COIN-OR::LEMON - Graph Library

source: lemon-0.x/lemon/random.h

Last change on this file was 2618:6aa6fcaeaea5, checked in by Balazs Dezso, 11 years ago

G++-4.3 compatibility changes

File size: 26.1 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
69#include <ctime>
70#include <limits>
71
72#include <lemon/math.h>
73#include <lemon/dim2.h>
74///\ingroup misc
75///\file
76///\brief Mersenne Twister random number generator
77///
78///\author Balazs Dezso
79
80namespace lemon {
81
82  namespace _random_bits {
83   
84    template <typename _Word, int _bits = std::numeric_limits<_Word>::digits>
85    struct RandomTraits {};
86
87    template <typename _Word>
88    struct RandomTraits<_Word, 32> {
89
90      typedef _Word Word;
91      static const int bits = 32;
92
93      static const int length = 624;
94      static const int shift = 397;
95     
96      static const Word mul = 0x6c078965u;
97      static const Word arrayInit = 0x012BD6AAu;
98      static const Word arrayMul1 = 0x0019660Du;
99      static const Word arrayMul2 = 0x5D588B65u;
100
101      static const Word mask = 0x9908B0DFu;
102      static const Word loMask = (1u << 31) - 1;
103      static const Word hiMask = ~loMask;
104
105
106      static Word tempering(Word rnd) {
107        rnd ^= (rnd >> 11);
108        rnd ^= (rnd << 7) & 0x9D2C5680u;
109        rnd ^= (rnd << 15) & 0xEFC60000u;
110        rnd ^= (rnd >> 18);
111        return rnd;
112      }
113
114    };
115
116    template <typename _Word>
117    struct RandomTraits<_Word, 64> {
118
119      typedef _Word Word;
120      static const int bits = 64;
121
122      static const int length = 312;
123      static const int shift = 156;
124
125      static const Word mul = Word(0x5851F42Du) << 32 | Word(0x4C957F2Du);
126      static const Word arrayInit = Word(0x00000000u) << 32 |Word(0x012BD6AAu);
127      static const Word arrayMul1 = Word(0x369DEA0Fu) << 32 |Word(0x31A53F85u);
128      static const Word arrayMul2 = Word(0x27BB2EE6u) << 32 |Word(0x87B0B0FDu);
129
130      static const Word mask = Word(0xB5026F5Au) << 32 | Word(0xA96619E9u);
131      static const Word loMask = (Word(1u) << 31) - 1;
132      static const Word hiMask = ~loMask;
133
134      static Word tempering(Word rnd) {
135        rnd ^= (rnd >> 29) & (Word(0x55555555u) << 32 | Word(0x55555555u));
136        rnd ^= (rnd << 17) & (Word(0x71D67FFFu) << 32 | Word(0xEDA60000u));
137        rnd ^= (rnd << 37) & (Word(0xFFF7EEE0u) << 32 | Word(0x00000000u));
138        rnd ^= (rnd >> 43);
139        return rnd;
140      }
141
142    };
143
144    template <typename _Word>
145    class RandomCore {
146    public:
147
148      typedef _Word Word;
149
150    private:
151
152      static const int bits = RandomTraits<Word>::bits;
153
154      static const int length = RandomTraits<Word>::length;
155      static const int shift = RandomTraits<Word>::shift;
156
157    public:
158
159      void initState() {
160        static const Word seedArray[4] = {
161          0x12345u, 0x23456u, 0x34567u, 0x45678u
162        };
163   
164        initState(seedArray, seedArray + 4);
165      }
166
167      void initState(Word seed) {
168
169        static const Word mul = RandomTraits<Word>::mul;
170
171        current = state;
172
173        Word *curr = state + length - 1;
174        curr[0] = seed; --curr;
175        for (int i = 1; i < length; ++i) {
176          curr[0] = (mul * ( curr[1] ^ (curr[1] >> (bits - 2)) ) + i);
177          --curr;
178        }
179      }
180
181      template <typename Iterator>
182      void initState(Iterator begin, Iterator end) {
183
184        static const Word init = RandomTraits<Word>::arrayInit;
185        static const Word mul1 = RandomTraits<Word>::arrayMul1;
186        static const Word mul2 = RandomTraits<Word>::arrayMul2;
187
188
189        Word *curr = state + length - 1; --curr;
190        Iterator it = begin; int cnt = 0;
191        int num;
192
193        initState(init);
194
195        num = length > end - begin ? length : end - begin;
196        while (num--) {
197          curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul1))
198            + *it + cnt;
199          ++it; ++cnt;
200          if (it == end) {
201            it = begin; cnt = 0;
202          }
203          if (curr == state) {
204            curr = state + length - 1; curr[0] = state[0];
205          }
206          --curr;
207        }
208
209        num = length - 1; cnt = length - (curr - state) - 1;
210        while (num--) {
211          curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul2))
212            - cnt;
213          --curr; ++cnt;
214          if (curr == state) {
215            curr = state + length - 1; curr[0] = state[0]; --curr;
216            cnt = 1;
217          }
218        }
219       
220        state[length - 1] = Word(1) << (bits - 1);
221      }
222     
223      void copyState(const RandomCore& other) {
224        std::copy(other.state, other.state + length, state);
225        current = state + (other.current - other.state);
226      }
227
228      Word operator()() {
229        if (current == state) fillState();
230        --current;
231        Word rnd = *current;
232        return RandomTraits<Word>::tempering(rnd);
233      }
234
235    private:
236
237 
238      void fillState() {
239        static const Word mask[2] = { 0x0ul, RandomTraits<Word>::mask };
240        static const Word loMask = RandomTraits<Word>::loMask;
241        static const Word hiMask = RandomTraits<Word>::hiMask;
242
243        current = state + length;
244
245        register Word *curr = state + length - 1;
246        register long num;
247     
248        num = length - shift;
249        while (num--) {
250          curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^
251            curr[- shift] ^ mask[curr[-1] & 1ul];
252          --curr;
253        }
254        num = shift - 1;
255        while (num--) {
256          curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^
257            curr[length - shift] ^ mask[curr[-1] & 1ul];
258          --curr;
259        }
260        state[0] = (((state[0] & hiMask) | (curr[length - 1] & loMask)) >> 1) ^
261          curr[length - shift] ^ mask[int(curr[length - 1] & 1ul)];
262
263      }
264
265 
266      Word *current;
267      Word state[length];
268     
269    };
270
271
272    template <typename Result,
273              int shift = (std::numeric_limits<Result>::digits + 1) / 2>
274    struct Masker {
275      static Result mask(const Result& result) {
276        return Masker<Result, (shift + 1) / 2>::
277          mask(static_cast<Result>(result | (result >> shift)));
278      }
279    };
280   
281    template <typename Result>
282    struct Masker<Result, 1> {
283      static Result mask(const Result& result) {
284        return static_cast<Result>(result | (result >> 1));
285      }
286    };
287
288    template <typename Result, typename Word,
289              int rest = std::numeric_limits<Result>::digits, int shift = 0,
290              bool last = rest <= std::numeric_limits<Word>::digits>
291    struct IntConversion {
292      static const int bits = std::numeric_limits<Word>::digits;
293   
294      static Result convert(RandomCore<Word>& rnd) {
295        return static_cast<Result>(rnd() >> (bits - rest)) << shift;
296      }
297     
298    };
299
300    template <typename Result, typename Word, int rest, int shift>
301    struct IntConversion<Result, Word, rest, shift, false> {
302      static const int bits = std::numeric_limits<Word>::digits;
303
304      static Result convert(RandomCore<Word>& rnd) {
305        return (static_cast<Result>(rnd()) << shift) |
306          IntConversion<Result, Word, rest - bits, shift + bits>::convert(rnd);
307      }
308    };
309
310
311    template <typename Result, typename Word,
312              bool one_word = (std::numeric_limits<Word>::digits <
313                               std::numeric_limits<Result>::digits) >
314    struct Mapping {
315      static Result map(RandomCore<Word>& rnd, const Result& bound) {
316        Word max = Word(bound - 1);
317        Result mask = Masker<Result>::mask(bound - 1);
318        Result num;
319        do {
320          num = IntConversion<Result, Word>::convert(rnd) & mask;
321        } while (num > max);
322        return num;
323      }
324    };
325
326    template <typename Result, typename Word>
327    struct Mapping<Result, Word, false> {
328      static Result map(RandomCore<Word>& rnd, const Result& bound) {
329        Word max = Word(bound - 1);
330        Word mask = Masker<Word, (std::numeric_limits<Result>::digits + 1) / 2>
331          ::mask(max);
332        Word num;
333        do {
334          num = rnd() & mask;
335        } while (num > max);
336        return num;
337      }
338    };
339
340    template <typename Result, int exp, bool pos = (exp >= 0)>
341    struct ShiftMultiplier {
342      static const Result multiplier() {
343        Result res = ShiftMultiplier<Result, exp / 2>::multiplier();
344        res *= res;
345        if ((exp & 1) == 1) res *= static_cast<Result>(2.0);
346        return res;
347      }
348    };
349
350    template <typename Result, int exp>
351    struct ShiftMultiplier<Result, exp, false> {
352      static const Result multiplier() {
353        Result res = ShiftMultiplier<Result, exp / 2>::multiplier();
354        res *= res;
355        if ((exp & 1) == 1) res *= static_cast<Result>(0.5);
356        return res;
357      }
358    };
359
360    template <typename Result>
361    struct ShiftMultiplier<Result, 0, true> {
362      static const Result multiplier() {
363        return static_cast<Result>(1.0);
364      }
365    };
366
367    template <typename Result>
368    struct ShiftMultiplier<Result, -20, true> {
369      static const Result multiplier() {
370        return static_cast<Result>(1.0/1048576.0);
371      }
372    };
373   
374    template <typename Result>
375    struct ShiftMultiplier<Result, -32, true> {
376      static const Result multiplier() {
377        return static_cast<Result>(1.0/424967296.0);
378      }
379    };
380
381    template <typename Result>
382    struct ShiftMultiplier<Result, -53, true> {
383      static const Result multiplier() {
384        return static_cast<Result>(1.0/9007199254740992.0);
385      }
386    };
387
388    template <typename Result>
389    struct ShiftMultiplier<Result, -64, true> {
390      static const Result multiplier() {
391        return static_cast<Result>(1.0/18446744073709551616.0);
392      }
393    };
394
395    template <typename Result, int exp>
396    struct Shifting {
397      static Result shift(const Result& result) {
398        return result * ShiftMultiplier<Result, exp>::multiplier();
399      }
400    };
401
402    template <typename Result, typename Word,
403              int rest = std::numeric_limits<Result>::digits, int shift = 0,
404              bool last = rest <= std::numeric_limits<Word>::digits>
405    struct RealConversion{
406      static const int bits = std::numeric_limits<Word>::digits;
407
408      static Result convert(RandomCore<Word>& rnd) {
409        return Shifting<Result, - shift - rest>::
410          shift(static_cast<Result>(rnd() >> (bits - rest)));
411      }
412    };
413
414    template <typename Result, typename Word, int rest, int shift>
415    struct RealConversion<Result, Word, rest, shift, false> {
416      static const int bits = std::numeric_limits<Word>::digits;
417
418      static Result convert(RandomCore<Word>& rnd) {
419        return Shifting<Result, - shift - bits>::
420          shift(static_cast<Result>(rnd())) +
421          RealConversion<Result, Word, rest-bits, shift + bits>::
422          convert(rnd);
423      }
424    };
425
426    template <typename Result, typename Word>
427    struct Initializer {
428
429      template <typename Iterator>
430      static void init(RandomCore<Word>& rnd, Iterator begin, Iterator end) {
431        std::vector<Word> ws;
432        for (Iterator it = begin; it != end; ++it) {
433          ws.push_back(Word(*it));
434        }
435        rnd.initState(ws.begin(), ws.end());
436      }
437
438      static void init(RandomCore<Word>& rnd, Result seed) {
439        rnd.initState(seed);
440      }
441    };
442
443    template <typename Word>
444    struct BoolConversion {
445      static bool convert(RandomCore<Word>& rnd) {
446        return (rnd() & 1) == 1;
447      }
448    };
449
450    template <typename Word>
451    struct BoolProducer {
452      Word buffer;
453      int num;
454     
455      BoolProducer() : num(0) {}
456
457      bool convert(RandomCore<Word>& rnd) {
458        if (num == 0) {
459          buffer = rnd();
460          num = RandomTraits<Word>::bits;
461        }
462        bool r = (buffer & 1);
463        buffer >>= 1;
464        --num;
465        return r;
466      }
467    };
468
469  }
470
471  /// \ingroup misc
472  ///
473  /// \brief Mersenne Twister random number generator
474  ///
475  /// The Mersenne Twister is a twisted generalized feedback
476  /// shift-register generator of Matsumoto and Nishimura. The period
477  /// of this generator is \f$ 2^{19937} - 1 \f$ and it is
478  /// equi-distributed in 623 dimensions for 32-bit numbers. The time
479  /// performance of this generator is comparable to the commonly used
480  /// generators.
481  ///
482  /// This implementation is specialized for both 32-bit and 64-bit
483  /// architectures. The generators differ sligthly in the
484  /// initialization and generation phase so they produce two
485  /// completly different sequences.
486  ///
487  /// The generator gives back random numbers of serveral types. To
488  /// get a random number from a range of a floating point type you
489  /// can use one form of the \c operator() or the \c real() member
490  /// function. If you want to get random number from the {0, 1, ...,
491  /// n-1} integer range use the \c operator[] or the \c integer()
492  /// method. And to get random number from the whole range of an
493  /// integer type you can use the argumentless \c integer() or \c
494  /// uinteger() functions. After all you can get random bool with
495  /// equal chance of true and false or given probability of true
496  /// result with the \c boolean() member functions.
497  ///
498  ///\code
499  /// // The commented code is identical to the other
500  /// double a = rnd();                     // [0.0, 1.0)
501  /// // double a = rnd.real();             // [0.0, 1.0)
502  /// double b = rnd(100.0);                // [0.0, 100.0)
503  /// // double b = rnd.real(100.0);        // [0.0, 100.0)
504  /// double c = rnd(1.0, 2.0);             // [1.0, 2.0)
505  /// // double c = rnd.real(1.0, 2.0);     // [1.0, 2.0)
506  /// int d = rnd[100000];                  // 0..99999
507  /// // int d = rnd.integer(100000);       // 0..99999
508  /// int e = rnd[6] + 1;                   // 1..6
509  /// // int e = rnd.integer(1, 1 + 6);     // 1..6
510  /// int b = rnd.uinteger<int>();          // 0 .. 2^31 - 1
511  /// int c = rnd.integer<int>();           // - 2^31 .. 2^31 - 1
512  /// bool g = rnd.boolean();               // P(g = true) = 0.5
513  /// bool h = rnd.boolean(0.8);            // P(h = true) = 0.8
514  ///\endcode
515  ///
516  /// The lemon provides a global instance of the random number
517  /// generator which name is \ref lemon::rnd "rnd". Usually it is a
518  /// good programming convenience to use this global generator to get
519  /// random numbers.
520  ///
521  /// \author Balazs Dezso
522  class Random {
523  private:
524
525    // architecture word
526    typedef unsigned long Word;
527   
528    _random_bits::RandomCore<Word> core;
529    _random_bits::BoolProducer<Word> bool_producer;
530   
531
532  public:
533
534    /// \brief Constructor
535    ///
536    /// Constructor with constant seeding.
537    Random() { core.initState(); }
538
539    /// \brief Constructor
540    ///
541    /// Constructor with seed. The current number type will be converted
542    /// to the architecture word type.
543    template <typename Number>
544    Random(Number seed) {
545      _random_bits::Initializer<Number, Word>::init(core, seed);
546    }
547
548    /// \brief Constructor
549    ///
550    /// Constructor with array seeding. The given range should contain
551    /// any number type and the numbers will be converted to the
552    /// architecture word type.
553    template <typename Iterator>
554    Random(Iterator begin, Iterator end) {
555      typedef typename std::iterator_traits<Iterator>::value_type Number;
556      _random_bits::Initializer<Number, Word>::init(core, begin, end);
557    }
558
559    /// \brief Copy constructor
560    ///
561    /// Copy constructor. The generated sequence will be identical to
562    /// the other sequence. It can be used to save the current state
563    /// of the generator and later use it to generate the same
564    /// sequence.
565    Random(const Random& other) {
566      core.copyState(other.core);
567    }
568
569    /// \brief Assign operator
570    ///
571    /// Assign operator. The generated sequence will be identical to
572    /// the other sequence. It can be used to save the current state
573    /// of the generator and later use it to generate the same
574    /// sequence.
575    Random& operator=(const Random& other) {
576      if (&other != this) {
577        core.copyState(other.core);
578      }
579      return *this;
580    }
581
582    /// \brief Seeding random sequence
583    ///
584    /// Seeding the random sequence. The current number type will be
585    /// converted to the architecture word type.
586    template <typename Number>
587    void seed(Number seed) {
588      _random_bits::Initializer<Number, Word>::init(core, seed);
589    }
590
591    /// \brief Seeding random sequence
592    ///
593    /// Seeding the random sequence. The given range should contain
594    /// any number type and the numbers will be converted to the
595    /// architecture word type.
596    template <typename Iterator>
597    void seed(Iterator begin, Iterator end) {
598      typedef typename std::iterator_traits<Iterator>::value_type Number;
599      _random_bits::Initializer<Number, Word>::init(core, begin, end);
600    }
601
602    /// \brief Returns a random real number from the range [0, 1)
603    ///
604    /// It returns a random real number from the range [0, 1). The
605    /// default Number type is double.
606    template <typename Number>
607    Number real() {
608      return _random_bits::RealConversion<Number, Word>::convert(core);
609    }
610
611    double real() {
612      return real<double>();
613    }
614
615    /// \brief Returns a random real number the range [0, b)
616    ///
617    /// It returns a random real number from the range [0, b).
618    template <typename Number>
619    Number real(Number b) {
620      return real<Number>() * b;
621    }
622
623    /// \brief Returns a random real number from the range [a, b)
624    ///
625    /// It returns a random real number from the range [a, b).
626    template <typename Number>
627    Number real(Number a, Number b) {
628      return real<Number>() * (b - a) + a;
629    }
630
631    /// \brief Returns a random real number from the range [0, 1)
632    ///
633    /// It returns a random double from the range [0, 1).
634    double operator()() {
635      return real<double>();
636    }
637
638    /// \brief Returns a random real number from the range [0, b)
639    ///
640    /// It returns a random real number from the range [0, b).
641    template <typename Number>
642    Number operator()(Number b) {
643      return real<Number>() * b;
644    }
645
646    /// \brief Returns a random real number from the range [a, b)
647    ///
648    /// It returns a random real number from the range [a, b).
649    template <typename Number>
650    Number operator()(Number a, Number b) {
651      return real<Number>() * (b - a) + a;
652    }
653
654    /// \brief Returns a random integer from a range
655    ///
656    /// It returns a random integer from the range {0, 1, ..., b - 1}.
657    template <typename Number>
658    Number integer(Number b) {
659      return _random_bits::Mapping<Number, Word>::map(core, b);
660    }
661
662    /// \brief Returns a random integer from a range
663    ///
664    /// It returns a random integer from the range {a, a + 1, ..., b - 1}.
665    template <typename Number>
666    Number integer(Number a, Number b) {
667      return _random_bits::Mapping<Number, Word>::map(core, b - a) + a;
668    }
669
670    /// \brief Returns a random integer from a range
671    ///
672    /// It returns a random integer from the range {0, 1, ..., b - 1}.
673    template <typename Number>
674    Number operator[](Number b) {
675      return _random_bits::Mapping<Number, Word>::map(core, b);
676    }
677
678    /// \brief Returns a random non-negative integer
679    ///
680    /// It returns a random non-negative integer uniformly from the
681    /// whole range of the current \c Number type.  The default result
682    /// type of this function is unsigned int.
683    template <typename Number>
684    Number uinteger() {
685      return _random_bits::IntConversion<Number, Word>::convert(core);
686    }
687
688    unsigned int uinteger() {
689      return uinteger<unsigned int>();
690    }
691
692    /// \brief Returns a random integer
693    ///
694    /// It returns a random integer uniformly from the whole range of
695    /// the current \c Number type. The default result type of this
696    /// function is int.
697    template <typename Number>
698    Number integer() {
699      static const int nb = std::numeric_limits<Number>::digits +
700        (std::numeric_limits<Number>::is_signed ? 1 : 0);
701      return _random_bits::IntConversion<Number, Word, nb>::convert(core);
702    }
703
704    int integer() {
705      return integer<int>();
706    }
707   
708    /// \brief Returns a random bool
709    ///
710    /// It returns a random bool. The generator holds a buffer for
711    /// random bits. Every time when it become empty the generator makes
712    /// a new random word and fill the buffer up.
713    bool boolean() {
714      return bool_producer.convert(core);
715    }
716
717    ///\name Nonuniform distributions
718    ///
719   
720    ///@{
721   
722    /// \brief Returns a random bool
723    ///
724    /// It returns a random bool with given probability of true result
725    bool boolean(double p) {
726      return operator()() < p;
727    }
728
729    /// Standard Gauss distribution
730
731    /// Standard Gauss distribution.
732    /// \note The Cartesian form of the Box-Muller
733    /// transformation is used to generate a random normal distribution.
734    /// \todo Consider using the "ziggurat" method instead.
735    double gauss()
736    {
737      double V1,V2,S;
738      do {
739        V1=2*real<double>()-1;
740        V2=2*real<double>()-1;
741        S=V1*V1+V2*V2;
742      } while(S>=1);
743      return std::sqrt(-2*std::log(S)/S)*V1;
744    }
745    /// Gauss distribution with given standard deviation and mean 0
746
747    /// \sa gauss()
748    ///
749    double gauss(double std_dev)
750    {
751      return gauss()*std_dev;
752    }
753    /// Gauss distribution with given mean and standard deviation
754
755    /// \sa gauss()
756    ///
757    double gauss(double mean,double std_dev)
758    {
759      return gauss()*std_dev+mean;
760    }
761
762    /// Exponential distribution with given mean
763
764    /// This function generates an exponential distribution random number
765    /// with mean <tt>1/lambda</tt>.
766    ///
767    double exponential(double lambda=1.0)
768    {
769      return -std::log(real<double>())/lambda;
770    }
771
772    double gamma(int k)
773    {
774      double s = 0;
775      for(int i=0;i<k;i++) s-=std::log(1.0-real<double>());
776      return s;
777    }
778   
779    /// Gamma distribution with given shape and scale parameter
780
781    /// This function generates a gamma distribution random number.
782    ///
783    ///\param k shape parameter (<tt>k>0</tt>)
784    ///\param theta scale parameter
785    ///
786    double gamma(double k,double theta=1.0)
787    {
788      double xi,nu;
789      const double delta = k-std::floor(k);
790      const double v0=E/(E-delta);
791      do {
792        double V0=1.0-real<double>();
793        double V1=1.0-real<double>();
794        double V2=1.0-real<double>();
795        if(V2<=v0)
796          {
797            xi=std::pow(V1,1.0/delta);
798            nu=V0*std::pow(xi,delta-1.0);
799          }
800        else
801          {
802            xi=1.0-std::log(V1);
803            nu=V0*std::exp(-xi);
804          }
805      } while(nu>std::pow(xi,delta-1.0)*std::exp(-xi));
806      return theta*(xi+gamma(int(std::floor(k))));
807    }
808   
809     
810    ///@}
811   
812    ///\name Two dimensional distributions
813    ///
814
815    ///@{
816   
817    /// Uniform distribution on the full unit circle.
818    dim2::Point<double> disc()
819    {
820      double V1,V2;
821      do {
822        V1=2*real<double>()-1;
823        V2=2*real<double>()-1;
824       
825      } while(V1*V1+V2*V2>=1);
826      return dim2::Point<double>(V1,V2);
827    }
828    /// A kind of two dimensional Gauss distribution
829
830    /// This function provides a turning symmetric two-dimensional distribution.
831    /// Both coordinates are of standard normal distribution, but they are not
832    /// independent.
833    ///
834    /// \note The coordinates are the two random variables provided by
835    /// the Box-Muller method.
836    dim2::Point<double> gauss2()
837    {
838      double V1,V2,S;
839      do {
840        V1=2*real<double>()-1;
841        V2=2*real<double>()-1;
842        S=V1*V1+V2*V2;
843      } while(S>=1);
844      double W=std::sqrt(-2*std::log(S)/S);
845      return dim2::Point<double>(W*V1,W*V2);
846    }
847    /// A kind of two dimensional exponential distribution
848
849    /// This function provides a turning symmetric two-dimensional distribution.
850    /// The x-coordinate is of conditionally exponential distribution
851    /// with the condition that x is positive and y=0. If x is negative and
852    /// y=0 then, -x is of exponential distribution. The same is true for the
853    /// y-coordinate.
854    dim2::Point<double> exponential2()
855    {
856      double V1,V2,S;
857      do {
858        V1=2*real<double>()-1;
859        V2=2*real<double>()-1;
860        S=V1*V1+V2*V2;
861      } while(S>=1);
862      double W=-std::log(S)/S;
863      return dim2::Point<double>(W*V1,W*V2);
864    }
865
866    ///@}   
867  };
868
869
870  extern Random rnd;
871
872}
873
874#endif
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