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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals.luroth_expansion"></a><a class="link" href="luroth_expansion.html" title="Luroth Expansions">Luroth Expansions</a>
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</h3></div></div></div>
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<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">luroth_expansion</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Z</span> <span class="special">=</span> <span class="identifier">int64_t</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">luroth_expansion</span> <span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">luroth_expansion</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Z</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">digits</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">digit_geometric_mean</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Z_</span><span class="special">></span>
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<span class="keyword">friend</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="keyword">operator</span><span class="special"><<(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="identifier">out</span><span class="special">,</span> <span class="identifier">luroth_expansion</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Z_</span><span class="special">>&</span> <span class="identifier">luroth</span><span class="special">);</span>
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<span class="special">};</span>
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<span class="special">}</span>
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</pre>
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<p>
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The <code class="computeroutput"><span class="identifier">luroth_expansion</span></code> class
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provided by Boost expands a floating point number into a Lüroth representation,
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i.e.,
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/luroth_expansion.svg" width="457" height="55"></object></span>
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</p>
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<p>
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The numbers <span class="emphasis"><em>d</em></span><sub>i</sub> are called digits or denominators; we
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use the terminology digits, since technically in our notation <span class="emphasis"><em>d</em></span><sub>0</sub> is
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not a denominator.
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</p>
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<p>
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Here's a minimal working example:
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</p>
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<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">luroth_expansion</span><span class="special">;</span>
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<span class="keyword">auto</span> <span class="identifier">luroth</span> <span class="special">=</span> <span class="identifier">luroth_expansion</span><span class="special">(</span><span class="identifier">pi</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">>());</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"π ≈ "</span> <span class="special"><<</span> <span class="identifier">luroth</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="comment">// Prints:</span>
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<span class="comment">// π ≈ ((3; 7, 1, 1, 1, 2, 1, 4, 23, 4, 1, 1, 1, 1, 80, 1, 1, 5))</span>
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</pre>
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<p>
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The class computes denominators while simultaneously computing convergents.
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Once a convergent is within a few ulps of the input value, the computation
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stops.
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</p>
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<p>
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<span class="emphasis"><em>Nota bene:</em></span> There is an alternative definition of the
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Lüroth representation where every digit is shifted by 1. We follow the definition
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given in Kalpazidou; with the modification that we do not constrain the input
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to be in the interval [0,1] and let the first digit be the floor of the input.
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</p>
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<p>
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For almost all real numbers, the geometric mean of the digits converges to
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a constant which is approximately 2.2001610580. This is "Khinchin's
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constant" for the Lüroth representation.
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</p>
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<h5>
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<a name="math_toolkit.internals.luroth_expansion.h0"></a>
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<span class="phrase"><a name="math_toolkit.internals.luroth_expansion.references"></a></span><a class="link" href="luroth_expansion.html#math_toolkit.internals.luroth_expansion.references">References</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Kalpazidou, Sofia. "Khintchine's constant for Lüroth representation."
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Journal of Number Theory 29.2 (1988): 196-205.
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</li>
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<li class="listitem">
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Finch, Steven R. Mathematical constants. Cambridge university press,
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2003.
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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