1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Vector-valued Barycentric Rational 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="barycentric.html" title="Barycentric Rational Interpolation"> 10<link rel="next" href="catmull_rom.html" title="Catmull-Rom Splines"> 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="barycentric.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="catmull_rom.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.vector_barycentric"></a><a class="link" href="vector_barycentric.html" title="Vector-valued Barycentric Rational Interpolation">Vector-valued Barycentric 28 Rational Interpolation</a> 29</h2></div></div></div> 30<h4> 31<a name="math_toolkit.vector_barycentric.h0"></a> 32 <span class="phrase"><a name="math_toolkit.vector_barycentric.synopsis"></a></span><a class="link" href="vector_barycentric.html#math_toolkit.vector_barycentric.synopsis">Synopsis</a> 33 </h4> 34<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">vector_barycentric_rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 35 36<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> 37 38<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">TimeContainer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">SpaceContainer</span><span class="special">></span> 39<span class="keyword">class</span> <span class="identifier">vector_barycentric_rational</span> 40<span class="special">{</span> 41<span class="keyword">public</span><span class="special">:</span> 42 <span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">TimeContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span> 43 <span class="keyword">using</span> <span class="identifier">Point</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">SpaceContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span> 44 <span class="identifier">vector_barycentric_rational</span><span class="special">(</span><span class="identifier">TimeContainer</span><span class="special">&&</span> <span class="identifier">times</span><span class="special">,</span> <span class="identifier">SpaceContainer</span><span class="special">&&</span> <span class="identifier">points</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> 45 46 <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Point</span><span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 47 48 <span class="identifier">Point</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 49 50 <span class="keyword">void</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Point</span><span class="special">&</span> <span class="identifier">dxdt</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 51 52 <span class="identifier">Point</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">);</span> 53 54 <span class="keyword">void</span> <span class="identifier">eval_with_prime</span><span class="special">(</span><span class="identifier">Point</span><span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Point</span><span class="special">&</span> <span class="identifier">dxdt</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 55 56 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Point</span><span class="special">,</span> <span class="identifier">Point</span><span class="special">></span> <span class="identifier">eval_with_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span> 57<span class="special">};</span> 58 59<span class="special">}}</span> 60</pre> 61<h4> 62<a name="math_toolkit.vector_barycentric.h1"></a> 63 <span class="phrase"><a name="math_toolkit.vector_barycentric.description"></a></span><a class="link" href="vector_barycentric.html#math_toolkit.vector_barycentric.description">Description</a> 64 </h4> 65<p> 66 The <span class="emphasis"><em>n</em></span> dimensional vector-valued barycentric rational interpolator 67 is exactly the same as <span class="emphasis"><em>n</em></span> scalar-valued barycentric rational 68 interpolators. This is provided primarily for convenience and a slight improvement 69 in efficiency over using <span class="emphasis"><em>n</em></span> different rational interpolators 70 and combining their results. 71 </p> 72<p> 73 Use of the class requires a <code class="computeroutput"><span class="identifier">Point</span></code>-type 74 which has size known at compile-time. These requirements are satisfied by (for 75 example) <code class="computeroutput"><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span></code>s and <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">N</span><span class="special">></span></code> classes. The call to the constructor computes 76 the weights: 77 </p> 78<pre class="programlisting"><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">vector_barycentric_rational</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">t</span><span class="special">(</span><span class="number">100</span><span class="special">);</span> 80<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span><span class="special">></span> <span class="identifier">y</span><span class="special">(</span><span class="number">100</span><span class="special">);</span> 81<span class="comment">// initialize t and y . . .</span> 82<span class="identifier">vector_barycentric_rational</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">t</span><span class="special">),</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">y</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">t</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> 83</pre> 84<p> 85 To evaluate the interpolant, use 86 </p> 87<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">2.3</span><span class="special">;</span> 88<span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span> 89</pre> 90<p> 91 If you want to populate a vector passed into the interpolant, rather than get 92 it returned, that syntax is supported: 93 </p> 94<pre class="programlisting"><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector2d</span> <span class="identifier">y</span><span class="special">;</span> 95<span class="identifier">interpolant</span><span class="special">(</span><span class="identifier">y</span><span class="special">,</span> <span class="identifier">t</span><span class="special">);</span> 96</pre> 97<p> 98 We tested this with <code class="computeroutput"><span class="identifier">Eigen</span><span class="special">::</span><span class="identifier">Vector</span></code>s and found no performance benefit, 99 but other <code class="computeroutput"><span class="identifier">Point</span></code>-types might 100 not be the same. 101 </p> 102<p> 103 To evaluate the derivative of the interpolant use 104 </p> 105<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">y</span><span class="special">,</span> <span class="identifier">y_prime</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">interpolant</span><span class="special">.</span><span class="identifier">eval_with_prime</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> 106</pre> 107<p> 108 Computation of the derivative requires evaluation, so if you can try to use 109 both values at once. 110 </p> 111</div> 112<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 113<td align="left"></td> 114<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 115 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 116 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 117 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 118 Daryle Walker and Xiaogang Zhang<p> 119 Distributed under the Boost Software License, Version 1.0. (See accompanying 120 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>) 121 </p> 122</div></td> 123</tr></table> 124<hr> 125<div class="spirit-nav"> 126<a accesskey="p" href="barycentric.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="catmull_rom.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 127</div> 128</body> 129</html> 130
