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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.sf_poly.sph_harm"></a><a class="link" href="sph_harm.html" title="Spherical Harmonics">Spherical Harmonics</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.sf_poly.sph_harm.h0"></a>
        <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.synopsis"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.synopsis">Synopsis</a>
      </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">spherical_harmonic</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<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>

<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</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="special">&gt;</span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</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="special">&gt;</span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.sf_poly.sph_harm.h1"></a>
        <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.description"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.description">Description</a>
      </h5>
<p>
        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
        type calculation rules</em></span></a> when T1 and T2 are different types.
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</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="special">&gt;</span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</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="special">&gt;</span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the value of the Spherical Harmonic Y<sub>n</sub><sup>m</sup>(theta, phi):
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../equations/spherical_0.svg"></span>
      </p>
<p>
        The spherical harmonics Y<sub>n</sub><sup>m</sup>(theta, phi) are the angular portion of the solution
        to Laplace's equation in spherical coordinates where azimuthal symmetry is
        not present.
      </p>
<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>
          Care must be taken in correctly identifying the arguments to this function:
          &#952; &#160; is taken as the polar (colatitudinal) coordinate with &#952; &#160; in [0, &#960;], and &#966; &#160; as
          the azimuthal (longitudinal) coordinate with &#966; &#160; in [0,2&#960;). This is the convention
          used in Physics, and matches the definition used by <a href="http://documents.wolfram.com/mathematica/functions/SphericalHarmonicY" target="_top">Mathematica
          in the function SpericalHarmonicY</a>, but is opposite to the usual
          mathematical conventions.
        </p>
<p>
          Some other sources include an additional Condon-Shortley phase term of
          (-1)<sup>m</sup> in the definition of this function: note however that our definition
          of the associated Legendre polynomial already includes this term.
        </p>
<p>
          This implementation returns zero for m &gt; n
        </p>
<p>
          For &#952; &#160; outside [0, &#960;] and &#966; &#160; outside [0, 2&#960;] this implementation follows the convention
          used by Mathematica: the function is periodic with period &#960; &#160; in &#952; &#160; and 2&#960; &#160; in &#966;.
          Please note that this is not the behaviour one would get from a casual
          application of the function's definition. Cautious users should keep &#952; &#160; and
          &#966; &#160; to the range [0, &#960;] and [0, 2&#960;] respectively.
        </p>
<p>
          See: <a href="http://mathworld.wolfram.com/SphericalHarmonic.html" target="_top">Weisstein,
          Eric W. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource</a>.
        </p>
</td></tr>
</table></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the real part of Y<sub>n</sub><sup>m</sup>(theta, phi):
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../equations/spherical_1.svg"></span>
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</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">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</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">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the imaginary part of Y<sub>n</sub><sup>m</sup>(theta, phi):
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../equations/spherical_2.svg"></span>
      </p>
<h5>
<a name="math_toolkit.sf_poly.sph_harm.h2"></a>
        <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.accuracy"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.accuracy">Accuracy</a>
      </h5>
<p>
        The following table shows peak errors 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>. Peak errors are the same for both the real and imaginary
        parts, as the error is dominated by calculation of the associated Legendre
        polynomials: especially near the roots of the associated Legendre function.
      </p>
<p>
        All values are in units of epsilon.
      </p>
<div class="table">
<a name="math_toolkit.sf_poly.sph_harm.table_spherical_harmonic_r"></a><p class="title"><b>Table&#160;6.38.&#160;Error rates for spherical_harmonic_r</b></p>
<div class="table-contents"><table class="table" summary="Error rates for spherical_harmonic_r">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 12.0<br> Win32<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 5.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 5.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5130<br> Sun Solaris<br> long double
                </p>
              </th>
</tr></thead>
<tbody><tr>
<td>
                <p>
                  Spherical Harmonics
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.27e+004&#949; (Mean = 725&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.58&#949; (Mean = 0.0707&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.89e+03&#949; (Mean = 108&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.03e+04&#949; (Mean = 327&#949;)</span>
                </p>
              </td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_poly.sph_harm.table_spherical_harmonic_i"></a><p class="title"><b>Table&#160;6.39.&#160;Error rates for spherical_harmonic_i</b></p>
<div class="table-contents"><table class="table" summary="Error rates for spherical_harmonic_i">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 12.0<br> Win32<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 5.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 5.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5130<br> Sun Solaris<br> long double
                </p>
              </th>
</tr></thead>
<tbody><tr>
<td>
                <p>
                  Spherical Harmonics
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.27e+004&#949; (Mean = 725&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.36&#949; (Mean = 0.0765&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.89e+03&#949; (Mean = 108&#949;)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.03e+04&#949; (Mean = 327&#949;)</span>
                </p>
              </td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><p>
        Note that the worst errors occur when the degree increases, values greater
        than ~120 are very unlikely to produce sensible results, especially when
        the order 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.sph_harm.h3"></a>
        <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.testing"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.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.sph_harm.h4"></a>
        <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.implementation"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.implementation">Implementation</a>
      </h5>
<p>
        These functions are implemented fairly naively using the formulae given above.
        Some extra care is taken to prevent roundoff error when converting from polar
        coordinates (so for example the <span class="emphasis"><em>1-x<sup>2</sup></em></span> term used by the
        associated Legendre functions is calculated without roundoff error using
        <span class="emphasis"><em>x = cos(theta)</em></span>, and <span class="emphasis"><em>1-x<sup>2</sup> = sin<sup>2</sup>(theta)</em></span>).
        The limiting factor in the error rates for these functions is the need to
        calculate values near the roots of the associated Legendre functions.
      </p>
</div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
      Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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