ceph/src/boost/libs/math/doc/html/math_toolkit/sf_gamma/igamma_inv.html

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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_gamma.igamma_inv"></a><a class="link" href="igamma_inv.html" title="Incomplete Gamma Function Inverses">Incomplete Gamma Function
  28       Inverses</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.sf_gamma.igamma_inv.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.synopsis"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.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">gamma</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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">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>
  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">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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>
  43
  44 <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>
  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">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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">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>
  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">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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>
  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">&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">gamma_q_inva</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">q</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> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><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">gamma_q_inva</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">q</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>
  55
  56 <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>
  57 <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">gamma_p_inva</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">p</span><span class="special">);</span>
  58
  59 <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>
  60 <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">gamma_p_inva</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">p</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>
  61
  62 <span class="special">}}</span> <span class="comment">// namespaces</span>
  63 </pre>
  64 <h5>
  65 <a name="math_toolkit.sf_gamma.igamma_inv.h1"></a>
  66         <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.description"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.description">Description</a>
  67       </h5>
  68 <p>
  69         There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete
  70         gamma function</a> inverses which either compute <span class="emphasis"><em>x</em></span>
  71         given <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>,
  72         or else compute <span class="emphasis"><em>a</em></span> given <span class="emphasis"><em>x</em></span> and either
  73         <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>.
  74       </p>
  75 <p>
  76         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
  77         type calculation rules</em></span></a> when T1 and T2 are different types,
  78         otherwise the return type is simply T1.
  79       </p>
  80 <p>
  81         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
  82         be used to control the behaviour of the function: how it handles errors,
  83         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
  84         documentation for more details</a>.
  85       </p>
  86 <div class="tip"><table border="0" summary="Tip">
  87 <tr>
  88 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
  89 <th align="left">Tip</th>
  90 </tr>
  91 <tr><td align="left" valign="top">
  92 <p>
  93           When people normally talk about the inverse of the incomplete gamma function,
  94           they are talking about inverting on parameter <span class="emphasis"><em>x</em></span>. These
  95           are implemented here as gamma_p_inv and gamma_q_inv, and are by far the
  96           most efficient of the inverses presented here.
  97         </p>
  98 <p>
  99           The inverse on the <span class="emphasis"><em>a</em></span> parameter finds use in some statistical
 100           applications but has to be computed by rather brute force numerical techniques
 101           and is consequently several times slower. These are implemented here as
 102           gamma_p_inva and gamma_q_inva.
 103         </p>
 104 </td></tr>
 105 </table></div>
 106 <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>
 107 <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">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">);</span>
 108
 109 <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>
 110 <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">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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>
 111 </pre>
 112 <p>
 113         Returns a value x such that: <code class="computeroutput"><span class="identifier">q</span>
 114         <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 115         <span class="identifier">x</span><span class="special">);</span></code>
 116       </p>
 117 <p>
 118         Requires: <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 119       </p>
 120 <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>
 121 <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">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">);</span>
 122
 123 <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>
 124 <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">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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>
 125 </pre>
 126 <p>
 127         Returns a value x such that: <code class="computeroutput"><span class="identifier">p</span>
 128         <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 129         <span class="identifier">x</span><span class="special">);</span></code>
 130       </p>
 131 <p>
 132         Requires: <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 133       </p>
 134 <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>
 135 <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">gamma_q_inva</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">q</span><span class="special">);</span>
 136
 137 <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>
 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">gamma_q_inva</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">q</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>
 139 </pre>
 140 <p>
 141         Returns a value a such that: <code class="computeroutput"><span class="identifier">q</span>
 142         <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 143         <span class="identifier">x</span><span class="special">);</span></code>
 144       </p>
 145 <p>
 146         Requires: <span class="emphasis"><em>x &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 147       </p>
 148 <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>
 149 <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">gamma_p_inva</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">p</span><span class="special">);</span>
 150
 151 <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>
 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">gamma_p_inva</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">p</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>
 153 </pre>
 154 <p>
 155         Returns a value a such that: <code class="computeroutput"><span class="identifier">p</span>
 156         <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 157         <span class="identifier">x</span><span class="special">);</span></code>
 158       </p>
 159 <p>
 160         Requires: <span class="emphasis"><em>x &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 161       </p>
 162 <h5>
 163 <a name="math_toolkit.sf_gamma.igamma_inv.h2"></a>
 164         <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.accuracy"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.accuracy">Accuracy</a>
 165       </h5>
 166 <p>
 167         The accuracy of these functions doesn't vary much by platform or by the type
 168         T. Given that these functions are computed by iterative methods, they are
 169         deliberately "detuned" so as not to be too accurate: it is in any
 170         case impossible for these function to be more accurate than the regular forward
 171         incomplete gamma functions. In practice, the accuracy of these functions
 172         is very similar to that of <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
 173         and <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_q</a> functions:
 174       </p>
 175 <div class="table">
 176 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv"></a><p class="title"><b>Table&#160;6.13.&#160;Error rates for gamma_p_inv</b></p>
 177 <div class="table-contents"><table class="table" summary="Error rates for gamma_p_inv">
 178 <colgroup>
 179 <col>
 180 <col>
 181 <col>
 182 <col>
 183 <col>
 184 </colgroup>
 185 <thead><tr>
 186 <th>
 187               </th>
 188 <th>
 189                 <p>
 190                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 191                 </p>
 192               </th>
 193 <th>
 194                 <p>
 195                   GNU C++ version 5.1.0<br> linux<br> double
 196                 </p>
 197               </th>
 198 <th>
 199                 <p>
 200                   GNU C++ version 5.1.0<br> linux<br> long double
 201                 </p>
 202               </th>
 203 <th>
 204                 <p>
 205                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 206                 </p>
 207               </th>
 208 </tr></thead>
 209 <tbody>
 210 <tr>
 211 <td>
 212                 <p>
 213                   incomplete gamma inverse(a, z) medium values
 214                 </p>
 215               </td>
 216 <td>
 217                 <p>
 218                   <span class="blue">Max = 1.01&#949; (Mean = 0.307&#949;)</span>
 219                 </p>
 220               </td>
 221 <td>
 222                 <p>
 223                   <span class="blue">Max = 0.993&#949; (Mean = 0.15&#949;)</span><br> <br>
 224                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 4.88&#949; (Mean = 0.868&#949;))
 225                 </p>
 226               </td>
 227 <td>
 228                 <p>
 229                   <span class="blue">Max = 1.62&#949; (Mean = 0.365&#949;)</span>
 230                 </p>
 231               </td>
 232 <td>
 233                 <p>
 234                   <span class="blue">Max = 1.86&#949; (Mean = 0.405&#949;)</span>
 235                 </p>
 236               </td>
 237 </tr>
 238 <tr>
 239 <td>
 240                 <p>
 241                   incomplete gamma inverse(a, z) large values
 242                 </p>
 243               </td>
 244 <td>
 245                 <p>
 246                   <span class="blue">Max = 0.924&#949; (Mean = 0.118&#949;)</span>
 247                 </p>
 248               </td>
 249 <td>
 250                 <p>
 251                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 252                   3.0.2:</em></span> Max = 0.816&#949; (Mean = 0.0874&#949;))
 253                 </p>
 254               </td>
 255 <td>
 256                 <p>
 257                   <span class="blue">Max = 0.509&#949; (Mean = 0.0447&#949;)</span>
 258                 </p>
 259               </td>
 260 <td>
 261                 <p>
 262                   <span class="blue">Max = 0.509&#949; (Mean = 0.0447&#949;)</span>
 263                 </p>
 264               </td>
 265 </tr>
 266 <tr>
 267 <td>
 268                 <p>
 269                   incomplete gamma inverse(a, z) small values
 270                 </p>
 271               </td>
 272 <td>
 273                 <p>
 274                   <span class="blue">Max = 1.1e+003&#949; (Mean = 108&#949;)</span>
 275                 </p>
 276               </td>
 277 <td>
 278                 <p>
 279                   <span class="blue">Max = 441&#949; (Mean = 53.9&#949;)</span><br> <br>
 280                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 547&#949; (Mean = 61.6&#949;))
 281                 </p>
 282               </td>
 283 <td>
 284                 <p>
 285                   <span class="blue">Max = 9.17e+03&#949; (Mean = 1.32e+03&#949;)</span>
 286                 </p>
 287               </td>
 288 <td>
 289                 <p>
 290                   <span class="blue">Max = 1.09e+04&#949; (Mean = 1.46e+03&#949;)</span>
 291                 </p>
 292               </td>
 293 </tr>
 294 </tbody>
 295 </table></div>
 296 </div>
 297 <br class="table-break"><div class="table">
 298 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv"></a><p class="title"><b>Table&#160;6.14.&#160;Error rates for gamma_q_inv</b></p>
 299 <div class="table-contents"><table class="table" summary="Error rates for gamma_q_inv">
 300 <colgroup>
 301 <col>
 302 <col>
 303 <col>
 304 <col>
 305 <col>
 306 </colgroup>
 307 <thead><tr>
 308 <th>
 309               </th>
 310 <th>
 311                 <p>
 312                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 313                 </p>
 314               </th>
 315 <th>
 316                 <p>
 317                   GNU C++ version 5.1.0<br> linux<br> double
 318                 </p>
 319               </th>
 320 <th>
 321                 <p>
 322                   GNU C++ version 5.1.0<br> linux<br> long double
 323                 </p>
 324               </th>
 325 <th>
 326                 <p>
 327                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 328                 </p>
 329               </th>
 330 </tr></thead>
 331 <tbody>
 332 <tr>
 333 <td>
 334                 <p>
 335                   incomplete gamma inverse(a, z) medium values
 336                 </p>
 337               </td>
 338 <td>
 339                 <p>
 340                   <span class="blue">Max = 3.46&#949; (Mean = 0.475&#949;)</span>
 341                 </p>
 342               </td>
 343 <td>
 344                 <p>
 345                   <span class="blue">Max = 0.912&#949; (Mean = 0.154&#949;)</span><br> <br>
 346                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 4.66&#949; (Mean = 0.792&#949;))
 347                 </p>
 348               </td>
 349 <td>
 350                 <p>
 351                   <span class="blue">Max = 6.2&#949; (Mean = 0.659&#949;)</span>
 352                 </p>
 353               </td>
 354 <td>
 355                 <p>
 356                   <span class="blue">Max = 6.2&#949; (Mean = 0.661&#949;)</span>
 357                 </p>
 358               </td>
 359 </tr>
 360 <tr>
 361 <td>
 362                 <p>
 363                   incomplete gamma inverse(a, z) large values
 364                 </p>
 365               </td>
 366 <td>
 367                 <p>
 368                   <span class="blue">Max = 0.814&#949; (Mean = 0.0856&#949;)</span>
 369                 </p>
 370               </td>
 371 <td>
 372                 <p>
 373                   <span class="blue">Max = 0.894&#949; (Mean = 0.0915&#949;)</span><br>
 374                   <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.894&#949; (Mean = 0.106&#949;))
 375                 </p>
 376               </td>
 377 <td>
 378                 <p>
 379                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 380                 </p>
 381               </td>
 382 <td>
 383                 <p>
 384                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 385                 </p>
 386               </td>
 387 </tr>
 388 <tr>
 389 <td>
 390                 <p>
 391                   incomplete gamma inverse(a, z) small values
 392                 </p>
 393               </td>
 394 <td>
 395                 <p>
 396                   <span class="blue">Max = 451&#949; (Mean = 65&#949;)</span>
 397                 </p>
 398               </td>
 399 <td>
 400                 <p>
 401                   <span class="blue">Max = 292&#949; (Mean = 36.4&#949;)</span><br> <br>
 402                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 415&#949; (Mean = 48.7&#949;))
 403                 </p>
 404               </td>
 405 <td>
 406                 <p>
 407                   <span class="blue">Max = 8.28e+03&#949; (Mean = 963&#949;)</span>
 408                 </p>
 409               </td>
 410 <td>
 411                 <p>
 412                   <span class="blue">Max = 8.98e+03&#949; (Mean = 877&#949;)</span>
 413                 </p>
 414               </td>
 415 </tr>
 416 </tbody>
 417 </table></div>
 418 </div>
 419 <br class="table-break"><div class="table">
 420 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inva"></a><p class="title"><b>Table&#160;6.15.&#160;Error rates for gamma_p_inva</b></p>
 421 <div class="table-contents"><table class="table" summary="Error rates for gamma_p_inva">
 422 <colgroup>
 423 <col>
 424 <col>
 425 <col>
 426 <col>
 427 <col>
 428 </colgroup>
 429 <thead><tr>
 430 <th>
 431               </th>
 432 <th>
 433                 <p>
 434                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 435                 </p>
 436               </th>
 437 <th>
 438                 <p>
 439                   GNU C++ version 5.1.0<br> linux<br> double
 440                 </p>
 441               </th>
 442 <th>
 443                 <p>
 444                   GNU C++ version 5.1.0<br> linux<br> long double
 445                 </p>
 446               </th>
 447 <th>
 448                 <p>
 449                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 450                 </p>
 451               </th>
 452 </tr></thead>
 453 <tbody><tr>
 454 <td>
 455                 <p>
 456                   Incomplete gamma inverses.
 457                 </p>
 458               </td>
 459 <td>
 460                 <p>
 461                   <span class="blue">Max = 3.52&#949; (Mean = 0.997&#949;)</span>
 462                 </p>
 463               </td>
 464 <td>
 465                 <p>
 466                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 467                 </p>
 468               </td>
 469 <td>
 470                 <p>
 471                   <span class="blue">Max = 6.44&#949; (Mean = 1.1&#949;)</span>
 472                 </p>
 473               </td>
 474 <td>
 475                 <p>
 476                   <span class="blue">Max = 4.08&#949; (Mean = 1.12&#949;)</span>
 477                 </p>
 478               </td>
 479 </tr></tbody>
 480 </table></div>
 481 </div>
 482 <br class="table-break"><div class="table">
 483 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inva"></a><p class="title"><b>Table&#160;6.16.&#160;Error rates for gamma_q_inva</b></p>
 484 <div class="table-contents"><table class="table" summary="Error rates for gamma_q_inva">
 485 <colgroup>
 486 <col>
 487 <col>
 488 <col>
 489 <col>
 490 <col>
 491 </colgroup>
 492 <thead><tr>
 493 <th>
 494               </th>
 495 <th>
 496                 <p>
 497                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 498                 </p>
 499               </th>
 500 <th>
 501                 <p>
 502                   GNU C++ version 5.1.0<br> linux<br> double
 503                 </p>
 504               </th>
 505 <th>
 506                 <p>
 507                   GNU C++ version 5.1.0<br> linux<br> long double
 508                 </p>
 509               </th>
 510 <th>
 511                 <p>
 512                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 513                 </p>
 514               </th>
 515 </tr></thead>
 516 <tbody><tr>
 517 <td>
 518                 <p>
 519                   Incomplete gamma inverses.
 520                 </p>
 521               </td>
 522 <td>
 523                 <p>
 524                   <span class="blue">Max = 5.64&#949; (Mean = 1.09&#949;)</span>
 525                 </p>
 526               </td>
 527 <td>
 528                 <p>
 529                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 530                 </p>
 531               </td>
 532 <td>
 533                 <p>
 534                   <span class="blue">Max = 6.91&#949; (Mean = 1.17&#949;)</span>
 535                 </p>
 536               </td>
 537 <td>
 538                 <p>
 539                   <span class="blue">Max = 7.86&#949; (Mean = 1.25&#949;)</span>
 540                 </p>
 541               </td>
 542 </tr></tbody>
 543 </table></div>
 544 </div>
 545 <br class="table-break"><h5>
 546 <a name="math_toolkit.sf_gamma.igamma_inv.h3"></a>
 547         <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.testing"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.testing">Testing</a>
 548       </h5>
 549 <p>
 550         There are two sets of tests:
 551       </p>
 552 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 553 <li class="listitem">
 554             Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>a</em></span>
 555             and <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>
 556             and back again. These tests have quite generous tolerances: in general
 557             both the incomplete gamma, and its inverses, change so rapidly that round
 558             tripping to more than a couple of significant digits isn't possible.
 559             This is especially true when <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>
 560             is very near one: in this case there isn't enough "information content"
 561             in the input to the inverse function to get back where you started.
 562           </li>
 563 <li class="listitem">
 564             Accuracy checks using high precision test values. These measure the accuracy
 565             of the result, given exact input values.
 566           </li>
 567 </ul></div>
 568 <h5>
 569 <a name="math_toolkit.sf_gamma.igamma_inv.h4"></a>
 570         <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.implementation"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.implementation">Implementation</a>
 571       </h5>
 572 <p>
 573         The functions gamma_p_inv and <a href="http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/" target="_top">gamma_q_inv</a>
 574         share a common implementation.
 575       </p>
 576 <p>
 577         First an initial approximation is computed using the methodology described
 578         in:
 579       </p>
 580 <p>
 581         <a href="http://portal.acm.org/citation.cfm?id=23109&amp;coll=portal&amp;dl=ACM" target="_top">A.
 582         R. Didonato and A. H. Morris, Computation of the Incomplete Gamma Function
 583         Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.</a>
 584       </p>
 585 <p>
 586         Finally, the last few bits are cleaned up using Halley iteration, the iteration
 587         limit is set to 2/3 of the number of bits in T, which by experiment is sufficient
 588         to ensure that the inverses are at least as accurate as the normal incomplete
 589         gamma functions. In testing, no more than 3 iterations are required to produce
 590         a result as accurate as the forward incomplete gamma function, and in many
 591         cases only one iteration is required.
 592       </p>
 593 <p>
 594         The functions gamma_p_inva and gamma_q_inva also share a common implementation
 595         but are handled separately from gamma_p_inv and gamma_q_inv.
 596       </p>
 597 <p>
 598         An initial approximation for <span class="emphasis"><em>a</em></span> is computed very crudely
 599         so that <span class="emphasis"><em>gamma_p(a, x) ~ 0.5</em></span>, this value is then used
 600         as a starting point for a generic derivative-free root finding algorithm.
 601         As a consequence, these two functions are rather more expensive to compute
 602         than the gamma_p_inv or gamma_q_inv functions. Even so, the root is usually
 603         found in fewer than 10 iterations.
 604       </p>
 605 </div>
 606 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 607 <td align="left"></td>
 608 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 609       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 610       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 611       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 612         Distributed under the Boost Software License, Version 1.0. (See accompanying
 613         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>)
 614       </p>
 615 </div></td>
 616 </tr></table>
 617 <hr>
 618 <div class="spirit-nav">
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