# 29 Numerics library [numerics]

## 29.6 Random number generation [rand]

### 29.6.8 Random number distribution class templates [rand.dist]

#### 29.6.8.6.1 Class template discrete_­distribution[rand.dist.samp.discrete]

A discrete_­distribution random number distribution produces random integers i, 0i<n, distributed according to the discrete probability function

P(i|p0,,pn1)=pi.

Unless specified otherwise, the distribution parameters are calculated as: pk=wk/S for k=0,,n1 , in which the values wk, commonly known as the weights, shall be non-negative, non-NaN, and non-infinity. Moreover, the following relation shall hold: 0<S=w0++wn1.

```template<class IntType = int>
class discrete_distribution {
public:
// types
using result_type = IntType;
using param_type  = unspecified;

// constructor and reset functions
discrete_distribution();
template<class InputIterator>
discrete_distribution(InputIterator firstW, InputIterator lastW);
discrete_distribution(initializer_list<double> wl);
template<class UnaryOperation>
discrete_distribution(size_t nw, double xmin, double xmax, UnaryOperation fw);
explicit discrete_distribution(const param_type& parm);
void reset();

// generating functions
template<class URBG>
result_type operator()(URBG& g);
template<class URBG>
result_type operator()(URBG& g, const param_type& parm);

// property functions
vector<double> probabilities() const;
param_type param() const;
void param(const param_type& parm);
result_type min() const;
result_type max() const;
};```

```discrete_distribution(); ```

Effects: Constructs a discrete_­distribution object with n=1 and p0=1. [Note: Such an object will always deliver the value 0. end note]

```template<class InputIterator> discrete_distribution(InputIterator firstW, InputIterator lastW); ```

Requires: InputIterator shall satisfy the requirements of an input iterator. Moreover, iterator_­traits<InputIterator>​::​value_­type shall denote a type that is convertible to double. If firstW == lastW, let n=1 and w0=1. Otherwise, [firstW,lastW) shall form a sequence w of length n>0.

Effects: Constructs a discrete_­distribution object with probabilities given by the formula above.

```discrete_distribution(initializer_list<double> wl); ```

Effects: Same as discrete_­distribution(wl.begin(), wl.end()).

```template<class UnaryOperation> discrete_distribution(size_t nw, double xmin, double xmax, UnaryOperation fw); ```

Requires: Each instance of type UnaryOperation shall be a function object whose return type shall be convertible to double. Moreover, double shall be convertible to the type of UnaryOperation's sole parameter. If nw=0, let n=1, otherwise let n=nw. The relation 0<δ=(xmaxxmin)/n shall hold.

Effects: Constructs a discrete_­distribution object with probabilities given by the formula above, using the following values: If nw=0, let w0=1. Otherwise, let wk=fw(xmin+kδ+δ/2) for k=0,,n1.

Complexity: The number of invocations of fw shall not exceed n.

```vector<double> probabilities() const; ```

Returns: A vector<double> whose size member returns n and whose operator[] member returns pk when invoked with argument k for k=0,,n1.

#### 29.6.8.6.2 Class template piecewise_­constant_­distribution[rand.dist.samp.pconst]

A piecewise_­constant_­distribution random number distribution produces random numbers x, b0x<bn, uniformly distributed over each subinterval [bi,bi+1) according to the probability density function

p(x|b0,,bn,ρ0,,ρn1)=ρi, for bix<bi+1.

The n+1 distribution parameters bi, also known as this distribution's interval boundaries, shall satisfy the relation bi<bi+1 for i=0,,n1. Unless specified otherwise, the remaining n distribution parameters are calculated as:

ρk=wkS(bk+1bk) for k=0,,n1,

in which the values wk, commonly known as the weights, shall be non-negative, non-NaN, and non-infinity. Moreover, the following relation shall hold: 0<S=w0++wn1.

```template<class RealType = double>
class piecewise_constant_distribution {
public:
// types
using result_type = RealType;
using param_type  = unspecified;

// constructor and reset functions
piecewise_constant_distribution();
template<class InputIteratorB, class InputIteratorW>
piecewise_constant_distribution(InputIteratorB firstB, InputIteratorB lastB,
InputIteratorW firstW);
template<class UnaryOperation>
piecewise_constant_distribution(initializer_list<RealType> bl, UnaryOperation fw);
template<class UnaryOperation>
piecewise_constant_distribution(size_t nw, RealType xmin, RealType xmax,
UnaryOperation fw);
explicit piecewise_constant_distribution(const param_type& parm);
void reset();

// generating functions
template<class URBG>
result_type operator()(URBG& g);
template<class URBG>
result_type operator()(URBG& g, const param_type& parm);

// property functions
vector<result_type> intervals() const;
vector<result_type> densities() const;
param_type param() const;
void param(const param_type& parm);
result_type min() const;
result_type max() const;
};```

```piecewise_constant_distribution(); ```

Effects: Constructs a piecewise_­constant_­distribution object with n=1, ρ0=1, b0=0, and b1=1.

```template<class InputIteratorB, class InputIteratorW> piecewise_constant_distribution(InputIteratorB firstB, InputIteratorB lastB, InputIteratorW firstW); ```

Requires: InputIteratorB and InputIteratorW shall each satisfy the requirements of an input iterator type. Moreover, iterator_­traits<InputIteratorB>​::​value_­type and iterator_­traits<InputIteratorW>​::​value_­type shall each denote a type that is convertible to double. If firstB == lastB or ++firstB == lastB, let n=1, w0=1, b0=0, and b1=1. Otherwise, [firstB,lastB) shall form a sequence b of length n+1, the length of the sequence w starting from firstW shall be at least n, and any wk for kn shall be ignored by the distribution.

Effects: Constructs a piecewise_­constant_­distribution object with parameters as specified above.

```template<class UnaryOperation> piecewise_constant_distribution(initializer_list<RealType> bl, UnaryOperation fw); ```

Requires: Each instance of type UnaryOperation shall be a function object whose return type shall be convertible to double. Moreover, double shall be convertible to the type of UnaryOperation's sole parameter.

Effects: Constructs a piecewise_­constant_­distribution object with parameters taken or calculated from the following values: If bl.size()<2, let n=1, w0=1, b0=0, and b1=1. Otherwise, let [bl.begin(),bl.end()) form a sequence b0,,bn, and let wk=fw((bk+1+bk)/2) for k=0,,n1.

Complexity: The number of invocations of fw shall not exceed n.

```template<class UnaryOperation> piecewise_constant_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw); ```

Requires: Each instance of type UnaryOperation shall be a function object whose return type shall be convertible to double. Moreover, double shall be convertible to the type of UnaryOperation's sole parameter. If nw=0, let n=1, otherwise let n=nw. The relation 0<δ=(xmaxxmin)/n shall hold.

Effects: Constructs a piecewise_­constant_­distribution object with parameters taken or calculated from the following values: Let bk=xmin+kδ for k=0,,n, and wk=fw(bk+δ/2) for k=0,,n1.

Complexity: The number of invocations of fw shall not exceed n.

```vector<result_type> intervals() const; ```

Returns: A vector<result_­type> whose size member returns n+1 and whose operator[] member returns bk when invoked with argument k for k=0,,n.

```vector<result_type> densities() const; ```

Returns: A vector<result_­type> whose size member returns n and whose operator[] member returns ρk when invoked with argument k for k=0,,n1.

#### 29.6.8.6.3 Class template piecewise_­linear_­distribution[rand.dist.samp.plinear]

A piecewise_­linear_­distribution random number distribution produces random numbers x, b0x<bn, distributed over each subinterval [bi,bi+1) according to the probability density function

p(x|b0,,bn,ρ0,,ρn)=ρibi+1xbi+1bi+ρi+1xbibi+1bi, for bix<bi+1.

The n+1 distribution parameters bi, also known as this distribution's interval boundaries, shall satisfy the relation bi<bi+1 for i=0,,n1. Unless specified otherwise, the remaining n+1 distribution parameters are calculated as ρk=wk/S for k=0,,n, in which the values wk, commonly known as the weights at boundaries, shall be non-negative, non-NaN, and non-infinity. Moreover, the following relation shall hold:

0<S=12n1k=0(wk+wk+1)(bk+1bk).

```template<class RealType = double>
class piecewise_linear_distribution {
public:
// types
using result_type = RealType;
using param_type  = unspecified;

// constructor and reset functions
piecewise_linear_distribution();
template<class InputIteratorB, class InputIteratorW>
piecewise_linear_distribution(InputIteratorB firstB, InputIteratorB lastB,
InputIteratorW firstW);
template<class UnaryOperation>
piecewise_linear_distribution(initializer_list<RealType> bl, UnaryOperation fw);
template<class UnaryOperation>
piecewise_linear_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw);
explicit piecewise_linear_distribution(const param_type& parm);
void reset();

// generating functions
template<class URBG>
result_type operator()(URBG& g);
template<class URBG>
result_type operator()(URBG& g, const param_type& parm);

// property functions
vector<result_type> intervals() const;
vector<result_type> densities() const;
param_type param() const;
void param(const param_type& parm);
result_type min() const;
result_type max() const;
};```

```piecewise_linear_distribution(); ```

Effects: Constructs a piecewise_­linear_­distribution object with n=1, ρ0=ρ1=1, b0=0, and b1=1.

```template<class InputIteratorB, class InputIteratorW> piecewise_linear_distribution(InputIteratorB firstB, InputIteratorB lastB, InputIteratorW firstW); ```

Requires: InputIteratorB and InputIteratorW shall each satisfy the requirements of an input iterator type. Moreover, iterator_­traits<InputIteratorB>​::​value_­type and iterator_­traits<InputIteratorW>​::​value_­type shall each denote a type that is convertible to double. If firstB == lastB or ++firstB == lastB, let n=1, ρ0=ρ1=1, b0=0, and b1=1. Otherwise, [firstB,lastB) shall form a sequence b of length n+1, the length of the sequence w starting from firstW shall be at least n+1, and any wk for kn+1 shall be ignored by the distribution.

Effects: Constructs a piecewise_­linear_­distribution object with parameters as specified above.

```template<class UnaryOperation> piecewise_linear_distribution(initializer_list<RealType> bl, UnaryOperation fw); ```

Requires: Each instance of type UnaryOperation shall be a function object whose return type shall be convertible to double. Moreover, double shall be convertible to the type of UnaryOperation's sole parameter.

Effects: Constructs a piecewise_­linear_­distribution object with parameters taken or calculated from the following values: If bl.size()<2, let n=1, ρ0=ρ1=1, b0=0, and b1=1. Otherwise, let [bl.begin(),bl.end()) form a sequence b0,,bn, and let wk=fw(bk) for k=0,,n.

Complexity: The number of invocations of fw shall not exceed n+1.

```template<class UnaryOperation> piecewise_linear_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw); ```

Requires: Each instance of type UnaryOperation shall be a function object whose return type shall be convertible to double. Moreover, double shall be convertible to the type of UnaryOperation's sole parameter. If nw=0, let n=1, otherwise let n=nw. The relation 0<δ=(xmaxxmin)/n shall hold.

Effects: Constructs a piecewise_­linear_­distribution object with parameters taken or calculated from the following values: Let bk=xmin+kδ for k=0,,n, and wk=fw(bk) for k=0,,n.

Complexity: The number of invocations of fw shall not exceed n+1.

```vector<result_type> intervals() const; ```

Returns: A vector<result_­type> whose size member returns n+1 and whose operator[] member returns bk when invoked with argument k for k=0,,n.

```vector<result_type> densities() const; ```

Returns: A vector<result_­type> whose size member returns n and whose operator[] member returns ρk when invoked with argument k for k=0,,n.