mapquadi.h

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/// @addtogroup unifexp
/// @{

/////////////////////////////////////////////////////////////////////////////
///
/// @file mapquadi.h
///
/// This file contains the definition of the quadratic map f(x)=x^2-a
/// with the use of interval arithmetic for rigorous computations.
///
/// @author Pawel Pilarczyk
///
/////////////////////////////////////////////////////////////////////////////

// Copyright (C) 2007 by Pawel Pilarczyk.
//
// This file is part of my research program package.  This 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 2 of the License, or (at your option) any later version.
//
// This software 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 software; see the file "license.txt".  If not, write to the
// Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
// MA 02111-1307, USA.

// Started on March 8, 2007. Last revision: February 3, 2013.

#ifndef _mapquadi_h_
#define _mapquadi_h_

#include <string>
#include <iostream>
#include "maptype.h"
#include "intvcapd.h"


namespace unifexp {

// --------------------------------------------------
// --------------- the quadratic map ----------------
// ------------ with interval arithmetic ------------
// --------------------------------------------------

/// This class defines the quadratic map with the use of interval arithmetic.
/// It is suitable for doing rigorous computations.
template <class numType>
class mapQuadrIntv: public mapType<numType>
{
public:
        /// The constructor.
        mapQuadrIntv ();

        /// Returns the name of the object.
        std::string name () const;

        /// Returns the number of critical points.
        int countCritical () const;

        /// Returns the subsequent critical points.
        numType criticalPoint (int n) const;

        /// Returns the left bound of the domain of the map.
        numType leftBound () const;

        /// Returns the right bound of the domain of the map.
        numType rightBound () const;

        /// Computes an enclosure of the image of the given interval.
        void image (const numType &x1, const numType &x2,
                numType &y1, numType &y2) const;

        /// Computes the minimal log of the derivative over those points
        /// in the interval [x1,x2] whose images may fall into [y1,y2]
        numType minLogDerivative (const numType &x1, const numType &x2,
                const numType &y1, const numType &y2) const;

}; /* mapQuadrIntv */

// --------------------------------------------------

template <class numType>
mapQuadrIntv<numType>::mapQuadrIntv ()
{
        return;
} /* mapQuadrIntv::mapQuadrIntv */

template <class numType>
std::string mapQuadrIntv<numType>::name () const
{
        return std::string ("quadratic-intv");
} /* mapQuadrIntv::name */

template <class numType>
inline int mapQuadrIntv<numType>::countCritical () const
{
        return 1;
} /* mapQuadrIntv::countCritical */

template <class numType>
inline numType mapQuadrIntv<numType>::criticalPoint (int n) const
{
        return 0.0;
} /* mapQuadrIntv::criticalPoint */

template <class numType>
inline numType mapQuadrIntv<numType>::leftBound () const
{
        return -2;
//      return -this -> paramMax;
} /* mapQuadrIntv::leftBound */

template <class numType>
inline numType mapQuadrIntv<numType>::rightBound () const
{
        return 2;
//      numType result = (power (IntervalType (this -> paramMax,
//              this -> paramMax), 2) - this -> paramMin). rightBound ();
//      resetRounding ();
//      return result;
} /* mapQuadrIntv::rightBound */

template <class numType>
void mapQuadrIntv<numType>::image (const numType &x1, const numType &x2,
        numType &y1, numType &y2) const
{
        if (x2 < x1)
                throw "Image computation: Wrong interval for 'x'.";
        IntervalType img = power (IntervalType (x1, x2), 2) -
                IntervalType (this -> paramMin, this -> paramMax);
        y1 = img. leftBound ();
        y2 = img. rightBound ();
        resetRounding ();
        return;
} /* mapQuadrIntv::image */

template <class numType>
numType mapQuadrIntv<numType>::minLogDerivative (const numType &x1,
        const numType &x2, const numType &y1, const numType &y2) const
{
        // make sure the input data is correct
        if (x2 < x1)
                throw "MinLogDerivative: Wrong interval for 'x'.";
        if (y2 < y1)
                throw "MinLogDerivative: Wrong interval for 'y'.";

        // compute the positive preimage of the interval [y1,y2],
        // intersect the computed preimage with the interval [x1,x2],
        // and return the log of the derivative at the minimal endpoint
        const numType sum1 = (IntervalType (y1, y1) + this -> paramMin).
                leftBound ();
        const numType left1 = (0 < sum1) ?
                (sqrt (IntervalType (sum1, sum1))). leftBound () : 0;
        numType result;
        if (0 < x1)
        {
                const numType left = (left1 > x1) ? left1 : x1;
                result = (log (2 * IntervalType (left, left))). leftBound ();
        }
        else if (x2 < 0)
        {
                const numType left = (left1 > -x2) ? left1 : -x2;
                result = (log (2 * IntervalType (left, left))). leftBound ();
        }
        else
        {
                throw "Log of interval containing zero.";
                result = 0;
        }
        resetRounding ();
        return result;
} /* mapQuadrIntv::minLogDerivative */


} // namespace unifexp

#endif // _mapquadi_h_

/// @}