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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.barycentric"></a><a class="link" href="barycentric.html" title="Barycentric Rational Interpolation">Barycentric Rational Interpolation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.barycentric.h0"></a>
      <span class="phrase"><a name="math_toolkit.barycentric.synopsis"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.synopsis">Synopsis</a>
    </h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<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">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
    <span class="keyword">class</span> <span class="identifier">barycentric_rational</span>
    <span class="special">{</span>
    <span class="keyword">public</span><span class="special">:</span>
        <span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
        <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>

        <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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;&amp;</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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;&amp;</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>

        <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>

        <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>

        <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>

        <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;&amp;</span> <span class="identifier">return_x</span><span class="special">();</span>

        <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;&amp;</span> <span class="identifier">return_y</span><span class="special">();</span>
    <span class="special">};</span>

<span class="special">}}</span>
</pre>
<h4>
<a name="math_toolkit.barycentric.h1"></a>
      <span class="phrase"><a name="math_toolkit.barycentric.description"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.description">Description</a>
    </h4>
<p>
      Barycentric rational interpolation is a high-accuracy interpolation method
      for non-uniformly spaced samples. It requires &#119926;(<span class="emphasis"><em>N</em></span>) time
      for construction, and &#119926;(<span class="emphasis"><em>N</em></span>) time for each evaluation. Linear
      time evaluation is not optimal; for instance the cubic B-spline can be evaluated
      in constant time. However, using the cubic B-spline requires uniformly-spaced
      samples, which are not always available.
    </p>
<p>
      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>,
      <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>
      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">&gt;</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span></code>,
      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>,
      <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>.
      The call is trivial:
    </p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">y</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
<span class="comment">// populate x, y, then:</span>
<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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>
</pre>
<p>
      This implicitly calls the constructor with approximation order 3, and hence
      the accuracy is &#119926;(<span class="emphasis"><em>h</em></span><sup>4</sup>). In general, if you require an approximation
      order <span class="emphasis"><em>d</em></span>, then the error is &#119926;(<span class="emphasis"><em>h</em></span><sup><span class="emphasis"><em>d</em></span>+1</sup>).
      A call to the constructor with an explicit approximation order is demonstrated
      below
    </p>
<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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>
</pre>
<p>
      To evaluate the interpolant, simply use
    </p>
<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>
<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>
</pre>
<p>
      and to evaluate its derivative use
    </p>
<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>
</pre>
<p>
      If you no longer require the interpolant, then you can get your data back:
    </p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">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>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">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>
</pre>
<p>
      Be aware that once you return your data, the interpolant is <span class="bold"><strong>dead</strong></span>.
    </p>
<h4>
<a name="math_toolkit.barycentric.h2"></a>
      <span class="phrase"><a name="math_toolkit.barycentric.caveats"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.caveats">Caveats</a>
    </h4>
<p>
      Although this algorithm is robust, it can surprise you. The main way this occurs
      is if the sample spacing at the endpoints is much larger than the spacing in
      the center. This is to be expected; all interpolants perform better in the
      opposite regime, where samples are clustered at the endpoints and somewhat
      uniformly spaced throughout the center.
    </p>
<p>
      A desirable property of any interpolator <span class="emphasis"><em>f</em></span> is that for
      all <span class="emphasis"><em>x</em></span><sub>min</sub> &#8804; <span class="emphasis"><em>x</em></span> &#8804; <span class="emphasis"><em>x</em></span><sub>max</sub>,
      also <span class="emphasis"><em>y</em></span><sub>min</sub> &#8804; <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>)
      &#8804; <span class="emphasis"><em>y</em></span><sub>max</sub>.
    </p>
<p>
      <span class="emphasis"><em>This property does not hold for the barycentric rational interpolator.</em></span>
      However, unless you deliberately try to antagonize this interpolator (by, for
      instance, placing the final value far from all the rest), you will probably
      not fall victim to this problem.
    </p>
<p>
      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
      rational interpolation with no poles and a high rate of interpolation</a>,
      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
      New Aspects of Rational Interpolation</a>.
    </p>
<h4>
<a name="math_toolkit.barycentric.h3"></a>
      <span class="phrase"><a name="math_toolkit.barycentric.examples"></a></span><a class="link" href="barycentric.html#math_toolkit.barycentric.examples">Examples</a>
    </h4>
<p>
      This example shows how to use barycentric rational interpolation, using Walter
      Kohn's classic paper "Solution of the Schrodinger Equation in Periodic
      Lattices with an Application to Metallic Lithium" In this paper, Kohn
      needs to repeatedly solve an ODE (the radial Schrodinger equation) given a
      potential which is only known at non-equally samples data.
    </p>
<p>
      If he'd only had the barycentric rational interpolant of Boost.Math!
    </p>
<p>
      References: Kohn, W., and N. Rostoker. "Solution of the Schrodinger equation
      in periodic lattices with an application to metallic lithium." Physical
      Review 94.5 (1954): 1111.
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
    <span class="comment">// The lithium potential is given in Kohn's paper, Table I:</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">r</span><span class="special">(</span><span class="number">45</span><span class="special">);</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">mrV</span><span class="special">(</span><span class="number">45</span><span class="special">);</span>

    <span class="comment">// We'll skip the code for filling the above vectors with data for now...</span>

    <span class="comment">// Now we want to interpolate this potential at any r:</span>
    <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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>

    <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">&lt;</span> <span class="number">8</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
    <span class="special">{</span>
        <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>
        <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span>  <span class="string">"(r, V) = ("</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="string">", "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">")\n"</span><span class="special">;</span>
    <span class="special">}</span>
<span class="special">}</span>
</pre>
<p>
      This further example shows how to use the iterator based constructor, and then
      uses the function object in our root finding algorithms to locate the points
      where the potential achieves a specific value.
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
    <span class="comment">// The lithium potential is given in Kohn's paper, Table I.</span>
    <span class="comment">// (We could equally easily use an unordered_map, a list of tuples or pairs, or a 2-dimentional array).</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">map</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">r</span><span class="special">;</span>

    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>
    <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>

    <span class="comment">// Let's discover the absissa that will generate a potential of exactly 3.0,</span>
    <span class="comment">// start by creating 2 ranges for the x and y values:</span>
    <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>
    <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>
    <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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>
    <span class="comment">//</span>
    <span class="comment">// We'll use a lamda expression to provide the functor to our root finder, since we want</span>
    <span class="comment">// the abscissa value that yields 3, not zero.  We pass the functor b by value to the</span>
    <span class="comment">// lambda expression since barycentric_rational is trivial to copy.</span>
    <span class="comment">// Here we're using simple bisection to find the root:</span>
    <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">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&gt;::</span><span class="identifier">max</span><span class="special">)();</span>
    <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;(),</span> <span class="identifier">iterations</span><span class="special">).</span><span class="identifier">first</span><span class="special">;</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Abscissa value that yields a potential of 3 = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">abscissa_3</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Root was found in "</span> <span class="special">&lt;&lt;</span> <span class="identifier">iterations</span> <span class="special">&lt;&lt;</span> <span class="string">" iterations."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    <span class="comment">//</span>
    <span class="comment">// However, we have a more efficient root finding algorithm than simple bisection:</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">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&gt;::</span><span class="identifier">max</span><span class="special">)();</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">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">&lt;</span><span class="keyword">double</span><span class="special">&gt;(),</span> <span class="identifier">iterations</span><span class="special">).</span><span class="identifier">first</span><span class="special">;</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Abscissa value that yields a potential of 3 = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">abscissa_3</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Root was found in "</span> <span class="special">&lt;&lt;</span> <span class="identifier">iterations</span> <span class="special">&lt;&lt;</span> <span class="string">" iterations."</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<pre class="programlisting"><span class="identifier">Program</span> <span class="identifier">output</span> <span class="identifier">is</span><span class="special">:</span>
</pre>
<pre class="programlisting">Abscissa value that yields a potential of 3 = 0.604728
Root was found in 54 iterations.
Abscissa value that yields a potential of 3 = 0.604728
Root was found in 10 iterations.
</pre>
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      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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      R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
      Daryle Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        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>)
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