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Module « scipy.sparse.linalg »

Fonction cg - module scipy.sparse.linalg

Signature de la fonction cg

def cg(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M=None, callback=None) 

Description

help(scipy.sparse.linalg.cg)

Use Conjugate Gradient iteration to solve ``Ax = b``.

Parameters
----------
A : {sparse array, ndarray, LinearOperator}
    The real or complex N-by-N matrix of the linear system.
    `A` must represent a hermitian, positive definite matrix.
    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 ``atol=0.`` and ``rtol=1e-5``.
maxiter : integer
    Maximum number of iterations.  Iteration will stop after maxiter
    steps even if the specified tolerance has not been achieved.
M : {sparse array, ndarray, LinearOperator}
    Preconditioner for `A`. `M` must represent a hermitian, positive definite
    matrix. It should approximate the inverse of `A` (see Notes).
    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.

Returns
-------
x : ndarray
    The converged solution.
info : integer
    Provides convergence information:
        0  : successful exit
        >0 : convergence to tolerance not achieved, number of iterations

Notes
-----
The preconditioner `M` should be a matrix such that ``M @ A`` has a smaller
condition number than `A`, see [2]_.

References
----------
.. [1] "Conjugate Gradient Method, Wikipedia, 
       https://en.wikipedia.org/wiki/Conjugate_gradient_method
.. [2] "Preconditioner", 
       Wikipedia, https://en.wikipedia.org/wiki/Preconditioner

Examples
--------
>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import cg
>>> P = np.array([[4, 0, 1, 0],
...               [0, 5, 0, 0],
...               [1, 0, 3, 2],
...               [0, 0, 2, 4]])
>>> A = csc_array(P)
>>> b = np.array([-1, -0.5, -1, 2])
>>> x, exit_code = cg(A, b, atol=1e-5)
>>> print(exit_code)    # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True

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