# 28 Numerics library [numerics]

## 28.5 Random number generation [rand]

### 28.5.4 Random number engine class templates [rand.eng]

#### 28.5.4.3 Class template mersenne_­twister_­engine[rand.eng.mers]

A mersenne_­twister_­engine random number engine232 produces unsigned integer random numbers in the closed interval .
The state x of a mersenne_­twister_­engine object x is of size n and consists of a sequence X of n values of the type delivered by x; all subscripts applied to X are to be taken modulo n.
The transition algorithm employs a twisted generalized feedback shift register defined by shift values n and m, a twist value r, and a conditional xor-mask a.
To improve the uniformity of the result, the bits of the raw shift register are additionally tempered (i.e., scrambled) according to a bit-scrambling matrix defined by values u, d, s, b, t, c, and .
The state transition is performed as follows:
• Concatenate the upper bits of with the lower r bits of to obtain an unsigned integer value Y.
• With , set to .
The sequence X is initialized with the help of an initialization multiplier f.
The generation algorithm determines the unsigned integer values as follows, then delivers as its result:
• Let .
• Let .
• Let .
• Let .
namespace std { template<class UIntType, size_t w, size_t n, size_t m, size_t r, UIntType a, size_t u, UIntType d, size_t s, UIntType b, size_t t, UIntType c, size_t l, UIntType f> class mersenne_twister_engine { public: // types using result_type = UIntType; // engine characteristics static constexpr size_t word_size = w; static constexpr size_t state_size = n; static constexpr size_t shift_size = m; static constexpr size_t mask_bits = r; static constexpr UIntType xor_mask = a; static constexpr size_t tempering_u = u; static constexpr UIntType tempering_d = d; static constexpr size_t tempering_s = s; static constexpr UIntType tempering_b = b; static constexpr size_t tempering_t = t; static constexpr UIntType tempering_c = c; static constexpr size_t tempering_l = l; static constexpr UIntType initialization_multiplier = f; static constexpr result_type min() { return 0; } static constexpr result_type max() { return ; } static constexpr result_type default_seed = 5489u; // constructors and seeding functions mersenne_twister_engine() : mersenne_twister_engine(default_seed) {} explicit mersenne_twister_engine(result_type value); template<class Sseq> explicit mersenne_twister_engine(Sseq& q); void seed(result_type value = default_seed); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const mersenne_twister_engine& x, const mersenne_twister_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const mersenne_twister_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, mersenne_twister_engine& x); }; }
The following relations shall hold: 0 < m, m <= n, 2u < w, r <= w, u <= w, s <= w, t <= w, l <= w, w <= numeric_­limits<UIntType>​::​digits, a <= (1u<<w) - 1u, b <= (1u<<w) - 1u, c <= (1u<<w) - 1u, d <= (1u<<w) - 1u, and f <= (1u<<w) - 1u.
The textual representation of x consists of the values of , in that order.
```explicit mersenne_twister_engine(result_type value); ```
Effects: Sets to .
Then, iteratively for , sets to
Complexity: .
```template<class Sseq> explicit mersenne_twister_engine(Sseq& q); ```
Effects: With and a an array (or equivalent) of length , invokes q.generate(, ) and then, iteratively for , sets to .
Finally, if the most significant bits of are zero, and if each of the other resulting is 0, changes to .
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The name of this engine refers, in part, to a property of its period: For properly-selected values of the parameters, the period is closely related to a large Mersenne prime number.