29 Numerics library [numerics]

29.6 Random number generation [rand]

29.6.8 Random number distribution class templates [rand.dist]

29.6.8.6 Sampling distributions [rand.dist.samp]

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

A discrete_distribution random number distribution produces random integers i,

0 \leq i < n

, distributed according to the discrete probability function

$P (i | p_{0}, \dots, p_{n - 1}) = p_{i} .$

Unless specified otherwise, the distribution parameters are calculated as:

p_{k} = w_{k} / S for k = 0, \dots, n - 1

, in which the values

w_{k}

, commonly known as the weights, shall be non-negative, non-NaN, and non-infinity.

Moreover, the following relation shall hold:

0 < S = w_{0} + \dots + w_{n - 1}

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

p_{0} = 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 ([input.iterators]).

Moreover, iterator_traits<InputIterator>::value_type shall denote a type that is convertible to double.

If firstW == lastW, let

n = 1

and

w_{0} = 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 ([function.objects]) whose return type shall be convertible to double.

Moreover, double shall be convertible to the type of UnaryOperation's sole parameter.

nw = 0

, let

n = 1

, otherwise let

n = nw

The relation

0 < δ = (xmax - xmin) / n

shall hold.

Effects: Constructs a discrete_distribution object with probabilities given by the formula above, using the following values: If nw

= 0

, let

w_{0} = 1

Otherwise, let

w_{k} = fw (xmin + k \cdot δ + δ / 2)

for

k = 0, \dots, n - 1

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

p_{k}

when invoked with argument k for

k = 0, \dots, n - 1

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,

b_{0} \leq x < b_{n}

, uniformly distributed over each subinterval

[b_{i}, b_{i + 1})

according to the probability density function

$p (x | b_{0}, \dots, b_{n}, ρ_{0}, \dots, ρ_{n - 1}) = ρ_{i}, for b_{i} \leq x < b_{i + 1} .$

The

n + 1

distribution parameters

b_{i}

, also known as this distribution's interval boundaries, shall satisfy the relation

b_{i} < b_{i + 1}

for

i = 0, \dots, n - 1

Unless specified otherwise, the remaining n distribution parameters are calculated as:

$ρ_{k} = \frac{w_{k}}{S \cdot (b_{k + 1} - b_{k})} for k = 0, \dots, n - 1,$

in which the values $w_{k}$ , commonly known as the weights, shall be non-negative, non-NaN, and non-infinity.

Moreover, the following relation shall hold:

0 < S = w_{0} + \dots + w_{n - 1}

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

ρ_{0} = 1

b_{0} = 0

, and

b_{1} = 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 (Table 90) 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

w_{0} = 1

b_{0} = 0

, and

b_{1} = 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

w_{k}

for

k \geq n

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 ([function.objects]) 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

w_{0} = 1

b_{0} = 0

, and

b_{1} = 1

Otherwise, let

[bl.begin(), bl.end())

form a sequence

b_{0}, \dots, b_{n}

, and let

w_{k} = fw ((b_{k + 1} + b_{k}) / 2)

for

k = 0, \dots, n - 1

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 ([function.objects]) whose return type shall be convertible to double.

Moreover, double shall be convertible to the type of UnaryOperation's sole parameter.

nw = 0

, let

n = 1

, otherwise let

n = nw

The relation

0 < δ = (xmax - xmin) / n

shall hold.

Effects: Constructs a piecewise_constant_distribution object with parameters taken or calculated from the following values: Let

b_{k} = xmin + k \cdot δ

for

k = 0, \dots, n

, and

w_{k} = fw (b_{k} + δ / 2)

for

k = 0, \dots, n - 1

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

b_{k}

when invoked with argument k for

k = 0, \dots, n

🔗

vector<result_type> densities() const;

Returns: A vector<result_type> whose size member returns n and whose operator[] member returns

ρ_{k}

when invoked with argument k for

k = 0, \dots, n - 1

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,

b_{0} \leq x < b_{n}

, distributed over each subinterval

[b_{i}, b_{i + 1})

according to the probability density function

$p (x | b_{0}, \dots, b_{n}, ρ_{0}, \dots, ρ_{n}) = ρ_{i} \cdot \frac{b_{i + 1} - x}{b_{i + 1} - b_{i}} + ρ_{i + 1} \cdot \frac{x - b_{i}}{b_{i + 1} - b_{i}}, for b_{i} \leq x < b_{i + 1} .$

The

n + 1

distribution parameters

b_{i}

, also known as this distribution's interval boundaries, shall satisfy the relation

b_{i} < b_{i + 1}

for

i = 0, \dots, n - 1

Unless specified otherwise, the remaining

n + 1

distribution parameters are calculated as

ρ_{k} = w_{k} / S for k = 0, \dots, n

, in which the values

w_{k}

, commonly known as the weights at boundaries, shall be non-negative, non-NaN, and non-infinity.

Moreover, the following relation shall hold:

$0 < S = \frac{1}{2} \cdot n - 1 \sum k = 0 (w_{k} + w_{k + 1}) \cdot (b_{k + 1} - b_{k}) .$

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

ρ_{0} = ρ_{1} = 1

b_{0} = 0

, and

b_{1} = 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 (Table 90) 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

ρ_{0} = ρ_{1} = 1

b_{0} = 0

, and

b_{1} = 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

w_{k}

for

k \geq n + 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 ([function.objects]) 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

ρ_{0} = ρ_{1} = 1

b_{0} = 0

, and

b_{1} = 1

Otherwise, let

[bl.begin(), bl.end())

form a sequence

b_{0}, \dots, b_{n}

, and let

w_{k} = fw (b_{k})

for

k = 0, \dots, 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 ([function.objects]) whose return type shall be convertible to double.

Moreover, double shall be convertible to the type of UnaryOperation's sole parameter.

nw = 0

, let

n = 1

, otherwise let

n = nw

The relation

0 < δ = (xmax - xmin) / n

shall hold.

Effects: Constructs a piecewise_linear_distribution object with parameters taken or calculated from the following values: Let

b_{k} = xmin + k \cdot δ

for

k = 0, \dots, n

, and

w_{k} = fw (b_{k})

for

k = 0, \dots, 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

b_{k}

when invoked with argument k for

k = 0, \dots, n

🔗

vector<result_type> densities() const;

Returns: A vector<result_type> whose size member returns n and whose operator[] member returns

ρ_{k}

when invoked with argument k for

k = 0, \dots, n

29 Numerics library [numerics]

29.6 Random number generation [rand]

29.6.8 Random number distribution class templates [rand.dist]

29.6.8.6 Sampling distributions [rand.dist.samp]

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

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

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

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

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

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