path: root/cactus_utils.py

import h5py
import numpy as np
import pylab
import scipy.ndimage.interpolation
import scipy.interpolate
import scipy.integrate
import scipy.optimize

import collections
import os
import os.path
import re

axis_prop = { 'x' : 0, 'y' : 1, 'z' : 2 }
axis_data = { 'x' : 2, 'y' : 1, 'z' : 0 }

EPSILON   = 1e-3


def ndinterpolate(in_coords, values, out_coords):
    """
    A wrapper around scipy.image.interpolation.map_coordinates() providing more convenient API.

    Given the values of a function on an N-dimensional regular grid, interpolate
    its value on given coordinates.

    @param in_coords An N+1-dimensional array of the coordinates of the input values.
                     The first axis goes over the dimensions (i.e. in_coords[0] is
                     an N-dimensional array that gives the x-coordinate for each value).
    @param values An N-dimensional array of the values (at in_coords) of the function to
                  be interpolated
    @param out_coords A 2-dimensional array of the coordinates at which the output is
                      desired. The first axis goes over the dimensions (i.e. out_coords[0]
                      is a 1-dimensional array of length N of the x-coordinates for every
                      desired interpolation point).
    """
    N = len(values.shape)

    if len(in_coords.shape) != N + 1 or in_coords.shape[0] != N:
        raise ValueError('Invalid coords/values shapes: ' + str(in_coords.shape) + str(values.shape))

    out_coords = np.copy(out_coords)
    for i in xrange(N):
        max = np.max(in_coords[i])
        min = np.min(in_coords[i])
        out_coords[i] -= min
        out_coords[i] *= values.shape[i] / (max - min)

    return scipy.ndimage.interpolation.map_coordinates(values, out_coords)


def get_default_gridfunc(datafile):
    """
    Get the name of the default grid function from a given data file
    """
    funcs = datafile['Parameters and Global Attributes']['Datasets']

    if funcs.dtype.type != np.string_:
        funcs_str = ''.join(map(chr, funcs))
    else:
        funcs_str = funcs.value

    return funcs_str.strip().split('\n')[0]


def _get_iteration_from_time(datafile, time):
    """
    Get the simulation iteration with the given time.
    """
    for item in datafile.values():
        try:
            if item.attrs['time']  == time:
                return item.attrs['timestep']
        except KeyError:
            pass

    raise ValueError("No timelevel with time %d" % (time))


def _get_last_iteration(datafile, name, rl = 0):
    """
    Get the last iteration for which the data for the given gridfunction
    on the given refinement level exists in the data file.
    """
    last_it = -1

    for item in datafile.values():
        try:
            if (item.attrs['name']  == name and
                item.attrs['level'] == rl):
                last_it = max(last_it, item.attrs['timestep'])
        except KeyError:
            pass

    if last_it < 0:
        raise KeyError("No compatible datafile in the file")

    return last_it


def _extract_paren(string):
    """
    From a string starting with a parentheses-like pairwise construct --
    '(,[,{,<', extract the part inside it, taking into account nesting.
    """
    if string[0] == '[':
        parens = ('[', ']')
    elif string[0] == '(':
        parens = ('(', ')')
    elif string[0] == '{':
        parens = ('{', '}')
    elif string[0] == '<':
        parens = ('<', '>')
    else:
        raise ValueError("String '%s' does not start with a recognized pairwise delimiter" % string)

    cnt = 0
    for i in xrange(len(string)):
        if string[i] == parens[0]:
            cnt += 1
        elif string[i] == parens[1]:
            cnt -= 1

        if cnt == 0:
            return string[:i + 1]

    raise ValueError("Unbalanced parentheses")


def _parse_grid_struct(string):
    """
    Parse the grid structure v5 string into a dict.
    """
    ret = {}
    s = string
    while len(s) > 0:
        try:
            key, tail = s.split(':', 1)
        except ValueError:
            if len(ret) > 0:
                return ret
            else:
                return string

        val = _extract_paren(tail)
        ret[key] = val

        s = tail[len(val) + 1:]

    return ret


def _get_timestep(datafile):
    """
    Get the time step on the coarsest grid.
    """
    gridstruct = datafile['Parameters and Global Attributes']['Grid Structure v5']

    if gridstruct.dtype.type != np.string_:
        s = ''.join(map(chr, gridstruct))
    else:
        s = gridstruct.value

    match = re.search('grid_delta_times:\[(.*?)\],', s)
    if match is None:
        raise ValueError("Invalid grid structure entry")

    deltas = eval('list(%s)' % (match.group(1)))
    return deltas[0]


def _get_grid_spacing(datafile):
    """
    Get the numbers of grid points (not counting ghosts) in each direction.
    """
    gridstruct = datafile['Parameters and Global Attributes']['Grid Structure v5']

    if gridstruct.dtype.type != np.string_:
        s = ''.join(map(chr, gridstruct))
    else:
        s = gridstruct.value

    struct = _parse_grid_struct(s)
    match  = re.search('\[([0-9]*),([0-9]*),([0-9]*)\]/[0-9]*\)', struct['grid_structure'])
    gridpoints = np.array(map(int, (match.group(1), match.group(2), match.group(3))))

    match  = re.search('\[([0-9]*),([0-9]*),([0-9]*)\]', struct['grid_ghosts'])
    ghosts = np.array(map(int, (match.group(1), match.group(2), match.group(3))))

    gridpoints -= 2 * ghosts

    return gridpoints


def _get_data_axis(dataset, axis, axvalue):
    """
    Get a tuple of (coordinates, data) along the specified axis ('x', 'y', or 'z').
    axvalue is the value of the other two axes, it may be a number for the same
    value on both or an iterable with at least two elements for distinct values.
    """
    slicelist = [None] * 3
    try:
        if len(axvalue) > 2:
            raise ValueError("Need at most two axis values")
        elif len(axvalue) == 1:
            axvalue = [axvalue] * 2
    except TypeError:
        axvalue = [axvalue] * 2

    # data is stored in [z, y, x] order, properties are in normal [x, y, z] order
    axes = collections.OrderedDict([["x", {"data_idx" : 2, "idx" : 0 }],
                                    ["y", {"data_idx" : 1, "idx" : 1 }],
                                    ["z", {"data_idx" : 0, "idx" : 2 }]])

    # for the requested axis, just remove the ghost zones
    ax     = axes.pop(axis)
    ghosts = dataset.attrs["cctk_nghostzones"][ax["idx"]]
    delta  = dataset.attrs["delta"][ax["idx"]]
    origin = dataset.attrs["origin"][ax["idx"]]
    slicelist[ax["data_idx"]] = slice(ghosts, -ghosts)

    # for the other two axes, locate the proper values
    while len(axes):
        axname, ax = axes.popitem(last = False)

        o = dataset.attrs["origin"][ax["idx"]]
        d = dataset.attrs["delta"][ax["idx"]]
        idx = int((axvalue.pop(0) - o) / d)

        if idx >= dataset.shape[ax["data_idx"]]:
            raise ValueError("Axis %s is outside the domain [%f, %f]" % (axname, o, o + d * dataset.shape[ax["data_idx"]]))

        slicelist[ax["data_idx"]] = idx

    data = dataset[tuple(slicelist)]

    # now compute the corresponding distances from coordinate origin
    r = np.array([(origin + ghosts * delta + n * delta) for n in xrange(data.shape[0])])

    return r, data


def plot(datafile, axis = "x", axisvalue = 0.0, iteration = "last", time = -1, gridfunc = None, rl = 0, c = -1, mod = None, label = None):
    """
    Draw a 2D plot.
    """
    # guess dataset name
    if gridfunc is None:
        gridfunc = get_default_gridfunc(datafile)

    if time >= 0:
        iteration = _get_iteration_from_time(datafile, time)
    elif iteration == "last":
        iteration = _get_last_iteration(datafile, gridfunc, rl)

    dataset = datafile['%s it=%d tl=0 rl=%d%s' % (gridfunc, iteration, rl, " c=%d" % c if c >= 0 else "")]
    if axis == "x" or axis == "y" or axis == "z":
        r, data = _get_data_axis(dataset, axis, axisvalue)
#    elif len(axis) == 2 and re.match("[xyz]{2}", axis):
#        r, data = get_data_diagonal(dataset, axis)
    else:
        raise ValueError("Invalid axis specifier %s" % axis)

    if mod:
        data = mod(data)

    if not label:
        label = "%s time=%f level=%d" % (gridfunc, dataset.attrs['time'], rl)

    pylab.plot(r, data, label = label)
    pylab.legend()


def plot_diff(datafile, axis = "x", axisvalue = 0.0, it1 = 0, it2 = "last", time1 = -1, time2 = -1, gridfunc = None, rl = 0, absdiff = 0):
    """
    Plot a difference between two times or iterations.
    """
    # guess dataset name
    if gridfunc is None:
        gridfunc = get_default_gridfunc(datafile)

    if time1 >= 0:
        it1 = _get_iteration_from_time(datafile, time1)
    elif it1 == "last":
        it1 = _get_last_iteration(datafile, gridfunc, rl)

    if time2 >= 0:
        it2 = _get_iteration_from_time(datafile, time2)
    elif it2 == "last":
        it2 = _get_last_iteration(datafile, gridfunc, rl)

    dataset1 = datafile['%s it=%d tl=0 rl=%d' % (gridfunc, it1, rl)]
    dataset2 = datafile['%s it=%d tl=0 rl=%d' % (gridfunc, it2, rl)]
    if axis == "x" or axis == "y" or axis == "z":
        r, data1 = _get_data_axis(dataset1, axis, axisvalue)
        r, data2 = _get_data_axis(dataset2, axis, axisvalue)
#    elif len(axis) == 2 and re.match("[xyz]{2}", axis):
#        r, data = get_data_diagonal(dataset, axis)
    else:
        raise ValueError("Invalid axis specifier %s" % axis)

    data = data1 - data2
    if absdiff:
        data = np.abs(data)

    pylab.plot(r, data, label = "%s" % (gridfunc))
    pylab.legend()


def plot_convergence_time(datafiles, axis, gridfunc, rl):
    pass


def plot_convergence_space(datafiles, axis, gridfunc, rl):
    pass


def plot_convergence(datafiles, axis = "x", gridfunc = None, rl = 0, time = "last", mode = "selfconsistent"):
    if mode == "selfconsistent" and len(datafiles) < 3:
        raise ValueError("At least three data sets needed for self-consistent convergence plot")
    elif len(datafiles) < 2:
        raise ValueError("At least two data sets needed for convergence plot")

    if gridfunc is None:
        gridfunc = get_default_gridfunc(datafiles[0])

    # check if the datafiles have changing time steps
    prev_timestep = 0
    timestep_changing = 0
    for df in datafiles:
        timestep = _get_timestep(df)
        if prev_timestep == 0:
            prev_timestep = timestep
        elif prev_timestep != timestep:
            timestep_changing = 1
            break

    datasets = [f['%s it=0 tl=0 rl=%d' % (gridfunc, rl)] for f in datafiles]

    # check if our chosen datasets have different grid spacings
    prev_spacing = None
    spacing_changing = 0
    for ds in datasets:
        spacing = ds.attrs['delta']
        if prev_spacing is None:
            prev_spacing = spacing
        elif not np.all(spacing == prev_spacing):
            spacing_changing = 1
            break
    del datasets

    if timestep_changing and spacing_changing:
        raise ValueError("Both the time step and grid spacing change between files.")
    if not timestep_changing and not spacing_changing:
        raise ValueError("Neither the time step nor grid spacing change between files.")

    if timestep_changing:
        plot_convergence_time(datafiles, axis, gridfunc, rl)
    else:
        plot_convergence_space(datafiles, axis, gridfunc, rl)


def dminus(arr, dir, dx):
    """
    Fourth order centered derivative in the given direction.
    Second order centered derivatives are used for the points next to the
    border ones and first order for the border points.
    """
    ret = np.empty_like(arr)
    slicelist_ret      = [slice(None)] * len(arr.shape)
    slicelist_ret[dir] = slice(1, None)
    slicelist_m1       = [slice(None)] * len(arr.shape)
    slicelist_m1[dir]  = slice(0, -1)
    slicelist_0        = [slice(None)] * len(arr.shape)
    slicelist_0[dir]   = slice(1, None)

    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_0)] - arr[tuple(slicelist_m1)]) / dx

    slicelist_ret[dir] = slice(0, 1)
    slicelist_m1[dir] = slice(1, 2)
    slicelist_0[dir] = slice(0, 1)
    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_m1)] - arr[tuple(slicelist_0)]) / dx

    return ret


def dplus(arr, dir, dx):
    """
    Fourth order centered derivative in the given direction.
    Second order centered derivatives are used for the points next to the
    border ones and first order for the border points.
    """
    ret = np.empty_like(arr)
    slicelist_ret      = [slice(None)] * len(arr.shape)
    slicelist_ret[dir] = slice(0, -1)
    slicelist_p1       = [slice(None)] * len(arr.shape)
    slicelist_p1[dir]  = slice(1, None)
    slicelist_0        = [slice(None)] * len(arr.shape)
    slicelist_0[dir]   = slice(0, -1)

    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_p1)] - arr[tuple(slicelist_0)]) / dx

    slicelist_ret[dir] = slice(-1, None)
    slicelist_p1[dir] = slice(-2, -1)
    slicelist_0[dir] = slice(-1, None)
    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_0)] - arr[tuple(slicelist_p1)]) / dx

    return ret


def diff4(arr, dir, dx, coeff = None, div = 12.):
    """
    Fourth order centered derivative in the given direction.
    Second order centered derivatives are used for the points next to the
    border ones and first order for the border points.
    """
    ret = np.zeros_like(arr)
    slicelist_ret     = [slice(None)] * len(arr.shape)
    slicelist_ret[dir] = slice(2, -2)
    slicelist_0       = [slice(None)] * len(arr.shape)
    slicelist_0[dir]  = slice(2, -2)
    slicelist_p1      = [slice(None)] * len(arr.shape)
    slicelist_p1[dir] = slice(3, -1)
    slicelist_p2      = [slice(None)] * len(arr.shape)
    slicelist_p2[dir] = slice(4, None)
    slicelist_m1      = [slice(None)] * len(arr.shape)
    slicelist_m1[dir] = slice(1, -3)
    slicelist_m2      = [slice(None)] * len(arr.shape)
    slicelist_m2[dir] = slice(0, -4)

    sl = (slicelist_p2, slicelist_p1, slicelist_0, slicelist_m1, slicelist_m2)
    if coeff is None:
        coeff = (-1, 8, 0, -8, 1)

    for s, c in zip(sl, coeff):
        ret[tuple(slicelist_ret)] += c * arr[tuple(s)]
    ret[tuple(slicelist_ret)] /= (div * dx)

    slicelist_ret[dir] = slice(0, 1)
    slicelist_p1[dir] = slice(1, 2)
    slicelist_m1[dir] = slice(0, 1)
    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_p1)] - arr[tuple(slicelist_m1)]) / dx

    slicelist_ret[dir] = slice(-1, None)
    slicelist_p1[dir]  = slice(-1, None)
    slicelist_m1[dir]  = slice(-2, -1)
    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_p1)] - arr[tuple(slicelist_m1)]) / dx

    slicelist_ret[dir] = slice(1, 2)
    slicelist_p1[dir]  = slice(2, 3)
    slicelist_m1[dir]  = slice(0, 1)
    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_p1)] - arr[tuple(slicelist_m1)]) / (2 * dx)

    slicelist_ret[dir] = slice(-2, -1)
    slicelist_p1[dir] = slice(-1, None)
    slicelist_m1[dir] = slice(-3, -2)
    ret[tuple(slicelist_ret)] = (arr[tuple(slicelist_p1)] - arr[tuple(slicelist_m1)]) / (2 * dx)
    return ret


def get_meshgrid(datafile, gridfunc = None, rl = 0, c = -1, axis = "z"):
    """
    Get a tuple of coordinate mesh grids for the given data file.
    """
    if not gridfunc:
        gridfunc = get_default_gridfunc(datafile)
    dataset = datafile['%s it=0 tl=0 rl=%d%s' % (gridfunc, rl, " c=%d" % c if c >= 0 else "")]
    o = dataset.attrs['origin']
    d = dataset.attrs['delta']
    if axis == 'z':
        idx = (0, 1)
    elif axis == 'y':
        idx = (0, 2)
    elif axis == 'x':
        idx = (1, 2)
    else:
        raise ValueError('Invalid axis %s' % axis)

    shape = (dataset.shape[2], dataset.shape[1], dataset.shape[0])
    return np.meshgrid(np.array([o[idx[0]] + n * d[idx[0]] for n in xrange(shape[idx[0]])]), np.array([o[idx[1]] + n * d[idx[1]] for n in xrange(shape[idx[1]])]))


def _get_extrema(ar, rad = 3):
    """
    Find local extrema of a 1-D array.
    Returned is a tuple (maxima, minima) of the (coordinate, value) of the
    extrema.  rad determines the number of points in each direction over which
    the data must be monotonous for the center to be classified as an extremum.
    """
    maxima = []
    minima = []
    dar = diff4(ar, 0, 1)
    spl = scipy.interpolate.UnivariateSpline(range(ar.shape[0]), ar)

    for i in xrange(rad, len(ar) - rad):
        maximum = minimum = True
        for j in xrange(1, rad):
            for dir in (1, -1):
                idx = i + j * dir
                if ar[idx] > ar[idx - dir]:
                    maximum = False
                elif ar[idx] < ar[idx - dir]:
                    minimum = False

        if maximum or minimum:
            x = np.array(range(2 * rad + 1))
            y = dar[i - rad : i + rad + 1]
            roots = scipy.interpolate.UnivariateSpline(x, y).roots()
            if len(roots) > 0:
                root = roots[0] + i - rad
                (maxima if maximum else minima).append((root, spl(root)))

    return maxima, minima


def _get_movement(coord_old, val_old, ex_new):
    """
    Return the number of points an extremum i in val_old moves in val_new
    or None if no movement could be detected.
    """
    dist = None
    for coord_new, val_new in ex_new:
        ratio = val_new / val_old
        dist_new = coord_new - coord_old
        if ratio < 1.1 and ratio > 0.9 and (dist is None or abs(dist_new) < abs(dist)):
            dist = dist_new
    return dist


def calc_movement(ar1, ar2):
    """
    Find all local extrema in ar1 and calculate how much the move in ar2.
    Return the list of matched movements, in points.
    """
    maxima1, minima1 = _get_extrema(ar1)
    maxima2, minima2 = _get_extrema(ar2)

    ret = []
    for coord, val in maxima1:
        diff = _get_movement(coord, val, maxima2)
        if diff is not None:
            ret.append(diff)
    for coord, val in minima1:
        diff = _get_movement(coord, val, minima2)
        if diff is not None:
            ret.append(diff)

    return ret


def gf_itertime(datafile, rl = 0):
    """
    Find all iterations that are stored in the given datafile.
    Return an ordered list of (iteration, time) tuples.
    """
    ret = set()

    rl = ' rl=%d' % rl
    for key in datafile.keys():
        if not rl in key:
            continue

        match = re.search('it=([0-9]*)', key)
        if match:
            ret.add((int(match.group(1)), datafile[key].attrs['time']))

    return sorted(ret)


def gf_iter(gridfunc, rl = 0):
    """
    Find all iterations that are stored in the given datafile.
    Return an ordered list of iterations.
    """
    return zip(*gf_itertime(gridfunc, rl))[0]


def gf_time(gridfunc, rl = 0):
    """
    Find all iterations that are stored in the given datafile.
    Return an ordered list of times.
    """
    return zip(*gf_itertime(gridfunc, rl))[1]


def trumpet_r(R, M = 1.0):
    fac1 = ((2 * R + M + np.sqrt(4 * (R ** 2) + 4 * M * R + 3 * (M ** 2))) / 4)
    fac2 = ((4 + 3 * np.sqrt(2)) * (2 * R - 3 * M)) / (8 * R + 6 * M + 3 * np.sqrt(8 * (R ** 2) + 8 * M * R + 6 * (M ** 2)))
    return fac1 * (fac2 ** (1. / np.sqrt(2)))


def trumpet_log1_C2(n):
    s = np.sqrt(4 + 9 * (n ** 2))
    a_c = np.sqrt((s - 3 * n) / (s + 3 * n))
    C2 = ((3 * n + s) ** 3) / (128 * (n ** 3)) * np.exp(-2 * a_c / n)
    return C2


def _trumpet_log1_alphabar(alpha, R, M, n):
    num = 3 * M - 2 * R + 2 * R * (alpha ** 2)
    den = (R - 2 * M + n * R * alpha - R * (alpha ** 2))
    ret = - n * num / (R * den)
    return ret


def _trumpet_log1_alpha_implicit(alpha, R, M, n):
    C2 = trumpet_log1_C2(n)
    return alpha ** 2 - 1 + 2 * M / R - C2 * np.exp(2 * alpha / n) / (R ** 4)


def trumpet_log1_lapse(R, M = 1.0, n = 2.0):
    R_c = (3 * (n ** 2) * M + np.sqrt((2 * n * M) ** 2 + (3 * (n ** 2) * M) ** 2)) / (4 * (n ** 2))
    s = np.sqrt(4 + 9 * (n ** 2))
    a_c = np.sqrt((s - 3 * n) / (s + 3 * n))

    for i in xrange(R.shape[0]):
        if R[i] > R_c:
            break

    tmp = scipy.integrate.odeint(_trumpet_log1_alphabar, a_c, [R_c, R[i - 1]], args = (M, n))[1]
    ret1 = scipy.integrate.odeint(_trumpet_log1_alphabar, tmp, R[:i][::-1], args = (M, n))[::-1]
    ret1 = ret1.reshape(ret1.shape[0])

    tmp = scipy.integrate.odeint(_trumpet_log1_alphabar, a_c, [R_c, R[i]], args = (M, n))[1]
    ret2 = scipy.integrate.odeint(_trumpet_log1_alphabar, tmp, R[i:], args = (M, n))
    ret2 = ret2.reshape(ret2.shape[0])

    return np.concatenate((ret1, ret2))


def trumpet_log1_trk(R, M = 1.0, n = 2.0):
    a  = trumpet_log1_lapse(R, M, n)
    b = np.sqrt((a ** 2) - (1 - 2 * M / R))
    bbar = diff4(b, 0, R[1] - R[0])

    K = 2 * b / R + bbar
    return K


def trumpet_lapse(R, M = 1.0):
    C2 = 27. * (M ** 4) / 16
    return np.sqrt(1 - 2 * M / R + C2 / (R ** 4))


def trumpet_log1_r(R, M = 1.0, n = 2.0):
    R_aux = np.linspace(1.32, 100, 10000)
    alpha_aux = trumpet_log1_lapse(R_aux)

    R_alpha_aux = scipy.interpolate.UnivariateSpline(alpha_aux, R_aux, s = 0)
    alpha_s = alpha_aux[20]
    coeff = R_alpha_aux(alpha_s) ** (1. / alpha_s)

    def integrand_aux(a, R_alpha = R_alpha_aux, alpha = alpha_aux):
        if a > alpha[-1]:
            RR = 2 / (1 - a ** 2)
        else:
            RR = R_alpha(a)
        ret = np.log(RR) / (a ** 2)
        return ret
    C0, err = scipy.integrate.quad(integrand_aux, alpha_s, 1)

    alpha = trumpet_log1_lapse(R, M, n)
    R_alpha = scipy.interpolate.UnivariateSpline(alpha, R, s = 0)
    Rbar_alpha = scipy.interpolate.UnivariateSpline(alpha[1:-1], (R[2:] - R[:-2]) / (2 * (R[1] - R[0])), s = 0)

    def integrand(a, R_alpha = R_alpha, alpha = alpha):
        if a > alpha[-1]:
            RR = 2 / (1 - a ** 2)
        else:
            RR = R_alpha(a)
        ret = np.log(RR) / (a ** 2)
        return ret

    def integrand2(a, R_alpha = R_alpha, alpha = alpha):
        if a > alpha[-1]:
            RR = 2 / (1 - a)
            Rbar = - 4 * a / ((1 - a ** 2) ** 2)
        else:
            RR = R_alpha(a)
            Rbar = Rbar_alpha(a)
        return Rbar / (RR * a)

    r1 = []
    r2 = []

    if alpha[0] >= alpha_s:
        i = 0
    else:
        for i in xrange(alpha.shape[0]):
            if alpha[i] >= alpha_s:
                break

    for j in xrange(i):
        a = alpha[j]
        integral, err,  = scipy.integrate.quad(integrand2, a, alpha_s)
        r1.append(np.exp(-integral - C0) * coeff)

    int_prev = 0
    alpha_prev = alpha_s
    for j in xrange(i, alpha.shape[0]):
        a = alpha[j]
        integral, err, = scipy.integrate.quad(integrand, alpha_prev, a)
        integral += int_prev
        alpha_prev = a
        int_prev = integral

        r2.append(np.exp(integral - C0) * R_alpha(a) ** (1. / a))

    return np.array(r1 + r2)


def calc_kretschmann(s, axis = 2, xoffset = 0):
    rl = s.rl

    sliceidx       = [slice(0, 7), slice(0, 7), slice(xoffset, xoffset + 7)]
    sliceidx[axis] = slice(None)
    sliceidx       = tuple(sliceidx)

    offsetidx = [3] * 3
    offsetidx[axis] = slice(None)
    offsetidx = tuple(offsetidx)

    #sliceidx       = [slice(None)]*3
    #sliceidx       = tuple(sliceidx)

    #offsetidx       = [slice(None)]*3
    #offsetidx = tuple(offsetidx)

    # load grid functions
    phi = s['phi'][sliceidx]
    trK = s['trK'][sliceidx]
    gt  = np.array([[s['gt33'][sliceidx], s['gt23'][sliceidx], s['gt13'][sliceidx]],
                    [s['gt23'][sliceidx], s['gt22'][sliceidx], s['gt12'][sliceidx]],
                    [s['gt13'][sliceidx], s['gt12'][sliceidx], s['gt11'][sliceidx]]])
    At  = np.array([[s['At33'][sliceidx], s['At23'][sliceidx], s['At13'][sliceidx]],
                    [s['At23'][sliceidx], s['At22'][sliceidx], s['At12'][sliceidx]],
                    [s['At13'][sliceidx], s['At12'][sliceidx], s['At11'][sliceidx]]])

    # compute derivatives needed later
    dphi   = np.array([diff4(phi,  i, rl.dx[0]) for i in 0, 1, 2])
    d2phi  = np.array([diff4(dphi, i, rl.dx[0]) for i in 1, 2, 3])
    dgt    = np.array([diff4(gt,   i, rl.dx[0]) for i in 2, 3, 4])
    d2gt   = np.array([diff4(dgt,  i, rl.dx[0]) for i in 3, 4, 5])
    dAt    = np.array([diff4(At,   i, rl.dx[0]) for i in 2, 3, 4])
    dtrK   = np.array([diff4(trK,  i, rl.dx[0]) for i in 0, 1, 2])

    phi = phi[offsetidx]
    trK = trK[offsetidx]
    gt  = gt[(slice(None),) * 2 + offsetidx]
    At  = At[(slice(None),) * 2 + offsetidx]

    dphi   = dphi [(slice(None),) + offsetidx]
    d2phi  = d2phi[(slice(None),) * 2 + offsetidx]
    dgt    = dgt  [(slice(None),) * 3 + offsetidx]
    d2gt   = d2gt [(slice(None),) * 4 + offsetidx]
    dAt    = dAt  [(slice(None),) * 3 + offsetidx]
    dtrK   = dtrK [(slice(None),) + offsetidx]

    e4phi = 1. / phi ** 2
    gtu = np.zeros_like(gt)
    for i in xrange(gt.shape[2]):
        gtu[:, :, i] = np.linalg.inv(gt[:, :, i])
    #for i in xrange(gt.shape[2]):
    #    for j in xrange(gt.shape[3]):
    #        for k in xrange(gt.shape[4]):
    #            gtu[:, :, i, j, k] = np.linalg.inv(gt[:, :, i, j, k])

    g  = e4phi * gt
    gu = gtu / e4phi

    K = e4phi * At + (1. / 3.) * trK * g

    Dphi  = -0.5 * dphi / phi
    D2phi = - 0.5 * (d2phi - np.einsum('i...,j...->ij...', dphi, dphi) / phi) / phi

    Dg  = e4phi * (4 * np.einsum('i...,jk...->ijk...', Dphi, gt) + dgt)
    D2g = e4phi * (4 * np.einsum('ij...,kl...->ijkl...', D2phi + 4 * np.einsum('i...,j...->ij...', Dphi, Dphi), gt) + 4 * np.einsum('i...,jkl...->ijkl...', Dphi, dgt) + 4 * np.einsum('i...,jkl...->jikl...', Dphi, dgt) + d2gt)

    Gl = 0.5 * (np.einsum('cab...->abc...', Dg) + np.einsum('bac...->abc...', Dg) - Dg)
    G  = np.einsum('ij...,jkl...->ikl...', gu, Gl)

    R = (0.5 * (np.einsum('adbc...->abcd...', D2g) + np.einsum('bcad...->abcd...', D2g) - np.einsum('bdac...->abcd...', D2g) - np.einsum('acbd...->abcd...', D2g)) +
         np.einsum('ead...,ebc...->abcd...', Gl, G) - np.einsum('eac...,ebd...->abcd...', Gl, G))

    A   = R + np.einsum('ik...,jl...->ijkl...', K, K) - np.einsum('il...,jk...->ijkl...', K, K)
    trA = np.einsum('kl...,kilj...->ij...', gu, A)
    term1 = (np.einsum('ai...,bj...,ck...,dl...,abcd...,ijkl...', gu, gu, gu, gu, A, A))
    term3 = 4 * np.einsum('ai...,bj...,ab...,ij...', gu, gu, trA, trA)

    t1 = e4phi * (4 * np.einsum('i...,jk...->ijk...', Dphi, At) + dAt)
    t2 = (1. / 3.) * (np.einsum('i...,jk...->ijk...', dtrK, g) + trK * Dg)
    t3 = - np.einsum('iab...,ic...->bac...', G, K) - np.einsum('icb...,ia...->bac...', G, K)
    CDK = t1 + t2 + t3
    CDKu = np.einsum('ai...,bj...,ck...,ijk...->abc...', gu, gu, gu, CDK)

    term2 = 8 * (np.einsum('abc...,bac...', CDK, CDKu) - np.einsum('abc...,abc...', CDK, CDKu))

    return term1 + term2 + term3


def calc_kretschmann2(s):
    rl = s.rl

    sliceidx       = [slice(0, 7), slice(None), slice(None)]
    sliceidx       = tuple(sliceidx)

    offsetidx      = [3, slice(None), slice(None)]
    offsetidx      = tuple(offsetidx)

    #sliceidx       = [slice(None)]*3
    #sliceidx       = tuple(sliceidx)

    #offsetidx       = [slice(None)]*3
    #offsetidx = tuple(offsetidx)

    # load grid functions
    phi = s['phi'][sliceidx]
    trK = s['trK'][sliceidx]
    gt  = np.array([[s['gt33'][sliceidx], s['gt23'][sliceidx], s['gt13'][sliceidx]],
                    [s['gt23'][sliceidx], s['gt22'][sliceidx], s['gt12'][sliceidx]],
                    [s['gt13'][sliceidx], s['gt12'][sliceidx], s['gt11'][sliceidx]]])
    At  = np.array([[s['At33'][sliceidx], s['At23'][sliceidx], s['At13'][sliceidx]],
                    [s['At23'][sliceidx], s['At22'][sliceidx], s['At12'][sliceidx]],
                    [s['At13'][sliceidx], s['At12'][sliceidx], s['At11'][sliceidx]]])

    # compute derivatives needed later
    dphi   = np.array([diff4(phi,  i, rl.dx[0]) for i in 0, 1, 2])
    d2phi  = np.array([diff4(dphi, i, rl.dx[0]) for i in 1, 2, 3])
    dgt    = np.array([diff4(gt,   i, rl.dx[0]) for i in 2, 3, 4])
    d2gt   = np.array([diff4(dgt,  i, rl.dx[0]) for i in 3, 4, 5])
    dAt    = np.array([diff4(At,   i, rl.dx[0]) for i in 2, 3, 4])
    dtrK   = np.array([diff4(trK,  i, rl.dx[0]) for i in 0, 1, 2])

    phi = phi[offsetidx]
    trK = trK[offsetidx]
    gt  = gt[(slice(None),) * 2 + offsetidx]
    At  = At[(slice(None),) * 2 + offsetidx]

    dphi   = dphi [(slice(None),) + offsetidx]
    d2phi  = d2phi[(slice(None),) * 2 + offsetidx]
    dgt    = dgt  [(slice(None),) * 3 + offsetidx]
    d2gt   = d2gt [(slice(None),) * 4 + offsetidx]
    dAt    = dAt  [(slice(None),) * 3 + offsetidx]
    dtrK   = dtrK [(slice(None),) + offsetidx]

    e4phi = 1. / phi ** 2
    gtu = np.zeros_like(gt)
    for i in xrange(gt.shape[2]):
        for j in xrange(gt.shape[3]):
            gtu[:, :, i, j] = np.linalg.inv(gt[:, :, i, j])
    #for i in xrange(gt.shape[2]):
    #    for j in xrange(gt.shape[3]):
    #        for k in xrange(gt.shape[4]):
    #            gtu[:, :, i, j, k] = np.linalg.inv(gt[:, :, i, j, k])

    g  = e4phi * gt
    gu = gtu / e4phi

    K = e4phi * At + (1. / 3.) * trK * g

    Dphi  = -0.5 * dphi / phi
    D2phi = - 0.5 * (d2phi - np.einsum('i...,j...->ij...', dphi, dphi) / phi) / phi

    Dg  = e4phi * (4 * np.einsum('i...,jk...->ijk...', Dphi, gt) + dgt)
    D2g = e4phi * (4 * np.einsum('ij...,kl...->ijkl...', D2phi + 4 * np.einsum('i...,j...->ij...', Dphi, Dphi), gt) + 4 * np.einsum('i...,jkl...->ijkl...', Dphi, dgt) + 4 * np.einsum('i...,jkl...->jikl...', Dphi, dgt) + d2gt)

    Gl = 0.5 * (np.einsum('cab...->abc...', Dg) + np.einsum('bac...->abc...', Dg) - Dg)
    G  = np.einsum('ij...,jkl...->ikl...', gu, Gl)

    R = (0.5 * (np.einsum('adbc...->abcd...', D2g) + np.einsum('bcad...->abcd...', D2g) - np.einsum('bdac...->abcd...', D2g) - np.einsum('acbd...->abcd...', D2g)) +
         np.einsum('ead...,ebc...->abcd...', Gl, G) - np.einsum('eac...,ebd...->abcd...', Gl, G))

    A   = R + np.einsum('ik...,jl...->ijkl...', K, K) - np.einsum('il...,jk...->ijkl...', K, K)
    trA = np.einsum('kl...,kilj...->ij...', gu, A)
    term1 = (np.einsum('ai...,bj...,ck...,dl...,abcd...,ijkl...', gu, gu, gu, gu, A, A))
    term3 = 4 * np.einsum('ai...,bj...,ab...,ij...', gu, gu, trA, trA)

    t1 = e4phi * (4 * np.einsum('i...,jk...->ijk...', Dphi, At) + dAt)
    t2 = (1. / 3.) * (np.einsum('i...,jk...->ijk...', dtrK, g) + trK * Dg)
    t3 = - np.einsum('iab...,ic...->bac...', G, K) - np.einsum('icb...,ia...->bac...', G, K)
    CDK = t1 + t2 + t3
    CDKu = np.einsum('ai...,bj...,ck...,ijk...->abc...', gu, gu, gu, CDK)

    term2 = 8 * (np.einsum('abc...,bac...', CDK, CDKu) - np.einsum('abc...,abc...', CDK, CDKu))

    return term1 + term2 + term3


def l2norm(sd, it, gf, maxrl = None, minrl = None):
    results = []

    if maxrl is None:
        maxrl = len(sd.rl) - 1
    if minrl is None:
        minrl = 0

    prev_bounds_low  = []
    prev_bounds_high = []
    for rl in sd.rl[maxrl:minrl:-1]:
        s = rl.slice(it)
        trim = tuple([slice(rl.ghosts[i], -rl.ghosts[i]) for i in xrange(3)])
        data = s[gf][trim] ** 2
        x = s.x[trim[0]]
        y = s.y[trim[1]]
        z = s.z[trim[2]]

        if not prev_bounds_low:
            while len(data.shape) > 0:
                data = scipy.integrate.simps(data, dx = rl.dx[0])
            results.append(data)
        else:
            res = 0.0
            if prev_bounds_low[0] > x[0]:
                for i, xx in enumerate(x):
                    if prev_bounds_low[0] <= xx:
                        break

                tmp = scipy.integrate.simps(data[:, :, :i], dx = rl.dx[0])
                while len(tmp.shape) > 0:
                    tmp = scipy.integrate.simps(tmp, dx = rl.dx[0])
                res += tmp
            if prev_bounds_low[1] > y[0]:
                for i, yy in enumerate(y):
                    if prev_bounds_low[1] <= yy:
                        break

                tmp = scipy.integrate.simps(data[:, :i], axis = 1, dx = rl.dx[0])
                while len(tmp.shape) > 0:
                    tmp = scipy.integrate.simps(tmp, dx = rl.dx[0])
                res += tmp
            if prev_bounds_low[2] > z[0]:
                for i, zz in enumerate(z):
                    if prev_bounds_low[2] <= zz:
                        break

                tmp = scipy.integrate.simps(data[:i], axis = 0, dx = rl.dx[0])
                while len(tmp.shape) > 0:
                    tmp = scipy.integrate.simps(tmp, dx = rl.dx[0])
                res += tmp
            if prev_bounds_high[0] < x[-1]:
                for i, xx in enumerate(x):
                    if prev_bounds_high[0] >= xx:
                        i -= 1
                        break

                tmp = scipy.integrate.simps(data[:, :, i:], dx = rl.dx[0])
                while len(tmp.shape) > 0:
                    tmp = scipy.integrate.simps(tmp, dx = rl.dx[0])
                res += tmp
            if prev_bounds_high[1] < y[-1]:
                for i, yy in enumerate(y):
                    if prev_bounds_high[1] >= yy:
                        i -= 1
                        break

                tmp = scipy.integrate.simps(data[:, i:], axis = 1, dx = rl.dx[0])
                while len(tmp.shape) > 0:
                    tmp = scipy.integrate.simps(tmp, dx = rl.dx[0])
                res += tmp
            if prev_bounds_high[2] < z[-1]:
                for i, zz in enumerate(z):
                    if prev_bounds_high[2] >= zz:
                        i -= 1
                        break

                tmp = scipy.integrate.simps(data[i:], axis = 0, dx = rl.dx[0])
                while len(tmp.shape) > 0:
                    tmp = scipy.integrate.simps(tmp, dx = rl.dx[0])
                res += tmp

            results.append(res)

        prev_bounds_low  = [x[0],  y[0],  z[0]]
        prev_bounds_high = [x[-1], y[-1], z[-1]]

    return sum(results)


class RefinedSlice:
    # simulation time on this slice
    time = -1
    # simulation iteration on this slice
    it   = -1

    # [x, y, z] coordinates of the origin of the grid on this slice
    o  = None

    # a list of arrays of coordinate points along each direction [x, y, z]
    c = None

    # arrays of coordinate points along each direction
    x = None
    y = None
    z = None

    rl = None

    # private
    _sd = None
    _rl = None

    def __init__(self, sd, rl, time = -1, iteration = -1):
        self._sd = sd
        self._rl = rl
        self.rl  = rl

        if iteration == 'last':
            iteration = rl.itertimes[-1][0]

        if time >= 0:
            for it, t in rl.itertimes:
                if abs(t - time) < EPSILON:
                    self.time = time
                    self.it   = it
                    break
            if self.time < 0:
                raise IndexError('No such time: ' + str(time))
        elif iteration >= 0:
            for it, t in rl.itertimes:
                if it == iteration:
                    self.time = t
                    self.it   = it
                    break
            if self.it < 0:
                raise IndexError('No such iteration: ' + str(iteration))
        else:
            raise TypeError('Neither time nor iteration provided')

        # pick a representative datafile and get the grid properties from it
        if 'H' in self._sd.df:
            df = self._sd.df['H']
        else:
            df = self._sd.df.values()[0]

        gf = get_default_gridfunc(df)
        ds = df['%s it=%d tl=0 rl=%d' % (gf, self.it, rl.n)]

        self.o  = ds.attrs['origin']

        self.c             = [np.linspace(self.o[axis_prop[i]], self.o[axis_prop[i]] + rl.dx[axis_prop[i]] * ds.shape[axis_data[i]], ds.shape[axis_data[i]], endpoint = False) for i in ('x', 'y', 'z')]
        self.x, self.y, self.z = self.c

    def __getitem__(self, key):
        if key in self._sd.df:
            gf = get_default_gridfunc(self._sd.df[key])
            return self._sd.df[key]['%s it=%d tl=0 rl=%d' % (gf, self.it, self._rl.n)]
        elif key in self._sd.gf:
            df = self._sd.gf[key]
            return df['%s it=%d tl=0 rl=%d' % (key, self.it, self._rl.n)]
        raise IndexError


class RefinementLevel:
    # refinement level number
    n  = -1

    # [dx, dy, dz] grid spacings
    dx = None
    # number of ghost gridpoints in [x, y, z] dimensions
    ghosts = None

    # the time step
    dt = 0.0

    # a sorted tuple of all the (iteration, time) pairs for which the data
    # is present on this refinement level
    itertimes = None

    # private
    _sd = None

    def __init__(self, sd, n):
        self._sd = sd
        self.n   = n

    def slice(self, iteration = -1, time = -1):
        return RefinedSlice(self._sd, self, time, iteration)

    def __str__(self):
        return 'Refinement level %d; extent [%g, %g, %g] -- [%g, %g, %g] ([%g, %g, %g] -- [%g, %g, %g]); dx = [%g, %g, %g]; dt = %g' % \
               (self.n, self.x[self.ghosts[0]], self.y[self.ghosts[1]], self.z[self.ghosts[2]],
                self.x[-self.ghosts[0] - 1], self.y[-self.ghosts[1] - 1], self.z[-self.ghosts[2] - 1],
                self.x[0], self.y[0], self.z[0], self.x[-1], self.y[-1], self.z[-1],
                self.dx[0], self.dx[1], self.dx[2], self.dt)


class SimulationData:
    # directory name
    dirname = None

    # datafiles
    df = None

    # a dictionary of all the { gridfunction : datafile it is located in }
    # pairs for this set of data
    gf = None

    # courant factor of the time integration (dx / dt)
    courant = 0

    # per-refinement level parameters
    rl = None

    def __init__(self, dirname):
        self.dirname = os.path.abspath(dirname)
        self.df = {}
        self.gf = {}

        # open all the hdf5 files in the dir
        for f in os.listdir(dirname):
            if not f.endswith('.h5') or f.startswith('checkpoint'):
                continue

            self.df[f[:-3]] = h5py.File('%s/%s' % (dirname, f), 'r')
        if len(self.df) == 0:
            raise ValueError('No HDF5 data files in the directory.')

        for df in self.df.values():
            funcs = df['Parameters and Global Attributes']['Datasets']

            if funcs.dtype.type != np.string_:
                funcs_str = ''.join(map(chr, funcs)).strip('\x00')
            else:
                funcs_str = funcs.value

            for ds in funcs_str.strip().split():
                if ds in self.gf:
                    raise ValueError('Gridfunction %s present in more than one datafile: %s and %s' % (ds, self.gf[ds].filename, df.filename))
                self.gf[ds] = df

        # pick a representative datafile and get the grid properties from it
        if 'H' in self.df:
            df = self.df['H']
        else:
            df = self.df.values()[0]

        gf = get_default_gridfunc(df)

        # get the refinement levels, iterations and times
        self.rl = []
        while True:
            cur_rl = len(self.rl)
            try:
                ds = df['%s it=0 tl=0 rl=%d' % (gf, cur_rl)]
            except KeyError:
                break

            self.rl.append(RefinementLevel(self, len(self.rl)))
            rl = self.rl[-1]

            rl.itertimes = gf_itertime(df, cur_rl)
            rl.dx = ds.attrs['delta']
            rl.ghosts = ds.attrs["cctk_nghostzones"]
            rl.dt = _get_timestep(df) / (1 << (len(self.rl) - 1))

        self.courant = self.rl[0].dx[0] / self.rl[0].dt

    def __del__(self):
        if self.df:
            map(h5py.File.close, self.df.values())

    def calc_velocities(self, get_data, rl = 0, t_start = 0, t_end = float('inf'), offsets = None):
        rl = self.rl[rl]
        dt = rl.itertimes[1][1] - rl.itertimes[0][1]

        ret = []
        for idx, (iter, t) in enumerate(rl.itertimes):
            if t < t_start:
                continue
            if t > t_end:
                break

            data1 = get_data(self, iter)

            if offsets is None:
                step = int(self.courant)
                offsets = xrange(step, 10 * step, step)

            for offset in offsets:
                diff_t   = offset * dt
                try:
                    data2 = get_data(self, rl.itertimes[idx + offset][0])
                except IndexError:
                    break
                mov = calc_movement(data1, data2)

                for it in mov:
                    ret.append(it * rl.dx[0] / diff_t)
        return ret