1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Norms</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="../vector_functionals.html" title="Chapter 7. Vector Functionals - Norms"> 9<link rel="prev" href="../vector_functionals.html" title="Chapter 7. Vector Functionals - Norms"> 10<link rel="next" href="../special.html" title="Chapter 8. Special Functions"> 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="../vector_functionals.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../vector_functionals.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="../special.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.norms"></a><a class="link" href="norms.html" title="Norms">Norms</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.norms.h0"></a> 31 <span class="phrase"><a name="math_toolkit.norms.synopsis"></a></span><a class="link" href="norms.html#math_toolkit.norms.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">tools</span><span class="special">/</span><span class="identifier">norms</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> <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> 36 37 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 38 <span class="keyword">auto</span> <span class="identifier">l0_pseudo_norm</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span> 39 40 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 41 <span class="keyword">auto</span> <span class="identifier">l0_pseudo_norm</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span> 42 43 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 44 <span class="identifier">size_t</span> <span class="identifier">hamming_distance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">first2</span><span class="special">);</span> 45 46 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 47 <span class="identifier">size_t</span> <span class="identifier">hamming_distance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">);</span> 48 49 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 50 <span class="keyword">auto</span> <span class="identifier">l1_norm</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span> 51 52 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 53 <span class="keyword">auto</span> <span class="identifier">l1_norm</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span> 54 55 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 56 <span class="keyword">auto</span> <span class="identifier">l1_distance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v2</span><span class="special">);</span> 57 58 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 59 <span class="keyword">auto</span> <span class="identifier">l1_distance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">first2</span><span class="special">);</span> 60 61 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 62 <span class="keyword">auto</span> <span class="identifier">l2_norm</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span> 63 64 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 65 <span class="keyword">auto</span> <span class="identifier">l2_norm</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span> 66 67 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 68 <span class="keyword">auto</span> <span class="identifier">l2_distance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v2</span><span class="special">);</span> 69 70 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 71 <span class="keyword">auto</span> <span class="identifier">l2_distance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">first2</span><span class="special">);</span> 72 73 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 74 <span class="keyword">auto</span> <span class="identifier">sup_norm</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span> 75 76 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 77 <span class="keyword">auto</span> <span class="identifier">sup_norm</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span> 78 79 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 80 <span class="keyword">auto</span> <span class="identifier">sup_distance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v2</span><span class="special">);</span> 81 82 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 83 <span class="keyword">auto</span> <span class="identifier">sup_distance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">first2</span><span class="special">);</span> 84 85 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 86 <span class="keyword">auto</span> <span class="identifier">lp_norm</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">p</span><span class="special">);</span> 87 88 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 89 <span class="keyword">auto</span> <span class="identifier">lp_norm</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">p</span><span class="special">);</span> 90 91 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 92 <span class="keyword">auto</span> <span class="identifier">lp_distance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v2</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">p</span><span class="special">);</span> 93 94 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 95 <span class="keyword">auto</span> <span class="identifier">lp_distance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">first2</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">p</span><span class="special">);</span> 96 97 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span> 98 <span class="keyword">auto</span> <span class="identifier">total_variation</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span> 99 100 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span> 101 <span class="keyword">auto</span> <span class="identifier">total_variation</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span> 102 103<span class="special">}}}</span> 104</pre> 105<h4> 106<a name="math_toolkit.norms.h1"></a> 107 <span class="phrase"><a name="math_toolkit.norms.description"></a></span><a class="link" href="norms.html#math_toolkit.norms.description">Description</a> 108 </h4> 109<p> 110 The file <code class="computeroutput"><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">norms</span><span class="special">.</span><span class="identifier">hpp</span></code> is a set of facilities for computing 111 scalar values traditionally useful in numerical analysis from vectors. 112 </p> 113<p> 114 Our examples use <code class="computeroutput"><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></code> to 115 hold the data, but this not required. In general, you can store your data in 116 an Eigen array, an Armadillo vector, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>, 117 and for many of the routines, a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">forward_list</span></code>. 118 These routines are usable in float, double, long double, and Boost.Multiprecision 119 precision, as well as their complex extensions whenever the computation is 120 well-defined. Integral datatypes are supported for most routines. 121 </p> 122<h4> 123<a name="math_toolkit.norms.h2"></a> 124 <span class="phrase"><a name="math_toolkit.norms.u2113_super_u221e_norm"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_u221e_norm">ℓ<sup>∞</sup> norm</a> 125 </h4> 126<p> 127 Computes the supremum norm of a dataset: 128 </p> 129<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">v</span><span class="special">{-</span><span class="number">3</span><span class="special">,</span> <span class="number">2</span><span class="special">,</span> <span class="number">1</span><span class="special">};</span> 130<span class="keyword">double</span> <span class="identifier">sup</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">sup_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">cbegin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">cend</span><span class="special">());</span> 131<span class="comment">// sup = 3</span> 132 133<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">>></span> <span class="identifier">v</span><span class="special">{{</span><span class="number">0</span><span class="special">,</span> <span class="special">-</span><span class="number">8</span><span class="special">},</span> <span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">},</span> <span class="special">{-</span><span class="number">3</span><span class="special">,</span><span class="number">2</span><span class="special">}};</span> 134<span class="comment">// Range call:</span> 135<span class="keyword">double</span> <span class="identifier">sup</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">sup_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span> 136<span class="comment">// sup = 8</span> 137</pre> 138<p> 139 Supports real, integral, and complex arithmetic. Container must be forward 140 iterable and is not modified. 141 </p> 142<h4> 143<a name="math_toolkit.norms.h3"></a> 144 <span class="phrase"><a name="math_toolkit.norms.u2113_super_u221e_distance"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_u221e_distance">ℓ<sup>∞</sup> distance</a> 145 </h4> 146<p> 147 Computes the supremum norm distance between two vectors: 148 </p> 149<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">v</span><span class="special">{-</span><span class="number">3</span><span class="special">,</span> <span class="number">2</span><span class="special">,</span> <span class="number">1</span><span class="special">};</span> 150<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">w</span><span class="special">{</span><span class="number">6</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span><span class="special">,</span> <span class="number">1</span><span class="special">};</span> 151<span class="keyword">double</span> <span class="identifier">sup</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">sup_distance</span><span class="special">(</span><span class="identifier">w</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span> 152<span class="comment">// sup = 9</span> 153</pre> 154<p> 155 Supports real, integral, and complex arithmetic. Container must be forward 156 iterable and is not modified. If the input it integral, the output is a double 157 precision float. 158 </p> 159<h4> 160<a name="math_toolkit.norms.h4"></a> 161 <span class="phrase"><a name="math_toolkit.norms.u2113_super_p_norm"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_p_norm">ℓ<sup><span class="emphasis"><em>p</em></span></sup> norm</a> 162 </h4> 163<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">v</span><span class="special">{-</span><span class="number">8</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">};</span> 164<span class="keyword">double</span> <span class="identifier">sup</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">lp_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">cbegin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">cend</span><span class="special">(),</span> <span class="number">7</span><span class="special">);</span> 165<span class="comment">// sup = 8</span> 166 167<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">>></span> <span class="identifier">v</span><span class="special">{{</span><span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,-</span><span class="number">1</span><span class="special">}};</span> 168<span class="keyword">double</span> <span class="identifier">sup</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">lp_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">cbegin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">cend</span><span class="special">(),</span> <span class="number">3</span><span class="special">);</span> 169<span class="comment">// sup = cbrt(3)</span> 170</pre> 171<p> 172 Supports both real, integral, and complex arithmetic. If the input is integral, 173 the output is a double precision float. The container must be forward iterable 174 and the contents are not modified. 175 </p> 176<p> 177 Only supports integral <span class="emphasis"><em>p</em></span> for two reasons: The computation 178 is much slower for real <span class="emphasis"><em>p</em></span>, and the non-integral ℓ<sup><span class="emphasis"><em>p</em></span></sup> norm 179 is rarely used. 180 </p> 181<h4> 182<a name="math_toolkit.norms.h5"></a> 183 <span class="phrase"><a name="math_toolkit.norms.u2113_super_p_distance"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_p_distance">ℓ<sup><span class="emphasis"><em>p</em></span></sup> distance</a> 184 </h4> 185<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">v</span><span class="special">{-</span><span class="number">8</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">};</span> 186<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">w</span><span class="special">{</span><span class="number">8</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">};</span> 187<span class="keyword">double</span> <span class="identifier">dist</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">lp_distance</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">w</span><span class="special">,</span> <span class="number">7</span><span class="special">);</span> 188<span class="comment">// dist = 16</span> 189 190<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">>></span> <span class="identifier">v</span><span class="special">{{</span><span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,-</span><span class="number">1</span><span class="special">}};</span> 191<span class="keyword">double</span> <span class="identifier">dist</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">lp_distance</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">v</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span> 192<span class="comment">// dist = 0</span> 193</pre> 194<p> 195 Supports both real, integral, and complex arithmetic. If the input is integral, 196 the output is a double precision float. The container must be forward iterable 197 and the contents are not modified. 198 </p> 199<p> 200 Only supports integer <span class="emphasis"><em>p</em></span>. 201 </p> 202<h4> 203<a name="math_toolkit.norms.h6"></a> 204 <span class="phrase"><a name="math_toolkit.norms.u2113_super_0_pseudo_norm"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_0_pseudo_norm">ℓ<sup>0</sup> pseudo-norm</a> 205 </h4> 206<p> 207 Counts the number of non-zero elements in a container. 208 </p> 209<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">v</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 210<span class="identifier">size_t</span> <span class="identifier">count</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">l0_pseudo_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> 211<span class="comment">// count = 1</span> 212</pre> 213<p> 214 Supports real, integral, and complex numbers. The container must be forward 215 iterable and the contents are not modified. Note that this measure is not robust 216 against numerical noise and is therefore not as useful as (say) the Hoyer sparsity 217 in numerical applications. Works with real, complex, and integral inputs. 218 </p> 219<h4> 220<a name="math_toolkit.norms.h7"></a> 221 <span class="phrase"><a name="math_toolkit.norms.hamming_distance"></a></span><a class="link" href="norms.html#math_toolkit.norms.hamming_distance">Hamming 222 Distance</a> 223 </h4> 224<p> 225 Compute the number of non-equal elements between two vectors <span class="emphasis"><em>w</em></span> 226 and <span class="emphasis"><em>v</em></span>: 227 </p> 228<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">v</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 229<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">w</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span> 230<span class="identifier">size_t</span> <span class="identifier">count</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">hamming_distance</span><span class="special">(</span><span class="identifier">w</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span> 231<span class="comment">// count = 2</span> 232</pre> 233<p> 234 Works for any datatype for which the operator <code class="computeroutput"><span class="special">!=</span></code> 235 is defined. 236 </p> 237<h4> 238<a name="math_toolkit.norms.h8"></a> 239 <span class="phrase"><a name="math_toolkit.norms.u2113_super_1_norm"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_1_norm">ℓ<sup>1</sup> norm</a> 240 </h4> 241<p> 242 The ℓ<sup>1</sup> norm is a special case of the ℓ<sup><span class="emphasis"><em>p</em></span></sup> norm, 243 but is much faster: 244 </p> 245<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">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 246<span class="keyword">double</span> <span class="identifier">l1</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">l1_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> 247<span class="comment">// l1 = 3</span> 248</pre> 249<p> 250 Requires a forward iterable input, does not modify input data, and works with 251 real, integral, and complex numbers. 252 </p> 253<h4> 254<a name="math_toolkit.norms.h9"></a> 255 <span class="phrase"><a name="math_toolkit.norms.u2113_super_1_distance"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_1_distance">ℓ<sup>1</sup> distance</a> 256 </h4> 257<p> 258 Computes the ℓ<sup>1</sup> distance between two vectors: 259 </p> 260<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">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 261<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">w</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 262<span class="keyword">double</span> <span class="identifier">dist</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">l1_distance</span><span class="special">(</span><span class="identifier">w</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span> 263<span class="comment">// dist = 0</span> 264</pre> 265<p> 266 Requires a forward iterable inputs, does not modify input data, and works with 267 real, integral, and complex numbers. If the input type is integral, the output 268 is a double precision float. 269 </p> 270<h4> 271<a name="math_toolkit.norms.h10"></a> 272 <span class="phrase"><a name="math_toolkit.norms.u2113_super_2_norm"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_2_norm">ℓ<sup>2</sup> norm</a> 273 </h4> 274<p> 275 The ℓ<sup>2</sup> norm is again a special case of the ℓ<sup><span class="emphasis"><em>p</em></span></sup> norm, 276 but is much faster: 277 </p> 278<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">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 279<span class="keyword">double</span> <span class="identifier">l2</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">l2_norm</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> 280<span class="comment">// l2 = sqrt(3)</span> 281</pre> 282<p> 283 Requires a forward iterable input, does not modify input data, and works with 284 real, complex and integral data. If the input is integral, the output is a 285 double precision float. 286 </p> 287<h4> 288<a name="math_toolkit.norms.h11"></a> 289 <span class="phrase"><a name="math_toolkit.norms.u2113_super_2_distance"></a></span><a class="link" href="norms.html#math_toolkit.norms.u2113_super_2_distance">ℓ<sup>2</sup> distance</a> 290 </h4> 291<p> 292 Compute the ℓ<sup>2</sup> distance between two vectors <span class="emphasis"><em>w</em></span> and 293 <span class="emphasis"><em>v</em></span>: 294 </p> 295<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">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 296<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">w</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">1</span><span class="special">};</span> 297<span class="keyword">double</span> <span class="identifier">dist</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">l2_distance</span><span class="special">(</span><span class="identifier">w</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span> 298<span class="comment">// dist = 1</span> 299</pre> 300<p> 301 Requires a forward iterable input, does not modify input data, and works with 302 real, complex numbers, and integral data. If the input type is integral, the 303 output is a double precision float. 304 </p> 305<h4> 306<a name="math_toolkit.norms.h12"></a> 307 <span class="phrase"><a name="math_toolkit.norms.total_variation"></a></span><a class="link" href="norms.html#math_toolkit.norms.total_variation">Total 308 Variation</a> 309 </h4> 310<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">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 311<span class="keyword">double</span> <span class="identifier">tv</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">total_variation</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> 312<span class="comment">// no variation in v, so tv = 0.</span> 313<span class="identifier">v</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 314<span class="keyword">double</span> <span class="identifier">tv</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">total_variation</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span> 315<span class="comment">// variation is 1, so tv = 1.</span> 316<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span> 317<span class="keyword">double</span> <span class="identifier">tv</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">total_variation</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span> 318</pre> 319<p> 320 The total variation only supports real numbers and integers. If the input is 321 integral, the output is a double precision float. 322 </p> 323<p> 324 All the constituent operations to compute the total variation are well-defined 325 for complex numbers, but the computed result is not meaningful; a 2D total 326 variation is more appropriate. The container must be forward iterable, and 327 the contents are not modified. 328 </p> 329<p> 330 As an aside, the total variation is not technically a norm, since <span class="emphasis"><em>TV(v) 331 = 0</em></span> does not imply <span class="emphasis"><em>v = 0</em></span>. However, it satisfies 332 the triangle inequality and is absolutely 1-homogeneous, so it is a seminorm, 333 and hence is grouped with the other norms here. 334 </p> 335<h4> 336<a name="math_toolkit.norms.h13"></a> 337 <span class="phrase"><a name="math_toolkit.norms.references"></a></span><a class="link" href="norms.html#math_toolkit.norms.references">References</a> 338 </h4> 339<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 340<li class="listitem"> 341 Higham, Nicholas J. <span class="emphasis"><em>Accuracy and stability of numerical algorithms.</em></span> 342 Vol. 80. Siam, 2002. 343 </li> 344<li class="listitem"> 345 Mallat, Stephane. <span class="emphasis"><em>A wavelet tour of signal processing: the sparse 346 way.</em></span> Academic press, 2008. 347 </li> 348<li class="listitem"> 349 Hurley, Niall, and Scott Rickard. <span class="emphasis"><em>Comparing measures of sparsity.</em></span> 350 IEEE Transactions on Information Theory 55.10 (2009): 4723-4741. 351 </li> 352</ul></div> 353</div> 354<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 355<td align="left"></td> 356<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 357 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 358 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 359 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 360 Daryle Walker and Xiaogang Zhang<p> 361 Distributed under the Boost Software License, Version 1.0. (See accompanying 362 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>) 363 </p> 364</div></td> 365</tr></table> 366<hr> 367<div class="spirit-nav"> 368<a accesskey="p" href="../vector_functionals.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../vector_functionals.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="../special.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 369</div> 370</body> 371</html> 372
