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24 | </div> | |
25 | <div class="section"> | |
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"><</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 15. 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 | <p> | |
106 |   <span class="emphasis"><em>Ai(x<sub>m</sub>) = 0</em></span> | |
107 | </p> | |
108 | <p> | |
109 |   <span class="emphasis"><em>Bi(y<sub>m</sub>) = 0</em></span> | |
110 | </p> | |
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 | <p> | |
269 | <span class="inlinemediaobject"><img src="../../../graphs/airy_zeros.svg" align="middle"></span> | |
270 | </p> | |
271 | <h5> | |
272 | <a name="math_toolkit.airy.airy_root.h2"></a> | |
273 | <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 | |
274 | of finding Airy Zeros</a> | |
275 | </h5> | |
276 | <p> | |
277 | This example demonstrates calculating zeros of the Airy functions. It also | |
278 | shows how Boost.Math and Boost.Multiprecision can be combined to provide | |
279 | a many decimal digit precision. For 50 decimal digit precision we need to | |
280 | include | |
281 | </p> | |
282 | <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> | |
283 | </pre> | |
284 | <p> | |
285 | 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 | |
286 | a quick switch to re-compute at built-in <code class="computeroutput"><span class="keyword">double</span></code> | |
287 | or other precision) | |
288 | </p> | |
289 | <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> | |
290 | </pre> | |
291 | <p> | |
292 | To use the functions for finding zeros of the functions we need | |
293 | </p> | |
294 | <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> | |
295 | </pre> | |
296 | <p> | |
297 | This example shows obtaining both a single zero of the Airy functions, and | |
298 | 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> | |
299 | by providing an iterator. The signature of the single-value Airy Ai function | |
300 | is: | |
301 | </p> | |
302 | <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> | |
303 | <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> | |
304 | </pre> | |
305 | <p> | |
306 | The signature of multiple zeros Airy Ai function is: | |
307 | </p> | |
308 | <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> | |
309 | <span class="identifier">OutputIterator</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span> | |
310 | <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> | |
311 | <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> | |
312 | <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span> | |
313 | </pre> | |
314 | <p> | |
315 | There are also versions which allows control of the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policies</a> | |
316 | for error handling and precision. | |
317 | </p> | |
318 | <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> | |
319 | <span class="identifier">OutputIterator</span> <span class="identifier">airy_ai_zero</span><span class="special">(</span> | |
320 | <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> | |
321 | <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> | |
322 | <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span> <span class="comment">// Destination for zeros.</span> | |
323 | <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> | |
324 | </pre> | |
325 | <div class="tip"><table border="0" summary="Tip"> | |
326 | <tr> | |
327 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td> | |
328 | <th align="left">Tip</th> | |
329 | </tr> | |
330 | <tr><td align="left" valign="top"><p> | |
331 | 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; | |
332 | this will ensure that helpful error messages are shown when exceptional | |
333 | conditions arise. | |
334 | </p></td></tr> | |
335 | </table></div> | |
336 | <p> | |
337 | First, evaluate a single Airy zero. | |
338 | </p> | |
339 | <p> | |
340 | The precision is controlled by the template parameter <code class="computeroutput"><span class="identifier">T</span></code>, | |
341 | so this example has <code class="computeroutput"><span class="keyword">double</span></code> precision, | |
342 | at least 15 but up to 17 decimal digits (for the common 64-bit double). | |
343 | </p> | |
344 | <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> | |
345 | <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> | |
346 | <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> | |
347 | <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> | |
348 | <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> | |
349 | <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> | |
350 | </pre> | |
351 | <p> | |
352 | Other versions of <code class="computeroutput"><span class="identifier">airy_ai_zero</span></code> | |
353 | and <code class="computeroutput"><span class="identifier">airy_bi_zero</span></code> allow calculation | |
354 | of multiple zeros with one call, placing the results in a container, often | |
355 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>. For example, generate and display | |
356 | the first five <code class="computeroutput"><span class="keyword">double</span></code> roots | |
357 | <a href="http://mathworld.wolfram.com/AiryFunctionZeros.html" target="_top">Wolfram | |
358 | Airy Functions Zeros</a>. | |
359 | </p> | |
360 | <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> | |
361 | <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> | |
362 | <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> | |
363 | <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> | |
364 | <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> | |
365 | <span class="identifier">roots</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> | |
366 | <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> | |
367 | </pre> | |
368 | <p> | |
369 | The first few real roots of Ai(x) are approximately -2.33811, -4.08795, -5.52056, | |
370 | -6.7867144, -7.94413, -9.02265 ... | |
371 | </p> | |
372 | <p> | |
373 | Or we can use Boost.Multiprecision to generate 50 decimal digit roots. | |
374 | </p> | |
375 | <p> | |
376 | We set the precision of the output stream, and show trailing zeros to display | |
377 | a fixed 50 decimal digits. | |
378 | </p> | |
379 | <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> | |
380 | <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> | |
381 | ||
382 | <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> | |
383 | <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> | |
384 | <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> | |
385 | <span class="identifier">m</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> | |
386 | <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> | |
387 | <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> | |
388 | <span class="identifier">m</span> <span class="special">=</span> <span class="number">7U</span><span class="special">;</span> | |
389 | <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> | |
390 | <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> | |
391 | ||
392 | <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> | |
393 | <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> | |
394 | <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> | |
395 | <span class="comment">// Print the roots to the output stream.</span> | |
396 | <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> | |
397 | <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> | |
398 | </pre> | |
399 | <p> | |
400 | Produces the program output: | |
401 | </p> | |
402 | <pre class="programlisting">boost::math::airy_ai_zero<double>(1) = -2.33811 | |
403 | boost::math::airy_ai_zero<double>(2) = -4.08795 | |
404 | boost::math::airy_bi_zero<double>(3) = -4.83074 | |
405 | airy_ai_zeros: | |
406 | -2.33811 | |
407 | -4.08795 | |
408 | -5.52056 | |
409 | -6.78671 | |
410 | -7.94413 | |
411 | ||
412 | boost::math::airy_bi_zero<float_type>(1) = -2.3381074104597670384891972524467354406385401456711 | |
413 | boost::math::airy_bi_zero<float_type>(2) = -4.0879494441309706166369887014573910602247646991085 | |
414 | boost::math::airy_bi_zero<float_type>(7) = -9.5381943793462388866329885451560196208390720763825 | |
415 | airy_ai_zeros: | |
416 | -2.3381074104597670384891972524467354406385401456711 | |
417 | -4.0879494441309706166369887014573910602247646991085 | |
418 | -5.5205598280955510591298555129312935737972142806175 | |
419 | </pre> | |
420 | <p> | |
421 | The full code (and output) for this example is at <a href="../../../../example/airy_zeros_example.cpp" target="_top">airy_zeros_example.cpp</a>, | |
422 | </p> | |
423 | <h4> | |
424 | <a name="math_toolkit.airy.airy_root.h3"></a> | |
425 | <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> | |
426 | </h4> | |
427 | <p> | |
428 | Given the following function (A&S 10.4.105): | |
429 | </p> | |
430 | <p> | |
431 | <span class="inlinemediaobject"><img src="../../../equations/airy_zero_1.svg"></span> | |
432 | </p> | |
433 | <p> | |
434 | 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): | |
435 | </p> | |
436 | <p> | |
437 | <span class="inlinemediaobject"><img src="../../../equations/airy_zero_2.svg"></span> | |
438 | </p> | |
439 | <p> | |
440 | 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): | |
441 | </p> | |
442 | <p> | |
443 | <span class="inlinemediaobject"><img src="../../../equations/airy_zero_3.svg"></span> | |
444 | </p> | |
445 | <p> | |
446 | Thereafter the roots are refined using Newton iteration. | |
447 | </p> | |
448 | <h4> | |
449 | <a name="math_toolkit.airy.airy_root.h4"></a> | |
450 | <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> | |
451 | </h4> | |
452 | <p> | |
453 | The precision of evaluation of zeros was tested at 50 decimal digits using | |
454 | <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code> and found | |
455 | identical with spot values computed by <a href="http://www.wolframalpha.com/" target="_top">Wolfram | |
456 | Alpha</a>. | |
457 | </p> | |
458 | </div> | |
459 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
460 | <td align="left"></td> | |
461 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
462 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
463 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
464 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
465 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
466 | 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>) | |
467 | </p> | |
468 | </div></td> | |
469 | </tr></table> | |
470 | <hr> | |
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