81 lines
7.8 KiB
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81 lines
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<title>Setting the Maximum Interval Halvings and Memory Requirements</title>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.double_exponential.de_levels"></a><a class="link" href="de_levels.html" title="Setting the Maximum Interval Halvings and Memory Requirements">Setting the
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Maximum Interval Halvings and Memory Requirements</a>
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</h3></div></div></div>
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<p>
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The max interval halvings is the maximum number of times the interval can
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be cut in half before giving up. If the accuracy is not met at that time,
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the routine returns its best estimate, along with the <code class="computeroutput"><span class="identifier">error</span></code>
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and <code class="computeroutput"><span class="identifier">L1</span></code>, which allows the
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user to decide if another quadrature routine should be employed.
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</p>
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<p>
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An example of this is
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
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<span class="identifier">size_t</span> <span class="identifier">max_halvings</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
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<span class="identifier">tanh_sinh</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">integrator</span><span class="special">(</span><span class="identifier">max_halvings</span><span class="special">);</span>
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<span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">5</span><span class="special">*</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">7</span><span class="special">;</span> <span class="special">};</span>
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<span class="keyword">double</span> <span class="identifier">error</span><span class="special">,</span> <span class="identifier">L1</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">0</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">1</span><span class="special">,</span> <span class="special">&</span><span class="identifier">error</span><span class="special">,</span> <span class="special">&</span><span class="identifier">L1</span><span class="special">);</span>
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<span class="keyword">if</span> <span class="special">(</span><span class="identifier">error</span><span class="special">*</span><span class="identifier">L1</span> <span class="special">></span> <span class="number">0.01</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">some_other_quadrature_method</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">0</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">1</span><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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It's important to remember that the number of sample points doubles with
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each new level, as does the memory footprint of the integrator object. Further,
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if the integral is smooth, then the precision will be doubling with each
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new level, so that for example, many integrals can achieve 100 decimal digit
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precision after just 7 levels. That said, abscissa-weight pairs for new levels
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are computed only when a new level is actually required (see thread safety),
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none the less, you should avoid setting the maximum arbitrarily high "just
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in case" as the time and space requirements for a large number of levels
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can quickly grow out of control.
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</p>
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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 © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle 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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