.. SPDX-FileCopyrightText: 2019-2020 Intel Corporation
..
.. SPDX-License-Identifier: CC-BY-4.0

.. _onemath_rng_gamma:

gamma
=====

Class is used for generation of gamma distributed real types random numbers.

.. _onemath_rng_gamma_description:

.. rubric:: Description

The class object is used in the :ref:`oneapi::math::rng::generate()<onemath_rng_generate>`
function to provide random numbers gamma distributed with shape :math:`\alpha`,
displacement :math:`a`, and scale parameter :math:`\beta`, where
:math:`a, \alpha, \beta \in R; \alpha > 0; \beta > 0`.

The probability distribution is given by:

.. math::

    f_{a, \alpha, \beta}(x) = \left\{ \begin{array}{rcl} \frac{1}{\Gamma(\alpha)\beta^{\alpha}}(x - a)^{\alpha - 1}e^{-(x - a) / \beta}, x \ge a \\ 0, x < a \end{array}\right.

The cumulative distribution function is as follows:

.. math::

    F_{a, \alpha, \beta}(x) = \left\{ \begin{array}{rcl} \int^x_a\frac{1}{\Gamma(\alpha)\beta^{\alpha}}(y - a)^{\alpha - 1}e^{-(y - a) / \beta}dy, x \ge a \\ 0, x < a \end{array}\right.

.. _onemath_rng_gamma_syntax:

class gamma
-----------

.. rubric:: Syntax

.. code-block:: cpp

    namespace oneapi::math::rng {
    template<typename RealType = float, typename Method = gamma_method::by_default>
    class gamma {
    public:
        using method_type = Method;
        using result_type = RealType;
        gamma();
        explicit gamma(RealType alpha, RealType a, RealType beta);
        RealType alpha() const;
        RealType a() const;
        RealType beta() const;
    };
    }

.. container:: section

    .. rubric:: Template parameters

    .. container:: section

        typename RealType
            Type of the produced values. Supported types:
                * ``float``
                * ``double``

    .. container:: section

        typename Method = oneapi::math::rng::gamma_method::by_default
            Transformation method, which will be used for generation. Supported types:

                * ``oneapi::math::rng::gamma_method::by_default``
                * ``oneapi::math::rng::gamma_method::marsaglia``
                * ``oneapi::math::rng::gamma_method::marsaglia_accurate``

            See description of the methods in :ref:`Distributions methods template parameter<onemath_rng_distributions_template_parameter_onemath_rng_method_values>`.

.. container:: section

    .. rubric:: Class Members

    .. list-table::
        :header-rows: 1

        * - Routine
          - Description
        * - `gamma()`_
          - Default constructor
        * - `explicit gamma(RealType alpha, RealType a, RealType beta)`_
          - Constructor with parameters
        * - `RealType alpha() const`_
          - Method to obtain shape value
        * - `RealType a() const`_
          - Method to obtain displacement value
        * - `RealType beta() const`_
          - Method to obtain scale value

.. container:: section

    .. rubric:: Member types

    .. container:: section

        .. code-block:: cpp

            gamma::method_type = Method

        .. container:: section

            .. rubric:: Description

            The type which defines transformation method for generation.

    .. container:: section

        .. code-block:: cpp

            gamma::result_type = RealType

        .. container:: section

            .. rubric:: Description

            The type which defines type of generated random numbers.

.. container:: section

    .. rubric:: Constructors

    .. container:: section

        .. _`gamma()`:

        .. code-block:: cpp

            gamma::gamma()

        .. container:: section

            .. rubric:: Description

            Default constructor for distribution, parameters set as `alpha` = 1.0, `a` = 0.0, and `beta` = 1.0.

    .. container:: section

        .. _`explicit gamma(RealType alpha, RealType a, RealType beta)`:

        .. code-block:: cpp

            explicit gamma::gamma(RealType alpha, RealType a, RealType beta)

        .. container:: section

            .. rubric:: Description

            Constructor with parameters. `alpha` is a shape value, `a` is a displacement value, `beta` is a scale parameter.

        .. container:: section

            .. rubric:: Throws

            oneapi::math::invalid_argument
                Exception is thrown when :math:`alpha \leq` static_cast<RealType>(0.0), or :math:`beta \leq` static_cast<RealType>(0.0)

.. container:: section

    .. rubric:: Characteristics

    .. container:: section

        .. _`RealType alpha() const`:

        .. code-block:: cpp

            RealType gamma::alpha() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter `alpha` - shape value.

    .. container:: section

        .. _`RealType a() const`:

        .. code-block:: cpp

            RealType gamma::a() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter `a` - displacement value.

    .. container:: section

        .. _`RealType beta() const`:

        .. code-block:: cpp

            RealType gamma::beta() const

        .. container:: section

            .. rubric:: Return Value

            Returns the distribution parameter `beta` - scale parameter.

**Parent topic:** :ref:`onemath_rng_distributions`