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Interpolation"> 9<link rel="prev" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation"> 10<link rel="next" href="vector_barycentric.html" title="Vector-valued Barycentric Rational Interpolation"> 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="whittaker_shannon.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.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="vector_barycentric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.barycentric"></a><a class="link" href="barycentric.html" title="Barycentric Rational Interpolation">Barycentric Rational Interpolation</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.barycentric.h0"></a> 31 <span class="phrase"><a name="math_toolkit.barycentric.synopsis"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.synopsis">Synopsis</a> 32 </h4> 33<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">interpolators</span><span class="special">/</span><span class="identifier">barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34 35<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> 36 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 37 <span class="keyword">class</span> <span class="identifier">barycentric_rational</span> 38 <span class="special">{</span> 39 <span class="keyword">public</span><span class="special">:</span> 40 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">InputIterator1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">InputIterator2</span><span class="special">></span> 41 <span class="identifier">barycentric_rational</span><span class="special">(</span><span class="identifier">InputIterator1</span> <span class="identifier">start_x</span><span class="special">,</span> <span class="identifier">InputIterator1</span> <span class="identifier">end_x</span><span class="special">,</span> <span class="identifier">InputIterator2</span> <span class="identifier">start_y</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">approximation_order</span> <span class="special">=</span> <span class="number">3</span><span class="special">);</span> 42 43 <span class="identifier">barycentric_rational</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&&</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&&</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">order</span> <span class="special">=</span> <span class="number">3</span><span class="special">);</span> 44 45 <span class="identifier">barycentric_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">approximation_order</span> <span class="special">=</span> <span class="number">3</span><span class="special">);</span> 46 47 <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 48 49 <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 50 51 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&&</span> <span class="identifier">return_x</span><span class="special">();</span> 52 53 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&&</span> <span class="identifier">return_y</span><span class="special">();</span> 54 <span class="special">};</span> 55 56<span class="special">}}</span> 57</pre> 58<h4> 59<a name="math_toolkit.barycentric.h1"></a> 60 <span class="phrase"><a name="math_toolkit.barycentric.description"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.description">Description</a> 61 </h4> 62<p> 63 Barycentric rational interpolation is a high-accuracy interpolation method 64 for non-uniformly spaced samples. It requires ��(<span class="emphasis"><em>N</em></span>) time 65 for construction, and ��(<span class="emphasis"><em>N</em></span>) time for each evaluation. Linear 66 time evaluation is not optimal; for instance the cubic B-spline can be evaluated 67 in constant time. However, using the cubic B-spline requires uniformly-spaced 68 samples, which are not always available. 69 </p> 70<p> 71 Use of the class requires a vector of independent variables <code class="computeroutput"><span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>, 72 <code class="computeroutput"><span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span></code>, .... <code class="computeroutput"><span class="identifier">x</span><span class="special">[</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">]</span></code> 73 where <code class="computeroutput"><span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">+</span><span class="number">1</span><span class="special">]</span> <span class="special">></span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span></code>, 74 and a vector of dependent variables <code class="computeroutput"><span class="identifier">y</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>, 75 <code class="computeroutput"><span class="identifier">y</span><span class="special">[</span><span class="number">1</span><span class="special">]</span></code>, ... , <code class="computeroutput"><span class="identifier">y</span><span class="special">[</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">]</span></code>. 76 The call is trivial: 77 </p> 78<pre class="programlisting"><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">x</span><span class="special">(</span><span class="number">500</span><span class="special">);</span> 79<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">y</span><span class="special">(</span><span class="number">500</span><span class="special">);</span> 80<span class="comment">// populate x, y, then:</span> 81<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">barycentric_rational</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">));</span> 82</pre> 83<p> 84 This implicitly calls the constructor with approximation order 3, and hence 85 the accuracy is ��(<span class="emphasis"><em>h</em></span><sup>4</sup>). In general, if you require an approximation 86 order <span class="emphasis"><em>d</em></span>, then the error is ��(<span class="emphasis"><em>h</em></span><sup><span class="emphasis"><em>d</em></span>+1</sup>). 87 A call to the constructor with an explicit approximation order is demonstrated 88 below 89 </p> 90<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">barycentric_rational</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">),</span> <span class="number">5</span><span class="special">);</span> 91</pre> 92<p> 93 To evaluate the interpolant, simply use 94 </p> 95<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">2.3</span><span class="special">;</span> 96<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> 97</pre> 98<p> 99 and to evaluate its derivative use 100 </p> 101<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> 102</pre> 103<p> 104 If you no longer require the interpolant, then you can get your data back: 105 </p> 106<pre class="programlisting"><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">xs</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">.</span><span class="identifier">return_x</span><span class="special">();</span> 107<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">ys</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">.</span><span class="identifier">return_y</span><span class="special">();</span> 108</pre> 109<p> 110 Be aware that once you return your data, the interpolant is <span class="bold"><strong>dead</strong></span>. 111 </p> 112<h4> 113<a name="math_toolkit.barycentric.h2"></a> 114 <span class="phrase"><a name="math_toolkit.barycentric.caveats"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.caveats">Caveats</a> 115 </h4> 116<p> 117 Although this algorithm is robust, it can surprise you. The main way this occurs 118 is if the sample spacing at the endpoints is much larger than the spacing in 119 the center. This is to be expected; all interpolants perform better in the 120 opposite regime, where samples are clustered at the endpoints and somewhat 121 uniformly spaced throughout the center. 122 </p> 123<p> 124 A desirable property of any interpolator <span class="emphasis"><em>f</em></span> is that for 125 all <span class="emphasis"><em>x</em></span><sub>min</sub> ≤ <span class="emphasis"><em>x</em></span> ≤ <span class="emphasis"><em>x</em></span><sub>max</sub>, 126 also <span class="emphasis"><em>y</em></span><sub>min</sub> ≤ <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>) 127 ≤ <span class="emphasis"><em>y</em></span><sub>max</sub>. 128 </p> 129<p> 130 <span class="emphasis"><em>This property does not hold for the barycentric rational interpolator.</em></span> 131 However, unless you deliberately try to antagonize this interpolator (by, for 132 instance, placing the final value far from all the rest), you will probably 133 not fall victim to this problem. 134 </p> 135<p> 136 The reference used for implementation of this algorithm is <a href="https://web.archive.org/save/_embed/http://www.mn.uio.no/math/english/people/aca/michaelf/papers/rational.pdf" target="_top">Barycentric 137 rational interpolation with no poles and a high rate of interpolation</a>, 138 and the evaluation of the derivative is given by <a href="http://www.ams.org/journals/mcom/1986-47-175/S0025-5718-1986-0842136-8/S0025-5718-1986-0842136-8.pdf" target="_top">Some 139 New Aspects of Rational Interpolation</a>. 140 </p> 141<h4> 142<a name="math_toolkit.barycentric.h3"></a> 143 <span class="phrase"><a name="math_toolkit.barycentric.examples"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.examples">Examples</a> 144 </h4> 145<p> 146 This example shows how to use barycentric rational interpolation, using Walter 147 Kohn's classic paper "Solution of the Schrodinger Equation in Periodic 148 Lattices with an Application to Metallic Lithium" In this paper, Kohn 149 needs to repeatedly solve an ODE (the radial Schrodinger equation) given a 150 potential which is only known at non-equally samples data. 151 </p> 152<p> 153 If he'd only had the barycentric rational interpolant of Boost.Math! 154 </p> 155<p> 156 References: Kohn, W., and N. Rostoker. "Solution of the Schrodinger equation 157 in periodic lattices with an application to metallic lithium." Physical 158 Review 94.5 (1954): 1111. 159 </p> 160<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">interpolators</span><span class="special">/</span><span class="identifier">barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 161 162<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> 163<span class="special">{</span> 164 <span class="comment">// The lithium potential is given in Kohn's paper, Table I:</span> 165 <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">r</span><span class="special">(</span><span class="number">45</span><span class="special">);</span> 166 <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">mrV</span><span class="special">(</span><span class="number">45</span><span class="special">);</span> 167 168 <span class="comment">// We'll skip the code for filling the above vectors with data for now...</span> 169 170 <span class="comment">// Now we want to interpolate this potential at any r:</span> 171 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">barycentric_rational</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">mrV</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">size</span><span class="special">());</span> 172 173 <span class="keyword">for</span> <span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="number">8</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> 174 <span class="special">{</span> 175 <span class="keyword">double</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">i</span><span class="special">*</span><span class="number">0.5</span><span class="special">;</span> 176 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"(r, V) = ("</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="special">-</span><span class="identifier">b</span><span class="special">(</span><span class="identifier">r</span><span class="special">)/</span><span class="identifier">r</span> <span class="special"><<</span> <span class="string">")\n"</span><span class="special">;</span> 177 <span class="special">}</span> 178<span class="special">}</span> 179</pre> 180<p> 181 This further example shows how to use the iterator based constructor, and then 182 uses the function object in our root finding algorithms to locate the points 183 where the potential achieves a specific value. 184 </p> 185<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">interpolators</span><span class="special">/</span><span class="identifier">barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 186<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">range</span><span class="special">/</span><span class="identifier">adaptors</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 187<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">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 188 189<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> 190<span class="special">{</span> 191 <span class="comment">// The lithium potential is given in Kohn's paper, Table I.</span> 192 <span class="comment">// (We could equally easily use an unordered_map, a list of tuples or pairs, or a 2-dimensional array).</span> 193 <span class="identifier">std</span><span class="special">::</span><span class="identifier">map</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">r</span><span class="special">;</span> 194 195 <span class="identifier">r</span><span class="special">[</span><span class="number">0.02</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.727</span><span class="special">;</span> 196 <span class="identifier">r</span><span class="special">[</span><span class="number">0.04</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.544</span><span class="special">;</span> 197 <span class="identifier">r</span><span class="special">[</span><span class="number">0.06</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.450</span><span class="special">;</span> 198 <span class="identifier">r</span><span class="special">[</span><span class="number">0.08</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.351</span><span class="special">;</span> 199 <span class="identifier">r</span><span class="special">[</span><span class="number">0.10</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.253</span><span class="special">;</span> 200 <span class="identifier">r</span><span class="special">[</span><span class="number">0.12</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.157</span><span class="special">;</span> 201 <span class="identifier">r</span><span class="special">[</span><span class="number">0.14</span><span class="special">]</span> <span class="special">=</span> <span class="number">5.058</span><span class="special">;</span> 202 <span class="identifier">r</span><span class="special">[</span><span class="number">0.16</span><span class="special">]</span> <span class="special">=</span> <span class="number">4.960</span><span class="special">;</span> 203 <span class="identifier">r</span><span class="special">[</span><span class="number">0.18</span><span class="special">]</span> <span class="special">=</span> <span class="number">4.862</span><span class="special">;</span> 204 <span class="identifier">r</span><span class="special">[</span><span class="number">0.20</span><span class="special">]</span> <span class="special">=</span> <span class="number">4.762</span><span class="special">;</span> 205 <span class="identifier">r</span><span class="special">[</span><span class="number">0.24</span><span class="special">]</span> <span class="special">=</span> <span class="number">4.563</span><span class="special">;</span> 206 <span class="identifier">r</span><span class="special">[</span><span class="number">0.28</span><span class="special">]</span> <span class="special">=</span> <span class="number">4.360</span><span class="special">;</span> 207 <span class="identifier">r</span><span class="special">[</span><span class="number">0.32</span><span class="special">]</span> <span class="special">=</span> <span class="number">4.1584</span><span class="special">;</span> 208 <span class="identifier">r</span><span class="special">[</span><span class="number">0.36</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.9463</span><span class="special">;</span> 209 <span class="identifier">r</span><span class="special">[</span><span class="number">0.40</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.7360</span><span class="special">;</span> 210 <span class="identifier">r</span><span class="special">[</span><span class="number">0.44</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.5429</span><span class="special">;</span> 211 <span class="identifier">r</span><span class="special">[</span><span class="number">0.48</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.3797</span><span class="special">;</span> 212 <span class="identifier">r</span><span class="special">[</span><span class="number">0.52</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.2417</span><span class="special">;</span> 213 <span class="identifier">r</span><span class="special">[</span><span class="number">0.56</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.1209</span><span class="special">;</span> 214 <span class="identifier">r</span><span class="special">[</span><span class="number">0.60</span><span class="special">]</span> <span class="special">=</span> <span class="number">3.0138</span><span class="special">;</span> 215 <span class="identifier">r</span><span class="special">[</span><span class="number">0.68</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.8342</span><span class="special">;</span> 216 <span class="identifier">r</span><span class="special">[</span><span class="number">0.76</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.6881</span><span class="special">;</span> 217 <span class="identifier">r</span><span class="special">[</span><span class="number">0.84</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.5662</span><span class="special">;</span> 218 <span class="identifier">r</span><span class="special">[</span><span class="number">0.92</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.4242</span><span class="special">;</span> 219 <span class="identifier">r</span><span class="special">[</span><span class="number">1.00</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.3766</span><span class="special">;</span> 220 <span class="identifier">r</span><span class="special">[</span><span class="number">1.08</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.3058</span><span class="special">;</span> 221 <span class="identifier">r</span><span class="special">[</span><span class="number">1.16</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.2458</span><span class="special">;</span> 222 <span class="identifier">r</span><span class="special">[</span><span class="number">1.24</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.2035</span><span class="special">;</span> 223 <span class="identifier">r</span><span class="special">[</span><span class="number">1.32</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.1661</span><span class="special">;</span> 224 <span class="identifier">r</span><span class="special">[</span><span class="number">1.40</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.1350</span><span class="special">;</span> 225 <span class="identifier">r</span><span class="special">[</span><span class="number">1.48</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.1090</span><span class="special">;</span> 226 <span class="identifier">r</span><span class="special">[</span><span class="number">1.64</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0697</span><span class="special">;</span> 227 <span class="identifier">r</span><span class="special">[</span><span class="number">1.80</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0466</span><span class="special">;</span> 228 <span class="identifier">r</span><span class="special">[</span><span class="number">1.96</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0325</span><span class="special">;</span> 229 <span class="identifier">r</span><span class="special">[</span><span class="number">2.12</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0288</span><span class="special">;</span> 230 <span class="identifier">r</span><span class="special">[</span><span class="number">2.28</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0292</span><span class="special">;</span> 231 <span class="identifier">r</span><span class="special">[</span><span class="number">2.44</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0228</span><span class="special">;</span> 232 <span class="identifier">r</span><span class="special">[</span><span class="number">2.60</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0124</span><span class="special">;</span> 233 <span class="identifier">r</span><span class="special">[</span><span class="number">2.76</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0065</span><span class="special">;</span> 234 <span class="identifier">r</span><span class="special">[</span><span class="number">2.92</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0031</span><span class="special">;</span> 235 <span class="identifier">r</span><span class="special">[</span><span class="number">3.08</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0015</span><span class="special">;</span> 236 <span class="identifier">r</span><span class="special">[</span><span class="number">3.24</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0008</span><span class="special">;</span> 237 <span class="identifier">r</span><span class="special">[</span><span class="number">3.40</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0004</span><span class="special">;</span> 238 <span class="identifier">r</span><span class="special">[</span><span class="number">3.56</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0002</span><span class="special">;</span> 239 <span class="identifier">r</span><span class="special">[</span><span class="number">3.72</span><span class="special">]</span> <span class="special">=</span> <span class="number">2.0001</span><span class="special">;</span> 240 241 <span class="comment">// Let's discover the absissa that will generate a potential of exactly 3.0,</span> 242 <span class="comment">// start by creating 2 ranges for the x and y values:</span> 243 <span class="keyword">auto</span> <span class="identifier">x_range</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">adaptors</span><span class="special">::</span><span class="identifier">keys</span><span class="special">(</span><span class="identifier">r</span><span class="special">);</span> 244 <span class="keyword">auto</span> <span class="identifier">y_range</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">adaptors</span><span class="special">::</span><span class="identifier">values</span><span class="special">(</span><span class="identifier">r</span><span class="special">);</span> 245 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">barycentric_rational</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">x_range</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">x_range</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">y_range</span><span class="special">.</span><span class="identifier">begin</span><span class="special">());</span> 246 <span class="comment">//</span> 247 <span class="comment">// We'll use a lambda expression to provide the functor to our root finder, since we want</span> 248 <span class="comment">// the abscissa value that yields 3, not zero. We pass the functor b by value to the</span> 249 <span class="comment">// lambda expression since barycentric_rational is trivial to copy.</span> 250 <span class="comment">// Here we're using simple bisection to find the root:</span> 251 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">iterations</span> <span class="special">=</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">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)();</span> 252 <span class="keyword">double</span> <span class="identifier">abscissa_3</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">bisect</span><span class="special">([=](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">-</span> <span class="number">3</span><span class="special">;</span> <span class="special">},</span> <span class="number">0.44</span><span class="special">,</span> <span class="number">1.24</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">eps_tolerance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="identifier">iterations</span><span class="special">).</span><span class="identifier">first</span><span class="special">;</span> 253 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Abscissa value that yields a potential of 3 = "</span> <span class="special"><<</span> <span class="identifier">abscissa_3</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 254 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Root was found in "</span> <span class="special"><<</span> <span class="identifier">iterations</span> <span class="special"><<</span> <span class="string">" iterations."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 255 <span class="comment">//</span> 256 <span class="comment">// However, we have a more efficient root finding algorithm than simple bisection:</span> 257 <span class="identifier">iterations</span> <span class="special">=</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">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)();</span> 258 <span class="identifier">abscissa_3</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">bracket_and_solve_root</span><span class="special">([=](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">-</span> <span class="number">3</span><span class="special">;</span> <span class="special">},</span> <span class="number">0.6</span><span class="special">,</span> <span class="number">1.2</span><span class="special">,</span> <span class="keyword">false</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">eps_tolerance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="identifier">iterations</span><span class="special">).</span><span class="identifier">first</span><span class="special">;</span> 259 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Abscissa value that yields a potential of 3 = "</span> <span class="special"><<</span> <span class="identifier">abscissa_3</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 260 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Root was found in "</span> <span class="special"><<</span> <span class="identifier">iterations</span> <span class="special"><<</span> <span class="string">" iterations."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 261<span class="special">}</span> 262</pre> 263<pre class="programlisting"><span class="identifier">Program</span> <span class="identifier">output</span> <span class="identifier">is</span><span class="special">:</span> 264</pre> 265<pre class="programlisting">Abscissa value that yields a potential of 3 = 0.604728 266Root was found in 54 iterations. 267Abscissa value that yields a potential of 3 = 0.604728 268Root was found in 10 iterations. 269</pre> 270</div> 271<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 272<td align="left"></td> 273<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 274 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 275 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 276 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 277 Daryle Walker and Xiaogang Zhang<p> 278 Distributed under the Boost Software License, Version 1.0. (See accompanying 279 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>) 280 </p> 281</div></td> 282</tr></table> 283<hr> 284<div class="spirit-nav"> 285<a accesskey="p" href="whittaker_shannon.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.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="vector_barycentric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 286</div> 287</body> 288</html> 289
