nifty_operators.py 144 KB
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## 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/>.

from __future__ import division
import numpy as np
from nifty.nifty_about import about
from nifty.nifty_core import space, \
                         point_space, \
                         nested_space, \
                         field
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from nifty_minimization import conjugate_gradient
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from nifty_probing import trace_probing, \
                            diagonal_probing

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

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

        NIFTY base class for (linear) operators

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

        Parameters
        ----------
        domain : space
            The space wherein valid arguments live.
        sym : bool, *optional*
            Indicates whether the operator is self-adjoint or not
            (default: False)
        uni : bool, *optional*
            Indicates whether the operator is unitary or not
            (default: False)
        imp : bool, *optional*
            Indicates whether the incorporation of volume weights in
            multiplications is already implemented in the `multiply`
            instance methods or not (default: False)
        target : space, *optional*
            The space wherein the operator output lives (default: domain)
        para : {single object, list of objects}, *optional*
            This is a freeform list of parameters that derivatives of the
            operator class can use. Not used in the base operators.
            (default: None)

        See Also
        --------
        diagonal_operator :  An operator class for handling purely diagonal
            operators.
        power_operator : Similar to diagonal_operator but with handy features
            for dealing with diagonal operators whose diagonal
            consists of a power spectrum.
        vecvec_operator : Operators constructed from the outer product of two
            fields
        response_operator : Implements a modeled instrument response which
            translates a signal into data space.
        projection_operator : An operator that projects out one or more
            components in a basis, e.g. a spectral band
            of Fourier components.

        Attributes
        ----------
        domain : space
            The space wherein valid arguments live.
        sym : bool
            Indicates whether the operator is self-adjoint or not
        uni : bool
            Indicates whether the operator is unitary or not
        imp : bool
            Indicates whether the incorporation of volume weights in
            multiplications is already implemented in the `multiply`
            instance methods or not
        target : space
            The space wherein the operator output lives
        para : {single object, list of objects}
            This is a freeform list of parameters that derivatives of the
            operator class can use. Not used in the base operators.
    """
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    def __init__(self, domain, sym=False, uni=False, imp=False, target=None,\
                para=None):
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        """
            Sets the attributes for an operator class instance.

            Parameters
            ----------
            domain : space
                The space wherein valid arguments live.
            sym : bool, *optional*
                Indicates whether the operator is self-adjoint or not
                (default: False)
            uni : bool, *optional*
                Indicates whether the operator is unitary or not
                (default: False)
            imp : bool, *optional*
                Indicates whether the incorporation of volume weights in
                multiplications is already implemented in the `multiply`
                instance methods or not (default: False)
            target : space, *optional*
                The space wherein the operator output lives (default: domain)
            para : {object, list of objects}, *optional*
                This is a freeform list of parameters that derivatives of the
                operator class can use. Not used in the base operators.
                (default: None)

            Returns
            -------
            None
        """
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        ## Check if the domain is realy a space
        if not isinstance(domain,space):
            raise TypeError(about._errors.cstring(
                "ERROR: invalid input. domain is not a space."))
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        self.domain = domain
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        ## Cast the symmetric and unitary input        
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        self.sym = bool(sym)
        self.uni = bool(uni)

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        ## If no target is supplied, we assume that the operator is square
        ## If the operator is symmetric or unitary, we know that the operator 
        ## must be square

        if self.sym == True or self.uni == True:
            target = self.domain 
            if target is not None:
                about.warnings.cprint("WARNING: Ignoring target.")
        
        elif target is None:
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            target = self.domain
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        elif not isinstance(target, space):
            raise TypeError(about._errors.cstring(
            "ERROR: invalid input. Target is not a space."))
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        self.target = target

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        if self.domain.discrete and self.target.discrete:
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            self.imp = True
        else:
            self.imp = bool(imp)

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        self.para = para
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def nrow(self):
        """
            Computes the number of rows.

            Returns
            -------
            nrow : int
                number of rows (equal to the dimension of the codomain)
        """
        return self.target.dim(split=False)

    def ncol(self):
        """
            Computes the number of columns

            Returns
            -------
            ncol : int
                number of columns (equal to the dimension of the domain)
        """
        return self.domain.dim(split=False)

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    def dim(self, axis=None):
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        """
            Computes the dimension of the space

            Parameters
            ----------
            axis : int, *optional*
                Axis along which the dimension is to be calculated.
                (default: None)

            Returns
            -------
            dim : {int, ndarray}
                The dimension(s) of the operator.

        """
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        if axis is None:
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            return np.array([self.nrow(),self.ncol()])
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        elif axis == 0:
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            return self.nrow()
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        elif axis == 1:
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            return self.ncol()
        else:
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            raise ValueError(about._errors.cstring(
                "ERROR: invalid input axis."))
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def set_para(self,newpara):
        """
            Sets the parameters and creates the `para` property if it does
            not exist

            Parameters
            ----------
            newpara : {object, list of objects}
                A single parameter or a list of parameters.

            Returns
            -------
            None

        """
        self.para = newpara

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

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    def _multiply(self, x, **kwargs): 
    ## > applies the operator to a given field
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'multiply'."))
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    def _adjoint_multiply(self, x, **kwargs): 
    ## > applies the adjoint operator to a given field
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'adjoint_multiply'."))
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    def _inverse_multiply(self, x, **kwargs): 
    ## > applies the inverse operator to a given field
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'inverse_multiply'."))
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    def _adjoint_inverse_multiply(self, x, **kwargs): 
    ## > applies the inverse adjoint operator to a given field
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'adjoint_inverse_multiply'."))
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    def _inverse_adjoint_multiply(self, x, **kwargs): 
    ## > applies the adjoint inverse operator to a given field
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'inverse_adjoint_multiply'."))
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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    def _briefing(self, x, domain, inverse): ## > prepares x for `multiply`
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        ## inspect x
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        if not isinstance(x, field):
            x_ = field(domain, val=x)
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        else:
            ## check x.domain
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            if domain == x.domain:
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                x_ = x
            ## transform
            else:
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                x_ = x.transform(target=domain)
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        ## weight if ...
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        if self.imp == False and domain.discrete == False and inverse == False:
            x_ = x_.weight(power=1)
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        return x_

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    def _debriefing(self, x, x_, target, inverse): 
    ## > evaluates x and x_ after `multiply`
        if x_ is None:
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            return None
        else:
            ## inspect x_
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            if not isinstance(x_, field):
                x_ = field(target, val=x_)
            elif x_.domain != target:
                raise ValueError(about._errors.cstring(
                    "ERROR: invalid output domain."))
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            ## weight if ...
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            if self.imp == False and target.discrete == False and\
                inverse == True:
                x_ = x_.weight(power=-1)
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            ## inspect x
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            if isinstance(x, field):
                ## repair if the originally field was living in the codomain 
                ## of the operators domain
                if self.domain == self.target != x.domain:
                    x_ = x_.transform(target=x.domain) 
                if x_.domain == x.domain and (x_.target is not x.target):
                    x_.set_target(newtarget = x.target) 
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            return x_

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

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    def times(self, x, **kwargs):
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        """
            Applies the operator to a given object

            Parameters
            ----------
            x : {scalar, ndarray, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the domain of the operator.

            Returns
            -------
            Ox : field
                Mapped field on the target domain of the operator.
        """
        ## prepare
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        x_ = self._briefing(x, self.domain, inverse=False)
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        ## apply operator
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        x_ = self._multiply(x_, **kwargs)
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        ## evaluate
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        return self._debriefing(x, x_, self.target, inverse=False)
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    def __call__(self, x, **kwargs):
        return self.times(x, **kwargs)
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    def adjoint_times(self, x, **kwargs):
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        """
            Applies the adjoint operator to a given object.

            Parameters
            ----------
            x : {scalar, ndarray, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the target space of the operator.

            Returns
            -------
            OAx : field
                Mapped field on the domain of the operator.

        """
        ## check whether self-adjoint
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        if self.sym == True:
            return self.times(x, **kwargs)
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        ## check whether unitary
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        if self.uni == True:
            return self.inverse_times(x, **kwargs)
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        ## prepare
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        x_ = self._briefing(x, self.target, inverse=False)
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        ## apply operator
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        x_ = self._adjoint_multiply(x_, **kwargs)
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        ## evaluate
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        return self._debriefing(x, x_, self.domain, inverse=False)
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    def inverse_times(self, x, **kwargs):
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        """
            Applies the inverse operator to a given object.

            Parameters
            ----------
            x : {scalar, ndarray, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the domain space of the operator.

            Returns
            -------
            OIx : field
                Mapped field on the target space of the operator.
        """
        ## check whether self-inverse
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        if self.sym == True and self.uni == True:
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            return self.times(x,**kwargs)

        ## prepare
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        x_ = self._briefing(x, self.target, inverse=True)
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        ## apply operator
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        x_ = self._inverse_multiply(x_, **kwargs)
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        ## evaluate
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        return self._debriefing(x, x_, self.domain, inverse=True)
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    def adjoint_inverse_times(self, x, **kwargs):
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        """
            Applies the inverse adjoint operator to a given object.

            Parameters
            ----------
            x : {scalar, ndarray, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the target space of the operator.

            Returns
            -------
            OAIx : field
                Mapped field on the domain of the operator.

        """
        ## check whether self-adjoint
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        if self.sym == True:
            return self.inverse_times(x, **kwargs)
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        ## check whether unitary
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        if self.uni == True:
            return self.times(x, **kwargs)
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        ## prepare
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        x_ = self._briefing(x, self.domain, inverse=True)
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        ## apply operator
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        x_ = self._adjoint_inverse_multiply(x_, **kwargs)
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        ## evaluate
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        return self._debriefing(x, x_, self.target, inverse=True)
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    def inverse_adjoint_times(self, x, **kwargs):
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        """
            Applies the adjoint inverse operator to a given object.

            Parameters
            ----------
            x : {scalar, ndarray, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the target space of the operator.

            Returns
            -------
            OIAx : field
                Mapped field on the domain of the operator.

        """
        ## check whether self-adjoint
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        if self.sym == True:
            return self.inverse_times(x, **kwargs)
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        ## check whether unitary
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        if self.uni == True:
            return self.times(x, **kwargs)
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        ## prepare
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        x_ = self._briefing(x, self.domain, inverse=True)
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        ## apply operator
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        x_ = self._inverse_adjoint_multiply(x_, **kwargs)
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        ## evaluate
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        return self._debriefing(x, x_, self.target, inverse=True)
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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    def tr(self, domain=None, target=None, random="pm1", ncpu=2, nrun=8,\
            nper=1, var=False, loop=False, **kwargs):
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        """
            Computes the trace of the operator

            Parameters
            ----------
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            tr : float
                Trace of the operator
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

            See Also
            --------
            probing : The class used to perform probing operations
        """
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        if domain is None:
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            domain = self.domain
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        return trace_probing(self, 
                             function=self.times, 
                             domain=domain,
                             target=target, 
                             random=random, 
                             ncpu=(ncpu,1)[bool(loop)],
                             nrun=nrun, 
                             nper=nper, 
                             var=var, 
                             **kwargs)(loop=loop)

    def inverse_tr(self, domain=None, target=None, random="pm1", ncpu=2, 
                   nrun=8, nper=1, var=False, loop=False, **kwargs):
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        """
            Computes the trace of the inverse operator

            Parameters
            ----------
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            tr : float
                Trace of the inverse operator
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

            See Also
            --------
            probing : The class used to perform probing operations
        """
        if(domain is None):
            domain = self.target
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        return trace_probing(self, 
                             function=self.inverse_times, 
                             domain=domain, 
                             target=target, 
                             random=random, 
                             ncpu=(ncpu,1)[bool(loop)],
                             nrun=nrun, 
                             nper=nper, 
                             var=var, **kwargs)(loop=loop)
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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    def diag(self, bare=False, domain=None, target=None, random="pm1", ncpu=2,
             nrun=8, nper=1, var=False, save=False, path="tmp", prefix="",
             loop=False, **kwargs):
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        """
            Computes the diagonal of the operator via probing.

            Parameters
            ----------
            bare : bool, *optional*
                Indicatese whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            diag : ndarray
                The matrix diagonal
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
        if(domain is None):
            domain = self.domain
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        diag = diagonal_probing(self, 
                                function=self.times, 
                                domain=domain, 
                                target=target, 
                                random=random, 
                                ncpu=(ncpu,1)[bool(loop)], 
                                nrun=nrun, 
                                nper=nper, 
                                var=var, 
                                save=save, 
                                path=path, 
                                prefix=prefix, 
                                **kwargs)(loop=loop)
        if diag is None:
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#            about.warnings.cprint("WARNING: forwarding 'NoneType'.")
            return None
        ## weight if ...
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        elif domain.discrete == False and bare == True:
            if isinstance(diag, tuple): ## diag == (diag,variance)
                return (domain.calc_weight(diag[0],power=-1),
                        domain.calc_weight(diag[1],power=-1))
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            else:
                return domain.calc_weight(diag,power=-1)
        else:
            return diag

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    def inverse_diag(self, bare=False, domain=None, target=None, random="pm1", 
                     ncpu=2, nrun=8, nper=1, var=False, save=False, path="tmp", 
                     prefix="", loop=False, **kwargs):
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        """
            Computes the diagonal of the inverse operator via probing.

            Parameters
            ----------
            bare : bool, *optional*
                Indicatese whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            diag : ndarray
                The diagonal of the inverse matrix
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
        if(domain is None):
            domain = self.target
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        diag = diagonal_probing(self, 
                                function=self.inverse_times, 
                                domain=domain, 
                                target=target, 
                                random=random, 
                                ncpu=(ncpu,1)[bool(loop)], 
                                nrun=nrun, 
                                nper=nper, 
                                var=var, 
                                save=save, 
                                path=path, 
                                prefix=prefix, 
                                **kwargs)(loop=loop)
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        if(diag is None):
#            about.warnings.cprint("WARNING: forwarding 'NoneType'.")
            return None
        ## weight if ...
        elif(not domain.discrete)and(bare):
            if(isinstance(diag,tuple)): ## diag == (diag,variance)
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                return (domain.calc_weight(diag[0],power=-1),
                        domain.calc_weight(diag[1],power=-1))
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            else:
                return domain.calc_weight(diag,power=-1)
        else:
            return diag

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

    def det(self):
        """
            Computes the determinant of the operator.

            Returns
            -------
            det : float
                The determinant

        """
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        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'det'."))
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    def inverse_det(self):
        """
            Computes the determinant of the inverse operator.

            Returns
            -------
            det : float
                The determinant

        """
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        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'inverse_det'."))
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    def log_det(self):
        """
            Computes the logarithm of the determinant of the operator (if applicable).

            Returns
            -------
            logdet : float
                The logarithm of the determinant

        """
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        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'log_det'."))
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    def tr_log(self):
        """
            Computes the trace of the logarithm of the operator (if applicable).

            Returns
            -------
            logdet : float
                The trace of the logarithm

        """
        return self.log_det()

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

    def hat(self,bare=False,domain=None,target=None,**kwargs):
        """
            Translates the operator's diagonal into a field

            Parameters
            ----------
            bare : bool, *optional*
                Indicatese whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            x : field
                The matrix diagonal as a field living on the operator
                domain space

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
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        if domain is None:
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            domain = self.domain
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        diag = self.diag(bare=bare, domain=domain, target=target, 
                         var=False, **kwargs)
        if diag is None:
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            about.warnings.cprint("WARNING: forwarding 'NoneType'.")
            return None
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        return field(domain, val=diag, target=target)
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    def inverse_hat(self,bare=False,domain=None,target=None,**kwargs):
        """
            Translates the inverse operator's diagonal into a field

            Parameters
            ----------
            bare : bool, *optional*
                Indicatese whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            x : field
                The matrix diagonal as a field living on the operator
                domain space

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
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            domain = self.target
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        diag = self.inverse_diag(bare=bare, domain=domain, target=target, 
                                 var=False, **kwargs)
        if diag is None:
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            about.warnings.cprint("WARNING: forwarding 'NoneType'.")
            return None
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        return field(domain, val=diag, target=target)
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    def hathat(self,domain=None,**kwargs):
        """
            Translates the operator's diagonal into a diagonal operator

            Parameters
            ----------
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            D : diagonal_operator
                The matrix diagonal as an operator

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
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            domain = self.domain
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        diag = self.diag(bare=False, domain=domain, var=False, **kwargs)
        if diag is None:
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            about.warnings.cprint("WARNING: forwarding 'NoneType'.")
            return None
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        return diagonal_operator(domain=domain, diag=diag, bare=False)
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    def inverse_hathat(self,domain=None,**kwargs):
        """
            Translates the inverse operator's diagonal into a diagonal
            operator

            Parameters
            ----------
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            D : diagonal_operator
                The diagonal of the inverse matrix as an operator

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
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            domain = self.target
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        diag = self.inverse_diag(bare=False, domain=domain, 
                                 var=False, **kwargs)
        if diag is None:
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            about.warnings.cprint("WARNING: forwarding 'NoneType'.")
            return None
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        return diagonal_operator(domain=domain, diag=diag, bare=False)
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def __repr__(self):
        return "<nifty_core.operator>"

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



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

class diagonal_operator(operator):
    """
        ..           __   __                                                     __
        ..         /  / /__/                                                   /  /
        ..    ____/  /  __   ____ __   ____ __   ______    __ ___    ____ __  /  /
        ..  /   _   / /  / /   _   / /   _   / /   _   | /   _   | /   _   / /  /
        .. /  /_/  / /  / /  /_/  / /  /_/  / /  /_/  / /  / /  / /  /_/  / /  /_
        .. \______| /__/  \______|  \___   /  \______/ /__/ /__/  \______|  \___/  operator class
        ..                         /______/

        NIFTY subclass for diagonal operators

        Parameters
        ----------
        domain : space, *optional*
            The space wherein valid arguments live. If no domain is given
            then the diag parameter *must* be a field and the domain
            of that field is assumed. (default: None)
        diag : {scalar, ndarray, field}
            The diagonal entries of the operator. For a scalar, a constant
            diagonal is defined having the value provided. If no domain
            is given, diag must be a field. (default: 1)
        bare : bool, *optional*
            whether the diagonal entries are `bare` or not
            (mandatory for the correct incorporation of volume weights)
            (default: False)

        Notes
        -----
        The ambiguity of `bare` or non-bare diagonal entries is based
        on the choice of a matrix representation of the operator in
        question. The naive choice of absorbing the volume weights
        into the matrix leads to a matrix-vector calculus with the
        non-bare entries which seems intuitive, though. The choice of
        keeping matrix entries and volume weights separate deals with the
        bare entries that allow for correct interpretation of the matrix
        entries; e.g., as variance in case of an covariance operator.

        The inverse applications of the diagonal operator feature a ``pseudo``
        flag indicating if zero divison shall be ignored and return zero
        instead of causing an error.

        Attributes
        ----------
        domain : space
            The space wherein valid arguments live.
        val : ndarray
            A field containing the diagonal entries of the matrix.
        sym : bool
            Indicates whether the operator is self-adjoint or not
        uni : bool
            Indicates whether the operator is unitary or not
        imp : bool
            Indicates whether the incorporation of volume weights in
            multiplications is already implemented in the `multiply`
            instance methods or not
        target : space
            The space wherein the operator output lives
    """
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    def __init__(self, domain=None, diag=1, bare=False):
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        """
            Sets the standard operator properties and `values`.

            Parameters
            ----------
            domain : space, *optional*
                The space wherein valid arguments live. If no domain is given
                then the diag parameter *must* be a field and the domain
                of that field is assumed. (default: None)
            diag : {scalar, ndarray, field}, *optional*
                The diagonal entries of the operator. For a scalar, a constant
                diagonal is defined having the value provided. If no domain
                is given, diag must be a field. (default: 1)
            bare : bool, *optional*
                whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)

            Returns
            -------
            None

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

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        """
        ## Set the domain
        if domain is None:
            try:
                self.domain = diag.domain
            except(AttributeError):
                raise TypeError(about._errors.cstring(
                "ERROR: Explicit or implicit (via diag) domain inupt needed!"))                
        
        else:
            self.domain = domain

        self.target = self.domain
            
        ## Set the diag-val
        self.val = self.domain.cast(diag)

        ## Weight if necessary
        if self.domain.discrete == False and bare == True:
            self.val = self.domain.calc_weight(self.val, power = 1)
        
        ## Check complexity attributes
        if self.domain.calc_real_Q(self.val) == True:
            self.sym = True
        else:
            self.sym = False
        
        ## Check if unitary, i.e. identity
        if (self.val == 1).all() == True:
            self.uni = True
        else:
            self.uni = False
        
        self.imp = True
        
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        """
        if(domain is None)and(isinstance(diag,field)):
            domain = diag.domain
        if(not isinstance(domain,space)):
            raise TypeError(about._errors.cstring("ERROR: invalid input."))
        self.domain = domain

        diag = self.domain.enforce_values(diag,extend=True)
        ## weight if ...
        if(not self.domain.discrete)and(bare):
            diag = self.domain.calc_weight(diag,power=1)
        ## check complexity
        if(np.all(np.imag(diag)==0)):
            self.val = np.real(diag)
            self.sym = True
        else:
            self.val = diag
#            about.infos.cprint("INFO: non-self-adjoint complex diagonal operator.")
            self.sym = False

        ## check whether identity
        if(np.all(diag==1)):
            self.uni = True
        else:
            self.uni = False

        self.imp = True ## correctly implemented for efficiency
        self.target = self.domain
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        """
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def set_diag(self,newdiag,bare=False):
        """
            Sets the diagonal of the diagonal operator

            Parameters
            ----------
            newdiag : {scalar, ndarray, field}
                The new diagonal entries of the operator. For a scalar, a
                constant diagonal is defined having the value provided. If
                no domain is given, diag must be a field.

            bare : bool, *optional*
                whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)

            Returns
            -------
            None
        """
        newdiag = self.domain.enforce_values(newdiag,extend=True)
        ## weight if ...
        if(not self.domain.discrete)and(bare):
            newdiag = self.domain.calc_weight(newdiag,power=1)
        ## check complexity
        if(np.all(np.imag(newdiag)==0)):
            self.val = np.real(newdiag)
            self.sym = True
        else:
            self.val = newdiag
#            about.infos.cprint("INFO: non-self-adjoint complex diagonal operator.")
            self.sym = False

        ## check whether identity
        if(np.all(newdiag==1)):
            self.uni = True
        else:
            self.uni = False

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

    def _multiply(self,x,**kwargs): ## > applies the operator to a given field
        x_ = field(self.target,val=None,target=x.target)
        x_.val = x.val*self.val ## bypasses self.domain.enforce_values
        return x_

    def _adjoint_multiply(self,x,**kwargs): ## > applies the adjoint operator to a given field
        x_ = field(self.domain,val=None,target=x.target)
        x_.val = x.val*np.conjugate(self.val) ## bypasses self.domain.enforce_values
        return x_

    def _inverse_multiply(self,x,pseudo=False,**kwargs): ## > applies the inverse operator to a given field
        if(np.any(self.val==0)):
            if(pseudo):
                x_ = field(self.domain,val=None,target=x.target)
                x_.val = np.ma.filled(x.val/np.ma.masked_where(self.val==0,self.val,copy=False),fill_value=0) ## bypasses self.domain.enforce_values
                return x_
            else:
                raise AttributeError(about._errors.cstring("ERROR: singular operator."))
        else:
            x_ = field(self.domain,val=None,target=x.target)
            x_.val = x.val/self.val ## bypasses self.domain.enforce_values
            return x_

    def _adjoint_inverse_multiply(self,x,pseudo=False,**kwargs): ## > applies the inverse adjoint operator to a given field
        if(np.any(self.val==0)):
            if(pseudo):
                x_ = field(self.domain,val=None,target=x.target)
                x_.val = np.ma.filled(x.val/np.ma.masked_where(self.val==0,np.conjugate(self.val),copy=False),fill_value=0) ## bypasses self.domain.enforce_values
                return x_
            else:
                raise AttributeError(about._errors.cstring("ERROR: singular operator."))
        else:
            x_ = field(self.target,val=None,target=x.target)
            x_.val = x.val/np.conjugate(self.val) ## bypasses self.domain.enforce_values
            return x_

    def _inverse_adjoint_multiply(self,x,pseudo=False,**kwargs): ## > applies the adjoint inverse operator to a given field
        if(np.any(self.val==0)):
            if(pseudo):
                x_ = field(self.domain,val=None,target=x.target)
                x_.val = np.ma.filled(x.val/np.conjugate(np.ma.masked_where(self.val==0,self.val,copy=False)),fill_value=0) ## bypasses self.domain.enforce_values
                return x_
            else:
                raise AttributeError(about._errors.cstring("ERROR: singular operator."))

        else:
            x_ = field(self.target,val=None,target=x.target)
            x_.val = x.val*np.conjugate(1/self.val) ## bypasses self.domain.enforce_values
            return x_

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

    def tr(self,domain=None,**kwargs):
        """
            Computes the trace of the operator

            Parameters
            ----------
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            tr : float
                Trace of the operator
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

        """
        if(domain is None)or(domain==self.domain):
            if(self.uni): ## identity
                return (self.domain.datatype(self.domain.dof())).real
            elif(self.domain.dim(split=False)<self.domain.dof()): ## hidden degrees of freedom
                return self.domain.calc_dot(np.ones(self.domain.dim(split=True),dtype=self.domain.datatype,order='C'),self.val) ## discrete inner product
            else:
                return np.sum(self.val,axis=None,dtype=None,out=None)
        else:
            if(self.uni): ## identity
                if(not isinstance(domain,space)):
                    raise TypeError(about._errors.cstring("ERROR: invalid input."))
                ## check degrees of freedom
                if(self.domain.dof()>domain.dof()):
                    about.infos.cprint("INFO: variant numbers of degrees of freedom ( "+str(self.domain.dof())+" / "+str(domain.dof())+" ).")
                return (domain.datatype(domain.dof())).real
            else:
                return super(diagonal_operator,self).tr(domain=domain,**kwargs) ## probing

    def inverse_tr(self,domain=None,**kwargs):
        """
            Computes the trace of the inverse operator

            Parameters
            ----------
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            tr : float
                Trace of the inverse operator
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

        """
        if(np.any(self.val==0)):
            raise AttributeError(about._errors.cstring("ERROR: singular operator."))

        if(domain is None)or(domain==self.target):
            if(self.uni): ## identity
                return np.real(self.domain.datatype(self.domain.dof()))
            elif(self.domain.dim(split=False)<self.domain.dof()): ## hidden degrees of freedom
                return self.domain.calc_dot(np.ones(self.domain.dim(split=True),dtype=self.domain.datatype,order='C'),1/self.val) ## discrete inner product
            else:
                return np.sum(1/self.val,axis=None,dtype=None,out=None)
        else:
            if(self.uni): ## identity
                if(not isinstance(domain,space)):
                    raise TypeError(about._errors.cstring("ERROR: invalid input."))
                ## check degrees of freedom
                if(self.domain.dof()>domain.dof()):
                    about.infos.cprint("INFO: variant numbers of degrees of freedom ( "+str(self.domain.dof())+" / "+str(domain.dof())+" ).")
                return np.real(domain.datatype(domain.dof()))
            else:
                return super(diagonal_operator,self).inverse_tr(domain=domain,**kwargs) ## probing

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

    def diag(self,bare=False,domain=None,**kwargs):
        """
            Computes the diagonal of the operator.

            Parameters
            ----------
            bare : bool, *optional*
                Indicatese whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            diag : ndarray
                The matrix diagonal
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
        if(domain is None)or(domain==self.domain):
            ## weight if ...
            if(not self.domain.discrete)and(bare):
                diag = self.domain.calc_weight(self.val,power=-1)
                ## check complexity
                if(np.all(np.imag(diag)==0)):
                    diag = np.real(diag)
                return diag
            else:
                return self.val
        else:
            if(self.uni): ## identity
                if(not isinstance(domain,space)):
                    raise TypeError(about._errors.cstring("ERROR: invalid input."))
                ## weight if ...
                if(not domain.discrete)and(bare):
                    return np.real(domain.calc_weight(domain.enforce_values(1,extend=True),power=-1))
                else:
                    return np.real(domain.enforce_values(1,extend=True))
            else:
                return super(diagonal_operator,self).diag(bare=bare,domain=domain,**kwargs) ## probing

    def inverse_diag(self,bare=False,domain=None,**kwargs):
        """
            Computes the diagonal of the inverse operator.

            Parameters
            ----------
            bare : bool, *optional*
                Indicatese whether the diagonal entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: False)
            domain : space, *optional*
                space wherein the probes live (default: self.domain)
            target : space, *optional*
                space wherein the transform of the probes live
                (default: None, applies target of the domain)
            random : string, *optional*
                Specifies the pseudo random number generator. Valid
                options are "pm1" for a random vector of +/-1, or "gau"
                for a random vector with entries drawn from a Gaussian
                distribution with zero mean and unit variance.
                (default: "pm1")
            ncpu : int, *optional*
                number of used CPUs to use. (default: 2)
            nrun : int, *optional*
                total number of probes (default: 8)
            nper : int, *optional*
                number of tasks performed by one process (default: 1)
            var : bool, *optional*
                Indicates whether to additionally return the probing variance
                or not (default: False).
            save : bool, *optional*
                whether all individual probing results are saved or not
                (default: False)
            path : string, *optional*
                path wherein the results are saved (default: "tmp")
            prefix : string, *optional*
                prefix for all saved files (default: "")
            loop : bool, *optional*
                Indicates whether or not to perform a loop i.e., to
                parallelise (default: False)

            Returns
            -------
            diag : ndarray
                The diagonal of the inverse matrix
            delta : float, *optional*
                Probing variance of the trace. Returned if `var` is True in
                of probing case.

            See Also
            --------
            probing : The class used to perform probing operations

            Notes
            -----
            The ambiguity of `bare` or non-bare diagonal entries is based
            on the choice of a matrix representation of the operator in
            question. The naive choice of absorbing the volume weights
            into the matrix leads to a matrix-vector calculus with the
            non-bare entries which seems intuitive, though. The choice of
            keeping matrix entries and volume weights separate deals with the
            bare entries that allow for correct interpretation of the matrix
            entries; e.g., as variance in case of an covariance operator.

        """
        if(domain is None)or(domain==self.target):
            ## weight if ...
            if(not self.domain.discrete)and(bare):
                diag = self.domain.calc_weight(1/self.val,power=-1)
                ## check complexity
                if(np.all(np.imag(diag)==0)):
                    diag = np.real(diag)
                return diag
            else:
                return 1/self.val
        else:
            if(self.uni): ## identity
                if(not isinstance(domain,space)):
                    raise TypeError(about._errors.cstring("ERROR: invalid input."))
                ## weight if ...
                if(not domain.discrete)and(bare):
                    return np.real(domain.calc_weight(domain.enforce_values(1,extend=True),power=-1))
                else:
                    return np.real(domain.enforce_values(1,extend=True))
            else:
                return super(diagonal_operator,self).inverse_diag(bare=bare,domain=domain,**kwargs) ## probing

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

    def det(self):
        """
            Computes the determinant of the matrix.

            Returns
            -------
            det : float
                The determinant

        """
        if(self.uni): ## identity
            return 1
        elif(self.domain.dim(split=False)<self.domain.dof()): ## hidden degrees of freedom
            return np.exp(self.domain.calc_dot(np.ones(self.domain.dim(split=True),dtype=self.domain.datatype,order='C'),np.log(self.val)))
        else:
            return np.prod(self.val,axis=None,dtype=None,out=None)

    def inverse_det(self):
        """
            Computes the determinant of the inverse operator.

            Returns
            -------
            det : float
                The determinant

        """
        if(self.uni): ## identity
            return 1
        det = self.det()
        if(det<>0):
            return 1/det
        else:
            raise ValueError(about._errors.cstring("ERROR: singular operator."))

    def log_det(self):
        """
            Computes the logarithm of the determinant of the operator.

            Returns
            -------
            logdet : float
                The logarithm of the determinant

        """
        if(self.uni): ## identity
            return 0
        elif(self.domain.dim(split=False)<self.domain.dof()): ## hidden degrees of freedom
            return self.domain.calc_dot(np.ones(self.domain.dim(split=True),dtype=self.domain.datatype,order='C'),np.log(self.val))
        else:
            return np.sum(np.log(self.val),axis=None,dtype=None,out=None)

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

    def get_random_field(self,domain=None,target=None):
        """
            Generates a Gaussian random field with variance equal to the
            diagonal.

            Parameters
            ----------
            domain : space, *optional*
                space wherein the field lives (default: None, indicates
                to use self.domain)
            target : space, *optional*
                space wherein the transform of the field lives
                (default: None, indicates to use target of domain)

            Returns
            -------
            x : field
                Random field.

        """
        ## weight if ...
        if(not self.domain.discrete):
            diag = self.domain.calc_weight(self.val,power=-1)
            ## check complexity
            if(np.all(np.imag(diag)==0)):
                diag = np.real(diag)
        else:
            diag = self.val

        if(domain is None)or(domain==self.domain):
            return field(self.domain,val=None,target=target,random="gau",var=self.diag(bare=True,domain=self.domain))
        else:
            return field(self.domain,val=None,target=domain,random="gau",var=self.diag(bare=True,domain=self.domain)).transform(target=domain,overwrite=False)

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

    def __repr__(self):
        return "<nifty_core.diagonal_operator>"

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

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

def identity(domain):
    """
        Returns an identity operator.

        The identity operator is represented by a `diagonal_operator` instance,
        which is applicable to a field-like object; i.e., a scalar, list,
        array or field. (The identity operator is unrelated to PYTHON's
        built-in function :py:func:`id`.)

        Parameters
        ----------
        domain : space
            The space wherein valid arguments live.

        Returns
        -------
        id : diagonal_operator
            The identity operator as a `diagonal_operator` instance.

        See Also
        --------
        diagonal_operator

        Examples
        --------
        >>> I = identity(rg_space(8,dist=0.2))
        >>> I.diag()
        array([ 1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.])
        >>> I.diag(bare=True)
        array([ 5.,  5.,  5.,  5.,  5.,  5.,  5.,  5.])
        >>> I.tr()
        8.0
        >>> I(3)
        <nifty.field>
        >>> I(3).val
        array([ 3.,  3.,  3.,  3.,  3.,  3.,  3.,  3.])
        >>> I(np.arange(8))[:]
        array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.])
        >>> f = I.get_random_field()
        >>> print(I(f) - f)
        nifty.field instance
        - domain      = <nifty.rg_space>
        - val         = [...]
          - min.,max. = [0.0, 0.0]
          - med.,mean = [0.0, 0.0]
        - target      = <nifty.rg_space>
        >>> I.times(f) ## equal to I(f)
        <nifty.field>
        >>> I.inverse_times(f)
        <nifty.field>

    """
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    return diagonal_operator(domain=domain, diag=1, bare=False)
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##-----------------------------------------------------------------------------

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

class power_operator(diagonal_operator):
    """
        ..      ______    ______   __     __   _______   _____
        ..    /   _   | /   _   | |  |/\/  / /   __  / /   __/
        ..   /  /_/  / /  /_/  /  |       / /  /____/ /  /
        ..  /   ____/  \______/   |__/\__/  \______/ /__/     operator class
        .. /__/

        NIFTY subclass for (signal-covariance-type) diagonal operators containing a power spectrum

        Parameters
        ----------
        domain : space, *optional*
            The space wherein valid arguments live. If no domain is given
            then the diag parameter *must* be a field and the domain
            of that field is assumed. (default: None)
        spec : {scalar, list, array, field, function}
            The power spectrum. For a scalar, a constant power
            spectrum is defined having the value provided. If no domain
            is given, diag must be a field. (default: 1)
        bare : bool, *optional*
            whether the entries are `bare` or not
            (mandatory for the correct incorporation of volume weights)
            (default: True)
        pindex : ndarray, *optional*
            indexing array, obtainable from domain.get_power_indices
            (default: None)

        Other Parameters
        ----------------
        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).

        Notes
        -----
        The ambiguity of `bare` or non-bare diagonal entries is based
        on the choice of a matrix representation of the operator in
        question. The naive choice of absorbing the volume weights
        into the matrix leads to a matrix-vector calculus with the
        non-bare entries which seems intuitive, though. The choice of
        keeping matrix entries and volume weights separate deals with the
        bare entries that allow for correct interpretation of the matrix
        entries; e.g., as variance in case of an covariance operator.

        Attributes
        ----------
        domain : space
            The space wherein valid arguments live.
        val : ndarray
            A field containing the diagonal entries of the matrix.
        sym : bool
            Indicates whether the operator is self-adjoint or not
        uni : bool
            Indicates whether the operator is unitary or not
        imp : bool
            Indicates whether the incorporation of volume weights in
            multiplications is already implemented in the `multiply`
            instance methods or not
        target : space
            The space wherein the operator output lives

    """
    def __init__(self,domain,spec=1,bare=True,pindex=None,**kwargs):
        """
            Sets the diagonal operator's standard properties

            Parameters
            ----------
            domain : space, *optional*
                The space wherein valid arguments live. If no domain is given
                then the diag parameter *must* be a field and the domain
                of that field is assumed. (default: None)
            spec : {scalar, list, array, field, function}
                The power spectrum. For a scalar, a constant power
                spectrum is defined having the value provided. If no domain
                is given, diag must be a field. (default: 1)
            bare : bool, *optional*
                whether the entries are `bare` or not
                (mandatory for the correct incorporation of volume weights)
                (default: True)
            pindex : ndarray, *optional*
                indexing array, obtainable from domain.get_power_indices
                (default: None)

            Returns
            -------
            None

            Other Parameters
            ----------------
            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).

        """
        if(not isinstance(domain,space)):
            raise TypeError(about._errors.cstring("ERROR: invalid input."))
        self.domain = domain
        ## check implicit pindex
        if(pindex is None):
            try:
                self.domain.set_power_indices(**kwargs)
            except:
                raise ValueError(about._errors.cstring("ERROR: invalid input."))
            else:
                pindex = self.domain.power_indices.get("pindex")
        ## check explicit pindex
        else:
            pindex = np.array(pindex,dtype=np.int)
            if(not np.all(np.array(np.shape(pindex))==self.domain.dim(split=True))):
                raise ValueError(about._errors.cstring("ERROR: shape mismatch ( "+str(np.array(np.shape(pindex)))+" <> "+str(self.domain.dim(split=True))+" )."))
        ## set diagonal
        try:
            #diag = self.domain.enforce_power(spec,size=np.max(pindex,axis=None,out=None)+1)[pindex]
            temp_spec = self.domain.enforce_power(
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                            spec,size=np.max(pindex,axis=None,out=None)+1)
            diag = pindex.apply_scalar_function(lambda x: temp_spec[x],
                                              dtype = temp_spec.dtype.type)
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