"""Compatible Relaxation"""

__docformat__ = "restructuredtext en"

import numpy as np
import scipy
from scipy.linalg import norm
from scipy.sparse import isspmatrix, csr_matrix, spdiags, isspmatrix_csr

from pyamg.relaxation import gauss_seidel, gauss_seidel_indexed

__all__ = ['CR', 'binormalize']


def CR(S, method='habituated', maxiter=20):
    """Use Compatible Relaxation to compute a C/F splitting

    Parameters
    ----------
    S : csr_matrix
        sparse matrix (n x n) usually matrix A of Ax=b
    method : {'habituated','concurrent'}
        Method used during relaxation:
            - concurrent: GS relaxation on F-points, leaving e_c = 0
            - habituated: full relaxation, setting e_c = 0
    maxiter : int
        maximum number of outer iterations (lambda)

    Returns
    -------
    splitting : array
        C/F list of 1's (coarse pt) and 0's (fine pt) (n x 1)

    References
    ----------
    .. [1] Livne, O.E., "Coarsening by compatible relaxation."
       Numer. Linear Algebra Appl. 11, No. 2-3, 205-227 (2004).

    Examples
    --------
    >>> from pyamg.gallery import poisson
    >>> from pyamg.classical.cr import CR
    >>> A = poisson((20,20),format='csr')
    >>> splitting = CR(A)

    """
    # parameters (paper notation)
    ntests = 3      # (nu) number of random tests to do per iteration
    nrelax = 4      # (eta) number of relaxation sweeps per test

    smagic = 1.0    # (s) parameter in [1,5] to account for fill-in
    gamma = 1.5     # (gamma) cycle index.  use 1.5 for 2d
    G = 30          # (G) number of equivalence classes (# of bins)
    tdepth = 1      # (t) drop depth on parse of L bins
    delta = 0       # (delta) drop threshold on parse of L bins
    alphai = 0.25   # (alpha_inc) quota increase

    # initializations
    alpha = 0.0     # coarsening ratio, quota
    beta = np.inf      # quality criterion
    beta1 = np.inf     # quality criterion, older
    beta2 = np.inf     # quality criterion, oldest
    n = S.shape[0]    # problem size
    nC = 0          # number of current Coarse points
    rhs = np.zeros((n, 1))  # rhs for Ae=0

    if not isspmatrix(S):
        raise TypeError('expecting sparse matrix')

    S = binormalize(S)

    splitting = np.zeros((S.shape[0], 1), dtype='intc')

    # out iterations ---------------
    for m in range(0, maxiter):

        mu = 0.0  # convergence rate
        E = np.zeros((n, 1))  # slowness measure

        # random iterations ---------------
        for k in range(0, ntests):

            e = 0.5*(1 + scipy.rand(n, 1))
            e[splitting > 0] = 0

            enorm = norm(e)

            # relaxation iterations ---------------
            for l in range(0, nrelax):

                if method == 'habituated':
                    gauss_seidel(S, e, np.zeros((n, 1)), iterations=1)
                    e[splitting > 0] = 0
                elif method == 'concurrent':
                    raise NotImplementedError('not implemented: need an \
                                                F-smoother')
                else:
                    raise NotImplementedError('method not recognized: need \
                                               habituated or concurrent')

                enorm_old = enorm
                enorm = norm(e)

                if enorm <= 1e-14:
                    # break out of loops
                    ntests = k
                    nrelax = l
                    maxiter = m
            # end relax

            # check slowness
            E = np.where(np.abs(e) > E, np.abs(e), E)

            # update convergence rate
            mu = mu + enorm/enorm_old
        # end random tests
        mu = mu/ntests

        # work
        alpha = float(nC)/n

        W = (1 + (smagic-1)*gamma*alpha)/(1-gamma*alpha)

        # quality criterion
        beta2 = beta1
        beta1 = beta
        beta = np.power(max([mu, 0.1]), 1.0 / W)

        # check if we're doing well
        if (beta > beta1 and beta1 > beta2) or \
           m == (maxiter-1) or max(E) < 1e-13:
            return splitting.ravel()

        # now add points
        #
        # update limit on additions to splitting (C)
        if alpha < 1e-13:
            alpha = 0.25
        else:
            alpha = (1-alphai) * alpha + alphai * (1/gamma)

        nCmax = np.ceil(alpha * n)

        L = np.ceil(G * E / E.max()).ravel()

        binid = G

        # add whole bins (and t-depth nodes) at a time
        u = np.zeros((n, 1))
        # TODO This loop may never halt...
        #      Perhaps loop over nC < nCmax and binid > 0 ?
        while nC < nCmax:
            if delta > 0:
                raise NotImplementedError
            if tdepth != 1:
                raise NotImplementedError

            (roots,) = np.where(L == binid)

            for root in roots:
                if L[root] >= 0:
                    cols = S[root, :].indices
                    splitting[root] = 1    # add roots
                    nC += 1
                    L[cols] = -1
            binid -= 1

            #L[troots] = -1          # mark t-rings visited
            #u[:]=0.0
            #u[roots] = 1.0
            #for depth in range(0,tdepth):
            #    u = np.abs(S) * u
            #(troots,tmp) = np.where(u>0)

    return splitting.ravel()


def _CRsweep(A, Findex, Cindex, nu, thetacr, method):

    n = A.shape[0]    # problem size
    numax = nu
    z = np.zeros((n,))
    e = np.ones((n,))
    e[Cindex] = 0.0
    enorm = norm(e)
    rhok = 1

    for it in range(1, numax+1):

        if method == 'habituated':
            gauss_seidel(A, e, z, iterations=1)
            e[Cindex] = 0.0
        elif method == 'concurrent':
            gauss_seidel_indexed(A, e, z, indices=Findex, iterations=1)
        else:
            raise NotImplementedError('method not recognized: need \
                                       habituated or concurrent')

        enorm_old = enorm
        enorm = norm(e)
        rhok_old = rhok
        rhok = enorm / enorm_old

        # criteria 1
        if (abs(rhok - rhok_old) / rhok < 0.1) and (it >= nu):
            return rhok, e

        # criteria 2
        if rhok < 0.1 * thetacr:
            return rhok, e


def CRalpha(A, method='habituated', nu=3, thetacr=0.7, thetacs=[0.3, 0.5],
            maxiter=20):
    """
    >>> from pyamg.gallery import poisson
    >>> from cr import CRalpha
    >>> A = poisson((20,20),format='csr')
    >>> splitting = CRalpha(A)
    """
    n = A.shape[0]    # problem size

    thetacs = list(thetacs)
    thetacs.reverse()

    if not isspmatrix_csr(A):
        raise TypeError('expecting csr sparse matrix A')

    if A.dtype == complex:
        raise NotImplementedError('complex A not implemented')

    # 3.1a
    splitting = np.zeros((n,), dtype='intc')
    gamma = np.zeros((n,))

    # 3.1b
    Cindex = np.where(splitting == 1)[0]
    Findex = np.where(splitting == 0)[0]
    rho, e = _CRsweep(A, Findex, Cindex, nu, thetacr, method=method)

    # 3.1c
    for it in range(0, maxiter):

        print it
        # 3.1d (assuming constant initial e in _CRsweep)
        # should already be zero at C pts (Cindex)
        gamma[Findex] = np.abs(e[Findex]) / np.abs(e[Findex]).max()

        # 3.1e
        Uindex = np.where(gamma > thetacs[0])[0]
        if len(thetacs) > 1:
            thetacs.pop()

        # 3.1f
        # first find the weights: omega_i = |N_i\C| + gamma_i
        omega = -np.inf * np.ones((n,))
        for i in Uindex:
            J = A.indices[np.arange(A.indptr[i], A.indptr[i+1])]
            J = np.where(splitting[J] == 0)[0]
            omega[i] = len(J) + gamma[i]

        # independent set
        Usize = len(Uindex)
        while Usize > 0:
            # step 1
            i = omega.argmax()
            splitting[i] = 1
            gamma[i] = 0.0
            # step 2
            J = A.indices[np.arange(A.indptr[i], A.indptr[i+1])]
            J = np.intersect1d(J, Uindex, assume_unique=True)
            omega[i] = -np.inf
            omega[J] = -np.inf

            # step 3
            for j in J:
                K = A.indices[np.arange(A.indptr[j], A.indptr[j+1])]
                K = np.intersect1d(K, Uindex, assume_unique=True)
                omega[K] = omega[K] + 1.0

            Usize -= 1

        Cindex = np.where(splitting == 1)[0]
        Findex = np.where(splitting == 0)[0]
        rho, e = _CRsweep(A, Findex, Cindex, nu, thetacr, method=method)

        print rho
        if rho < thetacr:
            break

    return splitting


def binormalize(A, tol=1e-5, maxiter=10):
    """Binormalize matrix A.  Attempt to create unit l_1 norm rows.

    Parameters
    ----------
    A : csr_matrix
        sparse matrix (n x n)
    tol : float
        tolerance
    x : array
        guess at the diagonal
    maxiter : int
        maximum number of iterations to try

    Returns
    -------
    C : csr_matrix
        diagonally scaled A, C=DAD

    Notes
    -----
        - Goal: Scale A so that l_1 norm of the rows are equal to 1:
        - B = DAD
        - want row sum of B = 1
        - easily done with tol=0 if B=DA, but this is not symmetric
        - algorithm is O(N log (1.0/tol))

    Examples
    --------
    >>> from pyamg.gallery import poisson
    >>> from pyamg.classical import binormalize
    >>> A = poisson((10,),format='csr')
    >>> C = binormalize(A)

    References
    ----------
    .. [1] Livne, Golub, "Scaling by Binormalization"
       Tech Report SCCM-03-12, SCCM, Stanford, 2003
       http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

    """
    if not isspmatrix(A):
        raise TypeError('expecting sparse matrix A')

    if A.dtype == complex:
        raise NotImplementedError('complex A not implemented')

    n = A.shape[0]
    it = 0
    x = np.ones((n, 1)).ravel()

    # 1.
    B = A.multiply(A).tocsc()  # power(A,2) inconsistent in numpy, scipy.sparse
    d = B.diagonal().ravel()

    # 2.
    beta = B * x
    betabar = (1.0/n) * np.dot(x, beta)
    stdev = rowsum_stdev(x, beta)

    #3
    while stdev > tol and it < maxiter:
        for i in range(0, n):
            # solve equation x_i, keeping x_j's fixed
            # see equation (12)
            c2 = (n-1)*d[i]
            c1 = (n-2)*(beta[i] - d[i]*x[i])
            c0 = -d[i]*x[i]*x[i] + 2*beta[i]*x[i] - n*betabar
            if (-c0 < 1e-14):
                print 'warning: A nearly un-binormalizable...'
                return A
            else:
                # see equation (12)
                xnew = (2*c0)/(-c1 - np.sqrt(c1*c1 - 4*c0*c2))
            dx = xnew - x[i]

            # here we assume input matrix is symmetric since we grab a row of B
            # instead of a column
            ii = B.indptr[i]
            iii = B.indptr[i+1]
            dot_Bcol = np.dot(x[B.indices[ii:iii]], B.data[ii:iii])

            betabar = betabar + (1.0/n)*dx*(dot_Bcol + beta[i] + d[i]*dx)
            beta[B.indices[ii:iii]] += dx*B.data[ii:iii]

            x[i] = xnew

        stdev = rowsum_stdev(x, beta)
        it += 1

    # rescale for unit 2-norm
    d = np.sqrt(x)
    D = spdiags(d.ravel(), [0], n, n)
    C = D * A * D
    C = C.tocsr()
    beta = C.multiply(C).sum(axis=1)
    scale = np.sqrt((1.0/n) * np.sum(beta))
    return (1/scale)*C


def rowsum_stdev(x, beta):
    """Compute row sum standard deviation

    Compute for approximation x, the std dev of the row sums
    s(x) = ( 1/n \sum_k  (x_k beta_k - betabar)^2 )^(1/2)
    with betabar = 1/n dot(beta,x)

    Parameters
    ----------
    x : array
    beta : array

    Returns
    -------
    s(x)/betabar : float

    Notes
    -----
    equation (7) in Livne/Golub

    """
    n = x.size
    betabar = (1.0/n) * np.dot(x, beta)
    stdev = np.sqrt((1.0/n) *
                    np.sum(np.power(np.multiply(x, beta) - betabar, 2)))
    return stdev/betabar
