nifty_tools.py 41.7 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/>.

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

    A nifty set of tools.

    ## TODO: *DESCRIPTION*

"""
from __future__ import division
#import numpy as np
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from nifty_core import *
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##-----------------------------------------------------------------------------

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

        NIFTY subclass for invertible, self-adjoint (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.
        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).
        para : {single object, tuple/list of objects}, *optional*
            This is a freeform tuple/list of parameters that derivatives of
            the operator class can use (default: None).

        See Also
        --------
        operator

        Notes
        -----
        Operator classes derived from this one only need a `_multiply` or
        `_inverse_multiply` instance method to perform the other. However, one
        of them needs to be defined.

        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 tuple/list of parameters that derivatives of
            the operator class can use. Not used in the base operators.

    """
    def __init__(self,domain,uni=False,imp=False,para=None):
        """
            Sets the standard operator properties.

            Parameters
            ----------
            domain : space
                The space wherein valid arguments live.
            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).
            para : {single object, tuple/list of objects}, *optional*
                This is a freeform tuple/list of parameters that derivatives of
                the operator class can use (default: None).

        """
        if(not isinstance(domain,space)):
            raise TypeError(about._errors.cstring("ERROR: invalid input."))
        self.domain = domain
        self.sym = True
        self.uni = bool(uni)

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

        self.target = target

        if(para is not None):
            self.para = para

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

    def times(self,x,force=False,W=None,spam=None,reset=None,note=False,x0=None,tol=1E-4,clevel=1,limii=None,**kwargs):
        """
            Applies the propagator to a given object.

            Parameters
            ----------
            x : {scalar, list, array, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the domain of the operator.
            force : bool
                Indicates wheter to return a field instead of ``None`` is
                forced incase the conjugate gradient fails.

            Returns
            -------
            Ox : field
                Mapped field with suitable domain.

            See Also
            --------
            conjugate_gradient

            Other Parameters
            ----------------
            W : {operator, function}, *optional*
                Operator `W` that is a preconditioner on `A` and is applicable to a
                field (default: None).
            spam : function, *optional*
                Callback function which is given the current `x` and iteration
                counter each iteration (default: None).
            reset : integer, *optional*
                Number of iterations after which to restart; i.e., forget previous
                conjugated directions (default: sqrt(b.dim())).
            note : bool, *optional*
                Indicates whether notes are printed or not (default: False).
            x0 : field, *optional*
                Starting guess for the minimization.
            tol : scalar, *optional*
                Tolerance specifying convergence; measured by current relative
                residual (default: 1E-4).
            clevel : integer, *optional*
                Number of times the tolerance should be undershot before
                exiting (default: 1).
            limii : integer, *optional*
                Maximum number of iterations performed (default: 10 * b.dim()).

        """
        ## prepare
        x_ = self._briefing(x,self.domain,False)
        ## apply operator
        if(self.imp):
            A = self._inverse_multiply
        else:
            A = self.inverse_times
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        x_,convergence = conjugate_gradient(A,x_,W=W,spam=spam,reset=reset,note=note)(x0=x0,tol=tol,clevel=clevel,limii=limii)
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        ## evaluate
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        if(not convergence):
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            if(not force):
                return None
            about.warnings.cprint("WARNING: conjugate gradient failed.")
        return self._debriefing(x,x_,self.target,False)

    def inverse_times(self,x,force=False,W=None,spam=None,reset=None,note=False,x0=None,tol=1E-4,clevel=1,limii=None,**kwargs):
        """
            Applies the propagator to a given object.

            Parameters
            ----------
            x : {scalar, list, array, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the domain of the operator.
            force : bool
                Indicates wheter to return a field instead of ``None`` is
                forced incase the conjugate gradient fails.

            Returns
            -------
            OIx : field
                Mapped field with suitable domain.

            See Also
            --------
            conjugate_gradient

            Other Parameters
            ----------------
            W : {operator, function}, *optional*
                Operator `W` that is a preconditioner on `A` and is applicable to a
                field (default: None).
            spam : function, *optional*
                Callback function which is given the current `x` and iteration
                counter each iteration (default: None).
            reset : integer, *optional*
                Number of iterations after which to restart; i.e., forget previous
                conjugated directions (default: sqrt(b.dim())).
            note : bool, *optional*
                Indicates whether notes are printed or not (default: False).
            x0 : field, *optional*
                Starting guess for the minimization.
            tol : scalar, *optional*
                Tolerance specifying convergence; measured by current relative
                residual (default: 1E-4).
            clevel : integer, *optional*
                Number of times the tolerance should be undershot before
                exiting (default: 1).
            limii : integer, *optional*
                Maximum number of iterations performed (default: 10 * b.dim()).

        """
        ## check whether self-inverse
        if(self.sym)and(self.uni):
            return self.times(x,**kwargs)

        ## prepare
        x_ = self._briefing(x,self.target,True)
        ## apply operator
        if(self.imp):
            A = self._multiply
        else:
            A = self.times
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        x_,convergence = conjugate_gradient(A,x_,W=W,spam=spam,reset=reset,note=note)(x0=x0,tol=tol,clevel=clevel,limii=limii)
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        ## evaluate
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        if(not convergence):
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            if(not force):
                return None
            about.warnings.cprint("WARNING: conjugate gradient failed.")
        return self._debriefing(x,x_,self.domain,True)

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

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

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

        NIFTY subclass for propagator operators (of a certain family)

        The propagator operators :math:`D` implemented here have an inverse
        formulation like :math:`S^{-1} + M`, :math:`S^{-1} + N^{-1}`, or
        :math:`S^{-1} + R^\dagger N^{-1} R` as appearing in Wiener filter
        theory.

        Parameters
        ----------
        S : operator
            Covariance of the signal prior.
        M : operator
            Likelihood contribution.
        R : operator
            Response operator translating signal to (noiseless) data.
        N : operator
            Covariance of the noise prior or the likelihood, respectively.

        See Also
        --------
        conjugate_gradient

        Notes
        -----
        The propagator will puzzle the operators `S` and `M` or `R`,`N` or
        only `N` together in the predefined from, a domain is set
        automatically. The application of the inverse is done by invoking a
        conjugate gradient.
        Note that changes to `S`, `M`, `R` or `N` auto-update the propagator.

        Examples
        --------
        >>> f = field(rg_space(4), val=[2, 4, 6, 8])
        >>> S = power_operator(f.target, spec=1)
        >>> N = diagonal_operator(f.domain, diag=1)
        >>> D = propagator_operator(S=S, N=N) # D^{-1} = S^{-1} + N^{-1}
        >>> D(f).val
        array([ 1.,  2.,  3.,  4.])

        Attributes
        ----------
        domain : space
            A space wherein valid arguments live.
        codomain : space
            An alternative space wherein valid arguments live; commonly the
            codomain of the `domain` attribute.
        sym : bool
            Indicates that the operator is self-adjoint.
        uni : bool
            Indicates that the operator is not unitary.
        imp : bool
            Indicates that volume weights are implemented in the `multiply`
            instance methods.
        target : space
            The space wherein the operator output lives.
        _A1 : {operator, function}
            Application of :math:`S^{-1}` to a field.
        _A2 : {operator, function}
            Application of all operations not included in `A1` to a field.
        RN : {2-tuple of operators}, *optional*
            Contains `R` and `N` if given.

    """
    def __init__(self,S=None,M=None,R=None,N=None):
        """
            Sets the standard operator properties and `codomain`, `_A1`, `_A2`,
            and `RN` if required.

            Parameters
            ----------
            S : operator
                Covariance of the signal prior.
            M : operator
                Likelihood contribution.
            R : operator
                Response operator translating signal to (noiseless) data.
            N : operator
                Covariance of the noise prior or the likelihood, respectively.

        """
        ## check signal prior covariance
        if(S is None):
            raise Exception(about._errors.cstring("ERROR: insufficient input."))
        elif(not isinstance(S,operator)):
            raise ValueError(about._errors.cstring("ERROR: invalid input."))
        space1 = S.domain

        ## check likelihood (pseudo) covariance
        if(M is None):
            if(N is None):
                raise Exception(about._errors.cstring("ERROR: insufficient input."))
            elif(not isinstance(N,operator)):
                raise ValueError(about._errors.cstring("ERROR: invalid input."))
            if(R is None):
                space2 = N.domain
            elif(not isinstance(R,operator)):
                raise ValueError(about._errors.cstring("ERROR: invalid input."))
            else:
                space2 = R.domain
        elif(not isinstance(M,operator)):
            raise ValueError(about._errors.cstring("ERROR: invalid input."))
        else:
            space2 = M.domain

        ## set spaces
        self.domain = space2
        if(self.domain.check_codomain(space1)):
            self.codomain = space1
        else:
            raise ValueError(about._errors.cstring("ERROR: invalid input."))
        self.target = self.domain

        ## define A1 == S_inverse
        if(isinstance(S,diagonal_operator)):
            self._A1 = S._inverse_multiply ## S.imp == True
        else:
            self._A1 = S.inverse_times

        ## define A2 == M == R_adjoint N_inverse R == N_inverse
        if(M is None):
            if(R is not None):
                self.RN = (R,N)
                if(isinstance(N,diagonal_operator)):
                    self._A2 = self._standard_M_times_1
                else:
                    self._A2 = self._standard_M_times_2
            elif(isinstance(N,diagonal_operator)):
                self._A2 = N._inverse_multiply ## N.imp == True
            else:
                self._A2 = N.inverse_times
        elif(isinstance(M,diagonal_operator)):
            self._A2 = M._multiply ## M.imp == True
        else:
            self._A2 = M.times

        self.sym = True
        self.uni = False
        self.imp = True

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

    def _standard_M_times_1(self,x): ## applies > R_adjoint N_inverse R assuming N is diagonal
        return self.RN[0].adjoint_times(self.RN[1]._inverse_multiply(self.RN[0].times(x))) ## N.imp = True

    def _standard_M_times_2(self,x): ## applies > R_adjoint N_inverse R
        return self.RN[0].adjoint_times(self.RN[1].inverse_times(self.RN[0].times(x)))

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

    def _inverse_multiply_1(self,x,**kwargs): ## > applies A1 + A2 in self.codomain
        return self._A1(x)+self._A2(x.transform(self.domain)).transform(self.codomain)

    def _inverse_multiply_2(self,x,**kwargs): ## > applies A1 + A2 in self.domain
        return self._A1(x.transform(self.codomain)).transform(self.domain)+self._A2(x)

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

    def _briefing(self,x): ## > prepares x for `multiply`
        ## inspect x
        if(not isinstance(x,field)):
            return field(self.domain,val=x,target=None),False
        ## check x.domain
        elif(x.domain==self.domain):
            return x,False
        elif(x.domain==self.codomain):
            return x,True
        ## transform
        else:
            return x.transform(target=self.codomain,overwrite=False),True

    def _debriefing(self,x,x_,in_codomain): ## > evaluates x and x_ after `multiply`
        if(x_ is None):
            return None
        ## inspect x
        elif(isinstance(x,field)):
            ## repair ...
            if(in_codomain)and(x.domain!=self.codomain):
                    x_ = x_.transform(target=x.domain,overwrite=False) ## ... domain
            if(x_.target!=x.target):
                x_.set_target(newtarget=x.target) ## ... codomain
        return x_

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

    def times(self,x,force=False,W=None,spam=None,reset=None,note=False,x0=None,tol=1E-4,clevel=1,limii=None,**kwargs):
        """
            Applies the propagator to a given object.

            Parameters
            ----------
            x : {scalar, list, array, field}
                Scalars are interpreted as constant arrays, and an array will
                be interpreted as a field on the domain of the operator.
            force : bool
                Indicates wheter to return a field instead of ``None`` is
                forced incase the conjugate gradient fails.

            Returns
            -------
            Dx : field
                Mapped field with suitable domain.

            See Also
            --------
            conjugate_gradient

            Other Parameters
            ----------------
            W : {operator, function}, *optional*
                Operator `W` that is a preconditioner on `A` and is applicable to a
                field (default: None).
            spam : function, *optional*
                Callback function which is given the current `x` and iteration
                counter each iteration (default: None).
            reset : integer, *optional*
                Number of iterations after which to restart; i.e., forget previous
                conjugated directions (default: sqrt(b.dim())).
            note : bool, *optional*
                Indicates whether notes are printed or not (default: False).
            x0 : field, *optional*
                Starting guess for the minimization.
            tol : scalar, *optional*
                Tolerance specifying convergence; measured by current relative
                residual (default: 1E-4).
            clevel : integer, *optional*
                Number of times the tolerance should be undershot before
                exiting (default: 1).
            limii : integer, *optional*
                Maximum number of iterations performed (default: 10 * b.dim()).

        """
        ## prepare
        x_,in_codomain = self._briefing(x)
        ## apply operator
        if(in_codomain):
            A = self._inverse_multiply_1
        else:
            A = self._inverse_multiply_2
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        x_,convergence = conjugate_gradient(A,x_,W=W,spam=spam,reset=reset,note=note)(x0=x0,tol=tol,clevel=clevel,limii=limii)
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        ## evaluate
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        if(not convergence):
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            if(not force):
                return None
            about.warnings.cprint("WARNING: conjugate gradient failed.")
        return self._debriefing(x,x_,in_codomain)

    def inverse_times(self,x,**kwargs):
        """
            Applies the inverse propagator to a given object.

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

            Returns
            -------
            DIx : field
                Mapped field with suitable domain.

        """
        ## prepare
        x_,in_codomain = self._briefing(x)
        ## apply operator
        if(in_codomain):
            x_ = self._inverse_multiply_1(x_)
        else:
            x_ = self._inverse_multiply_2(x_)
        ## evaluate
        return self._debriefing(x,x_,in_codomain)

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

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

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

        NIFTY tool class for conjugate gradient

        This tool minimizes :math:`A x = b` with respect to `x` given `A` and
        `b` using a conjugate gradient; i.e., a step-by-step minimization
        relying on conjugated gradient directions. Further, `A` is assumed to
        be a positive definite and self-adjoint operator. The use of a
        preconditioner `W` that is roughly the inverse of `A` is optional.
        For details on the methodology refer to [#]_, for details on usage and
        output, see the notes below.

        Parameters
        ----------
        A : {operator, function}
            Operator `A` applicable to a field.
        b : field
            Resulting field of the operation `A(x)`.
        W : {operator, function}, *optional*
            Operator `W` that is a preconditioner on `A` and is applicable to a
            field (default: None).
        spam : function, *optional*
            Callback function which is given the current `x` and iteration
            counter each iteration (default: None).
        reset : integer, *optional*
            Number of iterations after which to restart; i.e., forget previous
            conjugated directions (default: sqrt(b.dim())).
        note : bool, *optional*
            Indicates whether notes are printed or not (default: False).

        Notes
        -----
        After initialization by `__init__`, the minimizer is started by calling
        it using `__call__`, which takes additional parameters. Notifications,
        if enabled, will state the iteration number, current step widths
        `alpha` and `beta`, the current relative residual `delta` that is
        compared to the tolerance, and the convergence level if changed.
        The minimizer will exit in two states: QUIT if the maximum number of
        iterations is reached, or DONE if convergence is achieved. Returned
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        will be the latest `x` and the latest convergence level, which can
        evaluate ``True`` for all exit states.
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        References
        ----------
        .. [#] J. R. Shewchuk, 1994, `"An Introduction to the Conjugate
        Gradient Method Without the Agonizing Pain"
        `<http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf>`_

        Examples
        --------
        >>> b = field(point_space(2), val=[1, 9])
        >>> A = diagonal_operator(b.domain, diag=[4, 3])
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        >>> x,convergence = conjugate_gradient(A, b, note=True)(tol=1E-4, clevel=3)
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        iteration : 00000001   alpha = 3.3E-01   beta = 1.3E-03   delta = 3.6E-02
        iteration : 00000002   alpha = 2.5E-01   beta = 7.6E-04   delta = 1.0E-03
        iteration : 00000003   alpha = 3.3E-01   beta = 2.5E-04   delta = 1.6E-05   convergence level : 1
        iteration : 00000004   alpha = 2.5E-01   beta = 1.8E-06   delta = 2.1E-08   convergence level : 2
        iteration : 00000005   alpha = 2.5E-01   beta = 2.2E-03   delta = 1.0E-09   convergence level : 3
        ... done.
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        >>> bool(convergence)
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        True
        >>> x.val # yields 1/4 and 9/3
        array([ 0.25,  3.  ])

        Attributes
        ----------
        A : {operator, function}
            Operator `A` applicable to a field.
        x : field
            Current field.
        b : field
            Resulting field of the operation `A(x)`.
        W : {operator, function}
            Operator `W` that is a preconditioner on `A` and is applicable to a
            field; can be ``None``.
        spam : function
            Callback function which is given the current `x` and iteration
            counter each iteration; can be ``None``.
        reset : integer
            Number of iterations after which to restart; i.e., forget previous
            conjugated directions (default: sqrt(b.dim())).
        note : notification
            Notification instance.

    """
    def __init__(self,A,b,W=None,spam=None,reset=None,note=False):
        """
            Initializes the conjugate_gradient and sets the attributes (except
            for `x`).

            Parameters
            ----------
            A : {operator, function}
                Operator `A` applicable to a field.
            b : field
                Resulting field of the operation `A(x)`.
            W : {operator, function}, *optional*
                Operator `W` that is a preconditioner on `A` and is applicable to a
                field (default: None).
            spam : function, *optional*
                Callback function which is given the current `x` and iteration
                counter each iteration (default: None).
            reset : integer, *optional*
                Number of iterations after which to restart; i.e., forget previous
                conjugated directions (default: sqrt(b.dim())).
            note : bool, *optional*
                Indicates whether notes are printed or not (default: False).

        """
        if(hasattr(A,"__call__")):
            self.A = A ## applies A
        else:
            raise AttributeError(about._errors.cstring("ERROR: invalid input."))
        self.b = b

        if(W is None)or(hasattr(W,"__call__")):
            self.W = W ## applies W ~ A_inverse
        else:
            raise AttributeError(about._errors.cstring("ERROR: invalid input."))

        self.spam = spam ## serves as callback given x and iteration number
        if(reset is None): ## 2 < reset ~ sqrt(dim)
            self.reset = max(2,int(np.sqrt(b.domain.dim(split=False))))
        else:
            self.reset = max(2,int(reset))
        self.note = notification(default=bool(note))

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

    def __call__(self,x0=None,**kwargs): ## > runs cg with/without preconditioner
        """
            Runs the conjugate gradient minimization.

            Parameters
            ----------
            x0 : field, *optional*
                Starting guess for the minimization.
            tol : scalar, *optional*
                Tolerance specifying convergence; measured by current relative
                residual (default: 1E-4).
            clevel : integer, *optional*
                Number of times the tolerance should be undershot before
                exiting (default: 1).
            limii : integer, *optional*
                Maximum number of iterations performed (default: 10 * b.dim()).

            Returns
            -------
            x : field
                Latest `x` of the minimization.
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            convergence : integer
                Latest convergence level indicating whether the minimization
                has converged or not.
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        """
        self.x = field(self.b.domain,val=x0,target=self.b.target)

        if(self.W is None):
            return self._calc_without(**kwargs)
        else:
            return self._calc_with(**kwargs)

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

    def _calc_without(self,tol=1E-4,clevel=1,limii=None): ## > runs cg without preconditioner

        if(limii is None):
            limii = 10*self.b.domain.dim(split=False)

        d = r = self.b-self.A(self.x)
        gamma = r.dot(d)
        delta_ = np.absolute(gamma)**(-0.5)

        convergence = 0
        ii = 1
        while(True):
            q = self.A(d)
            alpha = gamma/d.dot(q) ## positive definite
            self.x += alpha*d
            if(ii%self.reset==0)or(np.signbit(np.real(alpha))):
                r = self.b-self.A(self.x)
            else:
                r -= alpha*q
            gamma_ = gamma
            gamma = r.dot(r)
            beta = max(0,gamma/gamma_) ## positive definite
            d = r+beta*d

            delta = delta_*np.absolute(gamma)**0.5
            self.note.cflush("\niteration : %08u   alpha = %3.1E   beta = %3.1E   delta = %3.1E"%(ii,alpha,beta,delta))
            if(ii==limii):
                self.note.cprint("\n... quit.")
                break
            if(gamma==0):
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                convergence = clevel+1
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                self.note.cprint("   convergence level : INF\n... done.")
                break
            elif(np.absolute(delta)<tol):
                convergence += 1
                self.note.cflush("   convergence level : %u"%convergence)
                if(convergence==clevel):
                    self.note.cprint("\n... done.")
                    break
            else:
                convergence = max(0,convergence-1)

            if(self.spam is not None):
                self.spam(self.x,ii)

            ii += 1

        if(self.spam is not None):
            self.spam(self.x,ii)

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        return self.x,convergence
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def _calc_with(self,tol=1E-4,clevel=1,limii=None): ## > runs cg with preconditioner

        if(limii is None):
            limii = 10*self.b.domain.dim(split=False)

        r = self.b-self.A(self.x)
        d = self.W(r)
        gamma = r.dot(d)
        delta_ = np.absolute(gamma)**(-0.5)

        convergence = 0
        ii = 1
        while(True):
            q = self.A(d)
            alpha = gamma/d.dot(q) ## positive definite
            self.x += alpha*d ## update
            if(ii%self.reset==0)or(np.signbit(np.real(alpha))):
                r = self.b-self.A(self.x)
            else:
                r -= alpha*q
            s = self.W(r)
            gamma_ = gamma
            gamma = r.dot(s)
            beta = max(0,gamma/gamma_) ## positive definite
            d = s+beta*d ## conjugated gradient

            delta = delta_*np.absolute(gamma)**0.5
            self.note.cflush("\niteration : %08u   alpha = %3.1E   beta = %3.1E   delta = %3.1E"%(ii,alpha,beta,delta))
            if(ii==limii):
                self.note.cprint("\n... quit.")
                break
            if(gamma==0):
                convergence = clevel
                self.note.cprint("   convergence level : INF\n... done.")
                break
            elif(np.absolute(delta)<tol):
                convergence += 1
                self.note.cflush("   convergence level : %u"%convergence)
                if(convergence==clevel):
                    self.note.cprint("\n... done.")
                    break
            else:
                convergence = max(0,convergence-1)

            if(self.spam is not None):
                self.spam(self.x,ii)

            ii += 1

        if(self.spam is not None):
            self.spam(self.x,ii)

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        return self.x,convergence
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##=============================================================================





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

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

        NIFTY tool class for steepest descent minimization

        This tool minimizes a scalar energy-function by steepest descent using
        the functions gradient. Steps and step widths are choosen according to
        the Wolfe conditions [#]_. For details on usage and output, see the
        notes below.

        Parameters
        ----------
        eggs : function
            Given the current `x` it returns the tuple of energy and gradient.
        spam : function, *optional*
            Callback function which is given the current `x` and iteration
            counter each iteration (default: None).
        a : {4-tuple}, *optional*
            Numbers obeying 0 < a1 ~ a2 < 1 ~ a3 < a4 that modify the step
            widths (default: (0.2,0.5,1,2)).
        c : {2-tuple}, *optional*
            Numbers obeying 0 < c1 < c2 < 1 that specify the Wolfe-conditions
            (default: (1E-4,0.9)).
        note : bool, *optional*
            Indicates whether notes are printed or not (default: False).

        Notes
        -----
        After initialization by `__init__`, the minimizer is started by calling
        it using `__call__`, which takes additional parameters. Notifications,
        if enabled, will state the iteration number, current step width `alpha`,
        current maximal change `delta` that is compared to the tolerance, and
        the convergence level if changed. The minimizer will exit in three
        states: DEAD if no step width above 1E-13 is accepted, QUIT if the
        maximum number of iterations is reached, or DONE if convergence is
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        achieved. Returned will be the latest `x` and the latest convergence
        level, which can evaluate ``True`` for all exit states.
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        References
        ----------
        .. [#] J. Nocedal and S. J. Wright, Springer 2006, "Numerical
            Optimization", ISBN: 978-0-387-30303-1 (print) / 978-0-387-40065-5
            `(online) <http://link.springer.com/book/10.1007/978-0-387-40065-5/page/1>`_

        Examples
        --------
        >>> def egg(x):
        ...     E = 0.5*x.dot(x) # energy E(x) -- a two-dimensional parabola
        ...     g = x # gradient
        ...     return E,g
        >>> x = field(point_space(2), val=[1, 3])
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        >>> x,convergence = steepest_descent(egg, note=True)(x0=x, tol=1E-4, clevel=3)
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        iteration : 00000001   alpha = 1.0E+00   delta = 6.5E-01
        iteration : 00000002   alpha = 2.0E+00   delta = 1.4E-01
        iteration : 00000003   alpha = 1.6E-01   delta = 2.1E-03
        iteration : 00000004   alpha = 2.6E-03   delta = 3.0E-04
        iteration : 00000005   alpha = 2.0E-04   delta = 5.3E-05   convergence level : 1
        iteration : 00000006   alpha = 8.2E-05   delta = 4.4E-06   convergence level : 2
        iteration : 00000007   alpha = 6.6E-06   delta = 3.1E-06   convergence level : 3
        ... done.
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        >>> bool(convergence)
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        True
        >>> x.val # approximately zero
        array([ -6.87299426e-07  -2.06189828e-06])

        Attributes
        ----------
        x : field
            Current field.
        eggs : function
            Given the current `x` it returns the tuple of energy and gradient.
        spam : function
            Callback function which is given the current `x` and iteration
            counter each iteration; can be ``None``.
        a : {4-tuple}
            Numbers obeying 0 < a1 ~ a2 < 1 ~ a3 < a4 that modify the step
            widths (default: (0.2,0.5,1,2)).
        c : {2-tuple}
            Numbers obeying 0 < c1 < c2 < 1 that specify the Wolfe-conditions
            (default: (1E-4,0.9)).
        note : notification
            Notification instance.

    """
    def __init__(self,eggs,spam=None,a=(0.2,0.5,1,2),c=(1E-4,0.9),note=False):
        """
            Initializes the steepest_descent and sets the attributes (except
            for `x`).

            Parameters
            ----------
            eggs : function
                Given the current `x` it returns the tuple of energy and gradient.
            spam : function, *optional*
                Callback function which is given the current `x` and iteration
                counter each iteration (default: None).
            a : {4-tuple}, *optional*
                Numbers obeying 0 < a1 ~ a2 < 1 ~ a3 < a4 that modify the step
                widths (default: (0.2,0.5,1,2)).
            c : {2-tuple}, *optional*
                Numbers obeying 0 < c1 < c2 < 1 that specify the Wolfe-conditions
                (default: (1E-4,0.9)).
            note : bool, *optional*
                Indicates whether notes are printed or not (default: False).

        """
        self.eggs = eggs ## returns energy and gradient

        self.spam = spam ## serves as callback given x and iteration number
        self.a = a ## 0 < a1 ~ a2 < 1 ~ a3 < a4
        self.c = c ## 0 < c1 < c2 < 1
        self.note = notification(default=bool(note))

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

    def __call__(self,x0,alpha=1,tol=1E-4,clevel=8,limii=100000):
        """
            Runs the steepest descent minimization.

            Parameters
            ----------
            x0 : field
                Starting guess for the minimization.
            alpha : scalar, *optional*
                Starting step width to be multiplied with normalized gradient
                (default: 1).
            tol : scalar, *optional*
                Tolerance specifying convergence; measured by maximal change in
                `x` (default: 1E-4).
            clevel : integer, *optional*
                Number of times the tolerance should be undershot before
                exiting (default: 8).
            limii : integer, *optional*
                Maximum number of iterations performed (default: 100,000).

            Returns
            -------
            x : field
                Latest `x` of the minimization.
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            convergence : integer
                Latest convergence level indicating whether the minimization
                has converged or not.
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        """
        if(not isinstance(x0,field)):
            raise TypeError(about._errors.cstring("ERROR: invalid input."))
        self.x = x0
        E,g = self.eggs(self.x) ## energy and gradient
        norm = g.norm() ## gradient norm

        convergence = 0
        ii = 1
        while(True):
            x_,E,g,alpha,a = self._get_alpha(E,g,norm,alpha) ## "news",alpha,a

            if(alpha is None):
                self.note.cprint("\niteration : %08u   alpha < 1.0E-13\n... dead."%ii)
                break
            else:
                delta = np.absolute(g.val).max()*(alpha/norm)
                self.note.cflush("\niteration : %08u   alpha = %3.1E   delta = %3.1E"%(ii,alpha,delta))
                ## update
                self.x = x_
                alpha *= a

            norm = g.norm() ## gradient norm
            if(ii==limii):
                self.note.cprint("\n... quit.")
                break
            elif(delta<tol):
                convergence += 1
                self.note.cflush("   convergence level : %u"%convergence)
                if(convergence==clevel):
                    self.note.cprint("\n... done.")
                    break
            else:
                convergence = max(0,convergence-1)

            if(self.spam is not None):
                self.spam(self.x,ii)

            ii += 1

        if(self.spam is not None):
            self.spam(self.x,ii)

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        return self.x,convergence
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    ##+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

    def _get_alpha(self,E,g,norm,alpha): ## > determines the new alpha

        while(True):
            ## Wolfe conditions
            wolfe,x_,E_,g_,a = self._check_wolfe(E,g,norm,alpha)
#            wolfe,x_,E_,g_,a = self._check_strong_wolfe(E,g,norm,alpha)
            if(wolfe):
                return x_,E_,g_,alpha,a
            else:
                alpha *= a
                if(alpha<1E-13):
                    return None,None,None,None,None

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

    def _check_wolfe(self,E,g,norm,alpha): ## > checks the Wolfe conditions

        x_ = self._get_x(g,norm,alpha)
        pg = norm
        E_,g_ = self.eggs(x_)
        if(E_>E+self.c[0]*alpha*pg):
            if(E_<E):
                return True,x_,E_,g_,self.a[1]
            return False,None,None,None,self.a[0]
        pg_ = g.dot(g_)/norm
        if(pg_<self.c[1]*pg):
            return True,x_,E_,g_,self.a[3]
        return True,x_,E_,g_,self.a[2]

#    def _check_strong_wolfe(self,E,g,norm,alpha): ## > checks the strong Wolfe conditions
#
#        x_ = self._get_x(g,norm,alpha)
#        pg = norm
#        E_,g_ = self.eggs(x_)
#        if(E_>E+self.c[0]*alpha*pg):
#            if(E_<E):
#                return True,x_,E_,g_,self.a[1]
#            return False,None,None,None,self.a[0]
#        apg_ = np.absolute(g.dot(g_))/norm
#        if(apg_>self.c[1]*np.absolute(pg)):
#            return True,x_,E_,g_,self.a[3]
#        return True,x_,E_,g_,self.a[2]

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

    def _get_x(self,g,norm,alpha): ## > updates x

        return self.x-g*(alpha/norm)

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