(A, b, x0=None, tol=1e-5, criteria='rr',
       maxiter=None, M=None,
       callback=None, residuals=None)
Conjugate Gradient algorithm.

    Solves the linear system Ax = b. Left preconditioning is supported.

    Parameters
    ----------
    A : array, matrix, sparse matrix, LinearOperator
        n x n, linear system to solve
    b : array, matrix
        right hand side, shape is (n,) or (n,1)
    x0 : array, matrix
        initial guess, default is a vector of zeros
    tol : float
        Tolerance for stopping criteria
    criteria : string
        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
        n x n, inverse preconditioner, 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
    -------
    (xk, info)
    xk : an updated guess after k iterations to the solution of Ax = b
    info : halting status

            ==  =======================================
            0   successful exit
            >0  convergence to tolerance not achieved,
                return iteration count instead.
            <0  numerical breakdown, or illegal input
            ==  =======================================

    Notes
    -----
    The LinearOperator class is in scipy.sparse.linalg.interface.
    Use this class if you prefer to define A or M as a mat-vec routine
    as opposed to explicitly constructing the matrix.

    Examples
    --------
    >>> from pyamg.krylov import cg
    >>> 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) = cg(A,b, maxiter=2, tol=1e-8)
    >>> print(f'{norm(b - A*x):.6}')
    10.9371

    References
    ----------
    .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
       Second Edition", SIAM, pp. 262-67, 2003
       http://www-users.cs.umn.edu/~saad/books.html
Conjugate Gradient algorithm.
Conjugate Gradient algorithm .
def cg(A, b, x0=None, tol=1e-5, criteria='rr',
       maxiter=None, M=None,
       callback=None, residuals=None):
    """Conjugate Gradient algorithm.

    Solves the linear system Ax = b. Left preconditioning is supported.

    Parameters
    ----------
    A : array, matrix, sparse matrix, LinearOperator
        n x n, linear system to solve
    b : array, matrix
        right hand side, shape is (n,) or (n,1)
    x0 : array, matrix
        initial guess, default is a vector of zeros
    tol : float
        Tolerance for stopping criteria
    criteria : string
        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
        n x n, inverse preconditioner, 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
    -------
    (xk, info)
    xk : an updated guess after k iterations to the solution of Ax = b
    info : halting status

            ==  =======================================
            0   successful exit
            >0  convergence to tolerance not achieved,
                return iteration count instead.
            <0  numerical breakdown, or illegal input
            ==  =======================================

    Notes
    -----
    The LinearOperator class is in scipy.sparse.linalg.interface.
    Use this class if you prefer to define A or M as a mat-vec routine
    as opposed to explicitly constructing the matrix.

    Examples
    --------
    >>> from pyamg.krylov import cg
    >>> 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) = cg(A,b, maxiter=2, tol=1e-8)
    >>> print(f'{norm(b - A*x):.6}')
    10.9371

    References
    ----------
    .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
       Second Edition", SIAM, pp. 262-67, 2003
       http://www-users.cs.umn.edu/~saad/books.html

    """
    # Convert inputs to linear system, with error checking
    A, M, x, b, postprocess = make_system(A, M, x0, b)

    # Ensure that warnings are always reissued from this function
    warnings.filterwarnings('always', module='pyamg.krylov._cg')

    # determine maxiter
    if maxiter is None:
        maxiter = int(1.3*len(b)) + 2
    elif maxiter < 1:
        raise ValueError('Number of iterations must be positive')

    # setup method
    r = b - A @ x
    z = M @ r
    p = z.copy()
    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.')

    if normr < rtol:
        return (postprocess(x), 0)

    # How often should r be recomputed
    recompute_r = 8

    it = 0

    while True:                                   # Step number in Saad's pseudocode
        Ap = A @ p

        rz_old = rz
        pAp = np.inner(Ap.conjugate(), p)         # check curvature of A
        if pAp < 0.0:
            warn('\nIndefinite matrix detected in CG, aborting\n')
            return (postprocess(x), -1)

        alpha = rz/pAp                            # 3
        x += alpha * p                            # 4

        if np.mod(it, recompute_r) and it > 0:    # 5
            r -= alpha * Ap
        else:
            r = b - A @ x

        z = M @ r                                 # 6
        rz = np.inner(r.conjugate(), z)

        if rz < 0.0:                             # check curvature of M
            warn('\nIndefinite preconditioner detected in CG, aborting\n')
            return (postprocess(x), -1)

        beta = rz/rz_old                          # 7
        p *= beta                                 # 8
        p += z

        it += 1

        normr = np.linalg.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 (postprocess(x), 0)

        if it == maxiter:
            return (postprocess(x), it)
def cg ( A , b , x0 = None , tol = 1e-5 , criteria = 'rr' , maxiter = None , M = None , callback = None , residuals = None ) : # Convert inputs to linear system, with error checking A , M , x , b , postprocess = make_system ( A , M , x0 , b ) # Ensure that warnings are always reissued from this function warnings . filterwarnings ( 'always' , module = 'pyamg.krylov._cg' ) # determine maxiter if maxiter is None : maxiter = int ( 1.3 * len ( b ) ) + 2 elif maxiter < 1 : raise ValueError ( 'Number of iterations must be positive' ) # setup method r = b - A @ x z = M @ r p = z . copy ( ) 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.' ) if normr < rtol : return ( postprocess ( x ) , 0 ) # How often should r be recomputed recompute_r = 8 it = 0 while True : # Step number in Saad's pseudocode Ap = A @ p rz_old = rz pAp = np . inner ( Ap . conjugate ( ) , p ) # check curvature of A if pAp < 0.0 : warn ( '\nIndefinite matrix detected in CG, aborting\n' ) return ( postprocess ( x ) , - 1 ) alpha = rz / pAp # 3 x += alpha * p # 4 if np . mod ( it , recompute_r ) and it > 0 : # 5 r -= alpha * Ap else : r = b - A @ x z = M @ r # 6 rz = np . inner ( r . conjugate ( ) , z ) if rz < 0.0 : # check curvature of M warn ( '\nIndefinite preconditioner detected in CG, aborting\n' ) return ( postprocess ( x ) , - 1 ) beta = rz / rz_old # 7 p *= beta # 8 p += z it += 1 normr = np . linalg . 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 ( postprocess ( x ) , 0 ) if it == maxiter : return ( postprocess ( x ) , it )