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  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">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">norms</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  39
  40     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
  48
  49     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  51
  52     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v2</span><span class="special">);</span>
  57
  58     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  63
  64     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v2</span><span class="special">);</span>
  69
  70     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  75
  76     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v2</span><span class="special">);</span>
  81
  82     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</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">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  99
 100     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&#8467;<sup>&#8734;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;&gt;</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">&#8467;<sup>&#8734;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;&gt;</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 &#8467;<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">&#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;&gt;</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">&#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&#8467;<sup>1</sup> norm</a>
 240     </h4>
 241 <p>
 242       The &#8467;<sup>1</sup> norm is a special case of the &#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&#8467;<sup>1</sup> distance</a>
 256     </h4>
 257 <p>
 258       Computes the &#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&#8467;<sup>2</sup> norm</a>
 273     </h4>
 274 <p>
 275       The &#8467;<sup>2</sup> norm is again a special case of the &#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&#8467;<sup>2</sup> distance</a>
 290     </h4>
 291 <p>
 292       Compute the &#8467;<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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;</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">&lt;</span><span class="keyword">int</span><span class="special">&gt;</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 &#169; 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       R&#229;de, 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
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