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   4 <title>Spherical Harmonics</title>
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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.sph_harm"></a><a class="link" href="sph_harm.html" title="Spherical Harmonics">Spherical Harmonics</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_poly.sph_harm.h0"></a>
  31         <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>
  32       </h5>
  33 <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>
  34 </pre>
  35 <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>
  36
  37 <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>
  38 <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>
  39
  40 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <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>
  44 <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>
  45
  46 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  48
  49 <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>
  50 <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>
  51
  52 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  53 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  54
  55 <span class="special">}}</span> <span class="comment">// namespaces</span>
  56 </pre>
  57 <h5>
  58 <a name="math_toolkit.sf_poly.sph_harm.h1"></a>
  59         <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>
  60       </h5>
  61 <p>
  62         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
  63         type calculation rules</em></span></a> when T1 and T2 are different types.
  64       </p>
  65 <p>
  66         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  67         be used to control the behaviour of the function: how it handles errors,
  68         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  69         documentation for more details</a>.
  70       </p>
  71 <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>
  72 <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>
  73
  74 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  75 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  76 </pre>
  77 <p>
  78         Returns the value of the Spherical Harmonic Y<sub>n</sub><sup>m</sup>(theta, phi):
  79       </p>
  80 <div class="blockquote"><blockquote class="blockquote"><p>
  81           <span class="inlinemediaobject"><img src="../../../equations/spherical_0.svg"></span>
  82
  83         </p></blockquote></div>
  84 <p>
  85         The spherical harmonics Y<sub>n</sub><sup>m</sup>(theta, phi) are the angular portion of the solution
  86         to Laplace's equation in spherical coordinates where azimuthal symmetry is
  87         not present.
  88       </p>
  89 <div class="caution"><table border="0" summary="Caution">
  90 <tr>
  91 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
  92 <th align="left">Caution</th>
  93 </tr>
  94 <tr><td align="left" valign="top">
  95 <p>
  96           Care must be taken in correctly identifying the arguments to this function:
  97           &#952; is taken as the polar (colatitudinal) coordinate with &#952; in [0, &#960;], and &#966;as
  98           the azimuthal (longitudinal) coordinate with &#966;in [0,2&#960;). This is the convention
  99           used in Physics, and matches the definition used by <a href="http://documents.wolfram.com/mathematica/functions/SphericalHarmonicY" target="_top">Mathematica
 100           in the function SpericalHarmonicY</a>, but is opposite to the usual
 101           mathematical conventions.
 102         </p>
 103 <p>
 104           Some other sources include an additional Condon-Shortley phase term of
 105           (-1)<sup>m</sup> in the definition of this function: note however that our definition
 106           of the associated Legendre polynomial already includes this term.
 107         </p>
 108 <p>
 109           This implementation returns zero for m &gt; n
 110         </p>
 111 <p>
 112           For &#952; outside [0, &#960;] and &#966; outside [0, 2&#960;] this implementation follows the convention
 113           used by Mathematica: the function is periodic with period &#960; in &#952; and 2&#960; in &#966;.
 114           Please note that this is not the behaviour one would get from a casual
 115           application of the function's definition. Cautious users should keep &#952; and
 116           &#966; to the range [0, &#960;] and [0, 2&#960;] respectively.
 117         </p>
 118 <p>
 119           See: <a href="http://mathworld.wolfram.com/SphericalHarmonic.html" target="_top">Weisstein,
 120           Eric W. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource</a>.
 121         </p>
 122 </td></tr>
 123 </table></div>
 124 <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>
 125 <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>
 126
 127 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 128 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 129 </pre>
 130 <p>
 131         Returns the real part of Y<sub>n</sub><sup>m</sup>(theta, phi):
 132       </p>
 133 <div class="blockquote"><blockquote class="blockquote"><p>
 134           <span class="inlinemediaobject"><img src="../../../equations/spherical_1.svg"></span>
 135
 136         </p></blockquote></div>
 137 <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>
 138 <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>
 139
 140 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 141 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 142 </pre>
 143 <p>
 144         Returns the imaginary part of Y<sub>n</sub><sup>m</sup>(theta, phi):
 145       </p>
 146 <div class="blockquote"><blockquote class="blockquote"><p>
 147           <span class="inlinemediaobject"><img src="../../../equations/spherical_2.svg"></span>
 148
 149         </p></blockquote></div>
 150 <h5>
 151 <a name="math_toolkit.sf_poly.sph_harm.h2"></a>
 152         <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>
 153       </h5>
 154 <p>
 155         The following table shows peak errors for various domains of input arguments.
 156         Note that only results for the widest floating point type on the system are
 157         given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 158         zero error</a>. Peak errors are the same for both the real and imaginary
 159         parts, as the error is dominated by calculation of the associated Legendre
 160         polynomials: especially near the roots of the associated Legendre function.
 161       </p>
 162 <p>
 163         All values are in units of epsilon.
 164       </p>
 165 <div class="table">
 166 <a name="math_toolkit.sf_poly.sph_harm.table_spherical_harmonic_r"></a><p class="title"><b>Table&#160;8.38.&#160;Error rates for spherical_harmonic_r</b></p>
 167 <div class="table-contents"><table class="table" summary="Error rates for spherical_harmonic_r">
 168 <colgroup>
 169 <col>
 170 <col>
 171 <col>
 172 <col>
 173 <col>
 174 </colgroup>
 175 <thead><tr>
 176 <th>
 177               </th>
 178 <th>
 179                 <p>
 180                   GNU C++ version 7.1.0<br> linux<br> double
 181                 </p>
 182               </th>
 183 <th>
 184                 <p>
 185                   GNU C++ version 7.1.0<br> linux<br> long double
 186                 </p>
 187               </th>
 188 <th>
 189                 <p>
 190                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 191                 </p>
 192               </th>
 193 <th>
 194                 <p>
 195                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 196                 </p>
 197               </th>
 198 </tr></thead>
 199 <tbody><tr>
 200 <td>
 201                 <p>
 202                   Spherical Harmonics
 203                 </p>
 204               </td>
 205 <td>
 206                 <p>
 207                   <span class="blue">Max = 1.58&#949; (Mean = 0.0707&#949;)</span>
 208                 </p>
 209               </td>
 210 <td>
 211                 <p>
 212                   <span class="blue">Max = 2.89e+03&#949; (Mean = 108&#949;)</span>
 213                 </p>
 214               </td>
 215 <td>
 216                 <p>
 217                   <span class="blue">Max = 1.03e+04&#949; (Mean = 327&#949;)</span>
 218                 </p>
 219               </td>
 220 <td>
 221                 <p>
 222                   <span class="blue">Max = 2.27e+04&#949; (Mean = 725&#949;)</span>
 223                 </p>
 224               </td>
 225 </tr></tbody>
 226 </table></div>
 227 </div>
 228 <br class="table-break"><div class="table">
 229 <a name="math_toolkit.sf_poly.sph_harm.table_spherical_harmonic_i"></a><p class="title"><b>Table&#160;8.39.&#160;Error rates for spherical_harmonic_i</b></p>
 230 <div class="table-contents"><table class="table" summary="Error rates for spherical_harmonic_i">
 231 <colgroup>
 232 <col>
 233 <col>
 234 <col>
 235 <col>
 236 <col>
 237 </colgroup>
 238 <thead><tr>
 239 <th>
 240               </th>
 241 <th>
 242                 <p>
 243                   GNU C++ version 7.1.0<br> linux<br> double
 244                 </p>
 245               </th>
 246 <th>
 247                 <p>
 248                   GNU C++ version 7.1.0<br> linux<br> long double
 249                 </p>
 250               </th>
 251 <th>
 252                 <p>
 253                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 254                 </p>
 255               </th>
 256 <th>
 257                 <p>
 258                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 259                 </p>
 260               </th>
 261 </tr></thead>
 262 <tbody><tr>
 263 <td>
 264                 <p>
 265                   Spherical Harmonics
 266                 </p>
 267               </td>
 268 <td>
 269                 <p>
 270                   <span class="blue">Max = 1.36&#949; (Mean = 0.0765&#949;)</span>
 271                 </p>
 272               </td>
 273 <td>
 274                 <p>
 275                   <span class="blue">Max = 2.89e+03&#949; (Mean = 108&#949;)</span>
 276                 </p>
 277               </td>
 278 <td>
 279                 <p>
 280                   <span class="blue">Max = 1.03e+04&#949; (Mean = 327&#949;)</span>
 281                 </p>
 282               </td>
 283 <td>
 284                 <p>
 285                   <span class="blue">Max = 2.27e+04&#949; (Mean = 725&#949;)</span>
 286                 </p>
 287               </td>
 288 </tr></tbody>
 289 </table></div>
 290 </div>
 291 <br class="table-break"><p>
 292         Note that the worst errors occur when the degree increases, values greater
 293         than ~120 are very unlikely to produce sensible results, especially when
 294         the order is also large. Further the relative errors are likely to grow arbitrarily
 295         large when the function is very close to a root.
 296       </p>
 297 <h5>
 298 <a name="math_toolkit.sf_poly.sph_harm.h3"></a>
 299         <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>
 300       </h5>
 301 <p>
 302         A mixture of spot tests of values calculated using functions.wolfram.com,
 303         and randomly generated test data are used: the test data was computed using
 304         <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
 305         precision.
 306       </p>
 307 <h5>
 308 <a name="math_toolkit.sf_poly.sph_harm.h4"></a>
 309         <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>
 310       </h5>
 311 <p>
 312         These functions are implemented fairly naively using the formulae given above.
 313         Some extra care is taken to prevent roundoff error when converting from polar
 314         coordinates (so for example the <span class="emphasis"><em>1-x<sup>2</sup></em></span> term used by the
 315         associated Legendre functions is calculated without roundoff error using
 316         <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>).
 317         The limiting factor in the error rates for these functions is the need to
 318         calculate values near the roots of the associated Legendre functions.
 319       </p>
 320 </div>
 321 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 322 <td align="left"></td>
 323 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 324       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 325       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 326       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 327       Daryle Walker and Xiaogang Zhang<p>
 328         Distributed under the Boost Software License, Version 1.0. (See accompanying
 329         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>)
 330       </p>
 331 </div></td>
 332 </tr></table>
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