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"""Iterative methods for solving linear systems"""
from __future__ import division, print_function, absolute_import
__all__ = ['bicg','bicgstab','cg','cgs','gmres','qmr']
import warnings
import numpy as np
from . import _iterative
from scipy.sparse.linalg.interface import LinearOperator
from .utils import make_system
from scipy._lib._util import _aligned_zeros
from scipy._lib._threadsafety import non_reentrant
_type_conv = {'f':'s', 'd':'d', 'F':'c', 'D':'z'}
# Part of the docstring common to all iterative solvers
common_doc1 = \
"""
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}"""
common_doc2 = \
"""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, atol : float, optional
Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.
The default for ``atol`` is ``'legacy'``, which emulates
a different legacy behavior.
.. warning::
The default value for `atol` will be changed in a future release.
For future compatibility, specify `atol` explicitly.
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.
"""
def _stoptest(residual, atol):
"""
Successful termination condition for the solvers.
"""
resid = np.linalg.norm(residual)
if resid <= atol:
return resid, 1
else:
return resid, 0
def _get_atol(tol, atol, bnrm2, get_residual, routine_name):
"""
Parse arguments for absolute tolerance in termination condition.
Parameters
----------
tol, atol : object
The arguments passed into the solver routine by user.
bnrm2 : float
2-norm of the rhs vector.
get_residual : callable
Callable ``get_residual()`` that returns the initial value of
the residual.
routine_name : str
Name of the routine.
"""
if atol is None:
warnings.warn("scipy.sparse.linalg.{name} called without specifying `atol`. "
"The default value will be changed in a future release. "
"For compatibility, specify a value for `atol` explicitly, e.g., "
"``{name}(..., atol=0)``, or to retain the old behavior "
"``{name}(..., atol='legacy')``".format(name=routine_name),
category=DeprecationWarning, stacklevel=4)
atol = 'legacy'
tol = float(tol)
if atol == 'legacy':
# emulate old legacy behavior
resid = get_residual()
if resid <= tol:
return 'exit'
if bnrm2 == 0:
return tol
else:
return tol * float(bnrm2)
else:
return max(float(atol), tol * float(bnrm2))
def set_docstring(header, Ainfo, footer='', atol_default='0'):
def combine(fn):
fn.__doc__ = '\n'.join((header, common_doc1,
' ' + Ainfo.replace('\n', '\n '),
common_doc2, footer))
return fn
return combine
@set_docstring('Use BIConjugate Gradient iteration to solve ``Ax = b``.',
'The real or complex N-by-N matrix of the linear system.\n'
'It is required that the linear operator can produce\n'
'``Ax`` and ``A^T x``.')
@non_reentrant()
def bicg(A, b, x0=None, tol=1e-5, maxiter=None, M=None, callback=None, atol=None):
A,M,x,b,postprocess = make_system(A, M, x0, b)
n = len(b)
if maxiter is None:
maxiter = n*10
matvec, rmatvec = A.matvec, A.rmatvec
psolve, rpsolve = M.matvec, M.rmatvec
ltr = _type_conv[x.dtype.char]
revcom = getattr(_iterative, ltr + 'bicgrevcom')
get_residual = lambda: np.linalg.norm(matvec(x) - b)
atol = _get_atol(tol, atol, np.linalg.norm(b), get_residual, 'bicg')
if atol == 'exit':
return postprocess(x), 0
resid = atol
ndx1 = 1
ndx2 = -1
# Use _aligned_zeros to work around a f2py bug in Numpy 1.9.1
work = _aligned_zeros(6*n,dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
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*matvec(work[slice1])
elif (ijob == 2):
work[slice2] *= sclr2
work[slice2] += sclr1*rmatvec(work[slice1])
elif (ijob == 3):
work[slice1] = psolve(work[slice2])
elif (ijob == 4):
work[slice1] = rpsolve(work[slice2])
elif (ijob == 5):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 6):
if ftflag:
info = -1
ftflag = False
resid, info = _stoptest(work[slice1], atol)
ijob = 2
if info > 0 and iter_ == maxiter and not (resid <= atol):
# info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
@set_docstring('Use BIConjugate Gradient STABilized iteration to solve '
'``Ax = b``.',
'The real or complex N-by-N matrix of the linear system.')
@non_reentrant()
def bicgstab(A, b, x0=None, tol=1e-5, maxiter=None, M=None, callback=None, atol=None):
A, M, x, b, postprocess = make_system(A, M, x0, b)
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')
get_residual = lambda: np.linalg.norm(matvec(x) - b)
atol = _get_atol(tol, atol, np.linalg.norm(b), get_residual, 'bicgstab')
if atol == 'exit':
return postprocess(x), 0
resid = atol
ndx1 = 1
ndx2 = -1
# Use _aligned_zeros to work around a f2py bug in Numpy 1.9.1
work = _aligned_zeros(7*n,dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
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*matvec(work[slice1])
elif (ijob == 2):
work[slice1] = psolve(work[slice2])
elif (ijob == 3):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
resid, info = _stoptest(work[slice1], atol)
ijob = 2
if info > 0 and iter_ == maxiter and not (resid <= atol):
# info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
@set_docstring('Use Conjugate Gradient iteration to solve ``Ax = b``.',
'The real or complex N-by-N matrix of the linear system.\n'
'``A`` must represent a hermitian, positive definite matrix.')
@non_reentrant()
def cg(A, b, x0=None, tol=1e-5, maxiter=None, M=None, callback=None, atol=None):
A, M, x, b, postprocess = make_system(A, M, x0, b)
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 + 'cgrevcom')
get_residual = lambda: np.linalg.norm(matvec(x) - b)
atol = _get_atol(tol, atol, np.linalg.norm(b), get_residual, 'cg')
if atol == 'exit':
return postprocess(x), 0
resid = atol
ndx1 = 1
ndx2 = -1
# Use _aligned_zeros to work around a f2py bug in Numpy 1.9.1
work = _aligned_zeros(4*n,dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
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*matvec(work[slice1])
elif (ijob == 2):
work[slice1] = psolve(work[slice2])
elif (ijob == 3):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
resid, info = _stoptest(work[slice1], atol)
if info == 1 and iter_ > 1:
# recompute residual and recheck, to avoid
# accumulating rounding error
work[slice1] = b - matvec(x)
resid, info = _stoptest(work[slice1], atol)
ijob = 2
if info > 0 and iter_ == maxiter and not (resid <= atol):
# info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
@set_docstring('Use Conjugate Gradient Squared iteration to solve ``Ax = b``.',
'The real-valued N-by-N matrix of the linear system.')
@non_reentrant()
def cgs(A, b, x0=None, tol=1e-5, maxiter=None, M=None, callback=None, atol=None):
A, M, x, b, postprocess = make_system(A, M, x0, b)
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 + 'cgsrevcom')
get_residual = lambda: np.linalg.norm(matvec(x) - b)
atol = _get_atol(tol, atol, np.linalg.norm(b), get_residual, 'cgs')
if atol == 'exit':
return postprocess(x), 0
resid = atol
ndx1 = 1
ndx2 = -1
# Use _aligned_zeros to work around a f2py bug in Numpy 1.9.1
work = _aligned_zeros(7*n,dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
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*matvec(work[slice1])
elif (ijob == 2):
work[slice1] = psolve(work[slice2])
elif (ijob == 3):
work[slice2] *= sclr2
work[slice2] += sclr1*matvec(x)
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
resid, info = _stoptest(work[slice1], atol)
if info == 1 and iter_ > 1:
# recompute residual and recheck, to avoid
# accumulating rounding error
work[slice1] = b - matvec(x)
resid, info = _stoptest(work[slice1], atol)
ijob = 2
if info == -10:
# termination due to breakdown: check for convergence
resid, ok = _stoptest(b - matvec(x), atol)
if ok:
info = 0
if info > 0 and iter_ == maxiter and not (resid <= atol):
# info isn't set appropriately otherwise
info = iter_
return postprocess(x), info
@non_reentrant()
def gmres(A, b, x0=None, tol=1e-5, restart=None, maxiter=None, M=None, callback=None,
restrt=None, atol=None):
"""
Use Generalized Minimal RESidual iteration to solve ``Ax = b``.
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).
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, atol : float, optional
Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.
The default for ``atol`` is ``'legacy'``, which emulates
a different legacy behavior.
.. warning::
The default value for `atol` will be changed in a future release.
For future compatibility, specify `atol` explicitly.
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 (restart cycles). 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.
restrt : int, optional
DEPRECATED - use `restart` instead.
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)
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import gmres
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = gmres(A, b)
>>> print(exitCode) # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True
"""
# 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)
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')
bnrm2 = np.linalg.norm(b)
Mb_nrm2 = np.linalg.norm(psolve(b))
get_residual = lambda: np.linalg.norm(matvec(x) - b)
atol = _get_atol(tol, atol, bnrm2, get_residual, 'gmres')
if atol == 'exit':
return postprocess(x), 0
if bnrm2 == 0:
return postprocess(b), 0
# Tolerance passed to GMRESREVCOM applies to the inner iteration
# and deals with the left-preconditioned residual.
ptol_max_factor = 1.0
ptol = Mb_nrm2 * min(ptol_max_factor, atol / bnrm2)
resid = np.nan
presid = np.nan
ndx1 = 1
ndx2 = -1
# Use _aligned_zeros to work around a f2py bug in Numpy 1.9.1
work = _aligned_zeros((6+restrt)*n,dtype=x.dtype)
work2 = _aligned_zeros((restrt+1)*(2*restrt+2),dtype=x.dtype)
ijob = 1
info = 0
ftflag = True
iter_ = maxiter
old_ijob = ijob
first_pass = True
resid_ready = False
iter_num = 1
while True:
x, iter_, presid, info, ndx1, ndx2, sclr1, sclr2, ijob = \
revcom(b, x, restrt, work, work2, iter_, presid, info, ndx1, ndx2, ijob, ptol)
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(presid / bnrm2)
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(presid / bnrm2)
resid_ready = False
iter_num = iter_num+1
elif (ijob == 4):
if ftflag:
info = -1
ftflag = False
resid, info = _stoptest(work[slice1], atol)
# Inner loop tolerance control
if info or presid > ptol:
ptol_max_factor = min(1.0, 1.5 * ptol_max_factor)
else:
# Inner loop tolerance OK, but outer loop not.
ptol_max_factor = max(1e-16, 0.25 * ptol_max_factor)
if resid != 0:
ptol = presid * min(ptol_max_factor, atol / resid)
else:
ptol = presid * ptol_max_factor
old_ijob = ijob
ijob = 2
if iter_num > maxiter:
info = maxiter
break
if info >= 0 and not (resid <= atol):
# info isn't set appropriately otherwise
info = maxiter
return postprocess(x), info
@non_reentrant()
def qmr(A, b, x0=None, tol=1e-5, maxiter=None, M1=None, M2=None, callback=None,
atol=None):
"""Use Quasi-Minimal Residual iteration to solve ``Ax = b``.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The real-valued N-by-N matrix of the linear system.
It is required that the linear operator can produce
``Ax`` and ``A^T x``.
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, atol : float, optional
Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.
The default for ``atol`` is ``'legacy'``, which emulates
a different legacy behavior.
.. warning::
The default value for `atol` will be changed in a future release.
For future compatibility, specify `atol` explicitly.
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.
See Also
--------
LinearOperator
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import qmr
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = qmr(A, b)
>>> print(exitCode) # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True
"""
A_ = A
A, M, x, b, postprocess = make_system(A, None, x0, b)
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')
get_residual = lambda: np.linalg.norm(A.matvec(x) - b)
atol = _get_atol(tol, atol, np.linalg.norm(b), get_residual, 'qmr')
if atol == 'exit':
return postprocess(x), 0
resid = atol
ndx1 = 1
ndx2 = -1
# Use _aligned_zeros to work around a f2py bug in Numpy 1.9.1
work = _aligned_zeros(11*n,x.dtype)
ijob = 1
info = 0
ftflag = True
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
resid, info = _stoptest(work[slice1], atol)
ijob = 2
if info > 0 and iter_ == maxiter and not (resid <= atol):
# info isn't set appropriately otherwise
info = iter_
return postprocess(x), info