RhodeCode

/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2007

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

///\file

///\brief Instantiation of the Random class.

#include <lemon/random.h>

namespace lemon {

  /// \brief Global random number generator instance

  ///

  Random rnd;

}

/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2007

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

/*

 * This file contains the reimplemented version of the Mersenne Twister

 * Generator of Matsumoto and Nishimura.

 *

 * See the appropriate copyright notice below.

 * 

 * Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura,

 * All rights reserved.                          

 *

 * Redistribution and use in source and binary forms, with or without

 * modification, are permitted provided that the following conditions

 * are met:

 *

 * 1. Redistributions of source code must retain the above copyright

 *    notice, this list of conditions and the following disclaimer.

 *

 * 2. Redistributions in binary form must reproduce the above copyright

 *    notice, this list of conditions and the following disclaimer in the

 *    documentation and/or other materials provided with the distribution.

 *

 * 3. The names of its contributors may not be used to endorse or promote 

 *    products derived from this software without specific prior written 

 *    permission.

 *

 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS

 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT

 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS

 * FOR A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE

 * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,

 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES

 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR

 * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)

 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,

 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)

 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED

 * OF THE POSSIBILITY OF SUCH DAMAGE.

 *

 *

 * Any feedback is very welcome.

 * http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html

 * email: m-mat @ math.sci.hiroshima-u.ac.jp (remove space)

 */

#ifndef LEMON_RANDOM_H

#define LEMON_RANDOM_H

#include <algorithm>

#include <iterator>

#include <vector>

#include <ctime>

#include <cmath>

#include <lemon/dim2.h>

///\ingroup misc

///\file

///\brief Mersenne Twister random number generator

namespace lemon {

  namespace _random_bits {

    template <typename _Word, int _bits = std::numeric_limits<_Word>::digits>

    struct RandomTraits {};

    template <typename _Word>

    struct RandomTraits<_Word, 32> {

      typedef _Word Word;

      static const int bits = 32;

      static const int length = 624;

      static const int shift = 397;

      static const Word mul = 0x6c078965u;

      static const Word arrayInit = 0x012BD6AAu;

      static const Word arrayMul1 = 0x0019660Du;

      static const Word arrayMul2 = 0x5D588B65u;

      static const Word mask = 0x9908B0DFu;

      static const Word loMask = (1u << 31) - 1;

      static const Word hiMask = ~loMask;

      static Word tempering(Word rnd) {

        rnd ^= (rnd >> 11);

        rnd ^= (rnd << 7) & 0x9D2C5680u;

        rnd ^= (rnd << 15) & 0xEFC60000u;

        rnd ^= (rnd >> 18);

        return rnd;

      }

    };

    template <typename _Word>

    struct RandomTraits<_Word, 64> {

      typedef _Word Word;

      static const int bits = 64;

      static const int length = 312;

      static const int shift = 156;

      static const Word mul = Word(0x5851F42Du) << 32 | Word(0x4C957F2Du);

      static const Word arrayInit = Word(0x00000000u) << 32 |Word(0x012BD6AAu);

      static const Word arrayMul1 = Word(0x369DEA0Fu) << 32 |Word(0x31A53F85u);

      static const Word arrayMul2 = Word(0x27BB2EE6u) << 32 |Word(0x87B0B0FDu);

      static const Word mask = Word(0xB5026F5Au) << 32 | Word(0xA96619E9u);

      static const Word loMask = (Word(1u) << 31) - 1;

      static const Word hiMask = ~loMask;

      static Word tempering(Word rnd) {

        rnd ^= (rnd >> 29) & (Word(0x55555555u) << 32 | Word(0x55555555u));

        rnd ^= (rnd << 17) & (Word(0x71D67FFFu) << 32 | Word(0xEDA60000u));

        rnd ^= (rnd << 37) & (Word(0xFFF7EEE0u) << 32 | Word(0x00000000u));

        rnd ^= (rnd >> 43);

        return rnd;

      }

    };

    template <typename _Word>

    class RandomCore {

    public:

      typedef _Word Word;

    private:

      static const int bits = RandomTraits<Word>::bits;

      static const int length = RandomTraits<Word>::length;

      static const int shift = RandomTraits<Word>::shift;

    public:

      void initState() {

        static const Word seedArray[4] = {

          0x12345u, 0x23456u, 0x34567u, 0x45678u

        };

        initState(seedArray, seedArray + 4);

      }

      void initState(Word seed) {

        static const Word mul = RandomTraits<Word>::mul;

        current = state; 

        Word *curr = state + length - 1;

        curr[0] = seed; --curr;

    };

    template <typename Result, typename Word>

    struct Initializer {

      template <typename Iterator>

      static void init(RandomCore<Word>& rnd, Iterator begin, Iterator end) {

        std::vector<Word> ws;

        for (Iterator it = begin; it != end; ++it) {

          ws.push_back(Word(*it));

        }

        rnd.initState(ws.begin(), ws.end());

      }

      static void init(RandomCore<Word>& rnd, Result seed) {

        rnd.initState(seed);

      }

    };

    template <typename Word>

    struct BoolConversion {

      static bool convert(RandomCore<Word>& rnd) {

        return (rnd() & 1) == 1;

      }

    };

    template <typename Word>

    struct BoolProducer {

      Word buffer;

      int num;

      BoolProducer() : num(0) {}

      bool convert(RandomCore<Word>& rnd) {

        if (num == 0) {

          buffer = rnd();

          num = RandomTraits<Word>::bits;

        }

        bool r = (buffer & 1);

        buffer >>= 1;

        --num;

        return r;

      }

    };

  }

  /// \ingroup misc

  ///

  /// \brief Mersenne Twister random number generator

  ///

  /// The Mersenne Twister is a twisted generalized feedback

  /// shift-register generator of Matsumoto and Nishimura. The period

  /// of this generator is \f$ 2^{19937} - 1 \f$ and it is

  /// equi-distributed in 623 dimensions for 32-bit numbers. The time

  /// performance of this generator is comparable to the commonly used

  /// generators.

  ///

  /// This implementation is specialized for both 32-bit and 64-bit

  /// architectures. The generators differ sligthly in the

  /// initialization and generation phase so they produce two

  /// completly different sequences.

  ///

  /// The generator gives back random numbers of serveral types. To

  /// get a random number from a range of a floating point type you

  /// can use one form of the \c operator() or the \c real() member

  /// function. If you want to get random number from the {0, 1, ...,

  /// n-1} integer range use the \c operator[] or the \c integer()

  /// method. And to get random number from the whole range of an

  /// integer type you can use the argumentless \c integer() or \c

  /// uinteger() functions. After all you can get random bool with

  /// equal chance of true and false or given probability of true

  /// result with the \c boolean() member functions.

  ///

  ///\code

  /// // The commented code is identical to the other

  /// double a = rnd();                     // [0.0, 1.0)

  /// // double a = rnd.real();             // [0.0, 1.0)

  /// double b = rnd(100.0);                // [0.0, 100.0)

  /// // double b = rnd.real(100.0);        // [0.0, 100.0)

  /// double c = rnd(1.0, 2.0);             // [1.0, 2.0)

  /// // double c = rnd.real(1.0, 2.0);     // [1.0, 2.0)

  /// int d = rnd[100000];                  // 0..99999

  /// // int d = rnd.integer(100000);       // 0..99999

  /// int e = rnd[6] + 1;                   // 1..6

  /// // int e = rnd.integer(1, 1 + 6);     // 1..6

  /// int b = rnd.uinteger<int>();          // 0 .. 2^31 - 1

  /// int c = rnd.integer<int>();           // - 2^31 .. 2^31 - 1

  /// bool g = rnd.boolean();               // P(g = true) = 0.5

  /// bool h = rnd.boolean(0.8);            // P(h = true) = 0.8

  ///\endcode

  ///

  /// The lemon provides a global instance of the random number

  /// generator which name is \ref lemon::rnd "rnd". Usually it is a

  /// good programming convenience to use this global generator to get

  /// random numbers.

  class Random {

  private:

    typedef unsigned long Word;

    _random_bits::RandomCore<Word> core;

    _random_bits::BoolProducer<Word> bool_producer;

  public:

    /// \brief Constructor

    ///

    /// Constructor with constant seeding.

    Random() { core.initState(); }

    /// \brief Constructor

    ///

    /// Constructor with seed. The current number type will be converted

    /// to the architecture word type.

    template <typename Number>

    Random(Number seed) { 

      _random_bits::Initializer<Number, Word>::init(core, seed);

    }

    /// \brief Constructor

    ///

    /// Constructor with array seeding. The given range should contain

    /// any number type and the numbers will be converted to the

    /// architecture word type.

    template <typename Iterator>

    Random(Iterator begin, Iterator end) { 

      typedef typename std::iterator_traits<Iterator>::value_type Number;

      _random_bits::Initializer<Number, Word>::init(core, begin, end);

    }

    /// \brief Copy constructor

    ///

    /// Copy constructor. The generated sequence will be identical to

    /// the other sequence. It can be used to save the current state

    /// of the generator and later use it to generate the same

    /// sequence.

    Random(const Random& other) {

      core.copyState(other.core);

    }

    /// \brief Assign operator

    ///

    /// Assign operator. The generated sequence will be identical to

    /// the other sequence. It can be used to save the current state

    /// of the generator and later use it to generate the same

    /// sequence.

    Random& operator=(const Random& other) {

      if (&other != this) {

        core.copyState(other.core);

      }

      return *this;

    }

    /// \brief Returns a random real number from the range [0, 1)

    ///

    /// It returns a random real number from the range [0, 1). The

    /// default Number type is double.

    template <typename Number>

    Number real() {

      return _random_bits::RealConversion<Number, Word>::convert(core);

    }

    double real() {

      return real<double>();

    }

    /// \brief Returns a random real number the range [0, b)

    ///

    /// It returns a random real number from the range [0, b).

    template <typename Number>

    Number real(Number b) { 

      return real<Number>() * b; 

    }

    /// \brief Returns a random real number from the range [a, b)

    ///

    /// It returns a random real number from the range [a, b).

    template <typename Number>

    Number real(Number a, Number b) { 

      return real<Number>() * (b - a) + a; 

    }

    /// \brief Returns a random real number from the range [0, 1)

    ///

    /// It returns a random double from the range [0, 1).

    double operator()() {

      return real<double>();

    }

    /// \brief Returns a random real number from the range [0, b)

    ///

    /// It returns a random real number from the range [0, b).

    template <typename Number>

      return real<Number>() * (b - a) + a; 

    }

    /// \brief Returns a random integer from a range

    ///

    /// It returns a random integer from the range {0, 1, ..., b - 1}.

    template <typename Number>

    Number integer(Number b) {

      return _random_bits::Mapping<Number, Word>::map(core, b);

    }

    /// \brief Returns a random integer from a range

    ///

    /// It returns a random integer from the range {a, a + 1, ..., b - 1}.

    template <typename Number>

    Number integer(Number a, Number b) {

      return _random_bits::Mapping<Number, Word>::map(core, b - a) + a;

    }

    /// \brief Returns a random integer from a range

    ///

    /// It returns a random integer from the range {0, 1, ..., b - 1}.

    template <typename Number>

    Number operator[](Number b) {

      return _random_bits::Mapping<Number, Word>::map(core, b);

    }

    /// \brief Returns a random non-negative integer

    ///

    /// It returns a random non-negative integer uniformly from the

    /// whole range of the current \c Number type.  The default result

    /// type of this function is unsigned int.

    template <typename Number>

    Number uinteger() {

      return _random_bits::IntConversion<Number, Word>::convert(core);

    }

    unsigned int uinteger() {

      return uinteger<unsigned int>();

    }

    /// \brief Returns a random integer

    ///

    /// It returns a random integer uniformly from the whole range of

    /// the current \c Number type. The default result type of this

    /// function is int.

    template <typename Number>

    Number integer() {

      static const int nb = std::numeric_limits<Number>::digits + 

        (std::numeric_limits<Number>::is_signed ? 1 : 0);

      return _random_bits::IntConversion<Number, Word, nb>::convert(core);

    }

    int integer() {

      return integer<int>();

    }

    /// \brief Returns a random bool

    ///

    /// It returns a random bool. The generator holds a buffer for

    /// random bits. Every time when it become empty the generator makes

    /// a new random word and fill the buffer up.

    bool boolean() {

      return bool_producer.convert(core);

    }

    ///\name Nonuniform distributions

    ///

    ///@{

    /// \brief Returns a random bool

    ///

    /// It returns a random bool with given probability of true result

    bool boolean(double p) {

      return operator()() < p;

    }

    /// Standard Gauss distribution

    /// Standard Gauss distribution.

    /// \note The Cartesian form of the Box-Muller

    /// transformation is used to generate a random normal distribution.

    /// \todo Consider using the "ziggurat" method instead.

    double gauss() 

    {

      double V1,V2,S;

      do {

	V1=2*real<double>()-1;

	V2=2*real<double>()-1;

	S=V1*V1+V2*V2;

      } while(S>=1);

      return std::sqrt(-2*std::log(S)/S)*V1;

    }

    /// Gauss distribution with given mean and standard deviation

    /// \sa gauss()

    double gauss(double mean,double std_dev)

    {

      return gauss()*std_dev+mean;

    }

    /// Exponential distribution with given mean

    /// This function generates an exponential distribution random number

    /// with mean <tt>1/lambda</tt>.

    ///

    double exponential(double lambda=1.0)

    {

      return -std::log(1.0-real<double>())/lambda;

    }

    /// Gamma distribution with given integer shape

    /// This function generates a gamma distribution random number.

    /// 

    ///\param k shape parameter (<tt>k>0</tt> integer)

    double gamma(int k) 

    {

      double s = 0;

      for(int i=0;i<k;i++) s-=std::log(1.0-real<double>());

      return s;

    }

    /// Gamma distribution with given shape and scale parameter

    /// This function generates a gamma distribution random number.

    /// 

    ///\param k shape parameter (<tt>k>0</tt>)

    ///\param theta scale parameter

    ///

    double gamma(double k,double theta=1.0)

    {

      double xi,nu;

      const double delta = k-std::floor(k);

      const double v0=M_E/(M_E-delta);

      do {

	double V0=1.0-real<double>();

	double V1=1.0-real<double>();

	double V2=1.0-real<double>();

	if(V2<=v0) 

	  {

	    xi=std::pow(V1,1.0/delta);

	    nu=V0*std::pow(xi,delta-1.0);

	  }

	else 

	  {

	    xi=1.0-std::log(V1);

	    nu=V0*std::exp(-xi);

	  }

      } while(nu>std::pow(xi,delta-1.0)*std::exp(-xi));

      return theta*(xi-gamma(int(std::floor(k))));

    }

    /// Weibull distribution

    /// This function generates a Weibull distribution random number.

    /// 

    ///\param k shape parameter (<tt>k>0</tt>)

    ///\param lambda scale parameter (<tt>lambda>0</tt>)

    ///

    double weibull(double k,double lambda)

    {

      return lambda*pow(-std::log(1.0-real<double>()),1.0/k);

    }  

    /// Pareto distribution

    /// This function generates a Pareto distribution random number.

    /// 

    ///\param k shape parameter (<tt>k>0</tt>)

    ///\param x_min location parameter (<tt>x_min>0</tt>)

    ///

    double pareto(double k,double x_min)

    {

      return exponential(gamma(k,1.0/x_min));

    }  

    ///@}

    ///\name Two dimensional distributions

    ///

    ///@{

    /// Uniform distribution on the full unit circle.

    dim2::Point<double> disc() 

    {

      double V1,V2;

      do {

	V1=2*real<double>()-1;

	V2=2*real<double>()-1;

      } while(V1*V1+V2*V2>=1);

      return dim2::Point<double>(V1,V2);

    }

    /// A kind of two dimensional Gauss distribution

    /// This function provides a turning symmetric two-dimensional distribution.

    /// Both coordinates are of standard normal distribution, but they are not

    /// independent.

    ///

    /// \note The coordinates are the two random variables provided by

    /// the Box-Muller method.

    dim2::Point<double> gauss2()

    {

      double V1,V2,S;

      do {

	V1=2*real<double>()-1;

	V2=2*real<double>()-1;

	S=V1*V1+V2*V2;

      } while(S>=1);

      double W=std::sqrt(-2*std::log(S)/S);

      return dim2::Point<double>(W*V1,W*V2);

    }

    /// A kind of two dimensional exponential distribution

    /// This function provides a turning symmetric two-dimensional distribution.

    /// The x-coordinate is of conditionally exponential distribution

    /// with the condition that x is positive and y=0. If x is negative and 

    /// y=0 then, -x is of exponential distribution. The same is true for the

    /// y-coordinate.

    dim2::Point<double> exponential2() 

    {

      double V1,V2,S;

      do {

	V1=2*real<double>()-1;

	V2=2*real<double>()-1;

	S=V1*V1+V2*V2;

      } while(S>=1);

      double W=-std::log(S)/S;

      return dim2::Point<double>(W*V1,W*V2);

    }

    ///@}    

  };

  extern Random rnd;

}

#endif

/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2007

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_TOLERANCE_H

#define LEMON_TOLERANCE_H

///\ingroup misc

///\file

///\brief A basic tool to handle the anomalies of calculation with

///floating point numbers.

///

///\todo It should be in a module like "Basic tools"

namespace lemon {

  /// \addtogroup misc

  /// @{

  ///\brief A class to provide a basic way to

  ///handle the comparison of numbers that are obtained

  ///as a result of a probably inexact computation.

  ///

  ///Tolerance is a class to provide a basic way to

  ///handle the comparison of numbers that are obtained

  ///as a result of a probably inexact computation.

  ///

  ///This is an abstract class, it should be specialized for all numerical

  ///data types. These specialized classes like \ref Tolerance\<double\>

  ///may offer additional tuning parameters.

  ///

  ///\sa Tolerance<float>

  ///\sa Tolerance<double>

  ///\sa Tolerance<long double>

  ///\sa Tolerance<int>

  ///\sa Tolerance<long long int>

  ///\sa Tolerance<unsigned int>

  ///\sa Tolerance<unsigned long long int>

  template<class T>

  class Tolerance

  {

  public:

    typedef T Value;

    ///\name Comparisons

    ///The concept is that these bool functions return with \c true only if

    ///the related comparisons hold even if some numerical error appeared

    ///during the computations.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) {return false;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) {return false;}

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) {return false;}

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) {return false;}

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) {return false;}

    ///@}

    ///Returns the zero value.

    static Value zero() {return T();}

    //   static bool finite(Value a) {}

    //   static Value big() {}

    //   static Value negativeBig() {}

  };

  ///Float specialization of \ref Tolerance.

  ///Float specialization of \ref Tolerance.

  ///\sa Tolerance

  ///\relates Tolerance

  template<>

  class Tolerance<float>

  {

    static float def_epsilon;

    float _epsilon;

  public:

    ///\e

    typedef float Value;

    ///Constructor setting the epsilon tolerance to the default value.

    Tolerance() : _epsilon(def_epsilon) {}

    ///Constructor setting the epsilon tolerance.

    Tolerance(float e) : _epsilon(e) {}

    ///Return the epsilon value.

    Value epsilon() const {return _epsilon;}

    ///Set the epsilon value.

    void epsilon(Value e) {_epsilon=e;}

    ///Return the default epsilon value.

    static Value defaultEpsilon() {return def_epsilon;}

    ///Set the default epsilon value.

    static void defaultEpsilon(Value e) {def_epsilon=e;}

    ///\name Comparisons

    ///See class Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    bool less(Value a,Value b) const {return a+_epsilon<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    bool different(Value a,Value b) const { return less(a,b)||less(b,a); }

    ///Returns \c true if \c a is \e surely positive

    bool positive(Value a) const { return _epsilon<a; }

    ///Returns \c true if \c a is \e surely negative

    bool negative(Value a) const { return -_epsilon>a; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Double specialization of \ref Tolerance.

  ///Double specialization of \ref Tolerance.

  ///\sa Tolerance

  ///\relates Tolerance

  template<>

  class Tolerance<double>

  {

    static double def_epsilon;

    double _epsilon;

  public:

    ///\e

    typedef double Value;

    ///Constructor setting the epsilon tolerance to the default value.

    Tolerance() : _epsilon(def_epsilon) {}

    ///Constructor setting the epsilon tolerance.

    Tolerance(double e) : _epsilon(e) {}

    ///Return the epsilon value.

    Value epsilon() const {return _epsilon;}

    ///Set the epsilon value.

    void epsilon(Value e) {_epsilon=e;}

    ///Return the default epsilon value.

    static Value defaultEpsilon() {return def_epsilon;}

    ///Set the default epsilon value.

    static void defaultEpsilon(Value e) {def_epsilon=e;}

    ///\name Comparisons

    ///See class Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    bool less(Value a,Value b) const {return a+_epsilon<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    bool different(Value a,Value b) const { return less(a,b)||less(b,a); }

    ///Returns \c true if \c a is \e surely positive

    bool positive(Value a) const { return _epsilon<a; }

    ///Returns \c true if \c a is \e surely negative

    bool negative(Value a) const { return -_epsilon>a; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Long double specialization of \ref Tolerance.

  ///Long double specialization of \ref Tolerance.

  ///\sa Tolerance

  ///\relates Tolerance

  template<>

  class Tolerance<long double>

  {

    static long double def_epsilon;

    long double _epsilon;

  public:

    ///\e

    typedef long double Value;

    ///Constructor setting the epsilon tolerance to the default value.

    Tolerance() : _epsilon(def_epsilon) {}

    ///Constructor setting the epsilon tolerance.

    Tolerance(long double e) : _epsilon(e) {}

    ///Return the epsilon value.

    Value epsilon() const {return _epsilon;}

    ///Set the epsilon value.

    void epsilon(Value e) {_epsilon=e;}

    ///Return the default epsilon value.

    static Value defaultEpsilon() {return def_epsilon;}

    ///Set the default epsilon value.

    static void defaultEpsilon(Value e) {def_epsilon=e;}

    ///\name Comparisons

    ///See class Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    bool less(Value a,Value b) const {return a+_epsilon<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    bool different(Value a,Value b) const { return less(a,b)||less(b,a); }

    ///Returns \c true if \c a is \e surely positive

    bool positive(Value a) const { return _epsilon<a; }

    ///Returns \c true if \c a is \e surely negative

    bool negative(Value a) const { return -_epsilon>a; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Integer specialization of \ref Tolerance.

  ///Integer specialization of \ref Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<int>

  {

  public:

    ///\e

    typedef int Value;

    ///\name Comparisons

    ///See \ref Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) { return 0>a; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Unsigned integer specialization of \ref Tolerance.

  ///Unsigned integer specialization of \ref Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<unsigned int>

  {

  public:

    ///\e

    typedef unsigned int Value;

    ///\name Comparisons

    ///See \ref Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value) { return false; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Long integer specialization of \ref Tolerance.

  ///Long integer specialization of \ref Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<long int>

  {

  public:

    ///\e

    typedef long int Value;

    ///\name Comparisons

    ///See \ref Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) { return 0>a; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Unsigned long integer specialization of \ref Tolerance.

  ///Unsigned long integer specialization of \ref Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<unsigned long int>

  {

  public:

    ///\e

    typedef unsigned long int Value;

    ///\name Comparisons

    ///See \ref Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value) { return false; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

#if defined __GNUC__ && !defined __STRICT_ANSI__

  ///Long long integer specialization of \ref Tolerance.

  ///Long long integer specialization of \ref Tolerance.

  ///\warning This class (more exactly, type <tt>long long</tt>)

  ///is not ansi compatible.

  ///\sa Tolerance

  template<>

  class Tolerance<long long int>

  {

  public:

    ///\e

    typedef long long int Value;

    ///\name Comparisons

    ///See \ref Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) { return 0>a; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Unsigned long long integer specialization of \ref Tolerance.

  ///Unsigned long long integer specialization of \ref Tolerance.

  ///\warning This class (more exactly, type <tt>unsigned long long</tt>)

  ///is not ansi compatible.

  ///\sa Tolerance

  template<>

  class Tolerance<unsigned long long int>

  {

  public:

    ///\e

    typedef unsigned long long int Value;

    ///\name Comparisons

    ///See \ref Tolerance for more details.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value) { return false; }

    ///Returns \c true if \c a is \e surely non-zero

    ///@}