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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.laguerre"></a><a class="link" href="laguerre.html" title="Laguerre (and Associated) Polynomials">Laguerre (and Associated)
  28       Polynomials</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.sf_poly.laguerre.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.synopsis"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.synopsis">Synopsis</a>
  33       </h5>
  34 <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">laguerre</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  35 </pre>
  36 <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>
  37
  38 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  39 <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">laguerre</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>
  40
  41 <span class="keyword">template</span> <span class="special">&lt;</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  42 <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">laguerre</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  43
  44 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  45 <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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  46
  47 <span class="keyword">template</span> <span class="special">&lt;</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  48 <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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  49
  50 <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> <span class="identifier">T3</span><span class="special">&gt;</span>
  51 <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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span>
  52
  53 <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> <span class="identifier">T3</span><span class="special">&gt;</span>
  54 <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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span>
  55
  56
  57 <span class="special">}}</span> <span class="comment">// namespaces</span>
  58 </pre>
  59 <h5>
  60 <a name="math_toolkit.sf_poly.laguerre.h1"></a>
  61         <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.description"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.description">Description</a>
  62       </h5>
  63 <p>
  64         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
  65         type calculation rules</em></span></a>: note than when there is a single
  66         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
  67         type.
  68       </p>
  69 <p>
  70         The final <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  71         be used to control the behaviour of the function: how it handles errors,
  72         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">policy
  73         documentation for more details</a>.
  74       </p>
  75 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  76 <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">laguerre</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>
  77
  78 <span class="keyword">template</span> <span class="special">&lt;</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  79 <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">laguerre</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  80 </pre>
  81 <p>
  82         Returns the value of the Laguerre Polynomial of order <span class="emphasis"><em>n</em></span>
  83         at point <span class="emphasis"><em>x</em></span>:
  84       </p>
  85 <p>
  86         <span class="inlinemediaobject"><img src="../../../equations/laguerre_0.png"></span>
  87       </p>
  88 <p>
  89         The following graph illustrates the behaviour of the first few Laguerre Polynomials:
  90       </p>
  91 <p>
  92         <span class="inlinemediaobject"><img src="../../../graphs/laguerre.png" align="middle"></span>
  93       </p>
  94 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  95 <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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  96
  97 <span class="keyword">template</span> <span class="special">&lt;</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  98 <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">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  99 </pre>
 100 <p>
 101         Returns the Associated Laguerre polynomial of degree <span class="emphasis"><em>n</em></span>
 102         and order <span class="emphasis"><em>m</em></span> at point <span class="emphasis"><em>x</em></span>:
 103       </p>
 104 <p>
 105         <span class="inlinemediaobject"><img src="../../../equations/laguerre_1.png"></span>
 106       </p>
 107 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 108 <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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span>
 109 </pre>
 110 <p>
 111         Implements the three term recurrence relation for the Laguerre polynomials,
 112         this function can be used to create a sequence of values evaluated at the
 113         same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>n</em></span>.
 114       </p>
 115 <p>
 116         <span class="inlinemediaobject"><img src="../../../equations/laguerre_2.png"></span>
 117       </p>
 118 <p>
 119         For example we could produce a vector of the first 10 polynomial values using:
 120       </p>
 121 <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>
 122 <span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">;</span>
 123 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</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">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 124 <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">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 125    <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre_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>
 126 </pre>
 127 <p>
 128         Formally the arguments are:
 129       </p>
 130 <div class="variablelist">
 131 <p class="title"><b></b></p>
 132 <dl class="variablelist">
 133 <dt><span class="term">n</span></dt>
 134 <dd><p>
 135               The degree <span class="emphasis"><em>n</em></span> of the last polynomial calculated.
 136             </p></dd>
 137 <dt><span class="term">x</span></dt>
 138 <dd><p>
 139               The abscissa value
 140             </p></dd>
 141 <dt><span class="term">Ln</span></dt>
 142 <dd><p>
 143               The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>.
 144             </p></dd>
 145 <dt><span class="term">Lnm1</span></dt>
 146 <dd><p>
 147               The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>.
 148             </p></dd>
 149 </dl>
 150 </div>
 151 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 152 <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">laguerre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</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">Ln</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Lnm1</span><span class="special">);</span>
 153 </pre>
 154 <p>
 155         Implements the three term recurrence relation for the Associated Laguerre
 156         polynomials, this function can be used to create a sequence of values evaluated
 157         at the same <span class="emphasis"><em>x</em></span>, and for rising degree <span class="emphasis"><em>n</em></span>.
 158       </p>
 159 <p>
 160         <span class="inlinemediaobject"><img src="../../../equations/laguerre_3.png"></span>
 161       </p>
 162 <p>
 163         For example we could produce a vector of the first 10 polynomial values using:
 164       </p>
 165 <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>
 166 <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>
 167 <span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">;</span>
 168 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre</span><span class="special">(</span><span class="number">0</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">push_back</span><span class="special">(</span><span class="identifier">laguerre</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">x</span><span class="special">));</span>
 169 <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">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 170    <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">laguerre_next</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> <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>
 171 </pre>
 172 <p>
 173         Formally the arguments are:
 174       </p>
 175 <div class="variablelist">
 176 <p class="title"><b></b></p>
 177 <dl class="variablelist">
 178 <dt><span class="term">n</span></dt>
 179 <dd><p>
 180               The degree of the last polynomial calculated.
 181             </p></dd>
 182 <dt><span class="term">m</span></dt>
 183 <dd><p>
 184               The order of the Associated Polynomial.
 185             </p></dd>
 186 <dt><span class="term">x</span></dt>
 187 <dd><p>
 188               The abscissa value.
 189             </p></dd>
 190 <dt><span class="term">Ln</span></dt>
 191 <dd><p>
 192               The value of the polynomial evaluated at degree <span class="emphasis"><em>n</em></span>.
 193             </p></dd>
 194 <dt><span class="term">Lnm1</span></dt>
 195 <dd><p>
 196               The value of the polynomial evaluated at degree <span class="emphasis"><em>n-1</em></span>.
 197             </p></dd>
 198 </dl>
 199 </div>
 200 <h5>
 201 <a name="math_toolkit.sf_poly.laguerre.h2"></a>
 202         <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.accuracy"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.accuracy">Accuracy</a>
 203       </h5>
 204 <p>
 205         The following table shows peak errors (in units of epsilon) for various domains
 206         of input arguments. Note that only results for the widest floating point
 207         type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 208         zero error</a>.
 209       </p>
 210 <div class="table">
 211 <a name="math_toolkit.sf_poly.laguerre.peak_errors_in_the_laguerre_poly"></a><p class="title"><b>Table&#160;6.17.&#160;Peak Errors In the Laguerre Polynomial</b></p>
 212 <div class="table-contents"><table class="table" summary="Peak Errors In the Laguerre Polynomial">
 213 <colgroup>
 214 <col>
 215 <col>
 216 <col>
 217 </colgroup>
 218 <thead><tr>
 219 <th>
 220                 <p>
 221                   Significand Size
 222                 </p>
 223               </th>
 224 <th>
 225                 <p>
 226                   Platform and Compiler
 227                 </p>
 228               </th>
 229 <th>
 230                 <p>
 231                   Errors in range
 232                 </p>
 233                 <p>
 234                   0 &lt; l &lt; 20
 235                 </p>
 236               </th>
 237 </tr></thead>
 238 <tbody>
 239 <tr>
 240 <td>
 241                 <p>
 242                   53
 243                 </p>
 244               </td>
 245 <td>
 246                 <p>
 247                   Win32, Visual C++ 8
 248                 </p>
 249               </td>
 250 <td>
 251                 <p>
 252                   Peak=3000 Mean=185
 253                 </p>
 254               </td>
 255 </tr>
 256 <tr>
 257 <td>
 258                 <p>
 259                   64
 260                 </p>
 261               </td>
 262 <td>
 263                 <p>
 264                   SUSE Linux IA32, g++ 4.1
 265                 </p>
 266               </td>
 267 <td>
 268                 <p>
 269                   Peak=1x10<sup>4</sup> Mean=828
 270                 </p>
 271               </td>
 272 </tr>
 273 <tr>
 274 <td>
 275                 <p>
 276                   64
 277                 </p>
 278               </td>
 279 <td>
 280                 <p>
 281                   Red Hat Linux IA64, g++ 3.4.4
 282                 </p>
 283               </td>
 284 <td>
 285                 <p>
 286                   Peak=1x10<sup>4</sup> Mean=828
 287                 </p>
 288               </td>
 289 </tr>
 290 <tr>
 291 <td>
 292                 <p>
 293                   113
 294                 </p>
 295               </td>
 296 <td>
 297                 <p>
 298                   HPUX IA64, aCC A.06.06
 299                 </p>
 300               </td>
 301 <td>
 302                 <p>
 303                   Peak=680 Mean=40
 304                 </p>
 305               </td>
 306 </tr>
 307 </tbody>
 308 </table></div>
 309 </div>
 310 <br class="table-break"><div class="table">
 311 <a name="math_toolkit.sf_poly.laguerre.peak_errors_in_the_associated_la"></a><p class="title"><b>Table&#160;6.18.&#160;Peak Errors In the Associated Laguerre Polynomial</b></p>
 312 <div class="table-contents"><table class="table" summary="Peak Errors In the Associated Laguerre Polynomial">
 313 <colgroup>
 314 <col>
 315 <col>
 316 <col>
 317 </colgroup>
 318 <thead><tr>
 319 <th>
 320                 <p>
 321                   Significand Size
 322                 </p>
 323               </th>
 324 <th>
 325                 <p>
 326                   Platform and Compiler
 327                 </p>
 328               </th>
 329 <th>
 330                 <p>
 331                   Errors in range
 332                 </p>
 333                 <p>
 334                   0 &lt; l &lt; 20
 335                 </p>
 336               </th>
 337 </tr></thead>
 338 <tbody>
 339 <tr>
 340 <td>
 341                 <p>
 342                   53
 343                 </p>
 344               </td>
 345 <td>
 346                 <p>
 347                   Win32, Visual C++ 8
 348                 </p>
 349               </td>
 350 <td>
 351                 <p>
 352                   Peak=433 Mean=11
 353                 </p>
 354               </td>
 355 </tr>
 356 <tr>
 357 <td>
 358                 <p>
 359                   64
 360                 </p>
 361               </td>
 362 <td>
 363                 <p>
 364                   SUSE Linux IA32, g++ 4.1
 365                 </p>
 366               </td>
 367 <td>
 368                 <p>
 369                   Peak=61.4 Mean=19.5
 370                 </p>
 371               </td>
 372 </tr>
 373 <tr>
 374 <td>
 375                 <p>
 376                   64
 377                 </p>
 378               </td>
 379 <td>
 380                 <p>
 381                   Red Hat Linux IA64, g++ 3.4.4
 382                 </p>
 383               </td>
 384 <td>
 385                 <p>
 386                   Peak=61.4 Mean=19.5
 387                 </p>
 388               </td>
 389 </tr>
 390 <tr>
 391 <td>
 392                 <p>
 393                   113
 394                 </p>
 395               </td>
 396 <td>
 397                 <p>
 398                   HPUX IA64, aCC A.06.06
 399                 </p>
 400               </td>
 401 <td>
 402                 <p>
 403                   Peak=540 Mean=13.94
 404                 </p>
 405               </td>
 406 </tr>
 407 </tbody>
 408 </table></div>
 409 </div>
 410 <br class="table-break"><p>
 411         Note that the worst errors occur when the degree increases, values greater
 412         than ~120 are very unlikely to produce sensible results, especially in the
 413         associated polynomial case when the order is also large. Further the relative
 414         errors are likely to grow arbitrarily large when the function is very close
 415         to a root.
 416       </p>
 417 <h5>
 418 <a name="math_toolkit.sf_poly.laguerre.h3"></a>
 419         <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.testing"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.testing">Testing</a>
 420       </h5>
 421 <p>
 422         A mixture of spot tests of values calculated using functions.wolfram.com,
 423         and randomly generated test data are used: the test data was computed using
 424         <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
 425         precision.
 426       </p>
 427 <h5>
 428 <a name="math_toolkit.sf_poly.laguerre.h4"></a>
 429         <span class="phrase"><a name="math_toolkit.sf_poly.laguerre.implementation"></a></span><a class="link" href="laguerre.html#math_toolkit.sf_poly.laguerre.implementation">Implementation</a>
 430       </h5>
 431 <p>
 432         These functions are implemented using the stable three term recurrence relations.
 433         These relations guarentee low absolute error but cannot guarentee low relative
 434         error near one of the roots of the polynomials.
 435       </p>
 436 </div>
 437 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 438 <td align="left"></td>
 439 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 440       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 441       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 442       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 443         Distributed under the Boost Software License, Version 1.0. (See accompanying
 444         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>)
 445       </p>
 446 </div></td>
 447 </tr></table>
 448 <hr>
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