COIN-OR::LEMON - Graph Library

source: lemon-1.0/lemon/random.h @ 135:6e7aee618f03

Last change on this file since 135:6e7aee618f03 was 116:b6bede534255, checked in by Alpar Juttner <alpar@…>, 11 years ago

Serious buxfixes in Random::gamma() and Random::pareto()

File size: 27.5 KB
Line 
1/* -*- C++ -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library
4 *
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19/*
20 * This file contains the reimplemented version of the Mersenne Twister
21 * Generator of Matsumoto and Nishimura.
22 *
23 * See the appropriate copyright notice below.
24 *
25 * Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura,
26 * All rights reserved.                         
27 *
28 * Redistribution and use in source and binary forms, with or without
29 * modification, are permitted provided that the following conditions
30 * are met:
31 *
32 * 1. Redistributions of source code must retain the above copyright
33 *    notice, this list of conditions and the following disclaimer.
34 *
35 * 2. Redistributions in binary form must reproduce the above copyright
36 *    notice, this list of conditions and the following disclaimer in the
37 *    documentation and/or other materials provided with the distribution.
38 *
39 * 3. The names of its contributors may not be used to endorse or promote
40 *    products derived from this software without specific prior written
41 *    permission.
42 *
43 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
44 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
45 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
46 * FOR A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE
47 * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
48 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
49 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
50 * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
51 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
52 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
53 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
54 * OF THE POSSIBILITY OF SUCH DAMAGE.
55 *
56 *
57 * Any feedback is very welcome.
58 * http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html
59 * email: m-mat @ math.sci.hiroshima-u.ac.jp (remove space)
60 */
61
62#ifndef LEMON_RANDOM_H
63#define LEMON_RANDOM_H
64
65#include <algorithm>
66#include <iterator>
67#include <vector>
68#include <limits>
69
70#include <lemon/math.h>
71#include <lemon/dim2.h>
72
73///\ingroup misc
74///\file
75///\brief Mersenne Twister random number generator
76
77namespace lemon {
78
79  namespace _random_bits {
80   
81    template <typename _Word, int _bits = std::numeric_limits<_Word>::digits>
82    struct RandomTraits {};
83
84    template <typename _Word>
85    struct RandomTraits<_Word, 32> {
86
87      typedef _Word Word;
88      static const int bits = 32;
89
90      static const int length = 624;
91      static const int shift = 397;
92     
93      static const Word mul = 0x6c078965u;
94      static const Word arrayInit = 0x012BD6AAu;
95      static const Word arrayMul1 = 0x0019660Du;
96      static const Word arrayMul2 = 0x5D588B65u;
97
98      static const Word mask = 0x9908B0DFu;
99      static const Word loMask = (1u << 31) - 1;
100      static const Word hiMask = ~loMask;
101
102
103      static Word tempering(Word rnd) {
104        rnd ^= (rnd >> 11);
105        rnd ^= (rnd << 7) & 0x9D2C5680u;
106        rnd ^= (rnd << 15) & 0xEFC60000u;
107        rnd ^= (rnd >> 18);
108        return rnd;
109      }
110
111    };
112
113    template <typename _Word>
114    struct RandomTraits<_Word, 64> {
115
116      typedef _Word Word;
117      static const int bits = 64;
118
119      static const int length = 312;
120      static const int shift = 156;
121
122      static const Word mul = Word(0x5851F42Du) << 32 | Word(0x4C957F2Du);
123      static const Word arrayInit = Word(0x00000000u) << 32 |Word(0x012BD6AAu);
124      static const Word arrayMul1 = Word(0x369DEA0Fu) << 32 |Word(0x31A53F85u);
125      static const Word arrayMul2 = Word(0x27BB2EE6u) << 32 |Word(0x87B0B0FDu);
126
127      static const Word mask = Word(0xB5026F5Au) << 32 | Word(0xA96619E9u);
128      static const Word loMask = (Word(1u) << 31) - 1;
129      static const Word hiMask = ~loMask;
130
131      static Word tempering(Word rnd) {
132        rnd ^= (rnd >> 29) & (Word(0x55555555u) << 32 | Word(0x55555555u));
133        rnd ^= (rnd << 17) & (Word(0x71D67FFFu) << 32 | Word(0xEDA60000u));
134        rnd ^= (rnd << 37) & (Word(0xFFF7EEE0u) << 32 | Word(0x00000000u));
135        rnd ^= (rnd >> 43);
136        return rnd;
137      }
138
139    };
140
141    template <typename _Word>
142    class RandomCore {
143    public:
144
145      typedef _Word Word;
146
147    private:
148
149      static const int bits = RandomTraits<Word>::bits;
150
151      static const int length = RandomTraits<Word>::length;
152      static const int shift = RandomTraits<Word>::shift;
153
154    public:
155
156      void initState() {
157        static const Word seedArray[4] = {
158          0x12345u, 0x23456u, 0x34567u, 0x45678u
159        };
160   
161        initState(seedArray, seedArray + 4);
162      }
163
164      void initState(Word seed) {
165
166        static const Word mul = RandomTraits<Word>::mul;
167
168        current = state;
169
170        Word *curr = state + length - 1;
171        curr[0] = seed; --curr;
172        for (int i = 1; i < length; ++i) {
173          curr[0] = (mul * ( curr[1] ^ (curr[1] >> (bits - 2)) ) + i);
174          --curr;
175        }
176      }
177
178      template <typename Iterator>
179      void initState(Iterator begin, Iterator end) {
180
181        static const Word init = RandomTraits<Word>::arrayInit;
182        static const Word mul1 = RandomTraits<Word>::arrayMul1;
183        static const Word mul2 = RandomTraits<Word>::arrayMul2;
184
185
186        Word *curr = state + length - 1; --curr;
187        Iterator it = begin; int cnt = 0;
188        int num;
189
190        initState(init);
191
192        num = length > end - begin ? length : end - begin;
193        while (num--) {
194          curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul1))
195            + *it + cnt;
196          ++it; ++cnt;
197          if (it == end) {
198            it = begin; cnt = 0;
199          }
200          if (curr == state) {
201            curr = state + length - 1; curr[0] = state[0];
202          }
203          --curr;
204        }
205
206        num = length - 1; cnt = length - (curr - state) - 1;
207        while (num--) {
208          curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul2))
209            - cnt;
210          --curr; ++cnt;
211          if (curr == state) {
212            curr = state + length - 1; curr[0] = state[0]; --curr;
213            cnt = 1;
214          }
215        }
216       
217        state[length - 1] = Word(1) << (bits - 1);
218      }
219     
220      void copyState(const RandomCore& other) {
221        std::copy(other.state, other.state + length, state);
222        current = state + (other.current - other.state);
223      }
224
225      Word operator()() {
226        if (current == state) fillState();
227        --current;
228        Word rnd = *current;
229        return RandomTraits<Word>::tempering(rnd);
230      }
231
232    private:
233
234 
235      void fillState() {
236        static const Word mask[2] = { 0x0ul, RandomTraits<Word>::mask };
237        static const Word loMask = RandomTraits<Word>::loMask;
238        static const Word hiMask = RandomTraits<Word>::hiMask;
239
240        current = state + length;
241
242        register Word *curr = state + length - 1;
243        register long num;
244     
245        num = length - shift;
246        while (num--) {
247          curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^
248            curr[- shift] ^ mask[curr[-1] & 1ul];
249          --curr;
250        }
251        num = shift - 1;
252        while (num--) {
253          curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^
254            curr[length - shift] ^ mask[curr[-1] & 1ul];
255          --curr;
256        }
257        state[0] = (((state[0] & hiMask) | (curr[length - 1] & loMask)) >> 1) ^
258          curr[length - shift] ^ mask[curr[length - 1] & 1ul];
259
260      }
261
262 
263      Word *current;
264      Word state[length];
265     
266    };
267
268
269    template <typename Result,
270              int shift = (std::numeric_limits<Result>::digits + 1) / 2>
271    struct Masker {
272      static Result mask(const Result& result) {
273        return Masker<Result, (shift + 1) / 2>::
274          mask(static_cast<Result>(result | (result >> shift)));
275      }
276    };
277   
278    template <typename Result>
279    struct Masker<Result, 1> {
280      static Result mask(const Result& result) {
281        return static_cast<Result>(result | (result >> 1));
282      }
283    };
284
285    template <typename Result, typename Word,
286              int rest = std::numeric_limits<Result>::digits, int shift = 0,
287              bool last = rest <= std::numeric_limits<Word>::digits>
288    struct IntConversion {
289      static const int bits = std::numeric_limits<Word>::digits;
290   
291      static Result convert(RandomCore<Word>& rnd) {
292        return static_cast<Result>(rnd() >> (bits - rest)) << shift;
293      }
294     
295    };
296
297    template <typename Result, typename Word, int rest, int shift>
298    struct IntConversion<Result, Word, rest, shift, false> {
299      static const int bits = std::numeric_limits<Word>::digits;
300
301      static Result convert(RandomCore<Word>& rnd) {
302        return (static_cast<Result>(rnd()) << shift) |
303          IntConversion<Result, Word, rest - bits, shift + bits>::convert(rnd);
304      }
305    };
306
307
308    template <typename Result, typename Word,
309              bool one_word = (std::numeric_limits<Word>::digits <
310                               std::numeric_limits<Result>::digits) >
311    struct Mapping {
312      static Result map(RandomCore<Word>& rnd, const Result& bound) {
313        Word max = Word(bound - 1);
314        Result mask = Masker<Result>::mask(bound - 1);
315        Result num;
316        do {
317          num = IntConversion<Result, Word>::convert(rnd) & mask;
318        } while (num > max);
319        return num;
320      }
321    };
322
323    template <typename Result, typename Word>
324    struct Mapping<Result, Word, false> {
325      static Result map(RandomCore<Word>& rnd, const Result& bound) {
326        Word max = Word(bound - 1);
327        Word mask = Masker<Word, (std::numeric_limits<Result>::digits + 1) / 2>
328          ::mask(max);
329        Word num;
330        do {
331          num = rnd() & mask;
332        } while (num > max);
333        return num;
334      }
335    };
336
337    template <typename Result, int exp, bool pos = (exp >= 0)>
338    struct ShiftMultiplier {
339      static const Result multiplier() {
340        Result res = ShiftMultiplier<Result, exp / 2>::multiplier();
341        res *= res;
342        if ((exp & 1) == 1) res *= static_cast<Result>(2.0);
343        return res;
344      }
345    };
346
347    template <typename Result, int exp>
348    struct ShiftMultiplier<Result, exp, false> {
349      static const Result multiplier() {
350        Result res = ShiftMultiplier<Result, exp / 2>::multiplier();
351        res *= res;
352        if ((exp & 1) == 1) res *= static_cast<Result>(0.5);
353        return res;
354      }
355    };
356
357    template <typename Result>
358    struct ShiftMultiplier<Result, 0, true> {
359      static const Result multiplier() {
360        return static_cast<Result>(1.0);
361      }
362    };
363
364    template <typename Result>
365    struct ShiftMultiplier<Result, -20, true> {
366      static const Result multiplier() {
367        return static_cast<Result>(1.0/1048576.0);
368      }
369    };
370   
371    template <typename Result>
372    struct ShiftMultiplier<Result, -32, true> {
373      static const Result multiplier() {
374        return static_cast<Result>(1.0/424967296.0);
375      }
376    };
377
378    template <typename Result>
379    struct ShiftMultiplier<Result, -53, true> {
380      static const Result multiplier() {
381        return static_cast<Result>(1.0/9007199254740992.0);
382      }
383    };
384
385    template <typename Result>
386    struct ShiftMultiplier<Result, -64, true> {
387      static const Result multiplier() {
388        return static_cast<Result>(1.0/18446744073709551616.0);
389      }
390    };
391
392    template <typename Result, int exp>
393    struct Shifting {
394      static Result shift(const Result& result) {
395        return result * ShiftMultiplier<Result, exp>::multiplier();
396      }
397    };
398
399    template <typename Result, typename Word,
400              int rest = std::numeric_limits<Result>::digits, int shift = 0,
401              bool last = rest <= std::numeric_limits<Word>::digits>
402    struct RealConversion{
403      static const int bits = std::numeric_limits<Word>::digits;
404
405      static Result convert(RandomCore<Word>& rnd) {
406        return Shifting<Result, - shift - rest>::
407          shift(static_cast<Result>(rnd() >> (bits - rest)));
408      }
409    };
410
411    template <typename Result, typename Word, int rest, int shift>
412    struct RealConversion<Result, Word, rest, shift, false> {
413      static const int bits = std::numeric_limits<Word>::digits;
414
415      static Result convert(RandomCore<Word>& rnd) {
416        return Shifting<Result, - shift - bits>::
417          shift(static_cast<Result>(rnd())) +
418          RealConversion<Result, Word, rest-bits, shift + bits>::
419          convert(rnd);
420      }
421    };
422
423    template <typename Result, typename Word>
424    struct Initializer {
425
426      template <typename Iterator>
427      static void init(RandomCore<Word>& rnd, Iterator begin, Iterator end) {
428        std::vector<Word> ws;
429        for (Iterator it = begin; it != end; ++it) {
430          ws.push_back(Word(*it));
431        }
432        rnd.initState(ws.begin(), ws.end());
433      }
434
435      static void init(RandomCore<Word>& rnd, Result seed) {
436        rnd.initState(seed);
437      }
438    };
439
440    template <typename Word>
441    struct BoolConversion {
442      static bool convert(RandomCore<Word>& rnd) {
443        return (rnd() & 1) == 1;
444      }
445    };
446
447    template <typename Word>
448    struct BoolProducer {
449      Word buffer;
450      int num;
451     
452      BoolProducer() : num(0) {}
453
454      bool convert(RandomCore<Word>& rnd) {
455        if (num == 0) {
456          buffer = rnd();
457          num = RandomTraits<Word>::bits;
458        }
459        bool r = (buffer & 1);
460        buffer >>= 1;
461        --num;
462        return r;
463      }
464    };
465
466  }
467
468  /// \ingroup misc
469  ///
470  /// \brief Mersenne Twister random number generator
471  ///
472  /// The Mersenne Twister is a twisted generalized feedback
473  /// shift-register generator of Matsumoto and Nishimura. The period
474  /// of this generator is \f$ 2^{19937} - 1 \f$ and it is
475  /// equi-distributed in 623 dimensions for 32-bit numbers. The time
476  /// performance of this generator is comparable to the commonly used
477  /// generators.
478  ///
479  /// This implementation is specialized for both 32-bit and 64-bit
480  /// architectures. The generators differ sligthly in the
481  /// initialization and generation phase so they produce two
482  /// completly different sequences.
483  ///
484  /// The generator gives back random numbers of serveral types. To
485  /// get a random number from a range of a floating point type you
486  /// can use one form of the \c operator() or the \c real() member
487  /// function. If you want to get random number from the {0, 1, ...,
488  /// n-1} integer range use the \c operator[] or the \c integer()
489  /// method. And to get random number from the whole range of an
490  /// integer type you can use the argumentless \c integer() or \c
491  /// uinteger() functions. After all you can get random bool with
492  /// equal chance of true and false or given probability of true
493  /// result with the \c boolean() member functions.
494  ///
495  ///\code
496  /// // The commented code is identical to the other
497  /// double a = rnd();                     // [0.0, 1.0)
498  /// // double a = rnd.real();             // [0.0, 1.0)
499  /// double b = rnd(100.0);                // [0.0, 100.0)
500  /// // double b = rnd.real(100.0);        // [0.0, 100.0)
501  /// double c = rnd(1.0, 2.0);             // [1.0, 2.0)
502  /// // double c = rnd.real(1.0, 2.0);     // [1.0, 2.0)
503  /// int d = rnd[100000];                  // 0..99999
504  /// // int d = rnd.integer(100000);       // 0..99999
505  /// int e = rnd[6] + 1;                   // 1..6
506  /// // int e = rnd.integer(1, 1 + 6);     // 1..6
507  /// int b = rnd.uinteger<int>();          // 0 .. 2^31 - 1
508  /// int c = rnd.integer<int>();           // - 2^31 .. 2^31 - 1
509  /// bool g = rnd.boolean();               // P(g = true) = 0.5
510  /// bool h = rnd.boolean(0.8);            // P(h = true) = 0.8
511  ///\endcode
512  ///
513  /// LEMON provides a global instance of the random number
514  /// generator which name is \ref lemon::rnd "rnd". Usually it is a
515  /// good programming convenience to use this global generator to get
516  /// random numbers.
517  class Random {
518  private:
519
520    // Architecture word
521    typedef unsigned long Word;
522   
523    _random_bits::RandomCore<Word> core;
524    _random_bits::BoolProducer<Word> bool_producer;
525   
526
527  public:
528
529    /// \brief Default constructor
530    ///
531    /// Constructor with constant seeding.
532    Random() { core.initState(); }
533
534    /// \brief Constructor with seed
535    ///
536    /// Constructor with seed. The current number type will be converted
537    /// to the architecture word type.
538    template <typename Number>
539    Random(Number seed) {
540      _random_bits::Initializer<Number, Word>::init(core, seed);
541    }
542
543    /// \brief Constructor with array seeding
544    ///
545    /// Constructor with array seeding. The given range should contain
546    /// any number type and the numbers will be converted to the
547    /// architecture word type.
548    template <typename Iterator>
549    Random(Iterator begin, Iterator end) {
550      typedef typename std::iterator_traits<Iterator>::value_type Number;
551      _random_bits::Initializer<Number, Word>::init(core, begin, end);
552    }
553
554    /// \brief Copy constructor
555    ///
556    /// Copy constructor. The generated sequence will be identical to
557    /// the other sequence. It can be used to save the current state
558    /// of the generator and later use it to generate the same
559    /// sequence.
560    Random(const Random& other) {
561      core.copyState(other.core);
562    }
563
564    /// \brief Assign operator
565    ///
566    /// Assign operator. The generated sequence will be identical to
567    /// the other sequence. It can be used to save the current state
568    /// of the generator and later use it to generate the same
569    /// sequence.
570    Random& operator=(const Random& other) {
571      if (&other != this) {
572        core.copyState(other.core);
573      }
574      return *this;
575    }
576
577    /// \brief Seeding random sequence
578    ///
579    /// Seeding the random sequence. The current number type will be
580    /// converted to the architecture word type.
581    template <typename Number>
582    void seed(Number seed) {
583      _random_bits::Initializer<Number, Word>::init(core, seed);
584    }
585
586    /// \brief Seeding random sequence
587    ///
588    /// Seeding the random sequence. The given range should contain
589    /// any number type and the numbers will be converted to the
590    /// architecture word type.
591    template <typename Iterator>
592    void seed(Iterator begin, Iterator end) {
593      typedef typename std::iterator_traits<Iterator>::value_type Number;
594      _random_bits::Initializer<Number, Word>::init(core, begin, end);
595    }
596
597    /// \brief Returns a random real number from the range [0, 1)
598    ///
599    /// It returns a random real number from the range [0, 1). The
600    /// default Number type is \c double.
601    template <typename Number>
602    Number real() {
603      return _random_bits::RealConversion<Number, Word>::convert(core);
604    }
605
606    double real() {
607      return real<double>();
608    }
609
610    /// \brief Returns a random real number the range [0, b)
611    ///
612    /// It returns a random real number from the range [0, b).
613    template <typename Number>
614    Number real(Number b) {
615      return real<Number>() * b;
616    }
617
618    /// \brief Returns a random real number from the range [a, b)
619    ///
620    /// It returns a random real number from the range [a, b).
621    template <typename Number>
622    Number real(Number a, Number b) {
623      return real<Number>() * (b - a) + a;
624    }
625
626    /// \brief Returns a random real number from the range [0, 1)
627    ///
628    /// It returns a random double from the range [0, 1).
629    double operator()() {
630      return real<double>();
631    }
632
633    /// \brief Returns a random real number from the range [0, b)
634    ///
635    /// It returns a random real number from the range [0, b).
636    template <typename Number>
637    Number operator()(Number b) {
638      return real<Number>() * b;
639    }
640
641    /// \brief Returns a random real number from the range [a, b)
642    ///
643    /// It returns a random real number from the range [a, b).
644    template <typename Number>
645    Number operator()(Number a, Number b) {
646      return real<Number>() * (b - a) + a;
647    }
648
649    /// \brief Returns a random integer from a range
650    ///
651    /// It returns a random integer from the range {0, 1, ..., b - 1}.
652    template <typename Number>
653    Number integer(Number b) {
654      return _random_bits::Mapping<Number, Word>::map(core, b);
655    }
656
657    /// \brief Returns a random integer from a range
658    ///
659    /// It returns a random integer from the range {a, a + 1, ..., b - 1}.
660    template <typename Number>
661    Number integer(Number a, Number b) {
662      return _random_bits::Mapping<Number, Word>::map(core, b - a) + a;
663    }
664
665    /// \brief Returns a random integer from a range
666    ///
667    /// It returns a random integer from the range {0, 1, ..., b - 1}.
668    template <typename Number>
669    Number operator[](Number b) {
670      return _random_bits::Mapping<Number, Word>::map(core, b);
671    }
672
673    /// \brief Returns a random non-negative integer
674    ///
675    /// It returns a random non-negative integer uniformly from the
676    /// whole range of the current \c Number type. The default result
677    /// type of this function is <tt>unsigned int</tt>.
678    template <typename Number>
679    Number uinteger() {
680      return _random_bits::IntConversion<Number, Word>::convert(core);
681    }
682
683    unsigned int uinteger() {
684      return uinteger<unsigned int>();
685    }
686
687    /// \brief Returns a random integer
688    ///
689    /// It returns a random integer uniformly from the whole range of
690    /// the current \c Number type. The default result type of this
691    /// function is \c int.
692    template <typename Number>
693    Number integer() {
694      static const int nb = std::numeric_limits<Number>::digits +
695        (std::numeric_limits<Number>::is_signed ? 1 : 0);
696      return _random_bits::IntConversion<Number, Word, nb>::convert(core);
697    }
698
699    int integer() {
700      return integer<int>();
701    }
702   
703    /// \brief Returns a random bool
704    ///
705    /// It returns a random bool. The generator holds a buffer for
706    /// random bits. Every time when it become empty the generator makes
707    /// a new random word and fill the buffer up.
708    bool boolean() {
709      return bool_producer.convert(core);
710    }
711
712    ///\name Non-uniform distributions
713    ///
714   
715    ///@{
716   
717    /// \brief Returns a random bool
718    ///
719    /// It returns a random bool with given probability of true result.
720    bool boolean(double p) {
721      return operator()() < p;
722    }
723
724    /// Standard Gauss distribution
725
726    /// Standard Gauss distribution.
727    /// \note The Cartesian form of the Box-Muller
728    /// transformation is used to generate a random normal distribution.
729    /// \todo Consider using the "ziggurat" method instead.
730    double gauss()
731    {
732      double V1,V2,S;
733      do {
734        V1=2*real<double>()-1;
735        V2=2*real<double>()-1;
736        S=V1*V1+V2*V2;
737      } while(S>=1);
738      return std::sqrt(-2*std::log(S)/S)*V1;
739    }
740    /// Gauss distribution with given mean and standard deviation
741
742    /// Gauss distribution with given mean and standard deviation.
743    /// \sa gauss()
744    double gauss(double mean,double std_dev)
745    {
746      return gauss()*std_dev+mean;
747    }
748
749    /// Exponential distribution with given mean
750
751    /// This function generates an exponential distribution random number
752    /// with mean <tt>1/lambda</tt>.
753    ///
754    double exponential(double lambda=1.0)
755    {
756      return -std::log(1.0-real<double>())/lambda;
757    }
758
759    /// Gamma distribution with given integer shape
760
761    /// This function generates a gamma distribution random number.
762    ///
763    ///\param k shape parameter (<tt>k>0</tt> integer)
764    double gamma(int k)
765    {
766      double s = 0;
767      for(int i=0;i<k;i++) s-=std::log(1.0-real<double>());
768      return s;
769    }
770   
771    /// Gamma distribution with given shape and scale parameter
772
773    /// This function generates a gamma distribution random number.
774    ///
775    ///\param k shape parameter (<tt>k>0</tt>)
776    ///\param theta scale parameter
777    ///
778    double gamma(double k,double theta=1.0)
779    {
780      double xi,nu;
781      const double delta = k-std::floor(k);
782      const double v0=E/(E-delta);
783      do {
784        double V0=1.0-real<double>();
785        double V1=1.0-real<double>();
786        double V2=1.0-real<double>();
787        if(V2<=v0)
788          {
789            xi=std::pow(V1,1.0/delta);
790            nu=V0*std::pow(xi,delta-1.0);
791          }
792        else
793          {
794            xi=1.0-std::log(V1);
795            nu=V0*std::exp(-xi);
796          }
797      } while(nu>std::pow(xi,delta-1.0)*std::exp(-xi));
798      return theta*(xi+gamma(int(std::floor(k))));
799    }
800   
801    /// Weibull distribution
802
803    /// This function generates a Weibull distribution random number.
804    ///
805    ///\param k shape parameter (<tt>k>0</tt>)
806    ///\param lambda scale parameter (<tt>lambda>0</tt>)
807    ///
808    double weibull(double k,double lambda)
809    {
810      return lambda*pow(-std::log(1.0-real<double>()),1.0/k);
811    } 
812     
813    /// Pareto distribution
814
815    /// This function generates a Pareto distribution random number.
816    ///
817    ///\param k shape parameter (<tt>k>0</tt>)
818    ///\param x_min location parameter (<tt>x_min>0</tt>)
819    ///
820    double pareto(double k,double x_min)
821    {
822      return exponential(gamma(k,1.0/x_min))+x_min;
823    } 
824     
825    /// Poisson distribution
826
827    /// This function generates a Poisson distribution random number with
828    /// parameter \c lambda.
829    ///
830    /// The probability mass function of this distribusion is
831    /// \f[ \frac{e^{-\lambda}\lambda^k}{k!} \f]
832    /// \note The algorithm is taken from the book of Donald E. Knuth titled
833    /// ''Seminumerical Algorithms'' (1969). Its running time is linear in the
834    /// return value.
835   
836    int poisson(double lambda)
837    {
838      const double l = std::exp(-lambda);
839      int k=0;
840      double p = 1.0;
841      do {
842        k++;
843        p*=real<double>();
844      } while (p>=l);
845      return k-1;
846    } 
847     
848    ///@}
849   
850    ///\name Two dimensional distributions
851    ///
852
853    ///@{
854   
855    /// Uniform distribution on the full unit circle
856
857    /// Uniform distribution on the full unit circle.
858    ///
859    dim2::Point<double> disc()
860    {
861      double V1,V2;
862      do {
863        V1=2*real<double>()-1;
864        V2=2*real<double>()-1;
865       
866      } while(V1*V1+V2*V2>=1);
867      return dim2::Point<double>(V1,V2);
868    }
869    /// A kind of two dimensional Gauss distribution
870
871    /// This function provides a turning symmetric two-dimensional distribution.
872    /// Both coordinates are of standard normal distribution, but they are not
873    /// independent.
874    ///
875    /// \note The coordinates are the two random variables provided by
876    /// the Box-Muller method.
877    dim2::Point<double> gauss2()
878    {
879      double V1,V2,S;
880      do {
881        V1=2*real<double>()-1;
882        V2=2*real<double>()-1;
883        S=V1*V1+V2*V2;
884      } while(S>=1);
885      double W=std::sqrt(-2*std::log(S)/S);
886      return dim2::Point<double>(W*V1,W*V2);
887    }
888    /// A kind of two dimensional exponential distribution
889
890    /// This function provides a turning symmetric two-dimensional distribution.
891    /// The x-coordinate is of conditionally exponential distribution
892    /// with the condition that x is positive and y=0. If x is negative and
893    /// y=0 then, -x is of exponential distribution. The same is true for the
894    /// y-coordinate.
895    dim2::Point<double> exponential2()
896    {
897      double V1,V2,S;
898      do {
899        V1=2*real<double>()-1;
900        V2=2*real<double>()-1;
901        S=V1*V1+V2*V2;
902      } while(S>=1);
903      double W=-std::log(S)/S;
904      return dim2::Point<double>(W*V1,W*V2);
905    }
906
907    ///@}   
908  };
909
910
911  extern Random rnd;
912
913}
914
915#endif
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