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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def cr(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M=None, callback=None, residuals=None):
 """Conjugate Residual algorithm. Solves the linear system Ax = b. Left preconditioning is supported. The matrix A must be Hermitian symmetric (but not necessarily definite). 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 relative convergence tolerance, i.e. tol is scaled by the preconditioner norm of r_0, or ||r_0||_M. maxiter : int maximum number of allowed iterations xtype : type dtype for the solution, default is automatic type detection 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 residuals contains the residual norm history, including the initial residual. The preconditioner norm is used, instead of the Euclidean norm. Returns ------- (xNew, info) xNew : an updated guess to the solution of Ax = b info : halting status of cr == ======================================= 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. A.psolve(..) is still supported as a legacy. The 2-norm of the preconditioned residual is used both for halting and returned in the residuals list. Examples -------- 10.9370700187 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 """

    A, M, x, b, postprocess = make_system(A, M, x0, b)
    # n = len(b)
    # Ensure that warnings are always reissued from this function
    import warnings
    warnings.filterwarnings('always', module='pyamg\.krylov\._cr')

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

    # choose tolerance for numerically zero values
    # t = A.dtype.char
    # eps = np.finfo(np.float).eps
    # feps = np.finfo(np.single).eps
    # geps = np.finfo(np.longfloat).eps
    # _array_precision = {'f': 0, 'd': 1, 'g': 2, 'F': 0, 'D': 1, 'G': 2}
    # numerically_zero = {0: feps*1e3, 1: eps*1e6,
    #                     2: geps*1e6}[_array_precision[t]]

    # setup method
    r = b - A*x
    z = M*r
    p = z.copy()
    zz = np.inner(z.conjugate(), z)

    # use preconditioner norm
    normr = np.sqrt(zz)

    if residuals is not None:
        residuals[:] = [normr]  # initial residual

    # Check initial guess ( scaling by b, if b != 0,
    #   must account for case when norm(b) is very small)
    normb = norm(b)
    if normb == 0.0:
        normb = 1.0
    if normr < tol*normb:
        return (postprocess(x), 0)

    # Scale tol by ||r_0||_M
    if normr != 0.0:
        tol = tol*normr

    # How often should r be recomputed
    recompute_r = 8

    iter = 0

    Az = A*z
    rAz = np.inner(r.conjugate(), Az)
    Ap = A*p

    while True:

        rAz_old = rAz

        alpha = rAz / np.inner(Ap.conjugate(), Ap)       # 3
        x += alpha * p                           # 4

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

        z = M*r

        Az = A*z
        rAz = np.inner(r.conjugate(), Az)

        beta = rAz/rAz_old                        # 6

        p *= beta                               # 7
        p += z

        Ap *= beta                               # 8
        Ap += Az

        iter += 1

        zz = np.inner(z.conjugate(), z)
        normr = np.sqrt(zz)                          # use preconditioner norm

        if residuals is not None:
            residuals.append(normr)

        if callback is not None:
            callback(x)

        if normr < tol:
            return (postprocess(x), 0)
        elif zz == 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 CR, ceasing \
                  iterations\n")
            return (postprocess(x), -1)

        if iter == maxiter:
            return (postprocess(x), iter)