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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.hypergeometric.hypergeometric_1f1"></a><a class="link" href="hypergeometric_1f1.html" title="Hypergeometric 1F1">Hypergeometric
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<sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub></a>
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</h3></div></div></div>
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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">hypergeometric_1F1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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">hypergeometric_1F1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. 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">hypergeometric_1F1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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">hypergeometric_1F1_regularized</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. 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">hypergeometric_1F1_regularized</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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">log_hypergeometric_1F1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. 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">log_hypergeometric_1F1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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">log_hypergeometric_1F1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. 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">log_hypergeometric_1F1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. 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.hypergeometric.hypergeometric_1f1.h0"></a>
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<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.description"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.description">Description</a>
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</h5>
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<p>
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</p>
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<div>
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<script type="text/javascript" src="../../../graphs/hypergeometric_1f1/plotlyjs-bundle.js"></script>
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</div>
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<p>
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</p>
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<p>
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The function <code class="computeroutput"><span class="identifier">hypergeometric_1F1</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
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<span class="identifier">b</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span></code> returns
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the non-singular solution to <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function" target="_top">Kummer's
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equation</a>
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_2.svg" width="181" height="34"></object></span>
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</p></blockquote></div>
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<p>
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which for |<span class="emphasis"><em>z</em></span>| < 1 has the hypergeometric series expansion
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_1.svg" width="234" height="38"></object></span>
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</p></blockquote></div>
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<p>
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where (<span class="emphasis"><em>a</em></span>)<sub><span class="emphasis"><em>n</em></span></sub> denotes rising factorial.
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This function has the same definition as Mathematica's <code class="computeroutput"><span class="identifier">Hypergeometric1F1</span><span class="special">[</span><span class="identifier">a</span><span class="special">,</span>
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<span class="identifier">b</span><span class="special">,</span> <span class="identifier">z</span><span class="special">]</span></code> and
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Maple's <code class="computeroutput"><span class="identifier">KummerM</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span></code>.
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</p>
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<p>
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The "regularized" versions of the function return:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_17.svg" width="313" height="44"></object></span>
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</p></blockquote></div>
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<p>
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The "log" versions of the function return:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_18.svg" width="119" height="20"></object></span>
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</p></blockquote></div>
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<p>
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When the optional <code class="computeroutput"><span class="identifier">sign</span></code> argument
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is supplied it is set on exit to either +1 or -1 depending on the sign of
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<sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub>(<span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>b</em></span>,
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<span class="emphasis"><em>z</em></span>).
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</p>
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<p>
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Both the regularized and the logarithmic versions of these functions return
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results without the spurious under/overflow that plague naive implementations.
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</p>
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<h5>
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<a name="math_toolkit.hypergeometric.hypergeometric_1f1.h1"></a>
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<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.known_issues"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.known_issues">Known
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Issues</a>
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</h5>
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<p>
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This function is still very much the subject of active research, and a full
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range of methods capable of calculating the function over its entire domain
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are not yet available. We have worked extremely hard to ensure that problem
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domains result in an exception being thrown (an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>)
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rather than return a garbage result. Domains that are still unsolved include:
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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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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<p>
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domain
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</p>
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</th>
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<th>
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<p>
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comment
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</p>
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</th>
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<th>
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<p>
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example
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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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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_13.svg" width="101" height="16"></object></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="emphasis"><em>a, b</em></span> and <span class="emphasis"><em>z</em></span> all large.
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</p>
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</td>
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<td>
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<p>
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<sub>1</sub>F<sub>1</sub>(-814723.75, -13586.87890625, -15.87335205078125)
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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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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_14.svg" width="89" height="16"></object></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="emphasis"><em>a < b</em></span>, <span class="emphasis"><em>b > z</em></span>, this
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is really the same domain as above.
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</p>
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</td>
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<td>
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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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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_15.svg" width="78" height="16"></object></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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There is a gap in between methods where no reliable implementation
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is possible, the issue becomes much worse for <span class="emphasis"><em>a</em></span>,
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<span class="emphasis"><em>b</em></span> and <span class="emphasis"><em>z</em></span> all large.
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</p>
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</td>
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<td>
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<p>
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<sub>1</sub>F<sub>1</sub>(9057.91796875, -1252.51318359375, 15.87335205078125)
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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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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_16.svg" width="98" height="18"></object></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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This domain is mostly handled by A&S 13.3.6 (see implementation
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notes below), but there are some values where either that series
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is non-convergent (most particularly for <span class="emphasis"><em>b</em></span>
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< 0) or where the series becomes divergent after a few terms
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limiting the precision that can be achieved.
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</p>
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</td>
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<td>
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<p>
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<sub>1</sub>F<sub>1</sub>(-5.9981750131794866e-15, 0.499999999999994, -240.42092034220695)
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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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<p>
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The following graph illustrates the problem area for <span class="emphasis"><em>b < 0</em></span>
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and <span class="emphasis"><em>a,z > 0</em></span>:
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</p>
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<p>
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</p>
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<div>
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<script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_b_incalculable.js"></script>
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<div id="fce5fa05-942c-4973-941f-2f3d25bcecb3" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
|
|
<script type="text/javascript">
|
|
(function(){
|
|
window.PLOTLYENV={'BASE_URL': 'https://plot.ly'};
|
|
|
|
var gd = document.getElementById('fce5fa05-942c-4973-941f-2f3d25bcecb3')
|
|
var resizeDebounce = null;
|
|
|
|
function resizePlot() {
|
|
var bb = gd.getBoundingClientRect();
|
|
Plotly.relayout(gd, {
|
|
width: bb.width,
|
|
height: bb.height
|
|
});
|
|
}
|
|
|
|
|
|
window.addEventListener('resize', function() {
|
|
if (resizeDebounce) {
|
|
window.clearTimeout(resizeDebounce);
|
|
}
|
|
resizeDebounce = window.setTimeout(resizePlot, 100);
|
|
});
|
|
|
|
|
|
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|
Plotly.plot(gd, {
|
|
data: negative_b_incalculable.data,
|
|
layout: negative_b_incalculable.layout,
|
|
frames: negative_b_incalculable.frames,
|
|
config: {"showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A"}
|
|
});
|
|
|
|
}());
|
|
</script>
|
|
</div>
|
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<p>
|
|
|
|
</p>
|
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<h5>
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<a name="math_toolkit.hypergeometric.hypergeometric_1f1.h2"></a>
|
|
<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.testing"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.testing">Testing</a>
|
|
</h5>
|
|
<p>
|
|
There are 3 main groups of tests:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
Spot tests for special inputs with known values.
|
|
</li>
|
|
<li class="listitem">
|
|
Sanity checks which use integrals of hypergeometric functions of known
|
|
value. These are particularly useful since for fixed <span class="emphasis"><em>a</em></span>
|
|
and <span class="emphasis"><em>b</em></span> they evaluate <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a,b,z)</em></span>
|
|
for all <span class="emphasis"><em>z</em></span>.
|
|
</li>
|
|
<li class="listitem">
|
|
Testing against values precomputed using arbitrary precision arithmetic
|
|
and the <span class="emphasis"><em>pFq</em></span> series.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
We also have a <a href="../../../../tools/hypergeometric_1F1_error_plot.cpp" target="_top">small
|
|
program</a> for testing random values over a user-specified domain of
|
|
<span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>b</em></span> and <span class="emphasis"><em>z</em></span>,
|
|
this program is also used for the error rate plots below and has been extremely
|
|
useful in fine-tuning the implementation.
|
|
</p>
|
|
<p>
|
|
It should be noted however, that there are some domains for large negative
|
|
<span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> that have poor test coverage
|
|
as we were simply unable to compute test values in reasonable time: it is
|
|
not uncommon for the <span class="emphasis"><em>pFq</em></span> series to cancel many hundreds
|
|
of digits and sometimes into the thousands of digits.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.hypergeometric.hypergeometric_1f1.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.errors"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.errors">Errors</a>
|
|
</h5>
|
|
<p>
|
|
We divide the domain of <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub> up into several domains to
|
|
explain the error rates.
|
|
</p>
|
|
<p>
|
|
In the following graphs we ran 100000 random test cases over each domain,
|
|
note that the scatter plots show only the very worst errors as otherwise
|
|
the graphs are both incomprehensible and virtually unplottable (as in sudden
|
|
browser death).
|
|
</p>
|
|
<p>
|
|
For 0 < a,b,z the error rates are trivially small unless we are forced
|
|
to take steps to avoid very-slow convergence and use a method other than
|
|
direct evaluation of the series for performance reasons. Even so the errors
|
|
rates are broadly acceptable, while the scatter graph shows where the worst
|
|
errors are located:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/positive_abz_bins.svg" width="567" height="320"></object></span>
|
|
</p>
|
|
<div>
|
|
<script type="text/javascript" src="../../../graphs/hypergeometric_1f1/positive_abz.js"></script>
|
|
|
|
<div id="87d8c26d-c743-4b69-8576-64f861bb7262" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
|
|
<script type="text/javascript">
|
|
(function () {
|
|
window.PLOTLYENV = { 'BASE_URL': 'https://plot.ly' };
|
|
|
|
var gd = document.getElementById('87d8c26d-c743-4b69-8576-64f861bb7262')
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|
var resizeDebounce = null;
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|
|
|
function resizePlot() {
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|
var bb = gd.getBoundingClientRect();
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|
Plotly.relayout(gd, {
|
|
width: bb.width,
|
|
height: bb.height
|
|
});
|
|
}
|
|
|
|
|
|
window.addEventListener('resize', function () {
|
|
if (resizeDebounce) {
|
|
window.clearTimeout(resizeDebounce);
|
|
}
|
|
resizeDebounce = window.setTimeout(resizePlot, 100);
|
|
});
|
|
|
|
|
|
|
|
Plotly.plot(gd, {
|
|
data: positive_abz.data,
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|
layout: positive_abz.layout,
|
|
frames: positive_abz.frames,
|
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config: { "showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A" }
|
|
});
|
|
|
|
}());
|
|
</script>
|
|
</div>
|
|
<p>
|
|
|
|
</p>
|
|
<p>
|
|
For -1000 < a < 0 and 0 < b,z < 1000 the maximum error rates
|
|
are higher, but most are still low, and the worst errors are fairly evenly
|
|
distributed:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/negative_a_bins.svg" width="567" height="331"></object></span>
|
|
</p>
|
|
<div>
|
|
<script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_a.js"></script>
|
|
|
|
<div id="cc677464-acba-4deb-b026-ef0ea03f1259" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
|
|
<script type="text/javascript">
|
|
(function () {
|
|
window.PLOTLYENV = { 'BASE_URL': 'https://plot.ly' };
|
|
|
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var gd = document.getElementById('cc677464-acba-4deb-b026-ef0ea03f1259')
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var resizeDebounce = null;
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function resizePlot() {
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var bb = gd.getBoundingClientRect();
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Plotly.relayout(gd, {
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width: bb.width,
|
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height: bb.height
|
|
});
|
|
}
|
|
|
|
|
|
window.addEventListener('resize', function () {
|
|
if (resizeDebounce) {
|
|
window.clearTimeout(resizeDebounce);
|
|
}
|
|
resizeDebounce = window.setTimeout(resizePlot, 100);
|
|
});
|
|
|
|
|
|
|
|
Plotly.plot(gd, {
|
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data: negative_a.data,
|
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layout: negative_a.layout,
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frames: negative_a.frames,
|
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config: { "showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A" }
|
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});
|
|
|
|
}());
|
|
</script>
|
|
</div>
|
|
<p>
|
|
|
|
</p>
|
|
<p>
|
|
For -1000 < <span class="emphasis"><em>b</em></span> < 0 and <span class="emphasis"><em>a</em></span>,<span class="emphasis"><em>z</em></span>
|
|
> 0 the errors are mostly low at double precision except near the "unimplementable
|
|
zone". Note however, that the some of the methods used here fail to
|
|
converge to much better than double precision.
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/negative_b_bins.svg" width="567" height="319"></object></span>
|
|
</p>
|
|
<div>
|
|
|
|
<script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_b.js"></script>
|
|
|
|
<div id="49ba2424-47a3-454d-ba72-b46ded00693f" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
|
|
<script type="text/javascript">
|
|
(function(){
|
|
window.PLOTLYENV={'BASE_URL': 'https://plot.ly'};
|
|
|
|
var gd = document.getElementById('49ba2424-47a3-454d-ba72-b46ded00693f')
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var resizeDebounce = null;
|
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|
function resizePlot() {
|
|
var bb = gd.getBoundingClientRect();
|
|
Plotly.relayout(gd, {
|
|
width: bb.width,
|
|
height: bb.height
|
|
});
|
|
}
|
|
|
|
|
|
window.addEventListener('resize', function() {
|
|
if (resizeDebounce) {
|
|
window.clearTimeout(resizeDebounce);
|
|
}
|
|
resizeDebounce = window.setTimeout(resizePlot, 100);
|
|
});
|
|
|
|
|
|
|
|
Plotly.plot(gd, {
|
|
data: negative_b.data,
|
|
layout: negative_b.layout,
|
|
frames: negative_b.frames,
|
|
config: {"showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A"}
|
|
});
|
|
|
|
}());
|
|
</script>
|
|
</div>
|
|
<p>
|
|
|
|
</p>
|
|
<p>
|
|
For both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> less than zero,
|
|
the errors the worst errors are clustered in a "difficult zone"
|
|
with <span class="emphasis"><em>b < a</em></span> and <span class="emphasis"><em>z</em></span> ≈ <span class="emphasis"><em>a</em></span>:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/negative_ab_bins.svg" width="461" height="280"></object></span>
|
|
<body>
|
|
<script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_ab.js"></script>
|
|
|
|
<div id="2867e84a-7d1d-4110-b28a-fb718dbd65ca" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
|
|
<script type="text/javascript">
|
|
(function(){
|
|
window.PLOTLYENV={'BASE_URL': 'https://plot.ly'};
|
|
|
|
var gd = document.getElementById('2867e84a-7d1d-4110-b28a-fb718dbd65ca')
|
|
var resizeDebounce = null;
|
|
|
|
function resizePlot() {
|
|
var bb = gd.getBoundingClientRect();
|
|
Plotly.relayout(gd, {
|
|
width: bb.width,
|
|
height: bb.height
|
|
});
|
|
}
|
|
|
|
|
|
window.addEventListener('resize', function() {
|
|
if (resizeDebounce) {
|
|
window.clearTimeout(resizeDebounce);
|
|
}
|
|
resizeDebounce = window.setTimeout(resizePlot, 100);
|
|
});
|
|
|
|
|
|
|
|
Plotly.plot(gd, {
|
|
data: negative_ab.data,
|
|
layout: negative_ab.layout,
|
|
frames: negative_ab.frames,
|
|
config: {"showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A"}
|
|
});
|
|
|
|
}());
|
|
</script>
|
|
</body>
|
|
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.hypergeometric.hypergeometric_1f1.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.implementation"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
The implementation for this function is remarkably complex; readers will
|
|
have to refer to the code for the transitions between methods, as we can
|
|
only give an overview here.
|
|
</p>
|
|
<p>
|
|
In almost all cases where <span class="emphasis"><em>z < 0</em></span> we use <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function" target="_top">Kummer's
|
|
relation</a> to make <span class="emphasis"><em>z</em></span> positive:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_12.svg" width="189" height="16"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
The main series representation
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_1.svg" width="234" height="38"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
is used only when
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
we are sure that it is convergent and not subject to excessive cancellation,
|
|
and
|
|
</li>
|
|
<li class="listitem">
|
|
there is no other better method available.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
The implementation of this series is complicated by the fact that for <span class="emphasis"><em>b</em></span>
|
|
< 0 then what appears to be a fully converged series can in fact suddenly
|
|
jump back to life again as <span class="emphasis"><em>b</em></span> crosses the origin.
|
|
</p>
|
|
<p>
|
|
A&S 13.3.6 gives
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_3.svg" width="614" height="39"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
which is particularly useful for <span class="emphasis"><em>a ≅ b</em></span> and <span class="emphasis"><em>z
|
|
> 0</em></span>, or <span class="emphasis"><em>a</em></span> ≪ 1, and <span class="emphasis"><em>z
|
|
< 0</em></span>.
|
|
</p>
|
|
<p>
|
|
Then we have Tricomi's approximation (given in simplified form in A&S
|
|
13.3.7) <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(7)</a>
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_4.svg" width="356" height="38"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
with
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_5.svg" width="390" height="33"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
and
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_6.svg" width="370" height="16"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
With <span class="emphasis"><em>E<sub>v</sub></em></span> defined as:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_7.svg" width="149" height="86"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
This approximation is especially effective when <span class="emphasis"><em>a < 0</em></span>.
|
|
</p>
|
|
<p>
|
|
For <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> both large and positive
|
|
and somewhat similar in size then an expansion in terms of gamma functions
|
|
can be used <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(6)</a>:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_8.svg" width="349" height="43"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
For <span class="emphasis"><em>z</em></span> large we have the asymptotic expansion:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_9.svg" width="261" height="43"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
which is a special case of the gamma function expansion above.
|
|
</p>
|
|
<p>
|
|
In the situation where <code class="computeroutput"><span class="identifier">ab</span><span class="special">/</span><span class="identifier">z</span></code> is reasonably
|
|
small then Luke's rational approximation is used - this is too complex to
|
|
document here, refer either to the code or the original paper <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(3)</a>.
|
|
</p>
|
|
<p>
|
|
The special case <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub>(1, <span class="emphasis"><em>b</em></span>, -<span class="emphasis"><em>z</em></span>)
|
|
is treated via Luke's Pade approximation <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(3)</a>.
|
|
</p>
|
|
<p>
|
|
That effectively completes the "direct" methods used, the remaining
|
|
areas are treated indirectly via recurrence relations. These require extreme
|
|
care in their use, as they often change the direction of stability, or else
|
|
are not stable at all. Sometimes this is a well defined and characterized
|
|
change in direction: for example with <span class="emphasis"><em>a,b</em></span> and <span class="emphasis"><em>z</em></span>
|
|
all positive recurrence on <span class="emphasis"><em>a</em></span> via
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_10.svg" width="429" height="16"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
abruptly changes from stable forwards recursion to stable backwards if <span class="emphasis"><em>2a-b+z</em></span>
|
|
changes sign. Thus recurrence on <span class="emphasis"><em>a</em></span>, even when <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub>(<span class="emphasis"><em>a</em></span>+<span class="emphasis"><em>N</em></span>,
|
|
<span class="emphasis"><em>b</em></span>, <span class="emphasis"><em>z</em></span>) is strictly increasing, needs
|
|
careful handling as <span class="emphasis"><em>a</em></span> → 0.
|
|
</p>
|
|
<p>
|
|
The transitory nature of the stable recurrence relations is well documented
|
|
in the literature, for example <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(10)</a>
|
|
gives many examples, including the anomalous behaviour of recurrence on
|
|
<span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> for large <span class="emphasis"><em>z</em></span>
|
|
as first documented by Gauchi <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(12)</a>.
|
|
Gauchi also provides the definitive reference on 3-term recurrence relations
|
|
in general in <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(11)</a>.
|
|
</p>
|
|
<p>
|
|
Unfortunately, not all recurrence stabilities are so well defined. For example,
|
|
when considering <sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, -<span class="emphasis"><em>b</em></span>, <span class="emphasis"><em>z</em></span>)
|
|
it would be convenient to use the continued fractions associated with the
|
|
recurrence relations to calculate <sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, -<span class="emphasis"><em>b</em></span>,
|
|
<span class="emphasis"><em>z</em></span>) / <sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, 1-<span class="emphasis"><em>b</em></span>,
|
|
<span class="emphasis"><em>z</em></span>) and then normalize either by iterating forwards on
|
|
<span class="emphasis"><em>b</em></span> until <span class="emphasis"><em>b > 0</em></span> and comparison
|
|
with a reference value, or by using the Wronskian
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_11.svg" width="564" height="33"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
which is of course the well known Miller's method <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(12)</a>.
|
|
</p>
|
|
<p>
|
|
Unfortunately, stable forwards recursion on <span class="emphasis"><em>b</em></span> is stable
|
|
only for <span class="emphasis"><em>|b| << |z|</em></span>, as <span class="emphasis"><em>|b|</em></span>
|
|
increases in magnitude it passes through a region where recursion is unstable
|
|
in both directions before eventually becoming stable in the backwards direction
|
|
(at least for a while!). This transition is moderated not by a change of
|
|
sign in the recurrence coefficients themselves, but by a change in the behaviour
|
|
of the <span class="emphasis"><em>values</em></span> of <sub>1</sub>F<sub>1</sub> - from alternating in sign when
|
|
<span class="emphasis"><em>|b|</em></span> is small to having the same sign when <span class="emphasis"><em>b</em></span>
|
|
is larger. During the actual transition, <sub>1</sub>F<sub>1</sub> will either pass through a zero
|
|
or a minima depending on whether b is even or odd (if there is a minima at
|
|
<sub>1</sub>F<sub>1</sub>(a, b, z) then there is necessarily a zero at <sub>1</sub>F<sub>1</sub>(a+1, b+1, z) as per
|
|
the <a href="https://dlmf.nist.gov/13.3#E15" target="_top">derivative of the function</a>).
|
|
As a result the behaviour of the recurrence relations is almost impossible
|
|
to reason about in this area, and we are left to using heuristic based approaches
|
|
to "guess" which region we are in.
|
|
</p>
|
|
<p>
|
|
In this implementation we use recurrence relations as follows:
|
|
</p>
|
|
<p>
|
|
For <span class="emphasis"><em>a,b,z > 0</em></span> and large we can either:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
Make <span class="emphasis"><em>0 < a < 1</em></span> and <span class="emphasis"><em>b > z</em></span>
|
|
and evaluate the defining series directly, or
|
|
</li>
|
|
<li class="listitem">
|
|
The gamma function approximation has decent convergence and accuracy
|
|
for <span class="emphasis"><em>2b - 1 < a < 2b < z</em></span>, so we can move
|
|
<span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> into this region, or
|
|
</li>
|
|
<li class="listitem">
|
|
We can recurse on <span class="emphasis"><em>b</em></span> alone so that <span class="emphasis"><em>b-1
|
|
< a < b</em></span> and use A&S 13.3.6 as long as <span class="emphasis"><em>z</em></span>
|
|
is not too large.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
For <span class="emphasis"><em>b < 0</em></span> and <span class="emphasis"><em>a</em></span> tiny we would
|
|
normally use A&S 13.3.6, but when that is non-convergent for some inputs
|
|
we can use the forward recurrence relation on <span class="emphasis"><em>b</em></span> to calculate
|
|
the ratio <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, -b, z)/<sub>1</sub>F<sub>1</sub>(a, 1-b, z)</em></span> and then iterate
|
|
forwards until <span class="emphasis"><em>b > 0</em></span> and compute a reference value
|
|
and normalize (Millers Method). In the same domain when <span class="emphasis"><em>b</em></span>
|
|
is near -1 we can use a single backwards recursion on <span class="emphasis"><em>b</em></span>
|
|
to compute <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, -b, z)</em></span> from <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, 2-b,
|
|
z)</em></span> and <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, 1-<span class="emphasis"><em>b</em></span>,
|
|
<span class="emphasis"><em>z</em></span>)</em></span> even though technically we are recursing
|
|
in the unstable direction.
|
|
</p>
|
|
<p>
|
|
For <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-N, b, z)</em></span> and integer <span class="emphasis"><em>N</em></span>
|
|
then backwards recursion from <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(0, b, z)</em></span> and <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-1,
|
|
b, z)</em></span> works well.
|
|
</p>
|
|
<p>
|
|
For <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> < 0 and if all the
|
|
direct methods have failed, then we use various fallbacks:
|
|
</p>
|
|
<p>
|
|
For <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-a, b, z)</em></span> we can use backwards recursion on
|
|
<span class="emphasis"><em>a</em></span> as long as <span class="emphasis"><em>b > z</em></span>, otherwise
|
|
a more complex scheme is required which starts from <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-a + N,
|
|
b + M, z)</em></span>, and recurses backwards in up to 3 stages: first on
|
|
<span class="emphasis"><em>a</em></span>, then on <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
|
|
together, and finally on <span class="emphasis"><em>b</em></span> alone.
|
|
</p>
|
|
<p>
|
|
For <span class="emphasis"><em>b < 0</em></span> we have no good methods in some domains
|
|
(see the unsolved issues above). However in some circumstances we can either
|
|
use:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
3-stage backwards recursion on both <span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>a</em></span>
|
|
and <span class="emphasis"><em>b</em></span> and then <span class="emphasis"><em>b</em></span> as above.
|
|
</li>
|
|
<li class="listitem">
|
|
Calculate the ratio <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, b, z) / <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a-1, b-1,
|
|
z)</em></span></em></span> via backwards recurrence when z is small, and
|
|
then normalize via the Wronskian above (Miller's method).
|
|
</li>
|
|
<li class="listitem">
|
|
Calculate the ratio <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, b, z) / <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a+1, b+1,
|
|
z)</em></span></em></span> via forwards recurrence when z is large, and
|
|
then normalize by iterating until b > 1 and comparing to a reference
|
|
value.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
The latter two methods use a lookup table to determine whether inputs are
|
|
in either of the domains or neither. Unfortunately the methods are basically
|
|
limited to double precision: calculation of the ratios require iteration
|
|
<span class="emphasis"><em>towards</em></span> the no-mans-land between the two methods where
|
|
iteration is unstable in both directions. As a result, only low-precision
|
|
results which require few iterations to calculate the ratio are available.
|
|
</p>
|
|
<p>
|
|
If all else fails, then we use a checked series summation which will raise
|
|
an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
|
|
if more than half the digits are destroyed by cancellation.
|
|
</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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, 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>
|
|
</div></td>
|
|
</tr></table>
|
|
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