1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>PCHIP interpolation</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="../interpolation.html" title="Chapter 12. Interpolation"> 9<link rel="prev" href="makima.html" title="Modified Akima interpolation"> 10<link rel="next" href="quintic_hermite.html" title="Quintic Hermite 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="makima.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="quintic_hermite.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.pchip"></a><a class="link" href="pchip.html" title="PCHIP interpolation">PCHIP interpolation</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.pchip.h0"></a> 31 <span class="phrase"><a name="math_toolkit.pchip.synopsis"></a></span><a class="link" href="pchip.html#math_toolkit.pchip.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">pchip</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="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span> <span class="special">{</span> 36 37<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span> 38<span class="keyword">class</span> <span class="identifier">pchip</span> 39<span class="special">{</span> 40<span class="keyword">public</span><span class="special">:</span> 41 42 <span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span> 43 44 <span class="identifier">pchip</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">abscissas</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">ordinates</span><span class="special">,</span> 45 <span class="identifier">Real</span> <span class="identifier">left_endpoint_derivative</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">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> 46 <span class="identifier">Real</span> <span class="identifier">right_endpoint_derivative</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">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">());</span> 47 48 <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> 49 50 <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> 51 52 <span class="keyword">void</span> <span class="identifier">push_back</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">y</span><span class="special">);</span> 53 54 <span class="keyword">friend</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="keyword">operator</span><span class="special"><<(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">pchip</span> <span class="special">&</span> <span class="identifier">m</span><span class="special">);</span> 55<span class="special">};</span> 56 57<span class="special">}</span> <span class="comment">// namespaces</span> 58</pre> 59<h4> 60<a name="math_toolkit.pchip.h1"></a> 61 <span class="phrase"><a name="math_toolkit.pchip.pchip_interpolation"></a></span><a class="link" href="pchip.html#math_toolkit.pchip.pchip_interpolation">PCHIP 62 Interpolation</a> 63 </h4> 64<p> 65 The PCHIP interpolant takes non-equispaced data and interpolates between them 66 via cubic Hermite polynomials whose slopes are chosen so that the resulting 67 interpolant is monotonic; see <a href="https://doi.org/10.1137/0717021" target="_top">Fritsch 68 and Carlson</a> for details. The interpolant is <span class="emphasis"><em>C</em></span><sup>1</sup> and 69 evaluation has ��(log(<span class="emphasis"><em>N</em></span>)) complexity. An example usage is 70 as follows: 71 </p> 72<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">1</span><span class="special">,</span> <span class="number">5</span><span class="special">,</span> <span class="number">9</span> <span class="special">,</span> <span class="number">12</span><span class="special">};</span> 73<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">8</span><span class="special">,</span><span class="number">17</span><span class="special">,</span> <span class="number">4</span><span class="special">,</span> <span class="special">-</span><span class="number">3</span><span class="special">};</span> 74<span class="keyword">using</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">pchip</span><span class="special">;</span> 75<span class="keyword">auto</span> <span class="identifier">spline</span> <span class="special">=</span> <span class="identifier">pchip</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> 76<span class="comment">// evaluate at a point:</span> 77<span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span> 78<span class="comment">// evaluate derivative at a point:</span> 79<span class="keyword">double</span> <span class="identifier">zprime</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span> 80</pre> 81<p> 82 Periodically, it is helpful to see what data the interpolator has, and the 83 slopes it has chosen. This can be achieved via 84 </p> 85<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">spline</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> 86</pre> 87<p> 88 Note that the interpolator is pimpl'd, so that copying the class is cheap, 89 and hence it can be shared between threads. (The call operator and <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span><span class="special">()</span></code> 90 are threadsafe; <code class="computeroutput"><span class="identifier">push_back</span></code> is 91 not.) 92 </p> 93<p> 94 This interpolant can be updated in constant time. Hence we can use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span></code> 95 to do real-time interpolation: 96 </p> 97<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">circular_buffer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 98<span class="special">...</span> 99<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">};</span> 100<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_y</span><span class="special">{</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">};</span> 101<span class="keyword">auto</span> <span class="identifier">circular_pchip</span> <span class="special">=</span> <span class="identifier">pchip</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">initial_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">initial_y</span><span class="special">));</span> 102<span class="comment">// interpolate via call operation:</span> 103<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_pchip</span><span class="special">(</span><span class="number">3.5</span><span class="special">);</span> 104<span class="comment">// add new data:</span> 105<span class="identifier">circular_pchip</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">8</span><span class="special">);</span> 106<span class="comment">// interpolate at 4.5:</span> 107<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_pchip</span><span class="special">(</span><span class="number">4.5</span><span class="special">);</span> 108</pre> 109<p> 110 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/pchip.svg"></object></span> 111 </p> 112<h4> 113<a name="math_toolkit.pchip.h2"></a> 114 <span class="phrase"><a name="math_toolkit.pchip.complexity_and_performance"></a></span><a class="link" href="pchip.html#math_toolkit.pchip.complexity_and_performance">Complexity 115 and Performance</a> 116 </h4> 117<p> 118 This interpolator chooses the slopes and forwards data to the cubic Hermite 119 interpolator, so the performance is stated in the documentation for <code class="computeroutput"><span class="identifier">cubic_hermite</span><span class="special">.</span><span class="identifier">hpp</span></code>. 120 </p> 121</div> 122<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 123<td align="left"></td> 124<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 125 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 126 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 127 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 128 Daryle Walker and Xiaogang Zhang<p> 129 Distributed under the Boost Software License, Version 1.0. (See accompanying 130 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>) 131 </p> 132</div></td> 133</tr></table> 134<hr> 135<div class="spirit-nav"> 136<a accesskey="p" href="makima.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="quintic_hermite.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 137</div> 138</body> 139</html> 140
