1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Modified Akima 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="cubic_hermite.html" title="Cubic Hermite interpolation"> 10<link rel="next" href="pchip.html" title="PCHIP 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="cubic_hermite.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="pchip.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.makima"></a><a class="link" href="makima.html" title="Modified Akima interpolation">Modified Akima interpolation</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.makima.h0"></a> 31 <span class="phrase"><a name="math_toolkit.makima.synopsis"></a></span><a class="link" href="makima.html#math_toolkit.makima.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">makima</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">makima</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">makima</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">makima</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.makima.h1"></a> 61 <span class="phrase"><a name="math_toolkit.makima.modified_akima_interpolation"></a></span><a class="link" href="makima.html#math_toolkit.makima.modified_akima_interpolation">Modified 62 Akima Interpolation</a> 63 </h4> 64<p> 65 The modified Akima interpolant takes non-equispaced data and interpolates between 66 them via cubic Hermite polynomials whose slopes are chosen by a modification 67 of a geometric construction proposed by <a href="https://doi.org/10.1145/321607.321609" target="_top">Akima</a>. 68 The modification is given by <a href="https://blogs.mathworks.com/cleve/2019/04/29/makima-piecewise-cubic-interpolation/" target="_top">Cosmin 69 Ionita</a> and agrees with Matlab's version. The interpolant is <span class="emphasis"><em>C</em></span><sup>1</sup> and 70 evaluation has ��(log(<span class="emphasis"><em>N</em></span>)) complexity. This is faster than 71 barycentric rational interpolation, but also less smooth. An example usage 72 is as follows: 73 </p> 74<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> 75<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> 76<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">makima</span><span class="special">;</span> 77<span class="keyword">auto</span> <span class="identifier">spline</span> <span class="special">=</span> <span class="identifier">makima</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> 78<span class="comment">// evaluate at a point:</span> 79<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> 80<span class="comment">// evaluate derivative at a point:</span> 81<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> 82</pre> 83<p> 84 Periodically, it is helpful to see what data the interpolator has, and the 85 slopes it has chosen. This can be achieved via 86 </p> 87<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> 88</pre> 89<p> 90 Note that the interpolator is pimpl'd, so that copying the class is cheap, 91 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> 92 are threadsafe.) 93 </p> 94<p> 95 One unique aspect of this interpolator is that it can be updated in constant 96 time. Hence we can use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span></code> 97 to do real-time interpolation: 98 </p> 99<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> 100<span class="special">...</span> 101<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> 102<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> 103<span class="keyword">auto</span> <span class="identifier">circular_akima</span> <span class="special">=</span> <span class="identifier">makima</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> 104<span class="comment">// interpolate via call operation:</span> 105<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_akima</span><span class="special">(</span><span class="number">3.5</span><span class="special">);</span> 106<span class="comment">// add new data:</span> 107<span class="identifier">circular_akima</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> 108<span class="comment">// interpolat at 4.5:</span> 109<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_akima</span><span class="special">(</span><span class="number">4.5</span><span class="special">);</span> 110</pre> 111<p> 112 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/makima_vs_cubic_b.svg"></object></span> 113 </p> 114<p> 115 The modified Akima spline compared to the cubic <span class="emphasis"><em>B</em></span>-spline. 116 The modified Akima spline oscillates less than the cubic spline, but has less 117 smoothness and is not exact on quadratic polynomials. 118 </p> 119<h4> 120<a name="math_toolkit.makima.h2"></a> 121 <span class="phrase"><a name="math_toolkit.makima.complexity_and_performance"></a></span><a class="link" href="makima.html#math_toolkit.makima.complexity_and_performance">Complexity 122 and Performance</a> 123 </h4> 124<p> 125 The complexity and performance is identical to that of the cubic Hermite interpolator, 126 since this object simply constructs derivatives and forwards the data to <code class="computeroutput"><span class="identifier">cubic_hermite</span><span class="special">.</span><span class="identifier">hpp</span></code>. 127 </p> 128</div> 129<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 130<td align="left"></td> 131<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 132 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 133 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 134 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 135 Daryle Walker and Xiaogang Zhang<p> 136 Distributed under the Boost Software License, Version 1.0. (See accompanying 137 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>) 138 </p> 139</div></td> 140</tr></table> 141<hr> 142<div class="spirit-nav"> 143<a accesskey="p" href="cubic_hermite.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="pchip.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 144</div> 145</body> 146</html> 147
