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- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.norms"></a><a class="link" href="norms.html" title="Norms">Norms</a>
- </h2></div></div></div>
- <h4>
- <a name="math_toolkit.norms.h0"></a>
- <span class="phrase"><a name="math_toolkit.norms.synopsis"></a></span><a class="link" href="norms.html#math_toolkit.norms.synopsis">Synopsis</a>
- </h4>
- <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>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
- <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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
- <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>
- <span class="special">}}}</span>
- </pre>
- <h4>
- <a name="math_toolkit.norms.h1"></a>
- <span class="phrase"><a name="math_toolkit.norms.description"></a></span><a class="link" href="norms.html#math_toolkit.norms.description">Description</a>
- </h4>
- <p>
- 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
- scalar values traditionally useful in numerical analysis from vectors.
- </p>
- <p>
- 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
- hold the data, but this not required. In general, you can store your data in
- an Eigen array, an Armadillo vector, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>,
- 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>.
- These routines are usable in float, double, long double, and Boost.Multiprecision
- precision, as well as their complex extensions whenever the computation is
- well-defined. Integral datatypes are supported for most routines.
- </p>
- <h4>
- <a name="math_toolkit.norms.h2"></a>
- <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>
- </h4>
- <p>
- Computes the supremum norm of a dataset:
- </p>
- <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>
- <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>
- <span class="comment">// sup = 3</span>
- <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>
- <span class="comment">// Range call:</span>
- <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="comment">// sup = 8</span>
- </pre>
- <p>
- Supports real, integral, and complex arithmetic. Container must be forward
- iterable and is not modified.
- </p>
- <h4>
- <a name="math_toolkit.norms.h3"></a>
- <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>
- </h4>
- <p>
- Computes the supremum norm distance between two vectors:
- </p>
- <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>
- <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>
- <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>
- <span class="comment">// sup = 9</span>
- </pre>
- <p>
- Supports real, integral, and complex arithmetic. Container must be forward
- iterable and is not modified. If the input it integral, the output is a double
- precision float.
- </p>
- <h4>
- <a name="math_toolkit.norms.h4"></a>
- <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>
- </h4>
- <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>
- <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>
- <span class="comment">// sup = 8</span>
- <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>
- <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>
- <span class="comment">// sup = cbrt(3)</span>
- </pre>
- <p>
- Supports both real, integral, and complex arithmetic. If the input is integral,
- the output is a double precision float. The container must be forward iterable
- and the contents are not modified.
- </p>
- <p>
- Only supports integral <span class="emphasis"><em>p</em></span> for two reasons: The computation
- 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
- is rarely used.
- </p>
- <h4>
- <a name="math_toolkit.norms.h5"></a>
- <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>
- </h4>
- <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>
- <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>
- <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>
- <span class="comment">// dist = 16</span>
- <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>
- <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>
- <span class="comment">// dist = 0</span>
- </pre>
- <p>
- Supports both real, integral, and complex arithmetic. If the input is integral,
- the output is a double precision float. The container must be forward iterable
- and the contents are not modified.
- </p>
- <p>
- Only supports integer <span class="emphasis"><em>p</em></span>.
- </p>
- <h4>
- <a name="math_toolkit.norms.h6"></a>
- <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>
- </h4>
- <p>
- Counts the number of non-zero elements in a container.
- </p>
- <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>
- <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>
- <span class="comment">// count = 1</span>
- </pre>
- <p>
- Supports real, integral, and complex numbers. The container must be forward
- iterable and the contents are not modified. Note that this measure is not robust
- against numerical noise and is therefore not as useful as (say) the Hoyer sparsity
- in numerical applications. Works with real, complex, and integral inputs.
- </p>
- <h4>
- <a name="math_toolkit.norms.h7"></a>
- <span class="phrase"><a name="math_toolkit.norms.hamming_distance"></a></span><a class="link" href="norms.html#math_toolkit.norms.hamming_distance">Hamming
- Distance</a>
- </h4>
- <p>
- Compute the number of non-equal elements between two vectors <span class="emphasis"><em>w</em></span>
- and <span class="emphasis"><em>v</em></span>:
- </p>
- <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>
- <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>
- <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>
- <span class="comment">// count = 2</span>
- </pre>
- <p>
- Works for any datatype for which the operator <code class="computeroutput"><span class="special">!=</span></code>
- is defined.
- </p>
- <h4>
- <a name="math_toolkit.norms.h8"></a>
- <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>
- </h4>
- <p>
- The ℓ<sup>1</sup> norm is a special case of the ℓ<sup><span class="emphasis"><em>p</em></span></sup> norm,
- but is much faster:
- </p>
- <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>
- <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>
- <span class="comment">// l1 = 3</span>
- </pre>
- <p>
- Requires a forward iterable input, does not modify input data, and works with
- real, integral, and complex numbers.
- </p>
- <h4>
- <a name="math_toolkit.norms.h9"></a>
- <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>
- </h4>
- <p>
- Computes the ℓ<sup>1</sup> distance between two vectors:
- </p>
- <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>
- <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>
- <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>
- <span class="comment">// dist = 0</span>
- </pre>
- <p>
- Requires a forward iterable inputs, does not modify input data, and works with
- real, integral, and complex numbers. If the input type is integral, the output
- is a double precision float.
- </p>
- <h4>
- <a name="math_toolkit.norms.h10"></a>
- <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>
- </h4>
- <p>
- The ℓ<sup>2</sup> norm is again a special case of the ℓ<sup><span class="emphasis"><em>p</em></span></sup> norm,
- but is much faster:
- </p>
- <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>
- <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>
- <span class="comment">// l2 = sqrt(3)</span>
- </pre>
- <p>
- Requires a forward iterable input, does not modify input data, and works with
- real, complex and integral data. If the input is integral, the output is a
- double precision float.
- </p>
- <h4>
- <a name="math_toolkit.norms.h11"></a>
- <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>
- </h4>
- <p>
- Compute the ℓ<sup>2</sup> distance between two vectors <span class="emphasis"><em>w</em></span> and
- <span class="emphasis"><em>v</em></span>:
- </p>
- <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>
- <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>
- <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>
- <span class="comment">// dist = 1</span>
- </pre>
- <p>
- Requires a forward iterable input, does not modify input data, and works with
- real, complex numbers, and integral data. If the input type is integral, the
- output is a double precision float.
- </p>
- <h4>
- <a name="math_toolkit.norms.h12"></a>
- <span class="phrase"><a name="math_toolkit.norms.total_variation"></a></span><a class="link" href="norms.html#math_toolkit.norms.total_variation">Total
- Variation</a>
- </h4>
- <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>
- <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>
- <span class="comment">// no variation in v, so tv = 0.</span>
- <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>
- <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>
- <span class="comment">// variation is 1, so tv = 1.</span>
- <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>
- <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>
- </pre>
- <p>
- The total variation only supports real numbers and integers. If the input is
- integral, the output is a double precision float.
- </p>
- <p>
- All the constituent operations to compute the total variation are well-defined
- for complex numbers, but the computed result is not meaningful; a 2D total
- variation is more appropriate. The container must be forward iterable, and
- the contents are not modified.
- </p>
- <p>
- As an aside, the total variation is not technically a norm, since <span class="emphasis"><em>TV(v)
- = 0</em></span> does not imply <span class="emphasis"><em>v = 0</em></span>. However, it satisfies
- the triangle inequality and is absolutely 1-homogeneous, so it is a seminorm,
- and hence is grouped with the other norms here.
- </p>
- <h4>
- <a name="math_toolkit.norms.h13"></a>
- <span class="phrase"><a name="math_toolkit.norms.references"></a></span><a class="link" href="norms.html#math_toolkit.norms.references">References</a>
- </h4>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- Higham, Nicholas J. <span class="emphasis"><em>Accuracy and stability of numerical algorithms.</em></span>
- Vol. 80. Siam, 2002.
- </li>
- <li class="listitem">
- Mallat, Stephane. <span class="emphasis"><em>A wavelet tour of signal processing: the sparse
- way.</em></span> Academic press, 2008.
- </li>
- <li class="listitem">
- Hurley, Niall, and Scott Rickard. <span class="emphasis"><em>Comparing measures of sparsity.</em></span>
- IEEE Transactions on Information Theory 55.10 (2009): 4723-4741.
- </li>
- </ul></div>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- 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>)
- </p>
- </div></td>
- </tr></table>
- <hr>
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