RhodeCode

/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * 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 <lemon/math.h>

#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;

        for (int i = 1; i < length; ++i) {

          curr[0] = (mul * ( curr[1] ^ (curr[1] >> (bits - 2)) ) + i);

          --curr;

        }

      }

      template <typename Iterator>

      void initState(Iterator begin, Iterator end) {

        static const Word init = RandomTraits<Word>::arrayInit;

        static const Word mul1 = RandomTraits<Word>::arrayMul1;

        static const Word mul2 = RandomTraits<Word>::arrayMul2;

        Word *curr = state + length - 1; --curr;

        Iterator it = begin; int cnt = 0;

        int num;

        initState(init);

        num = length > end - begin ? length : end - begin;

        while (num--) {

          curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul1)) 

            + *it + cnt;

          ++it; ++cnt;

          if (it == end) {

            it = begin; cnt = 0;

          }

          if (curr == state) {

            curr = state + length - 1; curr[0] = state[0];

          }

          --curr;

        }

        num = length - 1; cnt = length - (curr - state) - 1;

        while (num--) {

          curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul2))

            - cnt;

          --curr; ++cnt;

          if (curr == state) {

            curr = state + length - 1; curr[0] = state[0]; --curr;

            cnt = 1;

          }

        }

        state[length - 1] = Word(1) << (bits - 1);

      }

      void copyState(const RandomCore& other) {

        std::copy(other.state, other.state + length, state);

        current = state + (other.current - other.state);

      }

      Word operator()() {

        if (current == state) fillState();

        --current;

        Word rnd = *current;

        return RandomTraits<Word>::tempering(rnd);

      }

    private:

      void fillState() {

        static const Word mask[2] = { 0x0ul, RandomTraits<Word>::mask };

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

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

        current = state + length; 

        register Word *curr = state + length - 1;

        register long num;

        num = length - shift;

        while (num--) {

          curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^

            curr[- shift] ^ mask[curr[-1] & 1ul];

          --curr;

        }

        num = shift - 1;

        while (num--) {

          curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^

            curr[length - shift] ^ mask[curr[-1] & 1ul];

          --curr;

        }

        state[0] = (((state[0] & hiMask) | (curr[length - 1] & loMask)) >> 1) ^

          curr[length - shift] ^ mask[curr[length - 1] & 1ul];

      }

      Word *current;

      Word state[length];

    };

    template <typename Result, 

              int shift = (std::numeric_limits<Result>::digits + 1) / 2>

    struct Masker {

      static Result mask(const Result& result) {

        return Masker<Result, (shift + 1) / 2>::

          mask(static_cast<Result>(result | (result >> shift)));

      }

    };

    template <typename Result>

    struct Masker<Result, 1> {

      static Result mask(const Result& result) {

        return static_cast<Result>(result | (result >> 1));

      }

    };

    template <typename Result, typename Word, 

              int rest = std::numeric_limits<Result>::digits, int shift = 0, 

              bool last = rest <= std::numeric_limits<Word>::digits>

    struct IntConversion {

      static const int bits = std::numeric_limits<Word>::digits;

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

        return static_cast<Result>(rnd() >> (bits - rest)) << shift;

      }

    }; 

    template <typename Result, typename Word, int rest, int shift> 

    struct IntConversion<Result, Word, rest, shift, false> {

      static const int bits = std::numeric_limits<Word>::digits;

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

        return (static_cast<Result>(rnd()) << shift) | 

          IntConversion<Result, Word, rest - bits, shift + bits>::convert(rnd);

      }

    };

    template <typename Result, typename Word,

              bool one_word = (std::numeric_limits<Word>::digits < 

			       std::numeric_limits<Result>::digits) >

    struct Mapping {

      static Result map(RandomCore<Word>& rnd, const Result& bound) {

        Word max = Word(bound - 1);

        Result mask = Masker<Result>::mask(bound - 1);

        Result num;

        do {

          num = IntConversion<Result, Word>::convert(rnd) & mask; 

        } while (num > max);

        return num;

      }

    };

    template <typename Result, typename Word>

    struct Mapping<Result, Word, false> {

      static Result map(RandomCore<Word>& rnd, const Result& bound) {

        Word max = Word(bound - 1);

        Word mask = Masker<Word, (std::numeric_limits<Result>::digits + 1) / 2>

          ::mask(max);

        Word num;

        do {

          num = rnd() & mask;

        } while (num > max);

        return num;

      }

    };

    template <typename Result, int exp, bool pos = (exp >= 0)>

    struct ShiftMultiplier {

      static const Result multiplier() {

        Result res = ShiftMultiplier<Result, exp / 2>::multiplier();

        res *= res;

        if ((exp & 1) == 1) res *= static_cast<Result>(2.0);

        return res; 

      }

    };

    template <typename Result, int exp>

    struct ShiftMultiplier<Result, exp, false> {

      static const Result multiplier() {

        Result res = ShiftMultiplier<Result, exp / 2>::multiplier();

        res *= res;

        if ((exp & 1) == 1) res *= static_cast<Result>(0.5);

        return res; 

      }

    };

    template <typename Result>

    struct ShiftMultiplier<Result, 0, true> {

      static const Result multiplier() {

        return static_cast<Result>(1.0); 

      }

    };

    template <typename Result>

    struct ShiftMultiplier<Result, -20, true> {

      static const Result multiplier() {

        return static_cast<Result>(1.0/1048576.0); 

      }

    };

    template <typename Result>

    struct ShiftMultiplier<Result, -32, true> {

      static const Result multiplier() {

        return static_cast<Result>(1.0/424967296.0); 

      }

    };

    template <typename Result>

    struct ShiftMultiplier<Result, -53, true> {

      static const Result multiplier() {

        return static_cast<Result>(1.0/9007199254740992.0); 

      }

    };

    template <typename Result>

    struct ShiftMultiplier<Result, -64, true> {

      static const Result multiplier() {

        return static_cast<Result>(1.0/18446744073709551616.0); 

      }

    };

    template <typename Result, int exp>

    struct Shifting {

      static Result shift(const Result& result) {

        return result * ShiftMultiplier<Result, exp>::multiplier();

      }

    };

    template <typename Result, typename Word,

              int rest = std::numeric_limits<Result>::digits, int shift = 0, 

              bool last = rest <= std::numeric_limits<Word>::digits>

    struct RealConversion{ 

      static const int bits = std::numeric_limits<Word>::digits;

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

        return Shifting<Result, - shift - rest>::

          shift(static_cast<Result>(rnd() >> (bits - rest)));

      }

    };

    template <typename Result, typename Word, int rest, int shift>

    struct RealConversion<Result, Word, rest, shift, false> { 

      static const int bits = std::numeric_limits<Word>::digits;

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

        return Shifting<Result, - shift - bits>::

          shift(static_cast<Result>(rnd())) +

          RealConversion<Result, Word, rest-bits, shift + bits>::

          convert(rnd);

      }

    };

    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

  ///

  /// 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:

    // Architecture word

    typedef unsigned long Word;

    _random_bits::RandomCore<Word> core;

    _random_bits::BoolProducer<Word> bool_producer;

  public:

    /// \brief Default constructor

    ///

    /// Constructor with constant seeding.

    Random() { core.initState(); }

    /// \brief Constructor with seed

    ///

    /// 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 with array seeding

    ///

    /// 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 Seeding random sequence

    ///

    /// Seeding the random sequence. The current number type will be

    /// converted to the architecture word type.

    template <typename Number>

    void seed(Number seed) { 

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

    }

    /// \brief Seeding random sequence

    ///

    /// Seeding the random sequence. The given range should contain

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

    /// architecture word type.

    template <typename Iterator>

    void seed(Iterator begin, Iterator end) { 

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

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

    }

    /// \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 \c 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>

    Number operator()(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 operator()(Number a, Number b) { 

      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 <tt>unsigned int</tt>.

    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 \c 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 Non-uniform 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

    /// 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=E/(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));

    }  

    /// Poisson distribution

    /// This function generates a Poisson distribution random number with

    /// parameter \c lambda.

    /// 

    /// The probability mass function of this distribusion is

    /// \f[ \frac{e^{-\lambda}\lambda^k}{k!} \f]

    /// \note The algorithm is taken from the book of Donald E. Knuth titled

    /// ''Seminumerical Algorithms'' (1969). Its running time is linear in the

    /// return value.

    int poisson(double lambda)