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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.airy.airy_root"></a><a class="link" href="airy_root.html" title="Finding Zeros of Airy Functions">Finding Zeros of Airy Functions</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.airy.airy_root.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.airy.airy_root.synopsis"></a></span><a class="link" href="airy_root.html#math_toolkit.airy.airy_root.synopsis">Synopsis</a>
  32       </h5>
  33 <p>
  34         <code class="computeroutput"><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">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
  35       </p>
  36 <p>
  37         Functions for obtaining both a single zero or root of the Airy functions,
  38         and placing multiple zeros into a container like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
  39         by providing an output iterator.
  40       </p>
  41 <p>
  42         The signature of the single value functions are:
  43       </p>
  44 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  45 <span class="identifier">T</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span>
  46          <span class="keyword">int</span> <span class="identifier">m</span><span class="special">);</span>         <span class="comment">// 1-based index of zero.</span>
  47
  48 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  49 <span class="identifier">T</span> <span class="identifier">airy_bi_zero</span><span class="special">(</span>
  50          <span class="keyword">int</span> <span class="identifier">m</span><span class="special">);</span>         <span class="comment">// 1-based index of zero.</span>
  51 </pre>
  52 <p>
  53         and for multiple zeros:
  54       </p>
  55 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
  56 <span class="identifier">OutputIterator</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span>
  57                      <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>           <span class="comment">// 1-based index of first zero.</span>
  58                      <span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span>  <span class="comment">// How many zeros to generate.</span>
  59                      <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>    <span class="comment">// Destination for zeros.</span>
  60
  61 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
  62 <span class="identifier">OutputIterator</span> <span class="identifier">airy_bi_zero</span><span class="special">(</span>
  63                      <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>           <span class="comment">// 1-based index of zero.</span>
  64                      <span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span>  <span class="comment">// How many zeros to generate</span>
  65                      <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>    <span class="comment">// Destination for zeros.</span>
  66 </pre>
  67 <p>
  68         There are also versions which allow control of the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
  69         for error handling and precision.
  70       </p>
  71 <pre class="programlisting"> <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  72  <span class="identifier">T</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span>
  73           <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span>          <span class="comment">// 1-based index of zero.</span>
  74           <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;);</span> <span class="comment">// Policy to use.</span>
  75
  76  <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  77  <span class="identifier">T</span> <span class="identifier">airy_bi_zero</span><span class="special">(</span>
  78           <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span>          <span class="comment">// 1-based index of zero.</span>
  79           <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;);</span> <span class="comment">// Policy to use.</span>
  80
  81
  82 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
  83 <span class="identifier">OutputIterator</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span>
  84                      <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>           <span class="comment">// 1-based index of first zero.</span>
  85                      <span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span>  <span class="comment">// How many zeros to generate.</span>
  86                      <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span>     <span class="comment">// Destination for zeros.</span>
  87                      <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>        <span class="comment">// Policy to use.</span>
  88
  89 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
  90 <span class="identifier">OutputIterator</span> <span class="identifier">airy_bi_zero</span><span class="special">(</span>
  91                      <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>           <span class="comment">// 1-based index of zero.</span>
  92                      <span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span>  <span class="comment">// How many zeros to generate.</span>
  93                      <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span>     <span class="comment">// Destination for zeros.</span>
  94                      <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>        <span class="comment">// Policy to use.</span>
  95 </pre>
  96 <h5>
  97 <a name="math_toolkit.airy.airy_root.h1"></a>
  98         <span class="phrase"><a name="math_toolkit.airy.airy_root.description"></a></span><a class="link" href="airy_root.html#math_toolkit.airy.airy_root.description">Description</a>
  99       </h5>
 100 <p>
 101         The Airy Ai and Bi functions have an infinite number of zeros on the negative
 102         real axis. The real zeros on the negative real axis can be found by solving
 103         for the roots of
 104       </p>
 105 <div class="blockquote"><blockquote class="blockquote"><p>
 106           <span class="emphasis"><em>Ai(x<sub>m</sub>) = 0</em></span>
 107         </p></blockquote></div>
 108 <div class="blockquote"><blockquote class="blockquote"><p>
 109           <span class="emphasis"><em>Bi(y<sub>m</sub>) = 0</em></span>
 110         </p></blockquote></div>
 111 <p>
 112         Here, <span class="emphasis"><em>x<sub>m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root
 113         of the Airy Ai function, and <span class="emphasis"><em>y<sub>m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span>
 114         root of the Airy Bi function.
 115       </p>
 116 <p>
 117         The zeros or roots (values of <code class="computeroutput"><span class="identifier">x</span></code>
 118         where the function crosses the horizontal <code class="computeroutput"><span class="identifier">y</span>
 119         <span class="special">=</span> <span class="number">0</span></code>
 120         axis) of the Airy Ai and Bi functions are computed by two functions, <code class="computeroutput"><span class="identifier">airy_ai_zero</span></code> and <code class="computeroutput"><span class="identifier">airy_bi_zero</span></code>.
 121       </p>
 122 <p>
 123         In each case the index or rank of the zero returned is 1-based, which is
 124         to say:
 125       </p>
 126 <pre class="programlisting"><span class="identifier">airy_ai_zero</span><span class="special">(</span><span class="number">1</span><span class="special">);</span>
 127 </pre>
 128 <p>
 129         returns the first zero of Ai.
 130       </p>
 131 <p>
 132         Passing an <code class="computeroutput"><span class="identifier">start_index</span> <span class="special">&lt;=</span>
 133         <span class="number">0</span></code> results in a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 134         being raised.
 135       </p>
 136 <p>
 137         The first few zeros returned by these functions have approximate values as
 138         follows:
 139       </p>
 140 <div class="informaltable"><table class="table">
 141 <colgroup>
 142 <col>
 143 <col>
 144 <col>
 145 </colgroup>
 146 <thead><tr>
 147 <th>
 148                 <p>
 149                   m
 150                 </p>
 151               </th>
 152 <th>
 153                 <p>
 154                   Ai
 155                 </p>
 156               </th>
 157 <th>
 158                 <p>
 159                   Bi
 160                 </p>
 161               </th>
 162 </tr></thead>
 163 <tbody>
 164 <tr>
 165 <td>
 166                 <p>
 167                   1
 168                 </p>
 169               </td>
 170 <td>
 171                 <p>
 172                   -2.33811...
 173                 </p>
 174               </td>
 175 <td>
 176                 <p>
 177                   -1.17371...
 178                 </p>
 179               </td>
 180 </tr>
 181 <tr>
 182 <td>
 183                 <p>
 184                   2
 185                 </p>
 186               </td>
 187 <td>
 188                 <p>
 189                   -4.08795...
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   -3.27109...
 195                 </p>
 196               </td>
 197 </tr>
 198 <tr>
 199 <td>
 200                 <p>
 201                   3
 202                 </p>
 203               </td>
 204 <td>
 205                 <p>
 206                   -5.52056...
 207                 </p>
 208               </td>
 209 <td>
 210                 <p>
 211                   -4.83074...
 212                 </p>
 213               </td>
 214 </tr>
 215 <tr>
 216 <td>
 217                 <p>
 218                   4
 219                 </p>
 220               </td>
 221 <td>
 222                 <p>
 223                   -6.78671...
 224                 </p>
 225               </td>
 226 <td>
 227                 <p>
 228                   -6.16985...
 229                 </p>
 230               </td>
 231 </tr>
 232 <tr>
 233 <td>
 234                 <p>
 235                   5
 236                 </p>
 237               </td>
 238 <td>
 239                 <p>
 240                   -7.94413...
 241                 </p>
 242               </td>
 243 <td>
 244                 <p>
 245                   -7.37676...
 246                 </p>
 247               </td>
 248 </tr>
 249 <tr>
 250 <td>
 251                 <p>
 252                   6
 253                 </p>
 254               </td>
 255 <td>
 256                 <p>
 257                   -9.02265...
 258                 </p>
 259               </td>
 260 <td>
 261                 <p>
 262                   -8.49195...
 263                 </p>
 264               </td>
 265 </tr>
 266 </tbody>
 267 </table></div>
 268 <div class="blockquote"><blockquote class="blockquote"><p>
 269           <span class="inlinemediaobject"><img src="../../../graphs/airy_zeros.svg" align="middle"></span>
 270
 271         </p></blockquote></div>
 272 <h5>
 273 <a name="math_toolkit.airy.airy_root.h2"></a>
 274         <span class="phrase"><a name="math_toolkit.airy.airy_root.examples_of_finding_airy_zeros"></a></span><a class="link" href="airy_root.html#math_toolkit.airy.airy_root.examples_of_finding_airy_zeros">Examples
 275         of finding Airy Zeros</a>
 276       </h5>
 277 <p>
 278         This example demonstrates calculating zeros of the Airy functions. It also
 279         shows how Boost.Math and Boost.Multiprecision can be combined to provide
 280         a many decimal digit precision. For 50 decimal digit precision we need to
 281         include
 282       </p>
 283 <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">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 284 </pre>
 285 <p>
 286         and a <code class="computeroutput"><span class="keyword">typedef</span></code> for <code class="computeroutput"><span class="identifier">float_type</span></code> may be convenient (allowing
 287         a quick switch to re-compute at built-in <code class="computeroutput"><span class="keyword">double</span></code>
 288         or other precision)
 289       </p>
 290 <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">float_type</span><span class="special">;</span>
 291 </pre>
 292 <p>
 293         To use the functions for finding zeros of the functions we need
 294       </p>
 295 <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">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 296 </pre>
 297 <p>
 298         This example shows obtaining both a single zero of the Airy functions, and
 299         then placing multiple zeros into a container like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
 300         by providing an iterator. The signature of the single-value Airy Ai function
 301         is:
 302       </p>
 303 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 304 <span class="identifier">T</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">);</span> <span class="comment">// 1-based index of the zero.</span>
 305 </pre>
 306 <p>
 307         The signature of multiple zeros Airy Ai function is:
 308       </p>
 309 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">&gt;</span>
 310 <span class="identifier">OutputIterator</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span>
 311                          <span class="keyword">unsigned</span> <span class="identifier">start_index</span><span class="special">,</span> <span class="comment">// 1-based index of the zero.</span>
 312                          <span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span> <span class="comment">// How many zeros to generate.</span>
 313                          <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
 314 </pre>
 315 <p>
 316         There are also versions which allows control of the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a>
 317         for error handling and precision.
 318       </p>
 319 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
 320 <span class="identifier">OutputIterator</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span>
 321                          <span class="keyword">unsigned</span> <span class="identifier">start_index</span><span class="special">,</span> <span class="comment">// 1-based index of the zero.</span>
 322                          <span class="keyword">unsigned</span> <span class="identifier">number_of_zeros</span><span class="special">,</span> <span class="comment">// How many zeros to generate.</span>
 323                          <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span> <span class="comment">// Destination for zeros.</span>
 324                          <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>  <span class="comment">// Policy to use.</span>
 325 </pre>
 326 <div class="tip"><table border="0" summary="Tip">
 327 <tr>
 328 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
 329 <th align="left">Tip</th>
 330 </tr>
 331 <tr><td align="left" valign="top"><p>
 332           It is always wise to place code using Boost.Math inside <code class="computeroutput"><span class="keyword">try</span><span class="char">'n'</span><span class="keyword">catch</span></code> blocks;
 333           this will ensure that helpful error messages are shown when exceptional
 334           conditions arise.
 335         </p></td></tr>
 336 </table></div>
 337 <p>
 338         First, evaluate a single Airy zero.
 339       </p>
 340 <p>
 341         The precision is controlled by the template parameter <code class="computeroutput"><span class="identifier">T</span></code>,
 342         so this example has <code class="computeroutput"><span class="keyword">double</span></code> precision,
 343         at least 15 but up to 17 decimal digits (for the common 64-bit double).
 344       </p>
 345 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">aiz1</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_ai_zero</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">1</span><span class="special">);</span>
 346 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::airy_ai_zero&lt;double&gt;(1) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">aiz1</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 347 <span class="keyword">double</span> <span class="identifier">aiz2</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_ai_zero</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">2</span><span class="special">);</span>
 348 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::airy_ai_zero&lt;double&gt;(2) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">aiz2</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 349 <span class="keyword">double</span> <span class="identifier">biz3</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_bi_zero</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">3</span><span class="special">);</span>
 350 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::airy_bi_zero&lt;double&gt;(3) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">biz3</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 351 </pre>
 352 <p>
 353         Other versions of <code class="computeroutput"><span class="identifier">airy_ai_zero</span></code>
 354         and <code class="computeroutput"><span class="identifier">airy_bi_zero</span></code> allow calculation
 355         of multiple zeros with one call, placing the results in a container, often
 356         <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>. For example, generate and display
 357         the first five <code class="computeroutput"><span class="keyword">double</span></code> roots
 358         <a href="http://mathworld.wolfram.com/AiryFunctionZeros.html" target="_top">Wolfram
 359         Airy Functions Zeros</a>.
 360       </p>
 361 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="keyword">int</span> <span class="identifier">n_roots</span> <span class="special">=</span> <span class="number">5U</span><span class="special">;</span>
 362 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">roots</span><span class="special">;</span>
 363 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_ai_zero</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">1U</span><span class="special">,</span> <span class="identifier">n_roots</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">roots</span><span class="special">));</span>
 364 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"airy_ai_zeros:"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 365 <span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span>
 366           <span class="identifier">roots</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span>
 367           <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">"\n"</span><span class="special">));</span>
 368 </pre>
 369 <p>
 370         The first few real roots of Ai(x) are approximately -2.33811, -4.08795, -5.52056,
 371         -6.7867144, -7.94413, -9.02265 ...
 372       </p>
 373 <p>
 374         Or we can use Boost.Multiprecision to generate 50 decimal digit roots.
 375       </p>
 376 <p>
 377         We set the precision of the output stream, and show trailing zeros to display
 378         a fixed 50 decimal digits.
 379       </p>
 380 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span> <span class="comment">// float_type has 50 decimal digits.</span>
 381 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Show trailing zeros too.</span>
 382
 383 <span class="keyword">unsigned</span> <span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">1U</span><span class="special">;</span>
 384 <span class="identifier">float_type</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_ai_zero</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="number">1U</span><span class="special">);</span> <span class="comment">// 1st root.</span>
 385 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::airy_bi_zero&lt;float_type&gt;("</span> <span class="special">&lt;&lt;</span> <span class="identifier">m</span> <span class="special">&lt;&lt;</span> <span class="string">")  = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 386 <span class="identifier">m</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>
 387 <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_ai_zero</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="number">2U</span><span class="special">);</span> <span class="comment">// 2nd root.</span>
 388 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::airy_bi_zero&lt;float_type&gt;("</span> <span class="special">&lt;&lt;</span> <span class="identifier">m</span> <span class="special">&lt;&lt;</span> <span class="string">")  = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 389 <span class="identifier">m</span> <span class="special">=</span> <span class="number">7U</span><span class="special">;</span>
 390 <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_bi_zero</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="number">7U</span><span class="special">);</span> <span class="comment">// 7th root.</span>
 391 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"boost::math::airy_bi_zero&lt;float_type&gt;("</span> <span class="special">&lt;&lt;</span> <span class="identifier">m</span> <span class="special">&lt;&lt;</span> <span class="string">")  = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 392
 393 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;</span> <span class="identifier">zeros</span><span class="special">;</span>
 394 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">airy_ai_zero</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="number">1U</span><span class="special">,</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">zeros</span><span class="special">));</span>
 395 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"airy_ai_zeros:"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 396 <span class="comment">// Print the roots to the output stream.</span>
 397 <span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span><span class="identifier">zeros</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">zeros</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span>
 398           <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="identifier">float_type</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">"\n"</span><span class="special">));</span>
 399 </pre>
 400 <p>
 401         Produces the program output:
 402       </p>
 403 <pre class="programlisting">boost::math::airy_ai_zero&lt;double&gt;(1) = -2.33811
 404 boost::math::airy_ai_zero&lt;double&gt;(2) = -4.08795
 405 boost::math::airy_bi_zero&lt;double&gt;(3) = -4.83074
 406 airy_ai_zeros:
 407 -2.33811
 408 -4.08795
 409 -5.52056
 410 -6.78671
 411 -7.94413
 412
 413 boost::math::airy_bi_zero&lt;float_type&gt;(1)  = -2.3381074104597670384891972524467354406385401456711
 414 boost::math::airy_bi_zero&lt;float_type&gt;(2)  = -4.0879494441309706166369887014573910602247646991085
 415 boost::math::airy_bi_zero&lt;float_type&gt;(7)  = -9.5381943793462388866329885451560196208390720763825
 416 airy_ai_zeros:
 417 -2.3381074104597670384891972524467354406385401456711
 418 -4.0879494441309706166369887014573910602247646991085
 419 -5.5205598280955510591298555129312935737972142806175
 420 </pre>
 421 <p>
 422         The full code (and output) for this example is at <a href="../../../../example/airy_zeros_example.cpp" target="_top">airy_zeros_example.cpp</a>,
 423       </p>
 424 <h4>
 425 <a name="math_toolkit.airy.airy_root.h3"></a>
 426         <span class="phrase"><a name="math_toolkit.airy.airy_root.implementation"></a></span><a class="link" href="airy_root.html#math_toolkit.airy.airy_root.implementation">Implementation</a>
 427       </h4>
 428 <p>
 429         Given the following function (A&amp;S 10.4.105):
 430       </p>
 431 <div class="blockquote"><blockquote class="blockquote"><p>
 432           <span class="inlinemediaobject"><img src="../../../equations/airy_zero_1.svg"></span>
 433
 434         </p></blockquote></div>
 435 <p>
 436         Then an initial estimate for the n<sup>th</sup> zero a<sub>n</sub> of Ai is given by (A&amp;S 10.4.94):
 437       </p>
 438 <div class="blockquote"><blockquote class="blockquote"><p>
 439           <span class="inlinemediaobject"><img src="../../../equations/airy_zero_2.svg"></span>
 440
 441         </p></blockquote></div>
 442 <p>
 443         and an initial estimate for the n<sup>th</sup> zero b<sub>n</sub> of Bi is given by (A&amp;S 10.4.98):
 444       </p>
 445 <div class="blockquote"><blockquote class="blockquote"><p>
 446           <span class="inlinemediaobject"><img src="../../../equations/airy_zero_3.svg"></span>
 447
 448         </p></blockquote></div>
 449 <p>
 450         Thereafter the roots are refined using Newton iteration.
 451       </p>
 452 <h4>
 453 <a name="math_toolkit.airy.airy_root.h4"></a>
 454         <span class="phrase"><a name="math_toolkit.airy.airy_root.testing"></a></span><a class="link" href="airy_root.html#math_toolkit.airy.airy_root.testing">Testing</a>
 455       </h4>
 456 <p>
 457         The precision of evaluation of zeros was tested at 50 decimal digits using
 458         <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code> and found
 459         identical with spot values computed by <a href="http://www.wolframalpha.com/" target="_top">Wolfram
 460         Alpha</a>.
 461       </p>
 462 </div>
 463 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 464 <td align="left"></td>
 465 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 466       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 467       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 468       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 469       Daryle Walker and Xiaogang Zhang<p>
 470         Distributed under the Boost Software License, Version 1.0. (See accompanying
 471         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>)
 472       </p>
 473 </div></td>
 474 </tr></table>
 475 <hr>
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