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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.owens_t"></a><a class="link" href="owens_t.html" title="Owen's T function">Owen's T function</a>
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</h2></div></div></div>
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<h5>
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<a name="math_toolkit.owens_t.h0"></a>
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<span class="phrase"><a name="math_toolkit.owens_t.synopsis"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">owens_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">owens_t</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">owens_t</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.owens_t.h1"></a>
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<span class="phrase"><a name="math_toolkit.owens_t.description"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.description">Description</a>
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</h5>
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<p>
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Returns the <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owens_t
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function</a> of <span class="emphasis"><em>h</em></span> and <span class="emphasis"><em>a</em></span>.
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</p>
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<p>
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The final <a class="link" href="../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors, what
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level of precision to use etc. Refer to the <a class="link" href="../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy documentation
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for more details</a>.
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</p>
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<p>
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    <span class="inlinemediaobject"><img src="../../equations/owens_t.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/plot_owens_t.png"></span>
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</p>
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<p>
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The function <code class="computeroutput"><span class="identifier">owens_t</span><span class="special">(</span><span class="identifier">h</span><span class="special">,</span> <span class="identifier">a</span><span class="special">)</span></code> gives the probability of the event <span class="emphasis"><em>(X
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> h and 0 < Y < a * X)</em></span>, where <span class="emphasis"><em>X</em></span> and
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<span class="emphasis"><em>Y</em></span> are independent standard normal random variables.
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</p>
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<p>
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For h and a > 0, T(h,a), gives the volume of an uncorrelated bivariate normal
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distribution with zero means and unit variances over the area between <span class="emphasis"><em>y
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= ax</em></span> and <span class="emphasis"><em>y = 0</em></span> and to the right of <span class="emphasis"><em>x
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= h</em></span>.
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</p>
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<p>
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That is the area shaded in the figure below (Owens 1956).
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/owens_integration_area.svg" align="middle"></span>
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</p>
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<p>
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and is also illustrated by a 3D plot.
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/plot_owens_3d_xyp.png"></span>
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</p>
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<p>
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This function is used in the computation of the <a class="link" href="dist_ref/dists/skew_normal_dist.html" title="Skew Normal Distribution">Skew
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Normal Distribution</a>. It is also used in the computation of bivariate
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and multivariate normal distribution probabilities. The return type of this
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function is computed using the <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type T
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otherwise.
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</p>
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<p>
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Owen's original paper (page 1077) provides some additional corner cases.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>T(h, 0) = 0</em></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>T(0, a) = ½π arctan(a)</em></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>T(h, 1) = ½ G(h) [1 - G(h)]</em></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>T(h, ∞) = G(|h|)</em></span>
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</p></blockquote></div>
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<p>
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where G(h) is the univariate normal with zero mean and unit variance integral
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from -∞ to h.
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</p>
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<h5>
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<a name="math_toolkit.owens_t.h2"></a>
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<span class="phrase"><a name="math_toolkit.owens_t.accuracy"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.accuracy">Accuracy</a>
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</h5>
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<p>
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Over the built-in types and range tested, errors are less than 10 * std::numeric_limits<RealType>::epsilon().
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</p>
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<div class="table">
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<a name="math_toolkit.owens_t.table_owens_t"></a><p class="title"><b>Table 6.83. Error rates for owens_t</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for owens_t">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 12.0<br> Win32<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5130<br> Sun Solaris<br> long double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Owens T (medium small values)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 4.37ε (Mean = 0.973ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.34ε (Mean = 0.942ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.34ε (Mean = 0.91ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Owens T (large and diverse values)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.78ε (Mean = 0.621ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 49ε (Mean = 2.16ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 24.5ε (Mean = 1.38ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.owens_t.h3"></a>
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<span class="phrase"><a name="math_toolkit.owens_t.testing"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.testing">Testing</a>
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</h5>
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<p>
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Test data was generated by Patefield and Tandy algorithms T1 and T4, and also
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the suggested reference routine T7.
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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T1 was rejected if the result was too small compared to <code class="computeroutput"><span class="identifier">atan</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span></code>
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(ie cancellation),
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</li>
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<li class="listitem">
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T4 was rejected if there was no convergence,
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</li>
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<li class="listitem">
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Both were rejected if they didn't agree.
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</li>
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</ul></div>
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<p>
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Over the built-in types and range tested, errors are less than 10 std::numeric_limits<RealType>::epsilon().
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</p>
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<p>
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However, that there was a whole domain (large <span class="emphasis"><em>h</em></span>, small
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<span class="emphasis"><em>a</em></span>) where it was not possible to generate any reliable
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test values (all the methods got rejected for one reason or another).
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</p>
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<p>
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There are also two sets of sanity tests: spot values are computed using <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram Mathematica</a>
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and <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>.
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</p>
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<h5>
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<a name="math_toolkit.owens_t.h4"></a>
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<span class="phrase"><a name="math_toolkit.owens_t.implementation"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.implementation">Implementation</a>
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</h5>
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<p>
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The function was proposed and evaluated by <a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177728074" target="_top">Donald.
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B. Owen, Tables for computing bivariate normal probabilities, Ann. Math. Statist.,
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27, 1075-1090 (1956)</a>.
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</p>
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<p>
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The algorithms of Patefield, M. and Tandy, D. "Fast and accurate Calculation
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of Owen's T-Function", Journal of Statistical Software, 5 (5), 1 - 25
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(2000) are adapted for C++ with arbitrary RealType.
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</p>
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<p>
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The Patefield-Tandy algorithm provides six methods of evalualution (T1 to T6);
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the best method is selected according to the values of <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>h</em></span>. See the original paper and the source in <a href="../../../../../boost/math/special_functions/owens_t.hpp" target="_top">owens_t.hpp</a>
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for details.
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</p>
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<p>
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The Patefield-Tandy algorithm is accurate to approximately 20 decimal places,
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so for types with greater precision we use:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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A modified version of T1 which folds the calculation of <span class="emphasis"><em>atan(h)</em></span>
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into the T1 series (to avoid subtracting two values similar in magnitude),
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and then accelerates the resulting alternating series using method 1 from
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H. Cohen, F. Rodriguez Villegas, D. Zagier, "Convergence acceleration
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of alternating series", Bonn, (1991). The result is valid everywhere,
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but doesn't always converge, or may become too divergent in the first few
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terms to sum accurately. This is used for <span class="emphasis"><em>ah < 1</em></span>.
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</li>
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<li class="listitem">
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A modified version of T2 which is accelerated in the same manner as T1.
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This is used for <span class="emphasis"><em>h > 1</em></span>.
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||
|
|
</li>
|
||
|
|
<li class="listitem">
|
||
|
|
A version of T4 only when both T1 and T2 have failed to produce an accurate
|
||
|
|
answer.
|
||
|
|
</li>
|
||
|
|
<li class="listitem">
|
||
|
|
Fallback to the Patefiled Tandy algorithm when all the above methods fail:
|
||
|
|
this happens not at all for our test data at 100 decimal digits precision.
|
||
|
|
However, there is a difficult area when <span class="emphasis"><em>a</em></span> is very
|
||
|
|
close to 1 and the precision increases which may cause this to happen in
|
||
|
|
very exceptional circumstances.
|
||
|
|
</li>
|
||
|
|
</ul></div>
|
||
|
|
<p>
|
||
|
|
Using the above algorithm and a 100-decimal digit type, results accurate to
|
||
|
|
80 decimal places were obtained in the difficult area where <span class="emphasis"><em>a</em></span>
|
||
|
|
is close to 1, and greater than 95 decimal places elsewhere.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||
|
|
<td align="left"></td>
|
||
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||
|
|
Benjamin Sobotta, 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>
|
||
|
|
</div></td>
|
||
|
|
</tr></table>
|
||
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<hr>
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