"""Steepest Descent projection method."""
import warnings
from warnings import warn
import numpy as np
from scipy import sparse
from ..util.linalg import norm
from ..util import make_system
[docs]
def steepest_descent(A, b, x0=None, tol=1e-5, criteria='rr',
maxiter=None, M=None,
callback=None, residuals=None):
"""Steepest descent algorithm.
Solves the linear system Ax = b. Left preconditioning is supported.
Parameters
----------
A : array, matrix, sparse matrix, LinearOperator
Linear system of size (n,n) to solve.
b : array, matrix
Right hand side of size (n,) or (n,1).
x0 : array, matrix
Initial guess, default is a vector of zeros.
tol : float
Tolerance for stopping criteria.
criteria : str
Stopping criteria, let r=r_k, x=x_k
====== =========================================
'rr' ||r|| < tol ||b||
'rr+' ||r|| < tol (||b|| + ||A||_F ||x||)
'MrMr' ||M r|| < tol ||M b||
'rMr' <r, Mr>^1/2 < tol
====== =========================================
if ||b||=0, then set ||b||=1 for these tests.
maxiter : int
Maximum number of iterations allowed.
M : array, matrix, sparse matrix, LinearOperator
Inverted preconditioner of size (n,n), i.e. solve M A x = M b.
callback : function
User-supplied function is called after each iteration as
``callback(xk)``, where xk is the current solution vector.
residuals : list
Residual history in the 2-norm, including the initial residual.
Returns
-------
array
Updated guess after k iterations to the solution of Ax = b.
int
Halting status of cg
== =======================================
0 successful exit
>0 convergence to tolerance not achieved,
return iteration count instead.
<0 numerical breakdown, or illegal input
== =======================================
See Also
--------
_minimal_residual
Notes
-----
The LinearOperator class is in scipy.sparse.linalg.
Use this class if you prefer to define A or M as a mat-vec routine
as opposed to explicitly constructing the matrix.
References
----------
.. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
Second Edition", SIAM, pp. 137--142, 2003
http://www-users.cs.umn.edu/~saad/books.html
Examples
--------
>>> from pyamg.krylov import steepest_descent
>>> from pyamg.util.linalg import norm
>>> import numpy as np
>>> from pyamg.gallery import poisson
>>> A = poisson((10,10))
>>> b = np.ones((A.shape[0],))
>>> (x,flag) = steepest_descent(A,b, maxiter=2, tol=1e-8)
>>> print(f'{norm(b - A@x):.6}')
7.89436
"""
A, M, x, b = make_system(A, M, x0, b)
# Ensure that warnings are always reissued from this function
warnings.filterwarnings('always', module='pyamg.krylov._steepest_descent')
# determine maxiter
if maxiter is None:
maxiter = len(b)
elif maxiter < 1:
raise ValueError('Number of iterations must be positive')
# setup method
r = b - A @ x
z = M @ r
rz = np.inner(r.conjugate(), z)
normr = np.linalg.norm(r)
if residuals is not None:
residuals[:] = [normr] # initial residual
# Check initial guess if b != 0,
normb = norm(b)
if normb == 0.0:
normb = 1.0 # reset so that tol is unscaled
# set the stopping criteria (see the docstring)
if criteria == 'rr':
rtol = tol * normb
elif criteria == 'rr+':
if sparse.issparse(A.A):
normA = norm(A.A.data)
elif isinstance(A.A, np.ndarray):
normA = norm(np.ravel(A.A))
else:
raise ValueError('Unable to use ||A||_F with the current matrix format.')
rtol = tol * (normA * np.linalg.norm(x) + normb)
elif criteria == 'MrMr':
normr = norm(z)
normMb = norm(M @ b)
rtol = tol * normMb
elif criteria == 'rMr':
normr = np.sqrt(rz)
rtol = tol
else:
raise ValueError('Invalid stopping criteria.')
# How often should r be recomputed
recompute_r = 50
it = 0
while True:
q = A @ z
zAz = np.inner(z.conjugate(), q) # check curvature of A
if zAz < 0.0:
warn('\nIndefinite matrix detected in steepest descent, aborting\n')
return (x, -1)
alpha = rz / zAz # step size
x = x + alpha * z
it += 1
if np.mod(it, recompute_r) and it > 0:
r = b - A @ x
else:
r = r - alpha * q
z = M @ r
rz = np.inner(r.conjugate(), z)
if rz < 0.0: # check curvature of M
warn('\nIndefinite preconditioner detected in steepest descent, stopping.\n')
return (x, -1)
normr = norm(r)
if residuals is not None:
residuals.append(normr)
if callback is not None:
callback(x)
# set the stopping criteria (see the docstring)
if criteria == 'rr':
rtol = tol * normb
elif criteria == 'rr+':
rtol = tol * (normA * np.linalg.norm(x) + normb)
elif criteria == 'MrMr':
normr = norm(z)
rtol = tol * normMb
elif criteria == 'rMr':
normr = np.sqrt(rz)
rtol = tol
if normr < rtol:
return (x, 0)
if rz == 0.0:
# important to test after testing normr < tol. rz == 0.0 is an
# indicator of convergence when r = 0.0
warn('\nSingular preconditioner detected in steepest descent, stopping.\n')
return (x, -1)
if it == maxiter:
return (x, it)
