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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.internals.agm"></a><a class="link" href="agm.html" title="Arithmetic-Geometric Mean">Arithmetic-Geometric Mean</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.internals.agm.h0"></a>
<span class="phrase"><a name="math_toolkit.internals.agm.synopsis"></a></span><a class="link" href="agm.html#math_toolkit.internals.agm.synopsis">Synopsis</a>
</h5>
<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">agm</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="identifier">math</span><span class="special">::</span><span class="identifier">tools</span> <span class="special">{</span>
<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="identifier">Real</span> <span class="identifier">agm</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">a0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">g0</span><span class="special">);</span>
<span class="special">}</span> <span class="comment">// namespaces</span>
</pre>
<p>
The function <code class="computeroutput"><span class="identifier">agm</span></code> produces
the limiting value of the sequence
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/agm_sequence.svg" width="526" height="36"></object></span>
</p>
<p>
A basic usage is
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">G</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">agm</span><span class="special">(</span><span class="identifier">sqrt</span><span class="special">(</span><span class="number">2.0</span><span class="special">),</span> <span class="number">1.0</span><span class="special">);</span>
</pre>
<p>
The AGM inequality
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/agm_sequence.svg" width="526" height="36"></object></span>
</p>
<p>
shows that
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/agm_bound.svg" width="249" height="18"></object></span>
</p>
<p>
We use this condition internally to measure convergence; however, there is
no need to worry about putting arguments in the correct order since we extend
<code class="computeroutput"><span class="identifier">agm</span></code> to a symmetric function
by definition. Both arguments must be non-negative, as the sequence becomes
complex for negative arguments. (We have not implemented the complex-valued
AGM sequence.) The function <code class="computeroutput"><span class="identifier">agm</span></code>
is "essentially" one-dimensional, as the scale invariance <code class="computeroutput"><span class="identifier">agm</span><span class="special">(</span><span class="identifier">k</span><span class="special">*</span><span class="identifier">x</span><span class="special">,</span>
<span class="identifier">k</span><span class="special">*</span><span class="identifier">y</span><span class="special">)</span> <span class="special">==</span>
<span class="identifier">k</span><span class="special">*</span><span class="identifier">agm</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span><span class="identifier">y</span><span class="special">)</span></code>
always allows us to take one argument to be unity. The following ULP plot
has been generated with the function <code class="computeroutput"><span class="identifier">agm</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span>
<span class="identifier">Real</span><span class="special">(</span><span class="number">1</span><span class="special">))</span></code>:
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/agm_ulps_plot.svg"></object></span>
</p>
<p>
The graph above shows an ulps plot of the Boost implementation of <code class="computeroutput"><span class="identifier">agm</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">(</span><span class="number">1</span><span class="special">))</span></code>.
An ~2 ULP bound is to be expected.
</p>
<p>
A google benchmark for various types is available in <code class="computeroutput"><span class="identifier">boost</span><span class="special">/</span><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">reporting</span><span class="special">/</span><span class="identifier">performance</span><span class="special">/</span><span class="identifier">test_agm</span><span class="special">.</span><span class="identifier">cpp</span></code>; some
results on a consumer laptop are provided for convenience:
</p>
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="special">(</span><span class="number">16</span> <span class="identifier">X</span> <span class="number">2300</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span> <span class="identifier">s</span><span class="special">)</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">262</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">16777</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">2.02</span><span class="special">,</span> <span class="number">2.14</span><span class="special">,</span> <span class="number">2.00</span>
<span class="special">-------------------------------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span> <span class="identifier">Iterations</span>
<span class="special">-------------------------------------------------------------------------------</span>
<span class="identifier">AGM</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span> <span class="number">8.52</span> <span class="identifier">ns</span> <span class="number">8.51</span> <span class="identifier">ns</span> <span class="number">59654685</span>
<span class="identifier">AGM</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="number">13.5</span> <span class="identifier">ns</span> <span class="number">13.5</span> <span class="identifier">ns</span> <span class="number">51709746</span>
<span class="identifier">AGM</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">></span> <span class="number">30.6</span> <span class="identifier">ns</span> <span class="number">30.6</span> <span class="identifier">ns</span> <span class="number">18745247</span>
<span class="identifier">AGM</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">float128</span><span class="special">></span> <span class="number">2332</span> <span class="identifier">ns</span> <span class="number">2332</span> <span class="identifier">ns</span> <span class="number">299303</span>
</pre>
<p>
If any inputs are NaNs, the result is a NaN. If any inputs are +∞, the
result is +∞, unless the other argument fails NaN or negative validation.
</p>
<h5>
<a name="math_toolkit.internals.agm.h1"></a>
<span class="phrase"><a name="math_toolkit.internals.agm.references"></a></span><a class="link" href="agm.html#math_toolkit.internals.agm.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
Steven R. Finch. <span class="emphasis"><em>Mathematical Constants</em></span> Cambridge,
2003.
</li></ul></div>
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