Update INSTALL file.
1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
3 * This file is a part of LEMON, a generic C++ optimization library.
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
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.
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
20 * This file contains the reimplemented version of the Mersenne Twister
21 * Generator of Matsumoto and Nishimura.
23 * See the appropriate copyright notice below.
25 * Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura,
26 * All rights reserved.
28 * Redistribution and use in source and binary forms, with or without
29 * modification, are permitted provided that the following conditions
32 * 1. Redistributions of source code must retain the above copyright
33 * notice, this list of conditions and the following disclaimer.
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.
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
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.
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)
62 #ifndef LEMON_RANDOM_H
63 #define LEMON_RANDOM_H
71 #include <lemon/math.h>
72 #include <lemon/dim2.h>
77 #include <sys/types.h>
85 ///\brief Mersenne Twister random number generator
89 namespace _random_bits {
91 template <typename _Word, int _bits = std::numeric_limits<_Word>::digits>
92 struct RandomTraits {};
94 template <typename _Word>
95 struct RandomTraits<_Word, 32> {
98 static const int bits = 32;
100 static const int length = 624;
101 static const int shift = 397;
103 static const Word mul = 0x6c078965u;
104 static const Word arrayInit = 0x012BD6AAu;
105 static const Word arrayMul1 = 0x0019660Du;
106 static const Word arrayMul2 = 0x5D588B65u;
108 static const Word mask = 0x9908B0DFu;
109 static const Word loMask = (1u << 31) - 1;
110 static const Word hiMask = ~loMask;
113 static Word tempering(Word rnd) {
115 rnd ^= (rnd << 7) & 0x9D2C5680u;
116 rnd ^= (rnd << 15) & 0xEFC60000u;
123 template <typename _Word>
124 struct RandomTraits<_Word, 64> {
127 static const int bits = 64;
129 static const int length = 312;
130 static const int shift = 156;
132 static const Word mul = Word(0x5851F42Du) << 32 | Word(0x4C957F2Du);
133 static const Word arrayInit = Word(0x00000000u) << 32 |Word(0x012BD6AAu);
134 static const Word arrayMul1 = Word(0x369DEA0Fu) << 32 |Word(0x31A53F85u);
135 static const Word arrayMul2 = Word(0x27BB2EE6u) << 32 |Word(0x87B0B0FDu);
137 static const Word mask = Word(0xB5026F5Au) << 32 | Word(0xA96619E9u);
138 static const Word loMask = (Word(1u) << 31) - 1;
139 static const Word hiMask = ~loMask;
141 static Word tempering(Word rnd) {
142 rnd ^= (rnd >> 29) & (Word(0x55555555u) << 32 | Word(0x55555555u));
143 rnd ^= (rnd << 17) & (Word(0x71D67FFFu) << 32 | Word(0xEDA60000u));
144 rnd ^= (rnd << 37) & (Word(0xFFF7EEE0u) << 32 | Word(0x00000000u));
151 template <typename _Word>
159 static const int bits = RandomTraits<Word>::bits;
161 static const int length = RandomTraits<Word>::length;
162 static const int shift = RandomTraits<Word>::shift;
167 static const Word seedArray[4] = {
168 0x12345u, 0x23456u, 0x34567u, 0x45678u
171 initState(seedArray, seedArray + 4);
174 void initState(Word seed) {
176 static const Word mul = RandomTraits<Word>::mul;
180 Word *curr = state + length - 1;
181 curr[0] = seed; --curr;
182 for (int i = 1; i < length; ++i) {
183 curr[0] = (mul * ( curr[1] ^ (curr[1] >> (bits - 2)) ) + i);
188 template <typename Iterator>
189 void initState(Iterator begin, Iterator end) {
191 static const Word init = RandomTraits<Word>::arrayInit;
192 static const Word mul1 = RandomTraits<Word>::arrayMul1;
193 static const Word mul2 = RandomTraits<Word>::arrayMul2;
196 Word *curr = state + length - 1; --curr;
197 Iterator it = begin; int cnt = 0;
202 num = length > end - begin ? length : end - begin;
204 curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul1))
211 curr = state + length - 1; curr[0] = state[0];
216 num = length - 1; cnt = length - (curr - state) - 1;
218 curr[0] = (curr[0] ^ ((curr[1] ^ (curr[1] >> (bits - 2))) * mul2))
222 curr = state + length - 1; curr[0] = state[0]; --curr;
227 state[length - 1] = Word(1) << (bits - 1);
230 void copyState(const RandomCore& other) {
231 std::copy(other.state, other.state + length, state);
232 current = state + (other.current - other.state);
236 if (current == state) fillState();
239 return RandomTraits<Word>::tempering(rnd);
246 static const Word mask[2] = { 0x0ul, RandomTraits<Word>::mask };
247 static const Word loMask = RandomTraits<Word>::loMask;
248 static const Word hiMask = RandomTraits<Word>::hiMask;
250 current = state + length;
252 register Word *curr = state + length - 1;
255 num = length - shift;
257 curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^
258 curr[- shift] ^ mask[curr[-1] & 1ul];
263 curr[0] = (((curr[0] & hiMask) | (curr[-1] & loMask)) >> 1) ^
264 curr[length - shift] ^ mask[curr[-1] & 1ul];
267 state[0] = (((state[0] & hiMask) | (curr[length - 1] & loMask)) >> 1) ^
268 curr[length - shift] ^ mask[curr[length - 1] & 1ul];
279 template <typename Result,
280 int shift = (std::numeric_limits<Result>::digits + 1) / 2>
282 static Result mask(const Result& result) {
283 return Masker<Result, (shift + 1) / 2>::
284 mask(static_cast<Result>(result | (result >> shift)));
288 template <typename Result>
289 struct Masker<Result, 1> {
290 static Result mask(const Result& result) {
291 return static_cast<Result>(result | (result >> 1));
295 template <typename Result, typename Word,
296 int rest = std::numeric_limits<Result>::digits, int shift = 0,
297 bool last = rest <= std::numeric_limits<Word>::digits>
298 struct IntConversion {
299 static const int bits = std::numeric_limits<Word>::digits;
301 static Result convert(RandomCore<Word>& rnd) {
302 return static_cast<Result>(rnd() >> (bits - rest)) << shift;
307 template <typename Result, typename Word, int rest, int shift>
308 struct IntConversion<Result, Word, rest, shift, false> {
309 static const int bits = std::numeric_limits<Word>::digits;
311 static Result convert(RandomCore<Word>& rnd) {
312 return (static_cast<Result>(rnd()) << shift) |
313 IntConversion<Result, Word, rest - bits, shift + bits>::convert(rnd);
318 template <typename Result, typename Word,
319 bool one_word = (std::numeric_limits<Word>::digits <
320 std::numeric_limits<Result>::digits) >
322 static Result map(RandomCore<Word>& rnd, const Result& bound) {
323 Word max = Word(bound - 1);
324 Result mask = Masker<Result>::mask(bound - 1);
327 num = IntConversion<Result, Word>::convert(rnd) & mask;
333 template <typename Result, typename Word>
334 struct Mapping<Result, Word, false> {
335 static Result map(RandomCore<Word>& rnd, const Result& bound) {
336 Word max = Word(bound - 1);
337 Word mask = Masker<Word, (std::numeric_limits<Result>::digits + 1) / 2>
347 template <typename Result, int exp, bool pos = (exp >= 0)>
348 struct ShiftMultiplier {
349 static const Result multiplier() {
350 Result res = ShiftMultiplier<Result, exp / 2>::multiplier();
352 if ((exp & 1) == 1) res *= static_cast<Result>(2.0);
357 template <typename Result, int exp>
358 struct ShiftMultiplier<Result, exp, false> {
359 static const Result multiplier() {
360 Result res = ShiftMultiplier<Result, exp / 2>::multiplier();
362 if ((exp & 1) == 1) res *= static_cast<Result>(0.5);
367 template <typename Result>
368 struct ShiftMultiplier<Result, 0, true> {
369 static const Result multiplier() {
370 return static_cast<Result>(1.0);
374 template <typename Result>
375 struct ShiftMultiplier<Result, -20, true> {
376 static const Result multiplier() {
377 return static_cast<Result>(1.0/1048576.0);
381 template <typename Result>
382 struct ShiftMultiplier<Result, -32, true> {
383 static const Result multiplier() {
384 return static_cast<Result>(1.0/424967296.0);
388 template <typename Result>
389 struct ShiftMultiplier<Result, -53, true> {
390 static const Result multiplier() {
391 return static_cast<Result>(1.0/9007199254740992.0);
395 template <typename Result>
396 struct ShiftMultiplier<Result, -64, true> {
397 static const Result multiplier() {
398 return static_cast<Result>(1.0/18446744073709551616.0);
402 template <typename Result, int exp>
404 static Result shift(const Result& result) {
405 return result * ShiftMultiplier<Result, exp>::multiplier();
409 template <typename Result, typename Word,
410 int rest = std::numeric_limits<Result>::digits, int shift = 0,
411 bool last = rest <= std::numeric_limits<Word>::digits>
412 struct RealConversion{
413 static const int bits = std::numeric_limits<Word>::digits;
415 static Result convert(RandomCore<Word>& rnd) {
416 return Shifting<Result, - shift - rest>::
417 shift(static_cast<Result>(rnd() >> (bits - rest)));
421 template <typename Result, typename Word, int rest, int shift>
422 struct RealConversion<Result, Word, rest, shift, false> {
423 static const int bits = std::numeric_limits<Word>::digits;
425 static Result convert(RandomCore<Word>& rnd) {
426 return Shifting<Result, - shift - bits>::
427 shift(static_cast<Result>(rnd())) +
428 RealConversion<Result, Word, rest-bits, shift + bits>::
433 template <typename Result, typename Word>
436 template <typename Iterator>
437 static void init(RandomCore<Word>& rnd, Iterator begin, Iterator end) {
438 std::vector<Word> ws;
439 for (Iterator it = begin; it != end; ++it) {
440 ws.push_back(Word(*it));
442 rnd.initState(ws.begin(), ws.end());
445 static void init(RandomCore<Word>& rnd, Result seed) {
450 template <typename Word>
451 struct BoolConversion {
452 static bool convert(RandomCore<Word>& rnd) {
453 return (rnd() & 1) == 1;
457 template <typename Word>
458 struct BoolProducer {
462 BoolProducer() : num(0) {}
464 bool convert(RandomCore<Word>& rnd) {
467 num = RandomTraits<Word>::bits;
469 bool r = (buffer & 1);
480 /// \brief Mersenne Twister random number generator
482 /// The Mersenne Twister is a twisted generalized feedback
483 /// shift-register generator of Matsumoto and Nishimura. The period
484 /// of this generator is \f$ 2^{19937} - 1 \f$ and it is
485 /// equi-distributed in 623 dimensions for 32-bit numbers. The time
486 /// performance of this generator is comparable to the commonly used
489 /// This implementation is specialized for both 32-bit and 64-bit
490 /// architectures. The generators differ sligthly in the
491 /// initialization and generation phase so they produce two
492 /// completly different sequences.
494 /// The generator gives back random numbers of serveral types. To
495 /// get a random number from a range of a floating point type you
496 /// can use one form of the \c operator() or the \c real() member
497 /// function. If you want to get random number from the {0, 1, ...,
498 /// n-1} integer range use the \c operator[] or the \c integer()
499 /// method. And to get random number from the whole range of an
500 /// integer type you can use the argumentless \c integer() or \c
501 /// uinteger() functions. After all you can get random bool with
502 /// equal chance of true and false or given probability of true
503 /// result with the \c boolean() member functions.
506 /// // The commented code is identical to the other
507 /// double a = rnd(); // [0.0, 1.0)
508 /// // double a = rnd.real(); // [0.0, 1.0)
509 /// double b = rnd(100.0); // [0.0, 100.0)
510 /// // double b = rnd.real(100.0); // [0.0, 100.0)
511 /// double c = rnd(1.0, 2.0); // [1.0, 2.0)
512 /// // double c = rnd.real(1.0, 2.0); // [1.0, 2.0)
513 /// int d = rnd[100000]; // 0..99999
514 /// // int d = rnd.integer(100000); // 0..99999
515 /// int e = rnd[6] + 1; // 1..6
516 /// // int e = rnd.integer(1, 1 + 6); // 1..6
517 /// int b = rnd.uinteger<int>(); // 0 .. 2^31 - 1
518 /// int c = rnd.integer<int>(); // - 2^31 .. 2^31 - 1
519 /// bool g = rnd.boolean(); // P(g = true) = 0.5
520 /// bool h = rnd.boolean(0.8); // P(h = true) = 0.8
523 /// LEMON provides a global instance of the random number
524 /// generator which name is \ref lemon::rnd "rnd". Usually it is a
525 /// good programming convenience to use this global generator to get
531 typedef unsigned long Word;
533 _random_bits::RandomCore<Word> core;
534 _random_bits::BoolProducer<Word> bool_producer;
539 ///\name Initialization
543 ///\name Initialization
547 /// \brief Default constructor
549 /// Constructor with constant seeding.
550 Random() { core.initState(); }
552 /// \brief Constructor with seed
554 /// Constructor with seed. The current number type will be converted
555 /// to the architecture word type.
556 template <typename Number>
557 Random(Number seed) {
558 _random_bits::Initializer<Number, Word>::init(core, seed);
561 /// \brief Constructor with array seeding
563 /// Constructor with array seeding. The given range should contain
564 /// any number type and the numbers will be converted to the
565 /// architecture word type.
566 template <typename Iterator>
567 Random(Iterator begin, Iterator end) {
568 typedef typename std::iterator_traits<Iterator>::value_type Number;
569 _random_bits::Initializer<Number, Word>::init(core, begin, end);
572 /// \brief Copy constructor
574 /// Copy constructor. The generated sequence will be identical to
575 /// the other sequence. It can be used to save the current state
576 /// of the generator and later use it to generate the same
578 Random(const Random& other) {
579 core.copyState(other.core);
582 /// \brief Assign operator
584 /// Assign operator. The generated sequence will be identical to
585 /// the other sequence. It can be used to save the current state
586 /// of the generator and later use it to generate the same
588 Random& operator=(const Random& other) {
589 if (&other != this) {
590 core.copyState(other.core);
595 /// \brief Seeding random sequence
597 /// Seeding the random sequence. The current number type will be
598 /// converted to the architecture word type.
599 template <typename Number>
600 void seed(Number seed) {
601 _random_bits::Initializer<Number, Word>::init(core, seed);
604 /// \brief Seeding random sequence
606 /// Seeding the random sequence. The given range should contain
607 /// any number type and the numbers will be converted to the
608 /// architecture word type.
609 template <typename Iterator>
610 void seed(Iterator begin, Iterator end) {
611 typedef typename std::iterator_traits<Iterator>::value_type Number;
612 _random_bits::Initializer<Number, Word>::init(core, begin, end);
615 /// \brief Seeding from file or from process id and time
617 /// By default, this function calls the \c seedFromFile() member
618 /// function with the <tt>/dev/urandom</tt> file. If it does not success,
619 /// it uses the \c seedFromTime().
620 /// \return Currently always true.
623 if (seedFromFile("/dev/urandom", 0)) return true;
625 if (seedFromTime()) return true;
629 /// \brief Seeding from file
631 /// Seeding the random sequence from file. The linux kernel has two
632 /// devices, <tt>/dev/random</tt> and <tt>/dev/urandom</tt> which
633 /// could give good seed values for pseudo random generators (The
634 /// difference between two devices is that the <tt>random</tt> may
635 /// block the reading operation while the kernel can give good
636 /// source of randomness, while the <tt>urandom</tt> does not
637 /// block the input, but it could give back bytes with worse
639 /// \param file The source file
640 /// \param offset The offset, from the file read.
641 /// \return True when the seeding successes.
643 bool seedFromFile(const std::string& file = "/dev/urandom", int offset = 0)
645 bool seedFromFile(const std::string& file = "", int offset = 0)
648 std::ifstream rs(file.c_str());
651 if (offset != 0 && !rs.seekg(offset)) return false;
652 if (!rs.read(reinterpret_cast<char*>(buf), sizeof(buf))) return false;
653 seed(buf, buf + size);
657 /// \brief Seding from process id and time
659 /// Seding from process id and time. This function uses the
660 /// current process id and the current time for initialize the
662 /// \return Currently always true.
663 bool seedFromTime() {
666 gettimeofday(&tv, 0);
667 seed(getpid() + tv.tv_sec + tv.tv_usec);
670 GetSystemTimeAsFileTime(&time);
671 seed(GetCurrentProcessId() + time.dwHighDateTime + time.dwLowDateTime);
678 ///\name Uniform distributions
682 /// \brief Returns a random real number from the range [0, 1)
684 /// It returns a random real number from the range [0, 1). The
685 /// default Number type is \c double.
686 template <typename Number>
688 return _random_bits::RealConversion<Number, Word>::convert(core);
692 return real<double>();
695 /// \brief Returns a random real number the range [0, b)
697 /// It returns a random real number from the range [0, b).
698 template <typename Number>
699 Number real(Number b) {
700 return real<Number>() * b;
703 /// \brief Returns a random real number from the range [a, b)
705 /// It returns a random real number from the range [a, b).
706 template <typename Number>
707 Number real(Number a, Number b) {
708 return real<Number>() * (b - a) + a;
713 ///\name Uniform distributions
717 /// \brief Returns a random real number from the range [0, 1)
719 /// It returns a random double from the range [0, 1).
720 double operator()() {
721 return real<double>();
724 /// \brief Returns a random real number from the range [0, b)
726 /// It returns a random real number from the range [0, b).
727 template <typename Number>
728 Number operator()(Number b) {
729 return real<Number>() * b;
732 /// \brief Returns a random real number from the range [a, b)
734 /// It returns a random real number from the range [a, b).
735 template <typename Number>
736 Number operator()(Number a, Number b) {
737 return real<Number>() * (b - a) + a;
740 /// \brief Returns a random integer from a range
742 /// It returns a random integer from the range {0, 1, ..., b - 1}.
743 template <typename Number>
744 Number integer(Number b) {
745 return _random_bits::Mapping<Number, Word>::map(core, b);
748 /// \brief Returns a random integer from a range
750 /// It returns a random integer from the range {a, a + 1, ..., b - 1}.
751 template <typename Number>
752 Number integer(Number a, Number b) {
753 return _random_bits::Mapping<Number, Word>::map(core, b - a) + a;
756 /// \brief Returns a random integer from a range
758 /// It returns a random integer from the range {0, 1, ..., b - 1}.
759 template <typename Number>
760 Number operator[](Number b) {
761 return _random_bits::Mapping<Number, Word>::map(core, b);
764 /// \brief Returns a random non-negative integer
766 /// It returns a random non-negative integer uniformly from the
767 /// whole range of the current \c Number type. The default result
768 /// type of this function is <tt>unsigned int</tt>.
769 template <typename Number>
771 return _random_bits::IntConversion<Number, Word>::convert(core);
776 unsigned int uinteger() {
777 return uinteger<unsigned int>();
780 /// \brief Returns a random integer
782 /// It returns a random integer uniformly from the whole range of
783 /// the current \c Number type. The default result type of this
784 /// function is \c int.
785 template <typename Number>
787 static const int nb = std::numeric_limits<Number>::digits +
788 (std::numeric_limits<Number>::is_signed ? 1 : 0);
789 return _random_bits::IntConversion<Number, Word, nb>::convert(core);
793 return integer<int>();
796 /// \brief Returns a random bool
798 /// It returns a random bool. The generator holds a buffer for
799 /// random bits. Every time when it become empty the generator makes
800 /// a new random word and fill the buffer up.
802 return bool_producer.convert(core);
807 ///\name Non-uniform distributions
812 /// \brief Returns a random bool
814 /// It returns a random bool with given probability of true result.
815 bool boolean(double p) {
816 return operator()() < p;
819 /// Standard Gauss distribution
821 /// Standard Gauss distribution.
822 /// \note The Cartesian form of the Box-Muller
823 /// transformation is used to generate a random normal distribution.
824 /// \todo Consider using the "ziggurat" method instead.
829 V1=2*real<double>()-1;
830 V2=2*real<double>()-1;
833 return std::sqrt(-2*std::log(S)/S)*V1;
835 /// Gauss distribution with given mean and standard deviation
837 /// Gauss distribution with given mean and standard deviation.
839 double gauss(double mean,double std_dev)
841 return gauss()*std_dev+mean;
844 /// Exponential distribution with given mean
846 /// This function generates an exponential distribution random number
847 /// with mean <tt>1/lambda</tt>.
849 double exponential(double lambda=1.0)
851 return -std::log(1.0-real<double>())/lambda;
854 /// Gamma distribution with given integer shape
856 /// This function generates a gamma distribution random number.
858 ///\param k shape parameter (<tt>k>0</tt> integer)
862 for(int i=0;i<k;i++) s-=std::log(1.0-real<double>());
866 /// Gamma distribution with given shape and scale parameter
868 /// This function generates a gamma distribution random number.
870 ///\param k shape parameter (<tt>k>0</tt>)
871 ///\param theta scale parameter
873 double gamma(double k,double theta=1.0)
876 const double delta = k-std::floor(k);
877 const double v0=E/(E-delta);
879 double V0=1.0-real<double>();
880 double V1=1.0-real<double>();
881 double V2=1.0-real<double>();
884 xi=std::pow(V1,1.0/delta);
885 nu=V0*std::pow(xi,delta-1.0);
892 } while(nu>std::pow(xi,delta-1.0)*std::exp(-xi));
893 return theta*(xi+gamma(int(std::floor(k))));
896 /// Weibull distribution
898 /// This function generates a Weibull distribution random number.
900 ///\param k shape parameter (<tt>k>0</tt>)
901 ///\param lambda scale parameter (<tt>lambda>0</tt>)
903 double weibull(double k,double lambda)
905 return lambda*pow(-std::log(1.0-real<double>()),1.0/k);
908 /// Pareto distribution
910 /// This function generates a Pareto distribution random number.
912 ///\param k shape parameter (<tt>k>0</tt>)
913 ///\param x_min location parameter (<tt>x_min>0</tt>)
915 double pareto(double k,double x_min)
917 return exponential(gamma(k,1.0/x_min))+x_min;
920 /// Poisson distribution
922 /// This function generates a Poisson distribution random number with
923 /// parameter \c lambda.
925 /// The probability mass function of this distribusion is
926 /// \f[ \frac{e^{-\lambda}\lambda^k}{k!} \f]
927 /// \note The algorithm is taken from the book of Donald E. Knuth titled
928 /// ''Seminumerical Algorithms'' (1969). Its running time is linear in the
931 int poisson(double lambda)
933 const double l = std::exp(-lambda);
945 ///\name Two dimensional distributions
950 /// Uniform distribution on the full unit circle
952 /// Uniform distribution on the full unit circle.
954 dim2::Point<double> disc()
958 V1=2*real<double>()-1;
959 V2=2*real<double>()-1;
961 } while(V1*V1+V2*V2>=1);
962 return dim2::Point<double>(V1,V2);
964 /// A kind of two dimensional Gauss distribution
966 /// This function provides a turning symmetric two-dimensional distribution.
967 /// Both coordinates are of standard normal distribution, but they are not
970 /// \note The coordinates are the two random variables provided by
971 /// the Box-Muller method.
972 dim2::Point<double> gauss2()
976 V1=2*real<double>()-1;
977 V2=2*real<double>()-1;
980 double W=std::sqrt(-2*std::log(S)/S);
981 return dim2::Point<double>(W*V1,W*V2);
983 /// A kind of two dimensional exponential distribution
985 /// This function provides a turning symmetric two-dimensional distribution.
986 /// The x-coordinate is of conditionally exponential distribution
987 /// with the condition that x is positive and y=0. If x is negative and
988 /// y=0 then, -x is of exponential distribution. The same is true for the
990 dim2::Point<double> exponential2()
994 V1=2*real<double>()-1;
995 V2=2*real<double>()-1;
998 double W=-std::log(S)/S;
999 return dim2::Point<double>(W*V1,W*V2);