1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Finding Zeros of Airy Functions</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../airy.html" title="Airy Functions"> 9<link rel="prev" href="bip.html" title="Airy Bi' Function"> 10<link rel="next" href="../ellint.html" title="Elliptic Integrals"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a 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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"><</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">></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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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> 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"><</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> 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 21. 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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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">&);</span> <span class="comment">// Policy to use.</span> 75 76 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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">&);</span> <span class="comment">// Policy to use.</span> 80 81 82<span class="keyword">template</span> <span class="special"><</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> 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">&</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"><</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> 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">&</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"><=</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"><</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">></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"><</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">></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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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> 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 21. 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"><</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">></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">&</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"><</span><span class="keyword">double</span><span class="special">>(</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"><<</span> <span class="string">"boost::math::airy_ai_zero<double>(1) = "</span> <span class="special"><<</span> <span class="identifier">aiz1</span> <span class="special"><<</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"><</span><span class="keyword">double</span><span class="special">>(</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"><<</span> <span class="string">"boost::math::airy_ai_zero<double>(2) = "</span> <span class="special"><<</span> <span class="identifier">aiz2</span> <span class="special"><<</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"><</span><span class="keyword">double</span><span class="special">>(</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"><<</span> <span class="string">"boost::math::airy_bi_zero<double>(3) = "</span> <span class="special"><<</span> <span class="identifier">biz3</span> <span class="special"><<</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"><</span><span class="keyword">double</span><span class="special">></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"><</span><span class="keyword">double</span><span class="special">>(</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"><<</span> <span class="string">"airy_ai_zeros:"</span> <span class="special"><<</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"><</span><span class="keyword">double</span><span class="special">>(</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"><</span><span class="identifier">float_type</span><span class="special">>::</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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span> <span class="special"><<</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"><</span><span class="identifier">float_type</span><span class="special">>(</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"><<</span> <span class="string">"boost::math::airy_bi_zero<float_type>("</span> <span class="special"><<</span> <span class="identifier">m</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</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"><</span><span class="identifier">float_type</span><span class="special">>(</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"><<</span> <span class="string">"boost::math::airy_bi_zero<float_type>("</span> <span class="special"><<</span> <span class="identifier">m</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</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"><</span><span class="identifier">float_type</span><span class="special">>(</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"><<</span> <span class="string">"boost::math::airy_bi_zero<float_type>("</span> <span class="special"><<</span> <span class="identifier">m</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</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"><</span><span class="identifier">float_type</span><span class="special">></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"><</span><span class="identifier">float_type</span><span class="special">>(</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"><<</span> <span class="string">"airy_ai_zeros:"</span> <span class="special"><<</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"><</span><span class="identifier">float_type</span><span class="special">>(</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<double>(1) = -2.33811 404boost::math::airy_ai_zero<double>(2) = -4.08795 405boost::math::airy_bi_zero<double>(3) = -4.83074 406airy_ai_zeros: 407-2.33811 408-4.08795 409-5.52056 410-6.78671 411-7.94413 412 413boost::math::airy_bi_zero<float_type>(1) = -2.3381074104597670384891972524467354406385401456711 414boost::math::airy_bi_zero<float_type>(2) = -4.0879494441309706166369887014573910602247646991085 415boost::math::airy_bi_zero<float_type>(7) = -9.5381943793462388866329885451560196208390720763825 416airy_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&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&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&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 © 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 Råde, 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> 476<div class="spirit-nav"> 477<a accesskey="p" href="bip.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../ellint.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 478</div> 479</body> 480</html> 481