nifty_rg.py

# NIFTY (Numerical Information Field Theory) has been developed at the
# Max-Planck-Institute for Astrophysics.
##
# Copyright (C) 2015 Max-Planck-Society
##
# Author: Marco Selig
# Project homepage: <http://www.mpa-garching.mpg.de/ift/nifty/>
##
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
##
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
# See the GNU General Public License for more details.
##
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.

"""
    ..                  __   ____   __
    ..                /__/ /   _/ /  /_
    ..      __ ___    __  /  /_  /   _/  __   __
    ..    /   _   | /  / /   _/ /  /   /  / /  /
    ..   /  / /  / /  / /  /   /  /_  /  /_/  /
    ..  /__/ /__/ /__/ /__/    \___/  \___   /  rg
    ..                               /______/

    NIFTY submodule for regular Cartesian grids.

"""
from __future__ import division

import os
import numpy as np
from scipy.special import erf
import pylab as pl
from matplotlib.colors import LogNorm as ln
from matplotlib.ticker import LogFormatter as lf

from nifty.keepers import about,\
                          global_dependency_injector,\
                          global_configuration
from nifty.nifty_core import point_space,\
                             field
from nifty.nifty_random import random
from nifty.nifty_mpi_data import distributed_data_object
from nifty.nifty_mpi_data import STRATEGIES as DISTRIBUTION_STRATEGIES

from nifty.nifty_paradict import rg_space_paradict
import nifty.nifty_utilities as utilities

import nifty_fft

MPI = global_dependency_injector[global_configuration['mpi_module']]

RG_DISTRIBUTION_STRATEGIES = DISTRIBUTION_STRATEGIES['global']

'''
try:
    import gfft as gf
except(ImportError):
    about.infos.cprint('INFO: "plain" gfft version 0.1.0')
    import gfft_rg as gf
'''


# -----------------------------------------------------------------------------

class rg_space(point_space):
    """
        ..      _____   _______
        ..    /   __/ /   _   /
        ..   /  /    /  /_/  /
        ..  /__/     \____  /  space class
        ..          /______/

        NIFTY subclass for spaces of regular Cartesian grids.

        Parameters
        ----------
        num : {int, numpy.ndarray}
            Number of gridpoints or numbers of gridpoints along each axis.
        naxes : int, *optional*
            Number of axes (default: None).
        zerocenter : {bool, numpy.ndarray}, *optional*
            Whether the Fourier zero-mode is located in the center of the grid
            (or the center of each axis speparately) or not (default: True).
        hermitian : bool, *optional*
            Whether the fields living in the space follow hermitian symmetry or
            not (default: True).
        purelyreal : bool, *optional*
            Whether the field values are purely real (default: True).
        dist : {float, numpy.ndarray}, *optional*
            Distance between two grid points along each axis (default: None).
        fourier : bool, *optional*
            Whether the space represents a Fourier or a position grid
            (default: False).

        Notes
        -----
        Only even numbers of grid points per axis are supported.
        The basis transformations between position `x` and Fourier mode `k`
        rely on (inverse) fast Fourier transformations using the
        :math:`exp(2 \pi i k^\dagger x)`-formulation.

        Attributes
        ----------
        para : numpy.ndarray
            One-dimensional array containing information on the axes of the
            space in the following form: The first entries give the grid-points
            along each axis in reverse order; the next entry is 0 if the
            fields defined on the space are purely real-valued, 1 if they are
            hermitian and complex, and 2 if they are not hermitian, but
            complex-valued; the last entries hold the information on whether
            the axes are centered on zero or not, containing a one for each
            zero-centered axis and a zero for each other one, in reverse order.
        datatype : numpy.dtype
            Data type of the field values for a field defined on this space,
            either ``numpy.float64`` or ``numpy.complex128``.
        discrete : bool
            Whether or not the underlying space is discrete, always ``False``
            for regular grids.
        vol : numpy.ndarray
            One-dimensional array containing the distances between two grid
            points along each axis, in reverse order. By default, the total
            length of each axis is assumed to be one.
        fourier : bool
            Whether or not the grid represents a Fourier basis.
    """
    epsilon = 0.0001  # relative precision for comparisons

    def __init__(self, num, naxes=None, zerocenter=False,
                 complexity=None, hermitian=True, purelyreal=True,
                 dist=None, fourier=False, datamodel='fftw'):
        """
            Sets the attributes for an rg_space class instance.

            Parameters
            ----------
            num : {int, numpy.ndarray}
                Number of gridpoints or numbers of gridpoints along each axis.
            naxes : int, *optional*
                Number of axes (default: None).
            zerocenter : {bool, numpy.ndarray}, *optional*
                Whether the Fourier zero-mode is located in the center of the
                grid (or the center of each axis speparately) or not
                (default: False).
            hermitian : bool, *optional*
                Whether the fields living in the space follow hermitian
                symmetry or not (default: True).
            purelyreal : bool, *optional*
                Whether the field values are purely real (default: True).
            dist : {float, numpy.ndarray}, *optional*
                Distance between two grid points along each axis
                (default: None).
            fourier : bool, *optional*
                Whether the space represents a Fourier or a position grid
                (default: False).

            Returns
            -------
            None
        """
        if complexity is None:
            complexity = 2 - \
                (bool(hermitian) or bool(purelyreal)) - bool(purelyreal)

        if np.isscalar(num):
            num = (num,) * np.asscalar(np.array(naxes))

        self.paradict = rg_space_paradict(num=num,
                                          complexity=complexity,
                                          zerocenter=zerocenter)

        naxes = len(self.paradict['num'])

        # set datatype
        if self.paradict['complexity'] == 0:
            self.datatype = np.float64
        else:
            self.datatype = np.complex128

        # set datamodel
        if datamodel not in ['np'] + RG_DISTRIBUTION_STRATEGIES:
            about.warnings.cprint("WARNING: datamodel set to default.")
            self.datamodel = \
                global_configuration['default_distribution_strategy']
        else:
            self.datamodel = datamodel

        self.discrete = False

        # set volume
        if(dist is None):
            dist = 1 / np.array(self.paradict['num'], dtype=self.datatype)
        elif(np.isscalar(dist)):
            dist = self.datatype(dist) * np.ones(naxes, dtype=self.datatype,
                                                 order='C')
        else:
            dist = np.array(dist, dtype=self.datatype)
            if(np.size(dist) == 1):
                dist = dist * np.ones(naxes, dtype=self.datatype, order='C')
            if(np.size(dist) != naxes):
                raise ValueError(about._errors.cstring(
                    "ERROR: size mismatch ( " + str(np.size(dist)) + " <> " +
                    str(naxes) + " )."))
        if(np.any(dist <= 0)):
            raise ValueError(about._errors.cstring(
                "ERROR: nonpositive distance(s)."))
        self.vol = np.real(dist)

        # TODO: Deprecate self.fourier
        self.fourier = bool(fourier)
        self.harmonic = self.fourier
        # Initializes the fast-fourier-transform machine, which will be used
        # to transform the space
        self.fft_machine = nifty_fft.fft_factory()

        # Initialize the power_indices object which takes care of kindex,
        # pindex, rho and the pundex for a given set of parameters
        if self.harmonic:
            self.power_indices = power_indices(
                                    shape=self.get_shape(),
                                    dgrid=dist,
                                    zerocentered=self.paradict['zerocenter'],
                                    comm=MPI.COMM_WORLD, # TODO: insert output from about.configuration
                                    datamodel=self.datamodel)


    @property
    def para(self):
        temp = np.array(self.paradict['num'] +
                        [self.paradict['complexity']] +
                        self.paradict['zerocenter'], dtype=int)
        return temp

    @para.setter
    def para(self, x):
        self.paradict['num'] = x[:(np.size(x) - 1) // 2]
        self.paradict['zerocenter'] = x[(np.size(x) + 1) // 2:]
        self.paradict['complexity'] = x[(np.size(x) - 1) // 2]


    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
    def copy(self):
        return rg_space(num=self.paradict['num'],
                        complexity=self.paradict['complexity'],
                        zerocenter=self.paradict['zerocenter'],
                        dist=self.vol,
                        fourier=self.harmonic,
                        datamodel=self.datamodel)

    def num(self):
        np.prod(self.get_shape())

    def get_naxes(self):
        """
            Returns the number of axes of the grid.

            Returns
            -------
            naxes : int
                Number of axes of the regular grid.
        """
#        return (np.size(self.para)-1)//2
        return len(self.get_shape())

    def zerocenter(self):
        """
            Returns information on the centering of the axes.

            Returns
            -------
            zerocenter : numpy.ndarray
                Whether the grid is centered on zero for each axis or not.
        """
        # return self.para[-(np.size(self.para)-1)//2:][::-1].astype(np.bool)
        return self.paradict['zerocenter']

    def dist(self):
        """
            Returns the distances between grid points along each axis.

            Returns
            -------
            dist : np.ndarray
                Distances between two grid points on each axis.
        """
        return self.vol

    def get_shape(self):
        return np.array(self.paradict['num'])

    def get_dim(self, split=False):
        """
            Computes the dimension of the space, i.e.\  the number of pixels.

            Parameters
            ----------
            split : bool, *optional*
                Whether to return the dimension split up, i.e. the numbers of
                pixels along each axis, or their product (default: False).

            Returns
            -------
            dim : {int, numpy.ndarray}
                Dimension(s) of the space. If ``split==True``, a
                one-dimensional array with an entry for each axis is returned.
        """
        ## dim = product(n)
        if split == True:
            return self.get_shape()
        else:
            return np.prod(self.get_shape())

    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def get_dof(self):
        """
            Computes the number of degrees of freedom of the space, i.e.\  the
            number of grid points multiplied with one or two, depending on
            complex-valuedness and hermitian symmetry of the fields.

            Returns
            -------
            dof : int
                Number of degrees of freedom of the space.
        """
        # dof ~ dim
        if self.paradict['complexity'] < 2:
            return np.prod(self.paradict['num'])
        else:
            return 2 * np.prod(self.paradict['num'])

#        if(self.para[(np.size(self.para)-1)//2]<2):
#            return np.prod(self.para[:(np.size(self.para)-1)//2],axis=0,dtype=None,out=None)
#        else:
# return
# 2*np.prod(self.para[:(np.size(self.para)-1)//2],axis=0,dtype=None,out=None)

    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def enforce_power(self, spec, size=None, kindex=None, codomain=None,
                      log=False, nbin=None, binbounds=None):
        """
            Provides a valid power spectrum array from a given object.

            Parameters
            ----------
            spec : {float, list, numpy.ndarray, nifty.field, function}
                Fiducial power spectrum from which a valid power spectrum is to
                be calculated. Scalars are interpreted as constant power
                spectra.

            Returns
            -------
            spec : numpy.ndarray
                Valid power spectrum.

            Other parameters
            ----------------
            size : int, *optional*
                Number of bands the power spectrum shall have (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each band.
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on
                logarithmic scale or not; if set, the number of used bins is
                set automatically (if not given otherwise); by default no
                binning is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to
                ``False``; iintegers below the minimum of 3 induce an automatic
                setting; by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).
        """

        # Setting up the local variables: kindex
        # The kindex is only necessary if spec is a function or if
        # the size is not set explicitly
        if kindex is None and (size is None or callable(spec)):
            # Determine which space should be used to get the kindex
            if self.harmonic:
                kindex_supply_space = self
            else:
                # Check if the given codomain is compatible with the space
                try:
                    assert(self.check_codomain(codomain))
                    kindex_supply_space = codomain
                except(AssertionError):
                    about.warnings.cprint("WARNING: Supplied codomain is " +
                                          "incompatible. Generating a " +
                                          "generic codomain. This can " +
                                          "be expensive!")
                    kindex_supply_space = self.get_codomain()
            kindex = kindex_supply_space.\
                power_indices.get_index_dict(log=log, nbin=nbin,
                                             binbounds=binbounds)['kindex']

        # Now it's about to extract a powerspectrum from spec
        # First of all just extract a numpy array. The shape is cared about
        # later.

        # Case 1: spec is a function
        if callable(spec) == True:
            # Try to plug in the kindex array in the function directly
            try:
                spec = np.array(spec(kindex), dtype=self.datatype)
            except:
                # Second try: Use a vectorized version of the function.
                # This is slower, but better than nothing
                try:
                    spec = np.vectorize(spec)(kindex)
                except:
                    raise TypeError(about._errors.cstring(
                        "ERROR: invalid power spectra function."))

        # Case 2: spec is a field:
        elif isinstance(spec, field):
            spec = spec[:]
            spec = np.array(spec, dtype=self.datatype).flatten()

        # Case 3: spec is a scalar or something else:
        else:
            spec = np.array(spec, dtype=self.datatype).flatten()

        # Make some sanity checks
        # Drop imaginary part
        temp_spec = np.real(spec)
        try:
            np.testing.assert_allclose(spec, temp_spec)
        except(AssertionError):
            about.warnings.cprint("WARNING: Dropping imaginary part.")
        spec = temp_spec

        # check finiteness
        if not np.all(np.isfinite(spec)):
            about.warnings.cprint("WARNING: infinite value(s).")

        # check positivity (excluding null)
        if np.any(spec < 0):
            raise ValueError(about._errors.cstring(
                "ERROR: nonpositive value(s)."))
        if np.any(spec == 0):
            about.warnings.cprint("WARNING: nonpositive value(s).")

        # Set the size parameter
        if size == None:
            size = len(kindex)

        # Fix the size of the spectrum
        # If spec is singlevalued, expand it
        if np.size(spec) == 1:
            spec = spec * np.ones(size, dtype=spec.dtype, order='C')
        # If the size does not fit at all, throw an exception
        elif np.size(spec) < size:
            raise ValueError(about._errors.cstring("ERROR: size mismatch ( " +
                                                   str(np.size(spec)) + " < " + str(size) + " )."))
        elif np.size(spec) > size:
            about.warnings.cprint("WARNING: power spectrum cut to size ( == " +
                                  str(size) + " ).")
            spec = spec[:size]

        return spec

    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def set_power_indices(self, log=False, nbin=None, binbounds=None, **kwargs):
        """
            Sets the (un)indexing objects for spectral indexing internally.

            Parameters
            ----------
            log : bool
                Flag specifying if the binning is performed on logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer
                Number of used bins; if given `log` is set to ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).

            Returns
            -------
            None

            See Also
            --------
            get_power_indices

            Raises
            ------
            AttributeError
                If ``self.harmonic == False``.
            ValueError
                If the binning leaves one or more bins empty.

        """

        about.warnings.cflush("WARNING: set_power_indices is a deprecated" +
                              "function. Please use the interface of" +
                              "self.power_indices in future!")
        self.power_indices.set_default(log=log, nbin=nbin, binbounds=binbounds)
        return None

    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
    def _cast_to_d2o(self, x, dtype=None, ignore_complexity=False,
                     verbose=False, **kwargs):
        """
            Computes valid field values from a given object, trying
            to translate the given data into a valid form. Thereby it is as
            benevolent as possible.

            Parameters
            ----------
            x : {float, numpy.ndarray, nifty.field}
                Object to be transformed into an array of valid field values.

            Returns
            -------
            x : numpy.ndarray, distributed_data_object
                Array containing the field values, which are compatible to the
                space.

            Other parameters
            ----------------
            verbose : bool, *optional*
                Whether the method should raise a warning if information is
                lost during casting (default: False).
        """
        if dtype is None:
            dtype = self.datatype
        # Case 1: x is a field
        if isinstance(x, field):
            if verbose:
                # Check if the domain matches
                if(self != x.domain):
                    about.warnings.cflush(
                        "WARNING: Getting data from foreign domain!")
            # Extract the data, whatever it is, and cast it again
            return self.cast(x.val, dtype=dtype)

        # Case 2: x is a distributed_data_object
        if isinstance(x, distributed_data_object):
            # Check the shape
            if np.any(x.shape != self.get_shape()):
                # Check if at least the number of degrees of freedom is equal
                if x.get_dim() == self.get_dim():
                    # If the number of dof is equal or 1, use np.reshape...
                    about.warnings.cflush(
                        "WARNING: Trying to reshape the data. This operation is " +
                        "expensive as it consolidates the full data!\n")
                    temp = x.get_full_data()
                    temp = np.reshape(temp, self.get_shape())
                    # ... and cast again
                    return self.cast(temp, dtype=dtype)

                else:
                    raise ValueError(about._errors.cstring(
                        "ERROR: Data has incompatible shape!"))

            # Check the datatype
            if np.dtype(x.dtype) != np.dtype(dtype):
                if np.dtype(x.dtype) > np.dtype(dtype):
                    about.warnings.cflush(
                        "WARNING: Datatypes are uneqal/of conflicting precision (own: "
                        + str(dtype) + " <> foreign: " + str(x.dtype)
                        + ") and will be casted! "
                        + "Potential loss of precision!\n")
                temp = x.copy_empty(dtype=dtype)
                temp.set_local_data(x.get_local_data())
                temp.hermitian = x.hermitian
                x = temp

            if ignore_complexity == False:
                # Check hermitianity/reality
                if self.paradict['complexity'] == 0:
                    if x.iscomplex().any() == True:
                        about.warnings.cflush(
                            "WARNING: Data is not completely real. Imaginary part " +
                            "will be discarded!\n")
                        temp = x.copy_empty()
                        temp.set_local_data(np.real(x.get_local_data()))
                        x = temp

                elif self.paradict['complexity'] == 1:
                    if x.hermitian == False and about.hermitianize.status == True:
                        about.warnings.cflush(
                            "WARNING: Data gets hermitianized. This operation is " +
                            "extremely expensive\n")
                        #temp = x.copy_empty()
                        # temp.set_full_data(gp.nhermitianize_fast(x.get_full_data(),
                        #    (False, )*len(x.shape)))
                        x = utilities.hermitianize(x)

            return x

        # Case 3: x is something else
        # Use general d2o casting
        x = distributed_data_object(x, global_shape=self.get_shape(),
                                    dtype=dtype)
        # Cast the d2o
        return self.cast(x, dtype=dtype)

    def _cast_to_np(self, x, dtype=None, ignore_complexity=False,
                    verbose=False, **kwargs):
        """
            Computes valid field values from a given object, trying
            to translate the given data into a valid form. Thereby it is as
            benevolent as possible.

            Parameters
            ----------
            x : {float, numpy.ndarray, nifty.field}
                Object to be transformed into an array of valid field values.

            Returns
            -------
            x : numpy.ndarray, distributed_data_object
                Array containing the field values, which are compatible to the
                space.

            Other parameters
            ----------------
            verbose : bool, *optional*
                Whether the method should raise a warning if information is
                lost during casting (default: False).
        """
        if dtype is None:
            dtype = self.datatype
        # Case 1: x is a field
        if isinstance(x, field):
            if verbose:
                # Check if the domain matches
                if(self != x.domain):
                    about.warnings.cflush(
                        "WARNING: Getting data from foreign domain!")
            # Extract the data, whatever it is, and cast it again
            return self.cast(x.val, dtype=dtype)

        # Case 2: x is a distributed_data_object
        if isinstance(x, distributed_data_object):
            # Extract the data
            temp = x.get_full_data()
            # Cast the resulting numpy array again
            return self.cast(temp, dtype=dtype)

        elif isinstance(x, np.ndarray):
            # Check the shape
            if np.any(x.shape != self.get_shape()):
                # Check if at least the number of degrees of freedom is equal
                if x.size == self.get_dim():
                    # If the number of dof is equal or 1, use np.reshape...
                    temp = x.reshape(self.get_shape())
                    # ... and cast again
                    return self.cast(temp, dtype=dtype)
                elif x.size == 1:
                    temp = np.empty(shape=self.get_shape(),
                                    dtype=dtype)
                    temp[:] = x
                    return self.cast(temp, dtype=dtype)
                else:
                    raise ValueError(about._errors.cstring(
                        "ERROR: Data has incompatible shape!"))

            # Check the datatype
            if x.dtype != dtype:
                about.warnings.cflush(
                    "WARNING: Datatypes are uneqal/of conflicting precision (own: "
                    + str(dtype) + " <> foreign: " + str(x.dtype)
                    + ") and will be casted! "
                    + "Potential loss of precision!\n")
                # Fix the datatype...
                temp = x.astype(dtype)
                # ... and cast again
                return self.cast(temp, dtype=dtype)

            if ignore_complexity == False:
                # Check hermitianity/reality
                if self.paradict['complexity'] == 0:
                    if not np.all(np.isreal(x)) == True:
                        about.warnings.cflush(
                            "WARNING: Data is not completely real. Imaginary part " +
                            "will be discarded!\n")
                        x = np.real(x)

                elif self.paradict['complexity'] == 1:
                    if about.hermitianize.status == True:
                        about.warnings.cflush(
                            "WARNING: Data gets hermitianized. This operation is " +
                            "rather expensive.\n")
                        #temp = x.copy_empty()
                        # temp.set_full_data(gp.nhermitianize_fast(x.get_full_data(),
                        #    (False, )*len(x.shape)))
                        x = utilities.hermitianize(x)

            return x

        # Case 3: x is something else
        # Use general numpy casting
        else:
            temp = np.empty(self.get_shape(), dtype=dtype)
            temp[:] = x
            return temp

    def enforce_values(self, x, extend=True):
        """
            Computes valid field values from a given object, taking care of
            data types, shape, and symmetry.

            Parameters
            ----------
            x : {float, numpy.ndarray, nifty.field}
                Object to be transformed into an array of valid field values.

            Returns
            -------
            x : numpy.ndarray
                Array containing the valid field values.

            Other parameters
            ----------------
            extend : bool, *optional*
                Whether a scalar is extented to a constant array or not
                (default: True).
        """
        about.warnings.cflush(
            "WARNING: enforce_values is deprecated function. Please use self.cast")
        if(isinstance(x, field)):
            if(self == x.domain):
                if(self.datatype is not x.domain.datatype):
                    raise TypeError(about._errors.cstring("ERROR: inequal data types ( '" + str(
                        np.result_type(self.datatype)) + "' <> '" + str(np.result_type(x.domain.datatype)) + "' )."))
                else:
                    x = np.copy(x.val)
            else:
                raise ValueError(about._errors.cstring(
                    "ERROR: inequal domains."))
        else:
            if(np.size(x) == 1):
                if(extend):
                    x = self.datatype(
                        x) * np.ones(self.get_dim(split=True), dtype=self.datatype, order='C')
                else:
                    if(np.isscalar(x)):
                        x = np.array([x], dtype=self.datatype)
                    else:
                        x = np.array(x, dtype=self.datatype)
            else:
                x = self.enforce_shape(np.array(x, dtype=self.datatype))

        # hermitianize if ...
        if(about.hermitianize.status)and(np.size(x) != 1)and(self.para[(np.size(self.para) - 1) // 2] == 1):
            #x = gp.nhermitianize_fast(x,self.para[-((np.size(self.para)-1)//2):].astype(np.bool),special=False)
            x = utilities.hermitianize(x)
        # check finiteness
        if(not np.all(np.isfinite(x))):
            about.warnings.cprint("WARNING: infinite value(s).")

        return x

    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def get_random_values(self, **kwargs):
        """
            Generates random field values according to the specifications given
            by the parameters, taking into account possible complex-valuedness
            and hermitian symmetry.

            Returns
            -------
            x : numpy.ndarray
                Valid field values.

            Other parameters
            ----------------
            random : string, *optional*
                Specifies the probability distribution from which the random
                numbers are to be drawn.
                Supported distributions are:

                - "pm1" (uniform distribution over {+1,-1} or {+1,+i,-1,-i}
                - "gau" (normal distribution with zero-mean and a given standard
                    deviation or variance)
                - "syn" (synthesizes from a given power spectrum)
                - "uni" (uniform distribution over [vmin,vmax[)

                (default: None).
            dev : float, *optional*
                Standard deviation (default: 1).
            var : float, *optional*
                Variance, overriding `dev` if both are specified
                (default: 1).
            spec : {scalar, list, numpy.ndarray, nifty.field, function}, *optional*
                Power spectrum (default: 1).
            pindex : numpy.ndarray, *optional*
                Indexing array giving the power spectrum index of each band
                (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each band (default: None).
            codomain : nifty.rg_space, *optional*
                A compatible codomain (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution if ``random == "uni"``
                (default: 0).
            vmin : float, *optional*
                Lower limit for a uniform distribution (default: 0).
            vmax : float, *optional*
                Upper limit for a uniform distribution (default: 1).
        """
        # TODO: Without hermitianization the random-methods from pointspace
        # could be used.

        # Parse the keyword arguments
        arg = random.parse_arguments(self, **kwargs)

        # Prepare the empty distributed_data_object
        sample = distributed_data_object(global_shape=self.get_shape(),
                                         dtype=self.datatype)

        # Should the output be hermitianized? This does not depend on the
        # hermitianize boolean in about, as it would yield in wrong,
        # not recoverable results

        #hermitianizeQ = about.hermitianize.status and self.paradict['complexity']
        hermitianizeQ = self.paradict['complexity']

        # Case 1: uniform distribution over {-1,+1}/{1,i,-1,-i}
        if arg[0] == 'pm1' and hermitianizeQ == False:
            gen = lambda s: random.pm1(datatype=self.datatype,
                                       shape=s)
            sample.apply_generator(gen)

        elif arg[0] == 'pm1' and hermitianizeQ == True:
            sample = self.get_random_values(random='uni', vmin=-1, vmax=1)
            local_data = sample.get_local_data()
            if issubclass(sample.dtype, np.complexfloating):
                temp_data = local_data.copy()
                local_data[temp_data.real >= 0.5] = 1
                local_data[(temp_data.real >= 0) * (temp_data.real < 0.5)] = -1
                local_data[(temp_data.real < 0) * (temp_data.imag >= 0)] = 1j
                local_data[(temp_data.real < 0) * (temp_data.imag < 0)] = -1j
            else:
                local_data[local_data >= 0] = 1
                local_data[local_data < 0] = -1
            sample.set_local_data(local_data)

        # Case 2: normal distribution with zero-mean and a given standard
        ##         deviation or variance
        elif arg[0] == 'gau':
            var = arg[3]
            if np.isscalar(var) == True or var is None:
                processed_var = var
            else:
                try:
                    processed_var = sample.distributor.extract_local_data(var)
                except(AttributeError):
                    processed_var = var

            gen = lambda s: random.gau(datatype=self.datatype,
                                       shape=s,
                                       mean=arg[1],
                                       dev=arg[2],
                                       var=processed_var)
            sample.apply_generator(gen)

            if hermitianizeQ == True:
                sample = utilities.hermitianize(sample)

        # Case 3: uniform distribution
        elif arg[0] == "uni" and hermitianizeQ == False:
            gen = lambda s: random.uni(datatype=self.datatype,
                                       shape=s,
                                       vmin=arg[1],
                                       vmax=arg[2])
            sample.apply_generator(gen)

        elif arg[0] == "uni" and hermitianizeQ == True:
            # For a hermitian uniform sample, generate a gaussian one
            # and then convert it to a uniform one
            sample = self.get_random_values(random='gau')
            # Use the cummulative of the gaussian, the error function in order
            # to transform it to a uniform distribution.
            if issubclass(sample.dtype, np.complexfloating):
                temp_func = lambda x: erf(x.real) + 1j * erf(x.imag)
            else:
                temp_func = lambda x: erf(x / np.sqrt(2))
            sample.apply_scalar_function(function=temp_func,
                                         inplace=True)

            # Shift and stretch the uniform distribution into the given limits
            ## sample = (sample + 1)/2 * (vmax-vmin) + vmin
            vmin = arg[1]
            vmax = arg[2]
            sample *= (vmax - vmin) / 2.
            sample += 1 / 2. * (vmax + vmin)

        elif(arg[0] == "syn"):
            spec = arg[1]
            kpack = arg[2]
            harmonic_domain = arg[3]
            log = arg[4]
            nbin = arg[5]
            binbounds = arg[6]
            # Check whether there is a kpack available or not.
            # kpack is only used for computing kdict and extracting kindex
            # If not, take kdict and kindex from the fourier_domain
            if kpack == None:
                power_indices =\
                    harmonic_domain.power_indices.get_index_dict(log=log,
                                                                 nbin=nbin,
                                                                 binbounds=binbounds)

                kindex = power_indices['kindex']
                kdict = power_indices['kdict']
                kpack = [power_indices['pindex'], power_indices['kindex']]
            else:
                kindex = kpack[1]
                kdict = harmonic_domain.power_indices.\
                    _compute_kdict_from_pindex_kindex(kpack[0], kpack[1])

            # draw the random samples
            # Case 1: self is a harmonic space
            if self.harmonic:
                # subcase 1: self is real
                # -> simply generate a random field in fourier space and
                # weight the entries accordingly to the powerspectrum
                if self.paradict['complexity'] == 0:
                    # set up the sample object. Overwrite the default from
                    # above to be sure, that the distribution strategy matches
                    # with the one from kdict
                    sample = kdict.copy_empty(dtype=self.datatype)
                    # set up the random number generator
                    gen = lambda s: np.random.normal(loc=0, scale=1, size=s)
                    # apply the random number generator
                    sample.apply_generator(gen)

                # subcase 2: self is hermitian but probably complex
                # -> generate a real field (in position space) and transform
                # it to harmonic space -> field in harmonic space is
                # hermitian. Now weight the modes accordingly to the
                # powerspectrum.
                elif self.paradict['complexity'] == 1:
                    temp_codomain = self.get_codomain()
                    # set up the sample object. Overwrite the default from
                    # above to be sure, that the distribution strategy matches
                    # with the one from kdict
                    sample = kdict.copy_empty(
                        dtype=temp_codomain.datatype)
                    # set up the random number generator

                    gen = lambda s: np.random.normal(loc=0, scale=1, size=s)
                    # apply the random number generator
                    sample.apply_generator(gen)

                    # In order to get the normalisation right, the sqrt
                    # of self.dim must be divided out.
                    # Furthermore, the normalisation in the fft routine
                    # must be undone
                    # TODO: Insert explanation
                    sqrt_of_dim = np.sqrt(self.get_dim())
                    sample /= sqrt_of_dim
                    sample = temp_codomain.calc_weight(sample, power=-1)

                    # tronsform the random field to harmonic space
                    sample = temp_codomain.\
                        calc_transform(sample, codomain=self)

                    # ensure that the kdict and the harmonic_sample have the
                    # same distribution strategy
                    assert(kdict.distribution_strategy ==
                           sample.distribution_strategy)

                # subcase 3: self is fully complex
                # -> generate a complex random field in harmonic space and
                # weight the modes accordingly to the powerspectrum
                elif self.paradict['complexity'] == 2:
                    # set up the sample object. Overwrite the default from
                    # above to be sure, that the distribution strategy matches
                    # with the one from kdict
                    sample = kdict.copy_empty(dtype=self.datatype)
                    # set up the random number generator
                    gen = lambda s: (
                        np.random.normal(loc=0, scale=1 / np.sqrt(2), size=s) +
                        np.random.normal(loc=0, scale=1 /
                                         np.sqrt(2), size=s) * 1.j
                    )
                    # apply the random number generator
                    sample.apply_generator(gen)

                # apply the powerspectrum renormalization
                # therefore extract the local data from kdict
                local_kdict = kdict.get_local_data()
                rescaler = np.sqrt(
                    spec[np.searchsorted(kindex, local_kdict)])
                sample.apply_scalar_function(lambda x: x * rescaler,
                                             inplace=True)
            # Case 2: self is a position space
            else:
                # get a suitable codomain
                temp_codomain = self.get_codomain()

                # subcase 1: self is a real space.
                # -> generate a hermitian sample with the codomain in harmonic
                # space and make a fourier transformation.
                if self.paradict['complexity'] == 0:
                    # check that the codomain is hermitian
                    assert(temp_codomain.paradict['complexity'] == 1)

                # subcase 2: self is hermitian but probably complex
                # -> generate a real-valued random sample in fourier space
                # and transform it to real space
                elif self.paradict['complexity'] == 1:
                    # check that the codomain is real
                    assert(temp_codomain.paradict['complexity'] == 0)

                # subcase 3: self is fully complex
                # -> generate a complex-valued random sample in fourier space
                # and transform it to real space
                elif self.paradict['complexity'] == 2:
                    # check that the codomain is real
                    assert(temp_codomain.paradict['complexity'] == 2)

                # Get a hermitian/real/complex sample in harmonic space from
                # the codomain
                sample = temp_codomain.get_random_values(
                    random='syn',
                    pindex=kpack[0],
                    kindex=kpack[1],
                    spec=spec,
                    codomain=self,
                    log=log,
                    nbin=nbin,
                    binbounds=binbounds
                )
                # Correct the weighting
                #sample = self.calc_weight(sample, power=-1)

                # Take the fourier transform
                sample = temp_codomain.calc_transform(sample,
                                                      codomain=self)

            if self.paradict['complexity'] == 1:
                sample.hermitian = True

        else:
            raise KeyError(about._errors.cstring(
                "ERROR: unsupported random key '" + str(arg[0]) + "'."))

        return sample

    # +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def check_codomain(self, codomain):
        """
            Checks whether a given codomain is compatible to the space or not.

            Parameters
            ----------
            codomain : nifty.space
                Space to be checked for compatibility.

            Returns
            -------
            check : bool
                Whether or not the given codomain is compatible to the space.
        """
        if codomain is None:
            return False

        if not isinstance(codomain, rg_space):
            raise TypeError(about._errors.cstring("ERROR: invalid input."))

        if self.datamodel is not codomain.datamodel:
            return False
        # check number of number and size of axes
        if not np.all(self.paradict['num'] == codomain.paradict['num']):
            return False

        # check fourier flag
        if self.harmonic == codomain.fourier:
            return False

        # check complexity-type
        # prepare the shorthands
        dcomp = self.paradict['complexity']
        cocomp = codomain.paradict['complexity']