(A, b, x0=None, tol=1e-5, criteria='rr',
             maxiter=None, M=None,
             callback=None, residuals=None)
Biconjugate Gradient Algorithm with Stabilization.

    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||)
        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, inverted 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 bicgstab
    >>> 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) = bicgstab(A,b, maxiter=2, tol=1e-8)
    >>> print(f'{norm(b - A*x):.6}')
    4.68163

    References
    ----------
    .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
       Second Edition", SIAM, pp. 231-234, 2003
       http://www-users.cs.umn.edu/~saad/books.html
Biconjugate Gradient Algorithm with Stabilization.
Biconjugate Gradient Algorithm with Stabilization .
def bicgstab(A, b, x0=None, tol=1e-5, criteria='rr',
             maxiter=None, M=None,
             callback=None, residuals=None):
    """Biconjugate Gradient Algorithm with Stabilization.

    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||)
        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, inverted 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 bicgstab
    >>> 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) = bicgstab(A,b, maxiter=2, tol=1e-8)
    >>> print(f'{norm(b - A*x):.6}')
    4.68163

    References
    ----------
    .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
       Second Edition", SIAM, pp. 231-234, 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._bicgstab')

    # Check iteration numbers
    if maxiter is None:
        maxiter = len(x) + 5
    elif maxiter < 1:
        raise ValueError('Number of iterations must be positive')

    # Prep for method
    r = b - A @ x
    normr = norm(r)

    if residuals is not None:
        residuals[:] = [normr]

    # Check initial guess, if b != 0,
    normb = norm(b)
    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)
    else:
        raise ValueError('Invalid stopping criteria.')

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

    # Is thisAa one dimensional matrix?
    # Use a matvec to access A[0,0]
    if A.shape[0] == 1:
        entry = np.ravel(A @ np.array([1.0], dtype=x.dtype))
        return (postprocess(b/entry), 0)

    rstar = r.copy()
    p = r.copy()

    rrstarOld = np.inner(rstar.conjugate(), r)

    it = 0

    # Begin BiCGStab
    while True:
        Mp = M @ p
        AMp = A @ Mp

        # alpha = (r_j, rstar) / (A*M*p_j, rstar)
        alpha = rrstarOld/np.inner(rstar.conjugate(), AMp)

        # s_j = r_j - alpha*A*M*p_j
        s = r - alpha * AMp
        Ms = M @ s
        AMs = A @ Ms

        # omega = (A*M*s_j, s_j)/(A*M*s_j, A*M*s_j)
        omega = np.inner(AMs.conjugate(), s)/np.inner(AMs.conjugate(), AMs)

        # x_{j+1} = x_j +  alpha*M*p_j + omega*M*s_j
        x = x + alpha * Mp + omega * Ms

        # r_{j+1} = s_j - omega*A*M*s
        r = s - omega * AMs

        # beta_j = (r_{j+1}, rstar)/(r_j, rstar) * (alpha/omega)
        rrstarNew = np.inner(rstar.conjugate(), r)
        beta = (rrstarNew / rrstarOld) * (alpha / omega)
        rrstarOld = rrstarNew

        # p_{j+1} = r_{j+1} + beta*(p_j - omega*A*M*p)
        p = r + beta * (p - omega * AMp)

        it += 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)

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

        if it == maxiter:
            return (postprocess(x), it)
def bicgstab ( 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._bicgstab' ) # Check iteration numbers if maxiter is None : maxiter = len ( x ) + 5 elif maxiter < 1 : raise ValueError ( 'Number of iterations must be positive' ) # Prep for method r = b - A @ x normr = norm ( r ) if residuals is not None : residuals [ : ] = [ normr ] # Check initial guess, if b != 0, normb = norm ( b ) 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 ) else : raise ValueError ( 'Invalid stopping criteria.' ) if normr < rtol : return ( postprocess ( x ) , 0 ) # Is thisAa one dimensional matrix? # Use a matvec to access A[0,0] if A . shape [ 0 ] == 1 : entry = np . ravel ( A @ np . array ( [ 1.0 ] , dtype = x . dtype ) ) return ( postprocess ( b / entry ) , 0 ) rstar = r . copy ( ) p = r . copy ( ) rrstarOld = np . inner ( rstar . conjugate ( ) , r ) it = 0 # Begin BiCGStab while True : Mp = M @ p AMp = A @ Mp # alpha = (r_j, rstar) / (A*M*p_j, rstar) alpha = rrstarOld / np . inner ( rstar . conjugate ( ) , AMp ) # s_j = r_j - alpha*A*M*p_j s = r - alpha * AMp Ms = M @ s AMs = A @ Ms # omega = (A*M*s_j, s_j)/(A*M*s_j, A*M*s_j) omega = np . inner ( AMs . conjugate ( ) , s ) / np . inner ( AMs . conjugate ( ) , AMs ) # x_{j+1} = x_j +  alpha*M*p_j + omega*M*s_j x = x + alpha * Mp + omega * Ms # r_{j+1} = s_j - omega*A*M*s r = s - omega * AMs # beta_j = (r_{j+1}, rstar)/(r_j, rstar) * (alpha/omega) rrstarNew = np . inner ( rstar . conjugate ( ) , r ) beta = ( rrstarNew / rrstarOld ) * ( alpha / omega ) rrstarOld = rrstarNew # p_{j+1} = r_{j+1} + beta*(p_j - omega*A*M*p) p = r + beta * ( p - omega * AMp ) it += 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 ) if normr < rtol : return ( postprocess ( x ) , 0 ) if it == maxiter : return ( postprocess ( x ) , it )