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. 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 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 $0 < δ = (xmax - xmin) / n$ 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

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

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

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 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);

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);

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 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);

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);

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]