm, atol, rtol)
if b_norm == 0:
x = b
return (postprocess(x), 0)
if discard_C:
CU[:] = [(None, u) for c, u in CU]
# Reorthogonalize old vectors
if CU:
# Sort already existing vectors to the front
CU.sort(key=lambda cu: cu[0] is not None)
# Fill-in missing ones
C = np.empty((A.shape[0], len(CU)), dtype=r.dtype, order='F')
us = []
j = 0
while CU:
# More memory-efficient: throw away old vectors as we go
c, u = CU.pop(0)
if c is None:
c = matvec(u)
C[:,j] = c
j += 1
us.append(u)
# Orthogonalize
Q, R, P = qr(C, overwrite_a=True, mode='economic', pivoting=True)
del C
# C := Q
cs = list(Q.T)
# U := U P R^-1, back-substitution
new_us = []
for j in range(len(cs)):
u = us[P[j]]
for i in range(j):
u = axpy(us[P[i]], u, u.shape[0], -R[i,j])
if abs(R[j,j]) < 1e-12 * abs(R[0,0]):
# discard rest of the vectors
break
u = scal(1.0/R[j,j], u)
new_us.append(u)
# Form the new CU lists
CU[:] = list(zip(cs, new_us))[::-1]
if CU:
axpy, dot = get_blas_funcs(['axpy', 'dot'], (r,))
# Solve first the projection operation with respect to the CU
# vectors. This corresponds to modifying the initial guess to
# be
#
# x' = x + U y
# y = argmin_y || b - A (x + U y) ||^2
#
# The solution is y = C^H (b - A x)
for c, u in CU:
yc = dot(c, r)
x = axpy(u, x, x.shape[0], yc)
r = axpy(c, r, r.shape[0], -yc)
# GCROT main iteration
for j_outer in range(maxiter):
# -- callback
if callback is not None:
callback(x)
beta = nrm2(r)
# -- check stopping condition
beta_tol = max(atol, rtol * b_norm)
if beta <= beta_tol and (j_outer > 0 or CU):
# recompute residual to avoid rounding error
r = b - matvec(x)
beta = nrm2(r)
if beta <= beta_tol:
j_outer = -1
break
ml = m + max(k - len(CU), 0)
cs = [c for c, u in CU]
try:
Q, R, B, vs, zs, y, pres = _fgmres(matvec,
r/beta,
ml,
rpsolve=psolve,
atol=max(atol, rtol*b_norm)/beta,
cs=cs)
y *= beta
except LinAlgError:
# Floating point over/underflow, non-finite result from
# matmul etc. -- report failure.
break
#
# At this point,
#
# [A U, A Z] = [C, V] G; G = [ I B ]
# [ 0 H ]
#
# where [C, V] has orthonormal columns, and r = beta v_0. Moreover,
#
# || b - A (x + Z y + U q) ||_2 = || r - C B y - V H y - C q ||_2 = min!
#
# from which y = argmin_y || beta e_1 - H y ||_2, and q = -B y
#
#
# GCROT(m,k) update
#
# Define new outer vectors
# ux := (Z - U B) y
ux = zs[0]*y[0]
for z, yc in zip(zs[1:], y[1:]):
ux = axpy(z, ux, ux.shape[0], yc) # ux += z*yc
by = B.dot(y)
for cu, byc in zip(CU, by):
c, u = cu
ux = axpy(u, ux, ux.shape[0], -byc) # ux -= u*byc
# cx := V H y
hy = Q.dot(R.dot(y))
cx = vs[0] * hy[0]
for v, hyc in zip(vs[1:], hy[1:]):
cx = axpy(v, cx, cx.shape[0], hyc) # cx += v*hyc
# Normalize cx, maintaining cx = A ux
# This new cx is orthogonal to the previous C, by construction
try:
alpha = 1/nrm2(cx)
if not np.isfinite(alpha):
raise FloatingPointError()
except (FloatingPointError, ZeroDivisionError):
# Cannot update, so skip it
continue
cx = scal(alpha, cx)
ux = scal(alpha, ux)
# Update residual and solution
gamma = dot(cx, r)
r = axpy(cx, r, r.shape[0], -gamma) # r -= gamma*cx
x = axpy(ux, x, x.shape[0], gamma) # x += gamma*ux
# Truncate CU
if truncate == 'oldest':
while len(CU) >= k and CU:
del CU[0]
elif truncate == 'smallest':
if len(CU) >= k and CU:
# cf. [1,2]
D = solve(R[:-1,:].T, B.T).T
W, sigma, V = svd(D)
# C := C W[:,:k-1], U := U W[:,:k-1]
new_CU = []
for j, w in enumerate(W[:,:k-1].T):
c, u = CU[0]
c = c * w[0]
u = u * w[0]
for cup, wp in zip(CU[1:], w[1:]):
cp, up = cup
c = axpy(cp, c, c.shape[0], wp)
u = axpy(up, u, u.shape[0], wp)
# Reorthogonalize at the same time; not necessary
# in exact arithmetic, but floating point error
# tends to accumulate here
for cp, up in new_CU:
alpha = dot(cp, c)
c = axpy(cp, c, c.shape[0], -alpha)
u = axpy(up, u, u.shape[0], -alpha)
alpha = nrm2(c)
c = scal(1.0/alpha, c)
u = scal(1.0/alpha, u)
new_CU.append((c, u))
CU[:] = new_CU
# Add new vector to CU
CU.append((cx, ux))
# Include the solution vector to the span
CU.append((None, x.copy()))
if discard_C:
CU[:] = [(None, uz) for cz, uz in CU]
return postprocess(x), j_outer + 1
