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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.landau_dist"></a><a class="link" href="landau_dist.html" title="Landau Distribution">Landau Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">landau</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">landau_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">landau_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">landau</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">landau_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">landau_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">bias</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
The <a href="http://en.wikipedia.org/wiki/landau_distribution" target="_top">Landau
distribution</a> is named after Lev Landau. It is special case of a
<a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_top">stable distribution</a>
with shape parameter α=1, β=1.
</p>
<p>
<a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
distribution function PDF</a> given by:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/landau_ref1.svg"></span>
</p></blockquote></div>
<p>
The location parameter μ is the location of the distribution, while the scale
parameter [c] determines the width of the distribution, but unlike other
scalable distributions, it has a peculiarity that changes the location
of the distribution. If the location is zero, and the scale 1, then the
result is a standard landau distribution.
</p>
<p>
The distribution describe the statistical property of the energy loss by
charged particles as they traversing a thin layer of matter.
</p>
<p>
The following graph shows how the distributions moves as the location parameter
changes:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/landau_pdf1.svg" align="middle"></span>
</p></blockquote></div>
<p>
While the following graph shows how the shape (scale) parameter alters
the distribution:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/landau_pdf2.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.landau_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.member_functions"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.member_functions">Member
Functions</a>
</h5>
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">landau_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
Constructs a landau distribution, with location parameter <span class="emphasis"><em>location</em></span>
and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
their default values (location = 0, scale = 1) then the result is a Standard
landau Distribution.
</p>
<p>
Requires scale &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the location parameter of the distribution.
</p>
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the scale parameter of the distribution.
</p>
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">bias</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the amount of translation by the scale parameter.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">bias = - 2 / π log(c)</span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.landau_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.non_member_accessors"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.non_member_accessors">Non-member
Accessors</a>
</h5>
<p>
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
functions</a> that are generic to all distributions are supported:
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
__logcdf, __logpdf, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. For this distribution
all non-member accessor functions are marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
and can be run on both host and device.
</p>
<p>
Note however that the landau distribution does not have a mean, standard
deviation, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
undefined function</a> to control whether these should fail to compile
with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
</p>
<p>
Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>
and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
if called.
</p>
<p>
The domain of the random variable is [-[max_value], +[min_value]].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.landau_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.accuracy"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.accuracy">Accuracy</a>
</h5>
<p>
The error is within 4 epsilon except for the rapidly decaying left tail.
</p>
<p>
Errors in the PDF at 64-bit double precision:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/landau_pdf_accuracy_64.png"></span>
</p>
<p>
Errors in the CDF at 64-bit double precision:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/landau_cdf_accuracy_64.png"></span>
</p>
<p>
Errors in the CDF-complement at 64-bit double precision:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/landau_ccdf_accuracy_64.png"></span>
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.landau_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.implementation"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.implementation">Implementation</a>
</h5>
<p>
See references.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.landau_dist.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.references"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a href="http://en.wikipedia.org/wiki/landau_distribution" target="_top">landau
distribution</a>
</li>
<li class="listitem">
T. Yoshimura, Numerical Evaluation and High Precision Approximation
Formula for Landau Distribution, DOI: 10.36227/techrxiv.171822215.53612870/v2,
2024.
</li>
</ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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