1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Tools For 3-Term Recurrence Relations</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="../internals.html" title="Internal tools"> 9<link rel="prev" href="cf.html" title="Continued Fraction Evaluation"> 10<link rel="next" href="tuples.html" title="Tuples"> 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 href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="cf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.internals.recurrence"></a><a class="link" href="recurrence.html" title="Tools For 3-Term Recurrence Relations">Tools For 3-Term Recurrence 28 Relations</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.internals.recurrence.h0"></a> 32 <span class="phrase"><a name="math_toolkit.internals.recurrence.synopsis"></a></span><a class="link" href="recurrence.html#math_toolkit.internals.recurrence.synopsis">Synopsis</a> 33 </h5> 34<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">tools</span><span class="special">/</span><span class="identifier">recurrence</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 35</pre> 36<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span> 37 38<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 39<span class="identifier">T</span> <span class="identifier">function_ratio_from_backwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span> 40 41<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 42<span class="identifier">T</span> <span class="identifier">function_ratio_from_forwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span> 43 44<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">NextCoefs</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">apply_recurrence_relation_forward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 46 47<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">NextCoefs</span><span class="special">></span> 48<span class="identifier">T</span> <span class="identifier">apply_recurrence_relation_backward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 49 50<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span> 51<span class="keyword">struct</span> <span class="identifier">forward_recurrence_iterator</span><span class="special">;</span> 52 53<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span> 54<span class="keyword">struct</span> <span class="identifier">backward_recurrence_iterator</span><span class="special">;</span> 55 56<span class="special">}}}</span> <span class="comment">// namespaces</span> 57</pre> 58<h5> 59<a name="math_toolkit.internals.recurrence.h1"></a> 60 <span class="phrase"><a name="math_toolkit.internals.recurrence.description"></a></span><a class="link" href="recurrence.html#math_toolkit.internals.recurrence.description">Description</a> 61 </h5> 62<p> 63 All of the tools in this header require a description of the recurrence relation: 64 this takes the form of a functor that returns a tuple containing the 3 coefficients, 65 specifically, given a recurrence relation: 66 </p> 67<div class="blockquote"><blockquote class="blockquote"><p> 68 <span class="inlinemediaobject"><img src="../../../equations/three_term_recurrence.svg"></span> 69 70 </p></blockquote></div> 71<p> 72 And a functor <code class="computeroutput"><span class="identifier">F</span></code> then the 73 expression: 74 </p> 75<div class="blockquote"><blockquote class="blockquote"><p> 76 <span class="serif_italic">F(n);</span> 77 </p></blockquote></div> 78<p> 79 Returns a tuple containing <span class="serif_italic">{ a<sub>n</sub>, b<sub>n</sub>, c<sub>n</sub> }</span>. 80 </p> 81<p> 82 For example, the recurrence relation for the Bessel J and Y functions when 83 written in this form is: 84 </p> 85<p> 86 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_bessel_jy.svg" width="250" height="38"></object></span> 87 </p> 88<p> 89 Therefore, given local variables <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>v</em></span> 90 of type <code class="computeroutput"><span class="keyword">double</span></code> the recurrence 91 relation for Bessel J and Y can be encoded in a lambda expression like this: 92 </p> 93<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">recurrence_functor_jy</span> <span class="special">=</span> <span class="special">[&](</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">x</span><span class="special">,</span> <span class="number">1.0</span><span class="special">);</span> <span class="special">};</span> 94</pre> 95<p> 96 Similarly, the Bessel I and K recurrence relation differs just by the sign 97 of the final term: 98 </p> 99<p> 100 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_bessel_ik.svg" width="244" height="38"></object></span> 101 </p> 102<p> 103 And this could be encoded as: 104 </p> 105<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">recurrence_functor_ik</span> <span class="special">=</span> <span class="special">[&](</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">x</span><span class="special">,</span> <span class="special">-</span><span class="number">1.0</span><span class="special">);</span> <span class="special">};</span> 106</pre> 107<p> 108 The tools are then as follows: 109 </p> 110<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 111<span class="identifier">T</span> <span class="identifier">function_ratio_from_backwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span> 112</pre> 113<p> 114 Given a functor <code class="computeroutput"><span class="identifier">r</span></code> which encodes 115 the recurrence relation for function <code class="computeroutput"><span class="identifier">F</span></code> 116 at some location <span class="emphasis"><em>n</em></span>, then returns the ratio: 117 </p> 118<p> 119 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_backwards_ratio.svg" width="63" height="20"></object></span> 120 </p> 121<p> 122 This calculation is stable only if recurrence is stable in the backwards 123 direction. Further the ratio calculated is for the dominant solution (in 124 the backwards direction) of the recurrence relation, if there are multiple 125 solutions, then there is no guarantee that this will find the one you want 126 or expect. 127 </p> 128<p> 129 Argument <span class="emphasis"><em>factor</em></span> is the tolerance required for convergence 130 of the continued fraction associated with the recurrence relation, and should 131 be no smaller than machine epsilon. Argument <span class="emphasis"><em>max_iter</em></span> 132 sets the maximum number of permitted iterations in the associated continued 133 fraction. 134 </p> 135<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 136<span class="identifier">T</span> <span class="identifier">function_ratio_from_forwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span> 137</pre> 138<p> 139 Given a functor <code class="computeroutput"><span class="identifier">r</span></code> which encodes 140 the recurrence relation for function F at some location <span class="emphasis"><em>n</em></span>, 141 then returns the ratio: 142 </p> 143<p> 144 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_forwards_ratio.svg" width="63" height="20"></object></span> 145 </p> 146<p> 147 This calculation is stable only if recurrence is stable in the forwards direction. 148 Further the ratio calculated is for the dominant solution (in the forwards 149 direction) of the recurrence relation, if there are multiple solutions, then 150 there is no guarantee that this will find the one you want or expect. 151 </p> 152<p> 153 Argument <span class="emphasis"><em>factor</em></span> is the tolerance required for convergence 154 of the continued fraction associated with the recurrence relation, and should 155 be no smaller than machine epsilon. Argument <span class="emphasis"><em>max_iter</em></span> 156 sets the maximum number of permitted iterations in the associated continued 157 fraction. 158 </p> 159<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">NextCoefs</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 160<span class="identifier">T</span> <span class="identifier">apply_recurrence_relation_forward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 161</pre> 162<p> 163 Applies a recurrence relation in a stable forward direction, starting with 164 the values F<sub>n-1</sub> and F<sub>n</sub>. 165 </p> 166<div class="variablelist"> 167<p class="title"><b></b></p> 168<dl class="variablelist"> 169<dt><span class="term">get_coefs</span></dt> 170<dd><p> 171 Functor that returns the coefficients of the recurrence relation. The 172 coefficients should be centered on position <span class="emphasis"><em>second</em></span>. 173 </p></dd> 174<dt><span class="term">number_of_steps</span></dt> 175<dd><p> 176 The number of steps to apply the recurrence relation onwards from 177 <span class="emphasis"><em>second</em></span>. 178 </p></dd> 179<dt><span class="term">first</span></dt> 180<dd><p> 181 The value of F<sub>n-1</sub> 182 </p></dd> 183<dt><span class="term">second</span></dt> 184<dd><p> 185 The value of F<sub>n</sub> 186 </p></dd> 187<dt><span class="term">log_scaling</span></dt> 188<dd><p> 189 When provided, the recurrence relations may be rescaled internally 190 to avoid over/underflow issues. The result should be multiplied by 191 <code class="computeroutput"><span class="identifier">exp</span><span class="special">(*</span><span class="identifier">log_scaling</span><span class="special">)</span></code> 192 to get the true value of the result. 193 </p></dd> 194<dt><span class="term">previous</span></dt> 195<dd><p> 196 When provided, is set to the value of F<sub>n + number_of_steps - 1</sub> 197 </p></dd> 198</dl> 199</div> 200<p> 201 Returns F<sub>n + number_of_steps</sub>. 202 </p> 203<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">NextCoefs</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 204<span class="identifier">T</span> <span class="identifier">apply_recurrence_relation_backward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 205</pre> 206<p> 207 Applies a recurrence relation in a stable backward direction, starting with 208 the values F<sub>n+1</sub> and F<sub>n</sub>. 209 </p> 210<div class="variablelist"> 211<p class="title"><b></b></p> 212<dl class="variablelist"> 213<dt><span class="term">get_coefs</span></dt> 214<dd><p> 215 Functor that returns the coefficients of the recurrence relation. The 216 coefficients should be centered on position <span class="emphasis"><em>second</em></span>. 217 </p></dd> 218<dt><span class="term">number_of_steps</span></dt> 219<dd><p> 220 The number of steps to apply the recurrence relation backwards from 221 <span class="emphasis"><em>second</em></span>. 222 </p></dd> 223<dt><span class="term">first</span></dt> 224<dd><p> 225 The value of F<sub>n+1</sub> 226 </p></dd> 227<dt><span class="term">second</span></dt> 228<dd><p> 229 The value of F<sub>n</sub> 230 </p></dd> 231<dt><span class="term">log_scaling</span></dt> 232<dd><p> 233 When provided, the recurrence relations may be rescaled internally 234 to avoid over/underflow issues. The result should be multiplied by 235 <code class="computeroutput"><span class="identifier">exp</span><span class="special">(*</span><span class="identifier">log_scaling</span><span class="special">)</span></code> 236 to get the true value of the result. 237 </p></dd> 238<dt><span class="term">previous</span></dt> 239<dd><p> 240 When provided, is set to the value of F<sub>n - number_of_steps + 1</sub> 241 </p></dd> 242</dl> 243</div> 244<p> 245 Returns F<sub>n - number_of_steps</sub>. 246 </p> 247<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span> 248<span class="keyword">struct</span> <span class="identifier">forward_recurrence_iterator</span> 249<span class="special">{</span> 250 <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">remove_reference</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Recurrence</span><span class="special">&>()(</span><span class="number">0</span><span class="special">)))>::</span><span class="identifier">type</span> <span class="identifier">value_type</span><span class="special">;</span> 251 252 <span class="identifier">forward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n_minus_1</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span> 253 <span class="identifier">forward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span> 254 <span class="comment">/* Operators omitted for clarity */</span> 255<span class="special">};</span> 256</pre> 257<p> 258 Type <code class="computeroutput"><span class="identifier">forward_recurrence_iterator</span></code> 259 defines a forward-iterator for a recurrence relation stable in the forward 260 direction. The constructors take the recurrence relation, plus either one 261 or two values: if only one value is provided, then the second is computed 262 by using the recurrence relation to calculate the function ratio. 263 </p> 264<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span> 265<span class="keyword">struct</span> <span class="identifier">backward_recurrence_iterator</span> 266<span class="special">{</span> 267 <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">remove_reference</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Recurrence</span><span class="special">&>()(</span><span class="number">0</span><span class="special">)))>::</span><span class="identifier">type</span> <span class="identifier">value_type</span><span class="special">;</span> 268 269 <span class="identifier">backward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n_plus_1</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span> 270 <span class="identifier">backward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span> 271 <span class="comment">/* Operators omitted for clarity */</span> 272<span class="special">};</span> 273</pre> 274<p> 275 Type <code class="computeroutput"><span class="identifier">backward_recurrence_iterator</span></code> 276 defines a forward-iterator for a recurrence relation stable in the backward 277 direction. The constructors take the recurrence relation, plus either one 278 or two values: if only one value is provided, then the second is computed 279 by using the recurrence relation to calculate the function ratio. 280 </p> 281<p> 282 Note that <span class="emphasis"><em>incrementing</em></span> this iterator moves the value 283 returned successively to F<sub>n-1</sub>, F<sub>n-2</sub> etc. 284 </p> 285</div> 286<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 287<td align="left"></td> 288<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 289 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 290 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 291 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 292 Daryle Walker and Xiaogang Zhang<p> 293 Distributed under the Boost Software License, Version 1.0. (See accompanying 294 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>) 295 </p> 296</div></td> 297</tr></table> 298<hr> 299<div class="spirit-nav"> 300<a accesskey="p" href="cf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 301</div> 302</body> 303</html> 304
