nifty_core.py 469 KB
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## NIFTY (Numerical Information Field Theory) has been developed at the
## Max-Planck-Institute for Astrophysics.
##
## Copyright (C) 2013 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/>.

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

    .. The NIFTY project homepage is http://www.mpa-garching.mpg.de/ift/nifty/

    NIFTY [#]_, "Numerical Information Field Theory", is a versatile
    library designed to enable the development of signal inference algorithms
    that operate regardless of the underlying spatial grid and its resolution.
    Its object-oriented framework is written in Python, although it accesses
    libraries written in Cython, C++, and C for efficiency.

    NIFTY offers a toolkit that abstracts discretized representations of
    continuous spaces, fields in these spaces, and operators acting on fields
    into classes. Thereby, the correct normalization of operations on fields is
    taken care of automatically without concerning the user. This allows for an
    abstract formulation and programming of inference algorithms, including
    those derived within information field theory. Thus, NIFTY permits its user
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    to rapidly prototype algorithms in 1D and then apply the developed code in
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    higher-dimensional settings of real world problems. The set of spaces on
    which NIFTY operates comprises point sets, n-dimensional regular grids,
    spherical spaces, their harmonic counterparts, and product spaces
    constructed as combinations of those.

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    References
    ----------
    .. [#] Selig et al., "NIFTY -- Numerical Information Field Theory --
        a versatile Python library for signal inference",
        `A&A, vol. 554, id. A26 <http://dx.doi.org/10.1051/0004-6361/201321236>`_,
        2013; `arXiv:1301.4499 <http://www.arxiv.org/abs/1301.4499>`_

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    Class & Feature Overview
    ------------------------
    The NIFTY library features three main classes: **spaces** that represent
    certain grids, **fields** that are defined on spaces, and **operators**
    that apply to fields.

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    .. Overview of all (core) classes:
    ..
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    .. - switch
    .. - notification
    .. - _about
    .. - random
    .. - space
    ..     - point_space
    ..     - rg_space
    ..     - lm_space
    ..     - gl_space
    ..     - hp_space
    ..     - nested_space
    .. - field
    .. - operator
    ..     - diagonal_operator
    ..         - power_operator
    ..     - projection_operator
    ..     - vecvec_operator
    ..     - response_operator
    .. - probing
    ..     - trace_probing
    ..     - diagonal_probing

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    Overview of the main classes and functions:

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    .. automodule:: nifty

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    - :py:class:`space`
        - :py:class:`point_space`
        - :py:class:`rg_space`
        - :py:class:`lm_space`
        - :py:class:`gl_space`
        - :py:class:`hp_space`
        - :py:class:`nested_space`
    - :py:class:`field`
    - :py:class:`operator`
        - :py:class:`diagonal_operator`
            - :py:class:`power_operator`
        - :py:class:`projection_operator`
        - :py:class:`vecvec_operator`
        - :py:class:`response_operator`
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        .. currentmodule:: nifty.nifty_tools
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        - :py:class:`invertible_operator`
        - :py:class:`propagator_operator`
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        .. currentmodule:: nifty.nifty_explicit
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        - :py:class:`explicit_operator`
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    .. automodule:: nifty
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    - :py:class:`probing`
        - :py:class:`trace_probing`
        - :py:class:`diagonal_probing`
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        .. currentmodule:: nifty.nifty_explicit

        - :py:class:`explicit_probing`

    .. currentmodule:: nifty.nifty_tools

    - :py:class:`conjugate_gradient`
    - :py:class:`steepest_descent`

    .. currentmodule:: nifty.nifty_explicit

    - :py:func:`explicify`

    .. currentmodule:: nifty.nifty_power

    - :py:func:`weight_power`,
      :py:func:`smooth_power`,
      :py:func:`infer_power`,
      :py:func:`interpolate_power`
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"""
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## standard libraries
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from __future__ import division
import os
#import sys
from sys import stdout as so
import numpy as np
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import pylab as pl
from matplotlib.colors import LogNorm as ln
from matplotlib.ticker import LogFormatter as lf
from multiprocessing import Pool as mp
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from multiprocessing import Value as mv
from multiprocessing import Array as ma
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## third party libraries
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import gfft as gf
import healpy as hp
import libsharp_wrapper_gl as gl
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## internal libraries
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import smoothing as gs
import powerspectrum as gp


pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679

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__version__ = "0.9.7"
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##-----------------------------------------------------------------------------

class switch(object):
    """
        ..                            __   __               __
        ..                          /__/ /  /_            /  /
        ..     _______  __     __   __  /   _/  _______  /  /___
        ..   /  _____/ |  |/\/  / /  / /  /   /   ____/ /   _   |
        ..  /_____  /  |       / /  / /  /_  /  /____  /  / /  /
        .. /_______/   |__/\__/ /__/  \___/  \______/ /__/ /__/  class

        NIFTY support class for switches.

        Parameters
        ----------
        default : bool
            Default status of the switch (default: False).

        See Also
        --------
        notification : A derived class for displaying notifications.

        Examples
        --------
        >>> option = switch()
        >>> option.status
        False
        >>> option
        OFF
        >>> print(option)
        OFF
        >>> option.on()
        >>> print(option)
        ON

        Attributes
        ----------
        status : bool
            Status of the switch.

    """
    def __init__(self,default=False):
        """
            Initilizes the switch and sets the `status`

            Parameters
            ----------
            default : bool
                Default status of the switch (default: False).

        """
        self.status = bool(default)

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

    def on(self):
        """
            Switches the `status` to True.

        """
        self.status = True

    def off(self):
        """
            Switches the `status` to False.

        """
        self.status = False


    def toggle(self):
        """
            Switches the `status`.

        """
        self.status = not self.status

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

    def __repr__(self):
        if(self.status):
            return "ON"
        else:
            return "OFF"

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

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

class notification(switch):
    """
        ..                           __     __   ____   __                       __     __
        ..                         /  /_  /__/ /   _/ /__/                     /  /_  /__/
        ..     __ ___    ______   /   _/  __  /  /_   __   _______   ____ __  /   _/  __   ______    __ ___
        ..   /   _   | /   _   | /  /   /  / /   _/ /  / /   ____/ /   _   / /  /   /  / /   _   | /   _   |
        ..  /  / /  / /  /_/  / /  /_  /  / /  /   /  / /  /____  /  /_/  / /  /_  /  / /  /_/  / /  / /  /
        .. /__/ /__/  \______/  \___/ /__/ /__/   /__/  \______/  \______|  \___/ /__/  \______/ /__/ /__/  class

        NIFTY support class for notifications.

        Parameters
        ----------
        default : bool
            Default status of the switch (default: False).
        ccode : string
            Color code as string (default: "\033[0m"). The surrounding special
            characters are added if missing.

        Notes
        -----
        The color code is a special ANSI escape code, for a list of valid codes
        see [#]_. Multiple codes can be combined by seperating them with a
        semicolon ';'.

        References
        ----------
        .. [#] Wikipedia, `ANSI escape code <http://en.wikipedia.org/wiki/ANSI_escape_code#graphics>`_.

        Examples
        --------
        >>> note = notification()
        >>> note.status
        True
        >>> note.cprint("This is noteworthy.")
        This is noteworthy.
        >>> note.cflush("12"); note.cflush('3')
        123
        >>> note.off()
        >>> note.cprint("This is noteworthy.")
        >>>

        Raises
        ------
        TypeError
            If `ccode` is no string.

        Attributes
        ----------
        status : bool
            Status of the switch.
        ccode : string
            Color code as string.

    """
    _code = "\033[0m" ## "\033[39;49m"

    def __init__(self,default=True,ccode="\033[0m"):
        """
            Initializes the notification and sets `status` and `ccode`

            Parameters
            ----------
            default : bool
                Default status of the switch (default: False).
            ccode : string
                Color code as string (default: "\033[0m"). The surrounding
                special characters are added if missing.

            Raises
            ------
            TypeError
                If `ccode` is no string.

        """
        self.status = bool(default)

        ## check colour code
        if(not isinstance(ccode,str)):
            raise TypeError(about._errors.cstring("ERROR: invalid input."))
        if(ccode[0]!="\033"):
            ccode = "\033"+ccode
        if(ccode[1]!='['):
            ccode = ccode[0]+'['+ccode[1:]
        if(ccode[-1]!='m'):
            ccode = ccode+'m'
        self.ccode = ccode

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

    def set_ccode(self,newccode=None):
        """
            Resets the the `ccode` string.

            Parameters
            ----------
            newccode : string
                Color code as string (default: "\033[0m"). The surrounding
                characters "\033", '[', and 'm' are added if missing.

            Returns
            -------
            None

            Raises
            ------
            TypeError
                If `ccode` is no string.

            Examples
            --------
            >>> note = notification()
            >>> note.set_ccode("31;1") ## "31;1" corresponds to red and bright

        """
        if(newccode is None):
            newccode = self._code
        else:
            ## check colour code
            if(not isinstance(newccode,str)):
                raise TypeError(about._errors.cstring("ERROR: invalid input."))
            if(newccode[0]!="\033"):
                newccode = "\033"+newccode
            if(newccode[1]!='['):
                newccode = newccode[0]+'['+newccode[1:]
            if(newccode[-1]!='m'):
                newccode = newccode+'m'
        self.ccode = newccode

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

    def cstring(self,subject):
        """
            Casts an object to a string and augments that with a colour code.

            Parameters
            ----------
            subject : {string, object}
                String to be augmented with a color code. A given object is
                cast to its string representation by :py:func:`str`.

            Returns
            -------
            cstring : string
                String augmented with a color code.

        """
        return self.ccode+str(subject)+self._code

    def cflush(self,subject):
        """
            Flushes an object in its colour coded sting representation to the
            standard output (*without* line break).

            Parameters
            ----------
            subject : {string, object}
                String to be flushed. A given object is
                cast to a string by :py:func:`str`.

            Returns
            -------
            None

        """
        if(self.status):
            so.write(self.cstring(subject))
            so.flush()

    def cprint(self,subject):
        """
            Flushes an object in its colour coded sting representation to the
            standard output (*with* line break).

            Parameters
            ----------
            subject : {string, object}
                String to be flushed. A given object is
                cast to a string by :py:func:`str`.

            Returns
            -------
            None

        """
        if(self.status):
            so.write(self.cstring(subject)+"\n")
            so.flush()

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

    def __repr__(self):
        if(self.status):
            return self.cstring("ON")
        else:
            return self.cstring("OFF")

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

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

class _about(object): ## nifty support class for global settings
    """
        NIFTY support class for global settings.

        .. warning::
            Turning off the `_error` notification will suppress all NIFTY error
            strings (not recommended).

        Examples
        --------
        >>> from nifty import *
        >>> about
        nifty version 0.2.0
        >>> print(about)
        nifty version 0.2.0
        - errors          = ON (immutable)
        - warnings        = ON
        - infos           = OFF
        - multiprocessing = ON
        - hermitianize    = ON
        - lm2gl           = ON
        >>> about.infos.on()
        >>> about.about.save_config()

        >>> from nifty import *
        INFO: nifty version 0.2.0
        >>> print(about)
        nifty version 0.2.0
        - errors          = ON (immutable)
        - warnings        = ON
        - infos           = ON
        - multiprocessing = ON
        - hermitianize    = ON
        - lm2gl           = ON

        Attributes
        ----------
        warnings : notification
            Notification instance controlling whether warings shall be printed.
        infos : notification
            Notification instance controlling whether information shall be
            printed.
        multiprocessing : switch
            Switch instance controlling whether multiprocessing might be
            performed.
        hermitianize : switch
            Switch instance controlling whether hermitian symmetry for certain
            :py:class:`rg_space` instances is inforced.
        lm2gl : switch
            Switch instance controlling whether default target of a
            :py:class:`lm_space` instance is a :py:class:`gl_space` or a
            :py:class:`hp_space` instance.

    """
    def __init__(self):
        """
            Initializes the _about and sets the attributes.

        """
        ## version
519
        self._version = str(__version__)
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        ## switches and notifications
        self._errors = notification(default=True,ccode=notification._code)
        self.warnings = notification(default=True,ccode=notification._code)
        self.infos =  notification(default=False,ccode=notification._code)
        self.multiprocessing = switch(default=True)
        self.hermitianize = switch(default=True)
        self.lm2gl = switch(default=True)

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

    def load_config(self,force=True):
        """
            Reads the configuration file "~/.nifty/nifty_config".

            Parameters
            ----------
            force : bool
                Whether to cause an error if the file does not exsist or not.

            Returns
            -------
            None

            Raises
            ------
            ValueError
                If the configuration file is malformed.
            OSError
                If the configuration file does not exist.

        """
        nconfig = os.path.expanduser('~')+"/.nifty/nifty_config"
        if(os.path.isfile(nconfig)):
            rawconfig = []
            with open(nconfig,'r') as configfile:
                for ll in configfile:
                    if(not ll.startswith('#')):
                        rawconfig += ll.split()
            try:
                self._errors = notification(default=True,ccode=rawconfig[0])
                self.warnings = notification(default=int(rawconfig[1]),ccode=rawconfig[2])
                self.infos =  notification(default=int(rawconfig[3]),ccode=rawconfig[4])
                self.multiprocessing = switch(default=int(rawconfig[5]))
                self.hermitianize = switch(default=int(rawconfig[6]))
                self.lm2gl = switch(default=int(rawconfig[7]))
            except(IndexError):
                raise ValueError(about._errors.cstring("ERROR: '"+nconfig+"' damaged."))
        elif(force):
            raise OSError(about._errors.cstring("ERROR: '"+nconfig+"' nonexisting."))

    def save_config(self):
        """
            Writes to the configuration file "~/.nifty/nifty_config".

            Returns
            -------
            None

        """
        rawconfig = [self._errors.ccode[2:-1],str(int(self.warnings.status)),self.warnings.ccode[2:-1],str(int(self.infos.status)),self.infos.ccode[2:-1],str(int(self.multiprocessing.status)),str(int(self.hermitianize.status)),str(int(self.lm2gl.status))]

        nconfig = os.path.expanduser('~')+"/.nifty/nifty_config"
        if(os.path.isfile(nconfig)):
            rawconfig = [self._errors.ccode[2:-1],str(int(self.warnings.status)),self.warnings.ccode[2:-1],str(int(self.infos.status)),self.infos.ccode[2:-1],str(int(self.multiprocessing.status)),str(int(self.hermitianize.status)),str(int(self.lm2gl.status))]
            nconfig = os.path.expanduser('~')+"/.nifty/nifty_config"

            with open(nconfig,'r') as sourcefile:
                with open(nconfig+"_",'w') as targetfile:
                    for ll in sourcefile:
                        if(ll.startswith('#')):
                            targetfile.write(ll)
                        else:
                            ll = ll.replace(ll.split()[0],rawconfig[0]) ## one(!) per line
                            rawconfig = rawconfig[1:]
                            targetfile.write(ll)
            os.rename(nconfig+"_",nconfig) ## overwrite old congiguration
        else:
            if(not os.path.exists(os.path.expanduser('~')+"/.nifty")):
                os.makedirs(os.path.expanduser('~')+"/.nifty")
            with open(nconfig,'w') as targetfile:
                for rr in rawconfig:
                    targetfile.write(rr+"\n") ## one(!) per line

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

    def __repr__(self):
        return "nifty version "+self._version

    def __str__(self):
        return "nifty version "+self._version+"\n- errors          = "+self._errors.cstring("ON")+" (immutable)\n- warnings        = "+str(self.warnings)+"\n- infos           = "+str(self.infos)+"\n- multiprocessing = "+str(self.multiprocessing)+"\n- hermitianize    = "+str(self.hermitianize)+"\n- lm2gl           = "+str(self.lm2gl)

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

## set global instance
about = _about()
about.load_config(force=False)
about.infos.cprint("INFO: "+about.__repr__())





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

class random(object):
    """
        ..                                          __
        ..                                        /  /
        ..       _____   ____ __   __ ___    ____/  /  ______    __ ____ ___
        ..     /   __/ /   _   / /   _   | /   _   / /   _   | /   _    _   |
        ..    /  /    /  /_/  / /  / /  / /  /_/  / /  /_/  / /  / /  / /  /
        ..   /__/     \______| /__/ /__/  \______|  \______/ /__/ /__/ /__/  class

        NIFTY (static) class for pseudo random number generators.

    """
    __init__ = None

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

    @staticmethod
    def arguments(domain,**kwargs):
        """
            Analyses the keyword arguments for supported or necessary ones.

            Parameters
            ----------
            domain : space
                Space wherein the random field values live.
            random : string, *optional*
                Specifies a certain distribution to be drwan from using a
                pseudo random number generator. 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[)

            dev : {scalar, list, ndarray, field}, *optional*
                Standard deviation of the normal distribution if
                ``random == "gau"`` (default: None).
            var : {scalar, list, ndarray, field}, *optional*
                Variance of the normal distribution (outranks the standard
                deviation) if ``random == "gau"`` (default: None).
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            spec : {scalar, list, array, field, function}, *optional*
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                Power spectrum for ``random == "syn"`` (default: 1).
            size : integer, *optional*
                Number of irreducible bands for ``random == "syn"``
                (default: None).
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            pindex : numpy.ndarray, *optional*
                Indexing array giving the power spectrum index of each band
                (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each irreducible band (default: None).
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            vmax : {scalar, list, ndarray, field}, *optional*
                Upper limit of the uniform distribution if ``random == "uni"``
                (default: 1).

            Returns
            -------
            arg : list
                Ordered list of arguments (to be processed in
                ``get_random_values`` of the domain).

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            Other Parameters
            ----------------
            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``;
                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).

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            Raises
            ------
            KeyError
                If the `random` key is not supporrted.

        """
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        if("random" in kwargs):
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            key = kwargs.get("random")
        else:
            return None

        if(key=="pm1"):
            return [key]

        elif(key=="gau"):
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            if("mean" in kwargs):
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                mean = domain.enforce_values(kwargs.get("mean"),extend=False)
            else:
                mean = None
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            if("dev" in kwargs):
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                dev = domain.enforce_values(kwargs.get("dev"),extend=False)
            else:
                dev = None
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            if("var" in kwargs):
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                var = domain.enforce_values(kwargs.get("var"),extend=False)
            else:
                var = None
            return [key,mean,dev,var]

        elif(key=="syn"):
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            ## explicit power indices
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            if("pindex" in kwargs)and("kindex" in kwargs):
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                kindex = kwargs.get("kindex")
                if(kindex is None):
                    spec = domain.enforce_power(kwargs.get("spec",1),size=kwargs.get("size",None))
                    kpack = None
                else:
                    spec = domain.enforce_power(kwargs.get("spec",1),size=len(kindex),kindex=kindex)
                    pindex = kwargs.get("pindex",None)
                    if(pindex is None):
                        kpack = None
                    else:
                        kpack = [pindex,kindex]
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            ## implicit power indices
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            else:
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                try:
                    domain.set_power_indices(**kwargs)
                except:
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                    codomain = kwargs.get("codomain",None)
                    if(codomain is None):
                        spec = domain.enforce_power(kwargs.get("spec",1),size=kwargs.get("size",None))
                        kpack = None
                    else:
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                        domain.check_codomain(codomain)
                        codomain.set_power_indices(**kwargs)
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                        kindex = codomain.power_indices.get("kindex")
                        spec = domain.enforce_power(kwargs.get("spec",1),size=len(kindex),kindex=kindex,codomain=codomain)
                        kpack = [codomain.power_indices.get("pindex"),kindex]
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                else:
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                    kindex = domain.power_indices.get("kindex")
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                    spec = domain.enforce_power(kwargs.get("spec",1),size=len(kindex),kindex=kindex)
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                    kpack = [domain.power_indices.get("pindex"),kindex]
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            return [key,spec,kpack]
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        elif(key=="uni"):
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            if("vmin" in kwargs):
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                vmin = domain.enforce_values(kwargs.get("vmin"),extend=False)
            else:
                vmin = 0
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            if("vmax" in kwargs):
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                vmax = domain.enforce_values(kwargs.get("vmax"),extend=False)
            else:
                vmax = 1
            return [key,vmin,vmax]

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

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

    @staticmethod
    def pm1(datatype=np.int,shape=1):
        """
            Generates random field values according to an uniform distribution
            over {+1,-1} or {+1,+i,-1,-i}, respectively.

            Parameters
            ----------
            datatype : type, *optional*
                Data type of the field values (default: np.int).
            shape : {integer, tuple, list, ndarray}, *optional*
                Split up dimension of the space (default: 1).

            Returns
            -------
            x : ndarray
                Random field values (with correct dtype and shape).

        """
        size = np.prod(shape,axis=0,dtype=np.int,out=None)

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        if(issubclass(datatype,np.complexfloating)):
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            x = np.array([1+0j,0+1j,-1+0j,0-1j],dtype=datatype)[np.random.randint(4,high=None,size=size)]
        else:
            x = 2*np.random.randint(2,high=None,size=size)-1

        return x.astype(datatype).reshape(shape,order='C')

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

    @staticmethod
    def gau(datatype=np.float64,shape=1,mean=None,dev=None,var=None):
        """
            Generates random field values according to a normal distribution.

            Parameters
            ----------
            datatype : type, *optional*
                Data type of the field values (default: np.float64).
            shape : {integer, tuple, list, ndarray}, *optional*
                Split up dimension of the space (default: 1).
            mean : {scalar, ndarray}, *optional*
                Mean of the normal distribution (default: 0).
            dev : {scalar, ndarray}, *optional*
                Standard deviation of the normal distribution (default: 1).
            var : {scalar, ndarray}, *optional*
                Variance of the normal distribution (outranks the standard
                deviation) (default: None).

            Returns
            -------
            x : ndarray
                Random field values (with correct dtype and shape).

            Raises
            ------
            ValueError
                If the array dimension of `mean`, `dev` or `var` mismatch with
                `shape`.

        """
        size = np.prod(shape,axis=0,dtype=np.int,out=None)

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        if(issubclass(datatype,np.complexfloating)):
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            x = np.empty(size,dtype=datatype,order='C')
            x.real = np.random.normal(loc=0,scale=np.sqrt(0.5),size=size)
            x.imag = np.random.normal(loc=0,scale=np.sqrt(0.5),size=size)
        else:
            x = np.random.normal(loc=0,scale=1,size=size)

        if(var is not None):
            if(np.size(var)==1):
                x *= np.sqrt(np.abs(var))
            elif(np.size(var)==size):
                x *= np.sqrt(np.absolute(var).flatten(order='C'))
            else:
                raise ValueError(about._errors.cstring("ERROR: dimension mismatch ( "+str(np.size(var))+" <> "+str(size)+" )."))
        elif(dev is not None):
            if(np.size(dev)==1):
                x *= np.abs(dev)
            elif(np.size(dev)==size):
                x *= np.absolute(dev).flatten(order='C')
            else:
                raise ValueError(about._errors.cstring("ERROR: dimension mismatch ( "+str(np.size(dev))+" <> "+str(size)+" )."))
        if(mean is not None):
            if(np.size(mean)==1):
                x += mean
            elif(np.size(mean)==size):
                x += np.array(mean).flatten(order='C')
            else:
                raise ValueError(about._errors.cstring("ERROR: dimension mismatch ( "+str(np.size(mean))+" <> "+str(size)+" )."))

        return x.astype(datatype).reshape(shape,order='C')

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

    @staticmethod
    def uni(datatype=np.float64,shape=1,vmin=0,vmax=1):
        """
            Generates random field values according to an uniform distribution
            over [vmin,vmax[.

            Parameters
            ----------
            datatype : type, *optional*
                Data type of the field values (default: np.float64).
            shape : {integer, tuple, list, ndarray}, *optional*
                Split up dimension of the space (default: 1).

            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution (default: 0).
            vmax : {scalar, list, ndarray, field}, *optional*
                Upper limit of the uniform distribution (default: 1).

            Returns
            -------
            x : ndarray
                Random field values (with correct dtype and shape).

        """
        size = np.prod(shape,axis=0,dtype=np.int,out=None)
        if(np.size(vmin)>1):
            vmin = np.array(vmin).flatten(order='C')
        if(np.size(vmax)>1):
            vmax = np.array(vmax).flatten(order='C')

        if(datatype in [np.complex64,np.complex128]):
            x = np.empty(size,dtype=datatype,order='C')
            x.real = (vmax-vmin)*np.random.random(size=size)+vmin
            x.imag = (vmax-vmin)*np.random.random(size=size)+vmin
        elif(datatype in [np.int8,np.int16,np.int32,np.int64]):
            x = np.random.randint(min(vmin,vmax),high=max(vmin,vmax),size=size)
        else:
            x = (vmax-vmin)*np.random.random(size=size)+vmin

        return x.astype(datatype).reshape(shape,order='C')

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

    def __repr__(self):
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        return "<nifty_core.random>"
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##-----------------------------------------------------------------------------





##=============================================================================

class space(object):
    """
        ..     _______   ______    ____ __   _______   _______
        ..   /  _____/ /   _   | /   _   / /   ____/ /   __  /
        ..  /_____  / /  /_/  / /  /_/  / /  /____  /  /____/
        .. /_______/ /   ____/  \______|  \______/  \______/  class
        ..          /__/

        NIFTY base class for spaces and their discretizations.

        The base NIFTY space class is an abstract class from which other
        specific space subclasses, including those preimplemented in NIFTY
        (e.g. the regular grid class) must be derived.

        Parameters
        ----------
        para : {single object, list of objects}, *optional*
            This is a freeform list of parameters that derivatives of the space
            class can use (default: 0).
        datatype : numpy.dtype, *optional*
            Data type of the field values for a field defined on this space
            (default: numpy.float64).

        See Also
        --------
        point_space :  A class for unstructured lists of numbers.
        rg_space : A class for regular cartesian grids in arbitrary dimensions.
        hp_space : A class for the HEALPix discretization of the sphere
            [#]_.
        gl_space : A class for the Gauss-Legendre discretization of the sphere
            [#]_.
        lm_space : A class for spherical harmonic components.
        nested_space : A class for product spaces.

        References
        ----------
        .. [#] K.M. Gorski et al., 2005, "HEALPix: A Framework for
               High-Resolution Discretization and Fast Analysis of Data
               Distributed on the Sphere", *ApJ* 622..759G.
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        .. [#] M. Reinecke and D. Sverre Seljebotn, 2013, "Libsharp - spherical
               harmonic transforms revisited";
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               `arXiv:1303.4945 <http://www.arxiv.org/abs/1303.4945>`_
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        Attributes
        ----------
        para : {single object, list of objects}
            This is a freeform list of parameters that derivatives of the space class can use.
        datatype : numpy.dtype
            Data type of the field values for a field defined on this space.
        discrete : bool
            Whether the space is inherently discrete (true) or a discretization
            of a continuous space (false).
        vol : numpy.ndarray
            An array of pixel volumes, only one component if the pixels all
            have the same volume.
    """
    def __init__(self,para=0,datatype=None):
        """
            Sets the attributes for a space class instance.

            Parameters
            ----------
            para : {single object, list of objects}, *optional*
                This is a freeform list of parameters that derivatives of the
                space class can use (default: 0).