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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Legendre (and Associated) Polynomials</title>
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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.sf_poly.legendre"></a><a class="link" href="legendre.html" title="Legendre (and Associated) Polynomials">Legendre (and Associated)
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Polynomials</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_poly.legendre.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.legendre.synopsis"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.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">legendre</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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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">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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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">T</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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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</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">T</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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">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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><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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<p>
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The return type of these functions 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>: note than when there is a single
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template argument the result is the same type as that argument or <code class="computeroutput"><span class="keyword">double</span></code> if the template argument is an integer
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type.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. 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,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.sf_poly.legendre.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.legendre.description"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.description">Description</a>
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</h5>
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<pre class="programlisting"><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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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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</pre>
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<p>
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Returns the Legendre Polynomial of the first kind:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/legendre_0.svg"></span>
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</p></blockquote></div>
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<p>
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Requires -1 <= x <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<p>
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Negative orders are handled via the reflection formula:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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P<sub>-l-1</sub>(x) = P<sub>l</sub>(x)
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</p></blockquote></div>
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<p>
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The following graph illustrates the behaviour of the first few Legendre Polynomials:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/legendre_p.svg" align="middle"></span>
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</p></blockquote></div>
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<pre class="programlisting"><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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_p_prime</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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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</pre>
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<p>
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Returns the derivatives of the Legendre polynomials.
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</p>
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<pre class="programlisting"><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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</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">T</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">legendre_p_zeros</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">);</span>
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</pre>
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<p>
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The zeros of the Legendre polynomials are calculated by Newton's method using
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an initial guess given by Tricomi with root bracketing provided by Szego.
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</p>
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<p>
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Since the Legendre polynomials are alternatively even and odd, only the non-negative
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zeros are returned. For the odd Legendre polynomials, the first zero is always
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zero. The rest of the zeros are returned in increasing order.
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</p>
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<p>
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Note that the argument to the routine is an integer, and the output is a
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floating-point type. Hence the template argument is mandatory. The time to
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extract a single root is linear in <code class="computeroutput"><span class="identifier">l</span></code>
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(this is scaling to evaluate the Legendre polynomials), so recovering all
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roots is 𝑶(<code class="computeroutput"><span class="identifier">l</span></code><sup>2</sup>). Algorithms
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with linear scaling <a href="https://doi.org/10.1137/06067016X" target="_top">exist</a>
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for recovering all roots, but requires tooling not currently built into boost.math.
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This implementation proceeds under the assumption that calculating zeros
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of these functions will not be a bottleneck for any workflow.
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</p>
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<pre class="programlisting"><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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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 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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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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</pre>
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<p>
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Returns the associated Legendre polynomial of the first kind:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/legendre_1.svg"></span>
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</p></blockquote></div>
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<p>
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Requires -1 <= x <= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<p>
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Negative values of <span class="emphasis"><em>l</em></span> and <span class="emphasis"><em>m</em></span> are
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handled via the identity relations:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/legendre_3.svg"></span>
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</p></blockquote></div>
|
|
<div class="caution"><table border="0" summary="Caution">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
|
|
<th align="left">Caution</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
The definition of the associated Legendre polynomial used here includes
|
|
a leading Condon-Shortley phase term of (-1)<sup>m</sup>. This matches the definition
|
|
given by Abramowitz and Stegun (8.6.6) and that used by <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Mathworld</a>
|
|
and <a href="http://documents.wolfram.com/mathematica/functions/LegendreP" target="_top">Mathematica's
|
|
LegendreP function</a>. However, uses in the literature do not always
|
|
include this phase term, and strangely the specification for the associated
|
|
Legendre function in the C++ TR1 (assoc_legendre) also omits it, in spite
|
|
of stating that it uses Abramowitz and Stegun as the final arbiter on these
|
|
matters.
|
|
</p>
|
|
<p>
|
|
See:
|
|
</p>
|
|
<p>
|
|
<a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Weisstein,
|
|
Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web
|
|
Resource</a>.
|
|
</p>
|
|
<p>
|
|
Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions"
|
|
and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook
|
|
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
|
|
9th printing. New York: Dover, pp. 331-339 and 771-802, 1972.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
|
<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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
|
|
|
|
<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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</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>
|
|
</pre>
|
|
<p>
|
|
Returns the value of the Legendre polynomial that is the second solution
|
|
to the Legendre differential equation, for example:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/legendre_2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Requires -1 <= x <= 1, otherwise <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
is called.
|
|
</p>
|
|
<p>
|
|
The following graph illustrates the first few Legendre functions of the second
|
|
kind:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../graphs/legendre_q.svg" align="middle"></span>
|
|
|
|
</p></blockquote></div>
|
|
<pre class="programlisting"><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>
|
|
<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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
Implements the three term recurrence relation for the Legendre polynomials,
|
|
this function can be used to create a sequence of values evaluated at the
|
|
same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>. This
|
|
recurrence relation holds for Legendre Polynomials of both the first and
|
|
second kinds.
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/legendre_4.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
For example we could produce a vector of the first 10 polynomial values using:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value</span>
|
|
<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span>
|
|
<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
|
|
<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
|
|
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
|
|
<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span>
|
|
<span class="comment">// Double check values:</span>
|
|
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
|
|
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">]</span> <span class="special">==</span> <span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
|
|
</pre>
|
|
<p>
|
|
Formally the arguments are:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term">l</span></dt>
|
|
<dd><p>
|
|
The degree of the last polynomial calculated.
|
|
</p></dd>
|
|
<dt><span class="term">x</span></dt>
|
|
<dd><p>
|
|
The abscissa value
|
|
</p></dd>
|
|
<dt><span class="term">Pl</span></dt>
|
|
<dd><p>
|
|
The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">Plm1</span></dt>
|
|
<dd><p>
|
|
The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>.
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<pre class="programlisting"><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>
|
|
<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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
Implements the three term recurrence relation for the Associated Legendre
|
|
polynomials, this function can be used to create a sequence of values evaluated
|
|
at the same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>.
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/legendre_5.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
For example we could produce a vector of the first m+10 polynomial values
|
|
using:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value</span>
|
|
<span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> <span class="comment">// order</span>
|
|
<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span>
|
|
<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
|
|
<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span> <span class="special">+</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
|
|
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
|
|
<span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span> <span class="special">+</span> <span class="number">10</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span>
|
|
<span class="comment">// Double check values:</span>
|
|
<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
|
|
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">]</span> <span class="special">==</span> <span class="identifier">legendre_p</span><span class="special">(</span><span class="number">10</span> <span class="special">+</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
|
|
</pre>
|
|
<p>
|
|
Formally the arguments are:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term">l</span></dt>
|
|
<dd><p>
|
|
The degree of the last polynomial calculated.
|
|
</p></dd>
|
|
<dt><span class="term">m</span></dt>
|
|
<dd><p>
|
|
The order of the Associated Polynomial.
|
|
</p></dd>
|
|
<dt><span class="term">x</span></dt>
|
|
<dd><p>
|
|
The abscissa value
|
|
</p></dd>
|
|
<dt><span class="term">Pl</span></dt>
|
|
<dd><p>
|
|
The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>.
|
|
</p></dd>
|
|
<dt><span class="term">Plm1</span></dt>
|
|
<dd><p>
|
|
The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>.
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<h5>
|
|
<a name="math_toolkit.sf_poly.legendre.h2"></a>
|
|
<span class="phrase"><a name="math_toolkit.sf_poly.legendre.accuracy"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.accuracy">Accuracy</a>
|
|
</h5>
|
|
<p>
|
|
The following table shows peak errors (in units of epsilon) for various domains
|
|
of input arguments. Note that only results for the widest floating point
|
|
type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
|
|
zero error</a>.
|
|
</p>
|
|
<div class="table">
|
|
<a name="math_toolkit.sf_poly.legendre.table_legendre_p"></a><p class="title"><b>Table 8.32. Error rates for legendre_p</b></p>
|
|
<div class="table-contents"><table class="table" summary="Error rates for legendre_p">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 7.1.0<br> linux<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 7.1.0<br> linux<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Sun compiler version 0x5150<br> Sun Solaris<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Microsoft Visual C++ version 14.1<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Legendre Polynomials: Small Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.732ε (Mean = 0.0619ε)</span><br>
|
|
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 211ε (Mean = 20.4ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 69.2ε (Mean = 9.58ε)</span><br> <br>
|
|
(<span class="emphasis"><em><cmath>:</em></span> Max = 124ε (Mean = 13.2ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 69.2ε (Mean = 9.58ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 211ε (Mean = 20.4ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Legendre Polynomials: Large Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.632ε (Mean = 0.0693ε)</span><br>
|
|
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 300ε (Mean = 33.2ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 699ε (Mean = 59.6ε)</span><br> <br>
|
|
(<span class="emphasis"><em><cmath>:</em></span> Max = 343ε (Mean = 32.1ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 699ε (Mean = 59.6ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 300ε (Mean = 33.2ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><div class="table">
|
|
<a name="math_toolkit.sf_poly.legendre.table_legendre_q"></a><p class="title"><b>Table 8.33. Error rates for legendre_q</b></p>
|
|
<div class="table-contents"><table class="table" summary="Error rates for legendre_q">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 7.1.0<br> linux<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 7.1.0<br> linux<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Sun compiler version 0x5150<br> Sun Solaris<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Microsoft Visual C++ version 14.1<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Legendre Polynomials: Small Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.612ε (Mean = 0.0517ε)</span><br>
|
|
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 46.4ε (Mean = 7.46ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 50.9ε (Mean = 9ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 50.9ε (Mean = 8.98ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 46.4ε (Mean = 7.32ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Legendre Polynomials: Large Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.49ε (Mean = 0.202ε)</span><br> <br>
|
|
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 4.6e+03ε (Mean = 366ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 5.98e+03ε (Mean = 478ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 5.98e+03ε (Mean = 478ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.6e+03ε (Mean = 366ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><div class="table">
|
|
<a name="math_toolkit.sf_poly.legendre.table_legendre_p_associated_"></a><p class="title"><b>Table 8.34. Error rates for legendre_p (associated)</b></p>
|
|
<div class="table-contents"><table class="table" summary="Error rates for legendre_p (associated)">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 7.1.0<br> linux<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 7.1.0<br> linux<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Sun compiler version 0x5150<br> Sun Solaris<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Microsoft Visual C++ version 14.1<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody><tr>
|
|
<td>
|
|
<p>
|
|
Associated Legendre Polynomials: Small Values
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.999ε (Mean = 0.05ε)</span><br> <br>
|
|
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 121ε (Mean = 6.75ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_legendre_p_associated__GSL_2_1_Associated_Legendre_Polynomials_Small_Values">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 175ε (Mean = 9.88ε)</span><br> <br>
|
|
(<span class="emphasis"><em><cmath>:</em></span> Max = 175ε (Mean = 9.36ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_legendre_p_associated___cmath__Associated_Legendre_Polynomials_Small_Values">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 77.7ε (Mean = 5.59ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 121ε (Mean = 7.14ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr></tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><p>
|
|
Note that the worst errors occur when the order increases, values greater
|
|
than ~120 are very unlikely to produce sensible results, especially in the
|
|
associated polynomial case when the degree is also large. Further the relative
|
|
errors are likely to grow arbitrarily large when the function is very close
|
|
to a root.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.sf_poly.legendre.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.sf_poly.legendre.testing"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.testing">Testing</a>
|
|
</h5>
|
|
<p>
|
|
A mixture of spot tests of values calculated using functions.wolfram.com,
|
|
and randomly generated test data are used: the test data was computed using
|
|
<a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
|
|
precision.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.sf_poly.legendre.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.sf_poly.legendre.implementation"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
These functions are implemented using the stable three term recurrence relations.
|
|
These relations guarantee low absolute error but cannot guarantee low relative
|
|
error near one of the roots of the polynomials.
|
|
</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>
|
|
<hr>
|
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