.. _oneapi-mkl-rng-rayleigh:

oneapi::mkl::rng::rayleigh
==========================

Generates Rayleigh distributed random values.


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Description
***********

The ``oneapi::mkl::rng::rayleigh`` class object is used by the ``oneapi::mkl::rng::generate`` function to provide Rayleigh distributed random numbers with displacement (``a``) and scalefactor (``b``, ``β``), where :math:`a, \beta \in R ; \beta > 0`.

The Rayleigh distribution is a special case of the :ref:`Weibull<oneapi-mkl-rng-weibull>` distribution, where the shape parameter ``α = 2``.


The probability density function is given by:

.. math::

   f_{a, \beta}(x) =
   \begin{cases}
      \frac{2(x-a)}{\beta^2} \exp (-\frac{(x-a)^2}{\beta^2}), & x \geq a \\
      0, & x < a
   \end{cases},
   - \infty < x < + \infty

The cumulative distribution function is as follows:

.. math::

   F_{\alpha, \beta} (x) =
   \begin{cases}
      1 - \exp\left(-\frac{(x-\alpha)^2}{\beta^2}\right), & x \geq \alpha) \\
      0, & x < \alpha
   \end{cases},
   - \infty < x < + \infty


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   * - Product and Performance Information
   * - Performance varies by use, configuration and other factors. Learn more at `https://www.intel.com/PerformanceIndex <https://www.intel.com/PerformanceIndex>`__. Notice revision #20201201


API
***


Syntax
------

.. code-block:: cpp


   template<typename RealType = float, typename Method = rayleigh_method::by_default>
   class rayleigh {
   public:
   using method_type = Method;
   using result_type = RealType;
   rayleigh(): rayleigh((RealType)0.0, (RealType)1.0){}
   explicit rayleigh(RealType a, RealType b);
   explicit rayleigh(const param_type& pt);
   RealType a() const;
   RealType b() const;
   param_type param() const;
   void param(const param_type& pt);
   };

Devices supported: Host, CPU, and GPU.


Include Files
-------------

- ``oneapi/mkl/rng.hpp``


Template Parameters
-------------------

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   * - ``typename RealType = float``
     - Type of the produced values. The specific values are as follows:

       ``float``

       ``double``

   * - ``typename Method = oneapi::mkl::rng::rayleigh_method::by_default``
     - Generation method. The specific values are as follows:

       ``oneapi::mkl::rng::rayleigh_method::icdf``

       ``oneapi::mkl::rng::rayleigh_method::icdf_accurate``

       See brief descriptions of the methods in :ref:`distributions-template-parameter-method`.

Input Parameters
----------------

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   * - Name
     - Type
     - Description
   * - a
     - ``RealType (float, double)``
     - Displacement ``a``.
   * - b
     - ``RealType (float, double)``
     - Scalefactor ``b``.