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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.hypergeometric.hypergeometric_1f1"></a><a class="link" href="hypergeometric_1f1.html" title="Hypergeometric 1F1">Hypergeometric
  28       <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub></a>
  29 </h3></div></div></div>
  30 <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">hypergeometric_1F1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  31
  32 <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>
  33
  34 <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>
  35 <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">hypergeometric_1F1</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">);</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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>
  38 <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">hypergeometric_1F1</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</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>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  41 <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">hypergeometric_1F1_regularized</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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>
  44 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_1F1_regularized</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</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>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  47 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">log_hypergeometric_1F1</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">);</span>
  48
  49 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</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>
  50 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">log_hypergeometric_1F1</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</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>
  51
  52 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  53 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">log_hypergeometric_1F1</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</span><span class="special">);</span>
  54
  55 <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">,</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>
  56 <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">log_hypergeometric_1F1</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</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>
  57
  58 <span class="special">}}</span> <span class="comment">// namespaces</span>
  59 </pre>
  60 <h5>
  61 <a name="math_toolkit.hypergeometric.hypergeometric_1f1.h0"></a>
  62         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.description"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.description">Description</a>
  63       </h5>
  64 <p>
  65         </p>
  66 <div>
  67    <script type="text/javascript" src="../../../graphs/hypergeometric_1f1/plotlyjs-bundle.js"></script>
  68 </div>
  69 <p>
  70       </p>
  71 <p>
  72         The function <code class="computeroutput"><span class="identifier">hypergeometric_1F1</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
  73         <span class="identifier">b</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span></code> returns
  74         the non-singular solution to <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function" target="_top">Kummer's
  75         equation</a>
  76       </p>
  77 <div class="blockquote"><blockquote class="blockquote"><p>
  78           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_2.svg" width="181" height="34"></object></span>
  79         </p></blockquote></div>
  80 <p>
  81         which for |<span class="emphasis"><em>z</em></span>| &lt; 1 has the hypergeometric series expansion
  82       </p>
  83 <div class="blockquote"><blockquote class="blockquote"><p>
  84           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_1.svg" width="234" height="38"></object></span>
  85         </p></blockquote></div>
  86 <p>
  87         where (<span class="emphasis"><em>a</em></span>)<sub><span class="emphasis"><em>n</em></span></sub> denotes rising factorial.
  88         This function has the same definition as Mathematica's <code class="computeroutput"><span class="identifier">Hypergeometric1F1</span><span class="special">[</span><span class="identifier">a</span><span class="special">,</span>
  89         <span class="identifier">b</span><span class="special">,</span> <span class="identifier">z</span><span class="special">]</span></code> and
  90         Maple's <code class="computeroutput"><span class="identifier">KummerM</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span></code>.
  91       </p>
  92 <p>
  93         The "regularized" versions of the function return:
  94       </p>
  95 <div class="blockquote"><blockquote class="blockquote"><p>
  96           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_17.svg" width="313" height="44"></object></span>
  97         </p></blockquote></div>
  98 <p>
  99         The "log" versions of the function return:
 100       </p>
 101 <div class="blockquote"><blockquote class="blockquote"><p>
 102           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_18.svg" width="119" height="20"></object></span>
 103         </p></blockquote></div>
 104 <p>
 105         When the optional <code class="computeroutput"><span class="identifier">sign</span></code> argument
 106         is supplied it is set on exit to either +1 or -1 depending on the sign of
 107         <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub>(<span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>b</em></span>,
 108         <span class="emphasis"><em>z</em></span>).
 109       </p>
 110 <p>
 111         Both the regularized and the logarithmic versions of these functions return
 112         results without the spurious under/overflow that plague naive implementations.
 113       </p>
 114 <h5>
 115 <a name="math_toolkit.hypergeometric.hypergeometric_1f1.h1"></a>
 116         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.known_issues"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.known_issues">Known
 117         Issues</a>
 118       </h5>
 119 <p>
 120         This function is still very much the subject of active research, and a full
 121         range of methods capable of calculating the function over its entire domain
 122         are not yet available. We have worked extremely hard to ensure that problem
 123         domains result in an exception being thrown (an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>)
 124         rather than return a garbage result. Domains that are still unsolved include:
 125       </p>
 126 <div class="informaltable"><table class="table">
 127 <colgroup>
 128 <col>
 129 <col>
 130 <col>
 131 </colgroup>
 132 <thead><tr>
 133 <th>
 134                 <p>
 135                   domain
 136                 </p>
 137               </th>
 138 <th>
 139                 <p>
 140                   comment
 141                 </p>
 142               </th>
 143 <th>
 144                 <p>
 145                   example
 146                 </p>
 147               </th>
 148 </tr></thead>
 149 <tbody>
 150 <tr>
 151 <td>
 152                 <p>
 153                   <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_13.svg" width="101" height="16"></object></span>
 154                 </p>
 155               </td>
 156 <td>
 157                 <p>
 158                   <span class="emphasis"><em>a, b</em></span> and <span class="emphasis"><em>z</em></span> all large.
 159                 </p>
 160               </td>
 161 <td>
 162                 <p>
 163                   <sub>1</sub>F<sub>1</sub>(-814723.75, -13586.87890625, -15.87335205078125)
 164                 </p>
 165               </td>
 166 </tr>
 167 <tr>
 168 <td>
 169                 <p>
 170                   <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_14.svg" width="89" height="16"></object></span>
 171                 </p>
 172               </td>
 173 <td>
 174                 <p>
 175                   <span class="emphasis"><em>a &lt; b</em></span>, <span class="emphasis"><em>b &gt; z</em></span>, this
 176                   is really the same domain as above.
 177                 </p>
 178               </td>
 179 <td>
 180               </td>
 181 </tr>
 182 <tr>
 183 <td>
 184                 <p>
 185                   <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_15.svg" width="78" height="16"></object></span>
 186                 </p>
 187               </td>
 188 <td>
 189                 <p>
 190                   There is a gap in between methods where no reliable implementation
 191                   is possible, the issue becomes much worse for <span class="emphasis"><em>a</em></span>,
 192                   <span class="emphasis"><em>b</em></span> and <span class="emphasis"><em>z</em></span> all large.
 193                 </p>
 194               </td>
 195 <td>
 196                 <p>
 197                   <sub>1</sub>F<sub>1</sub>(9057.91796875, -1252.51318359375, 15.87335205078125)
 198                 </p>
 199               </td>
 200 </tr>
 201 <tr>
 202 <td>
 203                 <p>
 204                   <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_16.svg" width="98" height="18"></object></span>
 205                 </p>
 206               </td>
 207 <td>
 208                 <p>
 209                   This domain is mostly handled by A&amp;S 13.3.6 (see implementation
 210                   notes below), but there are some values where either that series
 211                   is non-convergent (most particularly for <span class="emphasis"><em>b</em></span>
 212                   &lt; 0) or where the series becomes divergent after a few terms
 213                   limiting the precision that can be achieved.
 214                 </p>
 215               </td>
 216 <td>
 217                 <p>
 218                   <sub>1</sub>F<sub>1</sub>(-5.9981750131794866e-15, 0.499999999999994, -240.42092034220695)
 219                 </p>
 220               </td>
 221 </tr>
 222 </tbody>
 223 </table></div>
 224 <p>
 225         The following graph illustrates the problem area for <span class="emphasis"><em>b &lt; 0</em></span>
 226         and <span class="emphasis"><em>a,z &gt; 0</em></span>:
 227       </p>
 228 <p>
 229         </p>
 230 <div>
 231         <script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_b_incalculable.js"></script>
 232
 233         <div id="fce5fa05-942c-4973-941f-2f3d25bcecb3" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
 234         <script type="text/javascript">
 235             (function(){
 236                 window.PLOTLYENV={'BASE_URL': 'https://plot.ly'};
 237
 238                 var gd = document.getElementById('fce5fa05-942c-4973-941f-2f3d25bcecb3')
 239                 var resizeDebounce = null;
 240
 241                 function resizePlot() {
 242                     var bb = gd.getBoundingClientRect();
 243                     Plotly.relayout(gd, {
 244                         width: bb.width,
 245                         height: bb.height
 246                     });
 247                 }
 248
 249
 250                 window.addEventListener('resize', function() {
 251                     if (resizeDebounce) {
 252                         window.clearTimeout(resizeDebounce);
 253                     }
 254                     resizeDebounce = window.setTimeout(resizePlot, 100);
 255                 });
 256
 257
 258
 259                 Plotly.plot(gd,  {
 260                    data: negative_b_incalculable.data,
 261                    layout: negative_b_incalculable.layout,
 262                    frames: negative_b_incalculable.frames,
 263                     config: {"showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A"}
 264                 });
 265
 266            }());
 267         </script>
 268     </div>
 269 <p>
 270
 271       </p>
 272 <h5>
 273 <a name="math_toolkit.hypergeometric.hypergeometric_1f1.h2"></a>
 274         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.testing"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.testing">Testing</a>
 275       </h5>
 276 <p>
 277         There are 3 main groups of tests:
 278       </p>
 279 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 280 <li class="listitem">
 281             Spot tests for special inputs with known values.
 282           </li>
 283 <li class="listitem">
 284             Sanity checks which use integrals of hypergeometric functions of known
 285             value. These are particularly useful since for fixed <span class="emphasis"><em>a</em></span>
 286             and <span class="emphasis"><em>b</em></span> they evaluate <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a,b,z)</em></span>
 287             for all <span class="emphasis"><em>z</em></span>.
 288           </li>
 289 <li class="listitem">
 290             Testing against values precomputed using arbitrary precision arithmetic
 291             and the <span class="emphasis"><em>pFq</em></span> series.
 292           </li>
 293 </ul></div>
 294 <p>
 295         We also have a <a href="../../../../tools/hypergeometric_1F1_error_plot.cpp" target="_top">small
 296         program</a> for testing random values over a user-specified domain of
 297         <span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>b</em></span> and <span class="emphasis"><em>z</em></span>,
 298         this program is also used for the error rate plots below and has been extremely
 299         useful in fine-tuning the implementation.
 300       </p>
 301 <p>
 302         It should be noted however, that there are some domains for large negative
 303         <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> that have poor test coverage
 304         as we were simply unable to compute test values in reasonable time: it is
 305         not uncommon for the <span class="emphasis"><em>pFq</em></span> series to cancel many hundreds
 306         of digits and sometimes into the thousands of digits.
 307       </p>
 308 <h5>
 309 <a name="math_toolkit.hypergeometric.hypergeometric_1f1.h3"></a>
 310         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.errors"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.errors">Errors</a>
 311       </h5>
 312 <p>
 313         We divide the domain of <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub> up into several domains to
 314         explain the error rates.
 315       </p>
 316 <p>
 317         In the following graphs we ran 100000 random test cases over each domain,
 318         note that the scatter plots show only the very worst errors as otherwise
 319         the graphs are both incomprehensible and virtually unplottable (as in sudden
 320         browser death).
 321       </p>
 322 <p>
 323         For 0 &lt; a,b,z the error rates are trivially small unless we are forced
 324         to take steps to avoid very-slow convergence and use a method other than
 325         direct evaluation of the series for performance reasons. Even so the errors
 326         rates are broadly acceptable, while the scatter graph shows where the worst
 327         errors are located:
 328       </p>
 329 <p>
 330         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/positive_abz_bins.svg" width="567" height="320"></object></span>
 331         </p>
 332 <div>
 333    <script type="text/javascript" src="../../../graphs/hypergeometric_1f1/positive_abz.js"></script>
 334
 335    <div id="87d8c26d-c743-4b69-8576-64f861bb7262" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
 336    <script type="text/javascript">
 337       (function () {
 338          window.PLOTLYENV = { 'BASE_URL': 'https://plot.ly' };
 339
 340          var gd = document.getElementById('87d8c26d-c743-4b69-8576-64f861bb7262')
 341          var resizeDebounce = null;
 342
 343          function resizePlot() {
 344             var bb = gd.getBoundingClientRect();
 345             Plotly.relayout(gd, {
 346                width: bb.width,
 347                height: bb.height
 348             });
 349          }
 350
 351
 352          window.addEventListener('resize', function () {
 353             if (resizeDebounce) {
 354                window.clearTimeout(resizeDebounce);
 355             }
 356             resizeDebounce = window.setTimeout(resizePlot, 100);
 357          });
 358
 359
 360
 361          Plotly.plot(gd, {
 362             data: positive_abz.data,
 363             layout: positive_abz.layout,
 364             frames: positive_abz.frames,
 365             config: { "showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A" }
 366          });
 367
 368       }());
 369    </script>
 370 </div>
 371 <p>
 372
 373       </p>
 374 <p>
 375         For -1000 &lt; a &lt; 0 and 0 &lt; b,z &lt; 1000 the maximum error rates
 376         are higher, but most are still low, and the worst errors are fairly evenly
 377         distributed:
 378       </p>
 379 <p>
 380         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/negative_a_bins.svg" width="567" height="331"></object></span>
 381         </p>
 382 <div>
 383    <script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_a.js"></script>
 384
 385    <div id="cc677464-acba-4deb-b026-ef0ea03f1259" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
 386    <script type="text/javascript">
 387       (function () {
 388          window.PLOTLYENV = { 'BASE_URL': 'https://plot.ly' };
 389
 390          var gd = document.getElementById('cc677464-acba-4deb-b026-ef0ea03f1259')
 391          var resizeDebounce = null;
 392
 393          function resizePlot() {
 394             var bb = gd.getBoundingClientRect();
 395             Plotly.relayout(gd, {
 396                width: bb.width,
 397                height: bb.height
 398             });
 399          }
 400
 401
 402          window.addEventListener('resize', function () {
 403             if (resizeDebounce) {
 404                window.clearTimeout(resizeDebounce);
 405             }
 406             resizeDebounce = window.setTimeout(resizePlot, 100);
 407          });
 408
 409
 410
 411          Plotly.plot(gd, {
 412             data: negative_a.data,
 413             layout: negative_a.layout,
 414             frames: negative_a.frames,
 415             config: { "showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A" }
 416          });
 417
 418       }());
 419    </script>
 420 </div>
 421 <p>
 422
 423       </p>
 424 <p>
 425         For -1000 &lt; <span class="emphasis"><em>b</em></span> &lt; 0 and <span class="emphasis"><em>a</em></span>,<span class="emphasis"><em>z</em></span>
 426         &gt; 0 the errors are mostly low at double precision except near the "unimplementable
 427         zone". Note however, that the some of the methods used here fail to
 428         converge to much better than double precision.
 429       </p>
 430 <p>
 431         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/negative_b_bins.svg" width="567" height="319"></object></span>
 432         </p>
 433 <div>
 434
 435         <script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_b.js"></script>
 436
 437         <div id="49ba2424-47a3-454d-ba72-b46ded00693f" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
 438         <script type="text/javascript">
 439             (function(){
 440                 window.PLOTLYENV={'BASE_URL': 'https://plot.ly'};
 441
 442                 var gd = document.getElementById('49ba2424-47a3-454d-ba72-b46ded00693f')
 443                 var resizeDebounce = null;
 444
 445                 function resizePlot() {
 446                     var bb = gd.getBoundingClientRect();
 447                     Plotly.relayout(gd, {
 448                         width: bb.width,
 449                         height: bb.height
 450                     });
 451                 }
 452
 453
 454                 window.addEventListener('resize', function() {
 455                     if (resizeDebounce) {
 456                         window.clearTimeout(resizeDebounce);
 457                     }
 458                     resizeDebounce = window.setTimeout(resizePlot, 100);
 459                 });
 460
 461
 462
 463                 Plotly.plot(gd,  {
 464                    data: negative_b.data,
 465                    layout: negative_b.layout,
 466                    frames: negative_b.frames,
 467                     config: {"showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A"}
 468                 });
 469
 470            }());
 471         </script>
 472     </div>
 473 <p>
 474
 475       </p>
 476 <p>
 477         For both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> less than zero,
 478         the errors the worst errors are clustered in a "difficult zone"
 479         with <span class="emphasis"><em>b &lt; a</em></span> and <span class="emphasis"><em>z</em></span> &#8776; <span class="emphasis"><em>a</em></span>:
 480       </p>
 481 <p>
 482         <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/hypergeometric_1f1/negative_ab_bins.svg" width="461" height="280"></object></span>
 483         <body>
 484         <script type="text/javascript" src="../../../graphs/hypergeometric_1f1/negative_ab.js"></script>
 485
 486         <div id="2867e84a-7d1d-4110-b28a-fb718dbd65ca" style="width: 800px; height: 600px;" class="plotly-graph-div"></div>
 487         <script type="text/javascript">
 488             (function(){
 489                 window.PLOTLYENV={'BASE_URL': 'https://plot.ly'};
 490
 491                 var gd = document.getElementById('2867e84a-7d1d-4110-b28a-fb718dbd65ca')
 492                 var resizeDebounce = null;
 493
 494                 function resizePlot() {
 495                     var bb = gd.getBoundingClientRect();
 496                     Plotly.relayout(gd, {
 497                         width: bb.width,
 498                         height: bb.height
 499                     });
 500                 }
 501
 502
 503                 window.addEventListener('resize', function() {
 504                     if (resizeDebounce) {
 505                         window.clearTimeout(resizeDebounce);
 506                     }
 507                     resizeDebounce = window.setTimeout(resizePlot, 100);
 508                 });
 509
 510
 511
 512                 Plotly.plot(gd,  {
 513                    data: negative_ab.data,
 514                    layout: negative_ab.layout,
 515                    frames: negative_ab.frames,
 516                     config: {"showLink": true, "linkText": "Export to plot.ly", "mapboxAccessToken": "pk.eyJ1IjoiY2hyaWRkeXAiLCJhIjoiY2lxMnVvdm5iMDA4dnhsbTQ5aHJzcGs0MyJ9.X9o_rzNLNesDxdra4neC_A"}
 517                 });
 518
 519            }());
 520         </script>
 521     </body>
 522
 523       </p>
 524 <h5>
 525 <a name="math_toolkit.hypergeometric.hypergeometric_1f1.h4"></a>
 526         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_1f1.implementation"></a></span><a class="link" href="hypergeometric_1f1.html#math_toolkit.hypergeometric.hypergeometric_1f1.implementation">Implementation</a>
 527       </h5>
 528 <p>
 529         The implementation for this function is remarkably complex; readers will
 530         have to refer to the code for the transitions between methods, as we can
 531         only give an overview here.
 532       </p>
 533 <p>
 534         In almost all cases where <span class="emphasis"><em>z &lt; 0</em></span> we use <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function" target="_top">Kummer's
 535         relation</a> to make <span class="emphasis"><em>z</em></span> positive:
 536       </p>
 537 <div class="blockquote"><blockquote class="blockquote"><p>
 538           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_12.svg" width="189" height="16"></object></span>
 539         </p></blockquote></div>
 540 <p>
 541         The main series representation
 542       </p>
 543 <div class="blockquote"><blockquote class="blockquote"><p>
 544           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_1.svg" width="234" height="38"></object></span>
 545         </p></blockquote></div>
 546 <p>
 547         is used only when
 548       </p>
 549 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 550 <li class="listitem">
 551             we are sure that it is convergent and not subject to excessive cancellation,
 552             and
 553           </li>
 554 <li class="listitem">
 555             there is no other better method available.
 556           </li>
 557 </ul></div>
 558 <p>
 559         The implementation of this series is complicated by the fact that for <span class="emphasis"><em>b</em></span>
 560         &lt; 0 then what appears to be a fully converged series can in fact suddenly
 561         jump back to life again as <span class="emphasis"><em>b</em></span> crosses the origin.
 562       </p>
 563 <p>
 564         A&amp;S 13.3.6 gives
 565       </p>
 566 <div class="blockquote"><blockquote class="blockquote"><p>
 567           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_3.svg" width="614" height="39"></object></span>
 568         </p></blockquote></div>
 569 <p>
 570         which is particularly useful for <span class="emphasis"><em>a &#8773; b</em></span> and <span class="emphasis"><em>z
 571         &gt; 0</em></span>, or <span class="emphasis"><em>a</em></span> &#8810; 1, and <span class="emphasis"><em>z
 572         &lt; 0</em></span>.
 573       </p>
 574 <p>
 575         Then we have Tricomi's approximation (given in simplified form in A&amp;S
 576         13.3.7) <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(7)</a>
 577       </p>
 578 <div class="blockquote"><blockquote class="blockquote"><p>
 579           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_4.svg" width="356" height="38"></object></span>
 580         </p></blockquote></div>
 581 <p>
 582         with
 583       </p>
 584 <div class="blockquote"><blockquote class="blockquote"><p>
 585           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_5.svg" width="390" height="33"></object></span>
 586         </p></blockquote></div>
 587 <p>
 588         and
 589       </p>
 590 <div class="blockquote"><blockquote class="blockquote"><p>
 591           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_6.svg" width="370" height="16"></object></span>
 592         </p></blockquote></div>
 593 <p>
 594         With <span class="emphasis"><em>E<sub>v</sub></em></span> defined as:
 595       </p>
 596 <div class="blockquote"><blockquote class="blockquote"><p>
 597           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_7.svg" width="149" height="86"></object></span>
 598         </p></blockquote></div>
 599 <p>
 600         This approximation is especially effective when <span class="emphasis"><em>a &lt; 0</em></span>.
 601       </p>
 602 <p>
 603         For <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> both large and positive
 604         and somewhat similar in size then an expansion in terms of gamma functions
 605         can be used <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(6)</a>:
 606       </p>
 607 <div class="blockquote"><blockquote class="blockquote"><p>
 608           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_8.svg" width="349" height="43"></object></span>
 609         </p></blockquote></div>
 610 <p>
 611         For <span class="emphasis"><em>z</em></span> large we have the asymptotic expansion:
 612       </p>
 613 <div class="blockquote"><blockquote class="blockquote"><p>
 614           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_9.svg" width="261" height="43"></object></span>
 615         </p></blockquote></div>
 616 <p>
 617         which is a special case of the gamma function expansion above.
 618       </p>
 619 <p>
 620         In the situation where <code class="computeroutput"><span class="identifier">ab</span><span class="special">/</span><span class="identifier">z</span></code> is reasonably
 621         small then Luke's rational approximation is used - this is too complex to
 622         document here, refer either to the code or the original paper <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(3)</a>.
 623       </p>
 624 <p>
 625         The special case <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub>(1, <span class="emphasis"><em>b</em></span>, -<span class="emphasis"><em>z</em></span>)
 626         is treated via Luke's Pade approximation <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(3)</a>.
 627       </p>
 628 <p>
 629         That effectively completes the "direct" methods used, the remaining
 630         areas are treated indirectly via recurrence relations. These require extreme
 631         care in their use, as they often change the direction of stability, or else
 632         are not stable at all. Sometimes this is a well defined and characterized
 633         change in direction: for example with <span class="emphasis"><em>a,b</em></span> and <span class="emphasis"><em>z</em></span>
 634         all positive recurrence on <span class="emphasis"><em>a</em></span> via
 635       </p>
 636 <div class="blockquote"><blockquote class="blockquote"><p>
 637           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_10.svg" width="429" height="16"></object></span>
 638         </p></blockquote></div>
 639 <p>
 640         abruptly changes from stable forwards recursion to stable backwards if <span class="emphasis"><em>2a-b+z</em></span>
 641         changes sign. Thus recurrence on <span class="emphasis"><em>a</em></span>, even when <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub>(<span class="emphasis"><em>a</em></span>+<span class="emphasis"><em>N</em></span>,
 642         <span class="emphasis"><em>b</em></span>, <span class="emphasis"><em>z</em></span>) is strictly increasing, needs
 643         careful handling as <span class="emphasis"><em>a</em></span> &#8594; 0.
 644       </p>
 645 <p>
 646         The transitory nature of the stable recurrence relations is well documented
 647         in the literature, for example <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(10)</a>
 648         gives many examples, including the anomalous behaviour of recurrence on
 649         <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> for large <span class="emphasis"><em>z</em></span>
 650         as first documented by Gauchi <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(12)</a>.
 651         Gauchi also provides the definitive reference on 3-term recurrence relations
 652         in general in <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(11)</a>.
 653       </p>
 654 <p>
 655         Unfortunately, not all recurrence stabilities are so well defined. For example,
 656         when considering <sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, -<span class="emphasis"><em>b</em></span>, <span class="emphasis"><em>z</em></span>)
 657         it would be convenient to use the continued fractions associated with the
 658         recurrence relations to calculate <sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, -<span class="emphasis"><em>b</em></span>,
 659         <span class="emphasis"><em>z</em></span>) / <sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, 1-<span class="emphasis"><em>b</em></span>,
 660         <span class="emphasis"><em>z</em></span>) and then normalize either by iterating forwards on
 661         <span class="emphasis"><em>b</em></span> until <span class="emphasis"><em>b &gt; 0</em></span> and comparison
 662         with a reference value, or by using the Wronskian
 663       </p>
 664 <div class="blockquote"><blockquote class="blockquote"><p>
 665           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_1f1_11.svg" width="564" height="33"></object></span>
 666         </p></blockquote></div>
 667 <p>
 668         which is of course the well known Miller's method <a class="link" href="hypergeometric_refs.html" title="Hypergeometric References">(12)</a>.
 669       </p>
 670 <p>
 671         Unfortunately, stable forwards recursion on <span class="emphasis"><em>b</em></span> is stable
 672         only for <span class="emphasis"><em>|b| &lt;&lt; |z|</em></span>, as <span class="emphasis"><em>|b|</em></span>
 673         increases in magnitude it passes through a region where recursion is unstable
 674         in both directions before eventually becoming stable in the backwards direction
 675         (at least for a while!). This transition is moderated not by a change of
 676         sign in the recurrence coefficients themselves, but by a change in the behaviour
 677         of the <span class="emphasis"><em>values</em></span> of <sub>1</sub>F<sub>1</sub> - from alternating in sign when
 678         <span class="emphasis"><em>|b|</em></span> is small to having the same sign when <span class="emphasis"><em>b</em></span>
 679         is larger. During the actual transition, <sub>1</sub>F<sub>1</sub> will either pass through a zero
 680         or a minima depending on whether b is even or odd (if there is a minima at
 681         <sub>1</sub>F<sub>1</sub>(a, b, z) then there is necessarily a zero at <sub>1</sub>F<sub>1</sub>(a+1, b+1, z) as per
 682         the <a href="https://dlmf.nist.gov/13.3#E15" target="_top">derivative of the function</a>).
 683         As a result the behaviour of the recurrence relations is almost impossible
 684         to reason about in this area, and we are left to using heuristic based approaches
 685         to "guess" which region we are in.
 686       </p>
 687 <p>
 688         In this implementation we use recurrence relations as follows:
 689       </p>
 690 <p>
 691         For <span class="emphasis"><em>a,b,z &gt; 0</em></span> and large we can either:
 692       </p>
 693 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 694 <li class="listitem">
 695             Make <span class="emphasis"><em>0 &lt; a &lt; 1</em></span> and <span class="emphasis"><em>b &gt; z</em></span>
 696             and evaluate the defining series directly, or
 697           </li>
 698 <li class="listitem">
 699             The gamma function approximation has decent convergence and accuracy
 700             for <span class="emphasis"><em>2b - 1 &lt; a &lt; 2b &lt; z</em></span>, so we can move
 701             <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> into this region, or
 702           </li>
 703 <li class="listitem">
 704             We can recurse on <span class="emphasis"><em>b</em></span> alone so that <span class="emphasis"><em>b-1
 705             &lt; a &lt; b</em></span> and use A&amp;S 13.3.6 as long as <span class="emphasis"><em>z</em></span>
 706             is not too large.
 707           </li>
 708 </ul></div>
 709 <p>
 710         For <span class="emphasis"><em>b &lt; 0</em></span> and <span class="emphasis"><em>a</em></span> tiny we would
 711         normally use A&amp;S 13.3.6, but when that is non-convergent for some inputs
 712         we can use the forward recurrence relation on <span class="emphasis"><em>b</em></span> to calculate
 713         the ratio <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, -b, z)/<sub>1</sub>F<sub>1</sub>(a, 1-b, z)</em></span> and then iterate
 714         forwards until <span class="emphasis"><em>b &gt; 0</em></span> and compute a reference value
 715         and normalize (Millers Method). In the same domain when <span class="emphasis"><em>b</em></span>
 716         is near -1 we can use a single backwards recursion on <span class="emphasis"><em>b</em></span>
 717         to compute <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, -b, z)</em></span> from <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, 2-b,
 718         z)</em></span> and <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(<span class="emphasis"><em>a</em></span>, 1-<span class="emphasis"><em>b</em></span>,
 719         <span class="emphasis"><em>z</em></span>)</em></span> even though technically we are recursing
 720         in the unstable direction.
 721       </p>
 722 <p>
 723         For <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-N, b, z)</em></span> and integer <span class="emphasis"><em>N</em></span>
 724         then backwards recursion from <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(0, b, z)</em></span> and <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-1,
 725         b, z)</em></span> works well.
 726       </p>
 727 <p>
 728         For <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> &lt; 0 and if all the
 729         direct methods have failed, then we use various fallbacks:
 730       </p>
 731 <p>
 732         For <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-a, b, z)</em></span> we can use backwards recursion on
 733         <span class="emphasis"><em>a</em></span> as long as <span class="emphasis"><em>b &gt; z</em></span>, otherwise
 734         a more complex scheme is required which starts from <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(-a + N,
 735         b + M, z)</em></span>, and recurses backwards in up to 3 stages: first on
 736         <span class="emphasis"><em>a</em></span>, then on <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 737         together, and finally on <span class="emphasis"><em>b</em></span> alone.
 738       </p>
 739 <p>
 740         For <span class="emphasis"><em>b &lt; 0</em></span> we have no good methods in some domains
 741         (see the unsolved issues above). However in some circumstances we can either
 742         use:
 743       </p>
 744 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 745 <li class="listitem">
 746             3-stage backwards recursion on both <span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>a</em></span>
 747             and <span class="emphasis"><em>b</em></span> and then <span class="emphasis"><em>b</em></span> as above.
 748           </li>
 749 <li class="listitem">
 750             Calculate the ratio <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, b, z) / <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a-1, b-1,
 751             z)</em></span></em></span> via backwards recurrence when z is small, and
 752             then normalize via the Wronskian above (Miller's method).
 753           </li>
 754 <li class="listitem">
 755             Calculate the ratio <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a, b, z) / <span class="emphasis"><em><sub>1</sub>F<sub>1</sub>(a+1, b+1,
 756             z)</em></span></em></span> via forwards recurrence when z is large, and
 757             then normalize by iterating until b &gt; 1 and comparing to a reference
 758             value.
 759           </li>
 760 </ul></div>
 761 <p>
 762         The latter two methods use a lookup table to determine whether inputs are
 763         in either of the domains or neither. Unfortunately the methods are basically
 764         limited to double precision: calculation of the ratios require iteration
 765         <span class="emphasis"><em>towards</em></span> the no-mans-land between the two methods where
 766         iteration is unstable in both directions. As a result, only low-precision
 767         results which require few iterations to calculate the ratio are available.
 768       </p>
 769 <p>
 770         If all else fails, then we use a checked series summation which will raise
 771         an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
 772         if more than half the digits are destroyed by cancellation.
 773       </p>
 774 </div>
 775 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 776 <td align="left"></td>
 777 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 778       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 779       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 780       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 781       Daryle Walker and Xiaogang Zhang<p>
 782         Distributed under the Boost Software License, Version 1.0. (See accompanying
 783         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>)
 784       </p>
 785 </div></td>
 786 </tr></table>
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