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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;20.&#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;20.&#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;20.&#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;20.&#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;20.&#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;20.&#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;20.&#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;20.&#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;20.&#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;20.&#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 <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
  96           and <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code>, and are
  97           by far the most efficient of the inverses presented here.
  98         </p>
  99 <p>
 100           The inverse on the <span class="emphasis"><em>a</em></span> parameter finds use in some statistical
 101           applications but has to be computed by rather brute force numerical techniques
 102           and is consequently several times slower. These are implemented here as
 103           <code class="computeroutput"><span class="identifier">gamma_p_inva</span></code> and <code class="computeroutput"><span class="identifier">gamma_q_inva</span></code>.
 104         </p>
 105 </td></tr>
 106 </table></div>
 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">&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">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>
 109
 110 <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>
 111 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 112 </pre>
 113 <p>
 114         Returns a value x such that: <code class="computeroutput"><span class="identifier">q</span>
 115         <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 116         <span class="identifier">x</span><span class="special">);</span></code>
 117       </p>
 118 <p>
 119         Requires: <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 120       </p>
 121 <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>
 122 <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>
 123
 124 <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>
 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">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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 126 </pre>
 127 <p>
 128         Returns a value x such that: <code class="computeroutput"><span class="identifier">p</span>
 129         <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 130         <span class="identifier">x</span><span class="special">);</span></code>
 131       </p>
 132 <p>
 133         Requires: <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 134       </p>
 135 <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>
 136 <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>
 137
 138 <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>
 139 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 140 </pre>
 141 <p>
 142         Returns a value a such that: <code class="computeroutput"><span class="identifier">q</span>
 143         <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 144         <span class="identifier">x</span><span class="special">);</span></code>
 145       </p>
 146 <p>
 147         Requires: <span class="emphasis"><em>x &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 148       </p>
 149 <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>
 150 <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>
 151
 152 <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>
 153 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 154 </pre>
 155 <p>
 156         Returns a value a such that: <code class="computeroutput"><span class="identifier">p</span>
 157         <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
 158         <span class="identifier">x</span><span class="special">);</span></code>
 159       </p>
 160 <p>
 161         Requires: <span class="emphasis"><em>x &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
 162       </p>
 163 <h5>
 164 <a name="math_toolkit.sf_gamma.igamma_inv.h2"></a>
 165         <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>
 166       </h5>
 167 <p>
 168         The accuracy of these functions doesn't vary much by platform or by the type
 169         T. Given that these functions are computed by iterative methods, they are
 170         deliberately "detuned" so as not to be too accurate: it is in any
 171         case impossible for these function to be more accurate than the regular forward
 172         incomplete gamma functions. In practice, the accuracy of these functions
 173         is very similar to that of <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
 174         and <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_q</a> functions:
 175       </p>
 176 <div class="table">
 177 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv"></a><p class="title"><b>Table&#160;8.13.&#160;Error rates for gamma_p_inv</b></p>
 178 <div class="table-contents"><table class="table" summary="Error rates for gamma_p_inv">
 179 <colgroup>
 180 <col>
 181 <col>
 182 <col>
 183 <col>
 184 <col>
 185 </colgroup>
 186 <thead><tr>
 187 <th>
 188               </th>
 189 <th>
 190                 <p>
 191                   GNU C++ version 7.1.0<br> linux<br> double
 192                 </p>
 193               </th>
 194 <th>
 195                 <p>
 196                   GNU C++ version 7.1.0<br> linux<br> long double
 197                 </p>
 198               </th>
 199 <th>
 200                 <p>
 201                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 202                 </p>
 203               </th>
 204 <th>
 205                 <p>
 206                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 207                 </p>
 208               </th>
 209 </tr></thead>
 210 <tbody>
 211 <tr>
 212 <td>
 213                 <p>
 214                   incomplete gamma inverse(a, z) medium values
 215                 </p>
 216               </td>
 217 <td>
 218                 <p>
 219                   <span class="blue">Max = 0.993&#949; (Mean = 0.15&#949;)</span><br> <br>
 220                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.88&#949; (Mean = 0.868&#949;))
 221                 </p>
 222               </td>
 223 <td>
 224                 <p>
 225                   <span class="blue">Max = 1.8&#949; (Mean = 0.406&#949;)</span>
 226                 </p>
 227               </td>
 228 <td>
 229                 <p>
 230                   <span class="blue">Max = 1.89&#949; (Mean = 0.466&#949;)</span>
 231                 </p>
 232               </td>
 233 <td>
 234                 <p>
 235                   <span class="blue">Max = 1.71&#949; (Mean = 0.34&#949;)</span>
 236                 </p>
 237               </td>
 238 </tr>
 239 <tr>
 240 <td>
 241                 <p>
 242                   incomplete gamma inverse(a, z) large values
 243                 </p>
 244               </td>
 245 <td>
 246                 <p>
 247                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 248                   3.2.3:</em></span> Max = 0.816&#949; (Mean = 0.0874&#949;))
 249                 </p>
 250               </td>
 251 <td>
 252                 <p>
 253                   <span class="blue">Max = 0.509&#949; (Mean = 0.0447&#949;)</span>
 254                 </p>
 255               </td>
 256 <td>
 257                 <p>
 258                   <span class="blue">Max = 0.509&#949; (Mean = 0.0447&#949;)</span>
 259                 </p>
 260               </td>
 261 <td>
 262                 <p>
 263                   <span class="blue">Max = 0.924&#949; (Mean = 0.108&#949;)</span>
 264                 </p>
 265               </td>
 266 </tr>
 267 <tr>
 268 <td>
 269                 <p>
 270                   incomplete gamma inverse(a, z) small values
 271                 </p>
 272               </td>
 273 <td>
 274                 <p>
 275                   <span class="blue">Max = 441&#949; (Mean = 53.9&#949;)</span><br> <br>
 276                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 547&#949; (Mean = 61.6&#949;))
 277                 </p>
 278               </td>
 279 <td>
 280                 <p>
 281                   <span class="blue">Max = 9.17e+03&#949; (Mean = 1.45e+03&#949;)</span>
 282                 </p>
 283               </td>
 284 <td>
 285                 <p>
 286                   <span class="blue">Max = 1.09e+04&#949; (Mean = 1.3e+03&#949;)</span>
 287                 </p>
 288               </td>
 289 <td>
 290                 <p>
 291                   <span class="blue">Max = 1.1e+03&#949; (Mean = 131&#949;)</span>
 292                 </p>
 293               </td>
 294 </tr>
 295 </tbody>
 296 </table></div>
 297 </div>
 298 <br class="table-break"><div class="table">
 299 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv"></a><p class="title"><b>Table&#160;8.14.&#160;Error rates for gamma_q_inv</b></p>
 300 <div class="table-contents"><table class="table" summary="Error rates for gamma_q_inv">
 301 <colgroup>
 302 <col>
 303 <col>
 304 <col>
 305 <col>
 306 <col>
 307 </colgroup>
 308 <thead><tr>
 309 <th>
 310               </th>
 311 <th>
 312                 <p>
 313                   GNU C++ version 7.1.0<br> linux<br> double
 314                 </p>
 315               </th>
 316 <th>
 317                 <p>
 318                   GNU C++ version 7.1.0<br> linux<br> long double
 319                 </p>
 320               </th>
 321 <th>
 322                 <p>
 323                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 324                 </p>
 325               </th>
 326 <th>
 327                 <p>
 328                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 329                 </p>
 330               </th>
 331 </tr></thead>
 332 <tbody>
 333 <tr>
 334 <td>
 335                 <p>
 336                   incomplete gamma inverse(a, z) medium values
 337                 </p>
 338               </td>
 339 <td>
 340                 <p>
 341                   <span class="blue">Max = 0.912&#949; (Mean = 0.154&#949;)</span><br> <br>
 342                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.66&#949; (Mean = 0.792&#949;))
 343                 </p>
 344               </td>
 345 <td>
 346                 <p>
 347                   <span class="blue">Max = 6.2&#949; (Mean = 0.627&#949;)</span>
 348                 </p>
 349               </td>
 350 <td>
 351                 <p>
 352                   <span class="blue">Max = 6.2&#949; (Mean = 0.683&#949;)</span>
 353                 </p>
 354               </td>
 355 <td>
 356                 <p>
 357                   <span class="blue">Max = 2.88&#949; (Mean = 0.469&#949;)</span>
 358                 </p>
 359               </td>
 360 </tr>
 361 <tr>
 362 <td>
 363                 <p>
 364                   incomplete gamma inverse(a, z) large values
 365                 </p>
 366               </td>
 367 <td>
 368                 <p>
 369                   <span class="blue">Max = 0.894&#949; (Mean = 0.0915&#949;)</span><br>
 370                   <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.894&#949; (Mean = 0.106&#949;))
 371                 </p>
 372               </td>
 373 <td>
 374                 <p>
 375                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 376                 </p>
 377               </td>
 378 <td>
 379                 <p>
 380                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 381                 </p>
 382               </td>
 383 <td>
 384                 <p>
 385                   <span class="blue">Max = 0.814&#949; (Mean = 0.0856&#949;)</span>
 386                 </p>
 387               </td>
 388 </tr>
 389 <tr>
 390 <td>
 391                 <p>
 392                   incomplete gamma inverse(a, z) small values
 393                 </p>
 394               </td>
 395 <td>
 396                 <p>
 397                   <span class="blue">Max = 292&#949; (Mean = 36.4&#949;)</span><br> <br>
 398                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 415&#949; (Mean = 48.7&#949;))
 399                 </p>
 400               </td>
 401 <td>
 402                 <p>
 403                   <span class="blue">Max = 8.28e+03&#949; (Mean = 1.09e+03&#949;)</span>
 404                 </p>
 405               </td>
 406 <td>
 407                 <p>
 408                   <span class="blue">Max = 8.98e+03&#949; (Mean = 877&#949;)</span>
 409                 </p>
 410               </td>
 411 <td>
 412                 <p>
 413                   <span class="blue">Max = 451&#949; (Mean = 64.7&#949;)</span>
 414                 </p>
 415               </td>
 416 </tr>
 417 </tbody>
 418 </table></div>
 419 </div>
 420 <br class="table-break"><div class="table">
 421 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inva"></a><p class="title"><b>Table&#160;8.15.&#160;Error rates for gamma_p_inva</b></p>
 422 <div class="table-contents"><table class="table" summary="Error rates for gamma_p_inva">
 423 <colgroup>
 424 <col>
 425 <col>
 426 <col>
 427 <col>
 428 <col>
 429 </colgroup>
 430 <thead><tr>
 431 <th>
 432               </th>
 433 <th>
 434                 <p>
 435                   GNU C++ version 7.1.0<br> linux<br> double
 436                 </p>
 437               </th>
 438 <th>
 439                 <p>
 440                   GNU C++ version 7.1.0<br> linux<br> long double
 441                 </p>
 442               </th>
 443 <th>
 444                 <p>
 445                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 446                 </p>
 447               </th>
 448 <th>
 449                 <p>
 450                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 451                 </p>
 452               </th>
 453 </tr></thead>
 454 <tbody><tr>
 455 <td>
 456                 <p>
 457                   Incomplete gamma inverses.
 458                 </p>
 459               </td>
 460 <td>
 461                 <p>
 462                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 463                 </p>
 464               </td>
 465 <td>
 466                 <p>
 467                   <span class="blue">Max = 7.87&#949; (Mean = 1.15&#949;)</span>
 468                 </p>
 469               </td>
 470 <td>
 471                 <p>
 472                   <span class="blue">Max = 4.08&#949; (Mean = 1.12&#949;)</span>
 473                 </p>
 474               </td>
 475 <td>
 476                 <p>
 477                   <span class="blue">Max = 4.92&#949; (Mean = 1.03&#949;)</span>
 478                 </p>
 479               </td>
 480 </tr></tbody>
 481 </table></div>
 482 </div>
 483 <br class="table-break"><div class="table">
 484 <a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inva"></a><p class="title"><b>Table&#160;8.16.&#160;Error rates for gamma_q_inva</b></p>
 485 <div class="table-contents"><table class="table" summary="Error rates for gamma_q_inva">
 486 <colgroup>
 487 <col>
 488 <col>
 489 <col>
 490 <col>
 491 <col>
 492 </colgroup>
 493 <thead><tr>
 494 <th>
 495               </th>
 496 <th>
 497                 <p>
 498                   GNU C++ version 7.1.0<br> linux<br> double
 499                 </p>
 500               </th>
 501 <th>
 502                 <p>
 503                   GNU C++ version 7.1.0<br> linux<br> long double
 504                 </p>
 505               </th>
 506 <th>
 507                 <p>
 508                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 509                 </p>
 510               </th>
 511 <th>
 512                 <p>
 513                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 514                 </p>
 515               </th>
 516 </tr></thead>
 517 <tbody><tr>
 518 <td>
 519                 <p>
 520                   Incomplete gamma inverses.
 521                 </p>
 522               </td>
 523 <td>
 524                 <p>
 525                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 526                 </p>
 527               </td>
 528 <td>
 529                 <p>
 530                   <span class="blue">Max = 8.42&#949; (Mean = 1.3&#949;)</span>
 531                 </p>
 532               </td>
 533 <td>
 534                 <p>
 535                   <span class="blue">Max = 7.86&#949; (Mean = 1.24&#949;)</span>
 536                 </p>
 537               </td>
 538 <td>
 539                 <p>
 540                   <span class="blue">Max = 5.05&#949; (Mean = 1.08&#949;)</span>
 541                 </p>
 542               </td>
 543 </tr></tbody>
 544 </table></div>
 545 </div>
 546 <br class="table-break"><h5>
 547 <a name="math_toolkit.sf_gamma.igamma_inv.h3"></a>
 548         <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>
 549       </h5>
 550 <p>
 551         There are two sets of tests:
 552       </p>
 553 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 554 <li class="listitem">
 555             Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>a</em></span>
 556             and <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>
 557             and back again. These tests have quite generous tolerances: in general
 558             both the incomplete gamma, and its inverses, change so rapidly that round
 559             tripping to more than a couple of significant digits isn't possible.
 560             This is especially true when <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>
 561             is very near one: in this case there isn't enough "information content"
 562             in the input to the inverse function to get back where you started.
 563           </li>
 564 <li class="listitem">
 565             Accuracy checks using high precision test values. These measure the accuracy
 566             of the result, given exact input values.
 567           </li>
 568 </ul></div>
 569 <h5>
 570 <a name="math_toolkit.sf_gamma.igamma_inv.h4"></a>
 571         <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>
 572       </h5>
 573 <p>
 574         The functions <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
 575         and <a href="http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/" target="_top"><code class="computeroutput"><span class="identifier">gamma_q_inv</span></code></a> share a common implementation.
 576       </p>
 577 <p>
 578         First an initial approximation is computed using the methodology described
 579         in:
 580       </p>
 581 <p>
 582         <a href="http://portal.acm.org/citation.cfm?id=23109&amp;coll=portal&amp;dl=ACM" target="_top">A.
 583         R. Didonato and A. H. Morris, Computation of the Incomplete Gamma Function
 584         Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.</a>
 585       </p>
 586 <p>
 587         Finally, the last few bits are cleaned up using Halley iteration, the iteration
 588         limit is set to 2/3 of the number of bits in T, which by experiment is sufficient
 589         to ensure that the inverses are at least as accurate as the normal incomplete
 590         gamma functions. In testing, no more than 3 iterations are required to produce
 591         a result as accurate as the forward incomplete gamma function, and in many
 592         cases only one iteration is required.
 593       </p>
 594 <p>
 595         The functions <code class="computeroutput"><span class="identifier">gamma_p_inva</span></code>
 596         and <code class="computeroutput"><span class="identifier">gamma_q_inva</span></code> also share
 597         a common implementation but are handled separately from <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
 598         and <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code>.
 599       </p>
 600 <p>
 601         An initial approximation for <span class="emphasis"><em>a</em></span> is computed very crudely
 602         so that <span class="emphasis"><em>gamma_p(a, x) ~ 0.5</em></span>, this value is then used
 603         as a starting point for a generic derivative-free root finding algorithm.
 604         As a consequence, these two functions are rather more expensive to compute
 605         than the <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code> or
 606         <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code> functions. Even
 607         so, the root is usually found in fewer than 10 iterations.
 608       </p>
 609 </div>
 610 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 611 <td align="left"></td>
 612 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 613       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 614       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 615       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 616       Daryle Walker and Xiaogang Zhang<p>
 617         Distributed under the Boost Software License, Version 1.0. (See accompanying
 618         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>)
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