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   4 <title>Setting the Maximum Interval Halvings and Memory Requirements</title>
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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <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
  28       Maximum Interval Halvings and Memory Requirements</a>
  29 </h3></div></div></div>
  30 <p>
  31         The max interval halvings is the maximum number of times the interval can
  32         be cut in half before giving up. If the accuracy is not met at that time,
  33         the routine returns its best estimate, along with the <code class="computeroutput"><span class="identifier">error</span></code>
  34         and <code class="computeroutput"><span class="identifier">L1</span></code>, which allows the
  35         user to decide if another quadrature routine should be employed.
  36       </p>
  37 <p>
  38         An example of this is
  39       </p>
  40 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">());</span>
  41 <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>
  42 <span class="identifier">tanh_sinh</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">integrator</span><span class="special">(</span><span class="identifier">max_halvings</span><span class="special">);</span>
  43 <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>
  44 <span class="keyword">double</span> <span class="identifier">error</span><span class="special">,</span> <span class="identifier">L1</span><span class="special">;</span>
  45 <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">&amp;</span><span class="identifier">error</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">L1</span><span class="special">);</span>
  46 <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">&gt;</span> <span class="number">0.01</span><span class="special">)</span>
  47 <span class="special">{</span>
  48     <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>
  49 <span class="special">}</span>
  50 </pre>
  51 <p>
  52         It's important to remember that the number of sample points doubles with
  53         each new level, as does the memory footprint of the integrator object. Further,
  54         if the integral is smooth, then the precision will be doubling with each
  55         new level, so that for example, many integrals can achieve 100 decimal digit
  56         precision after just 7 levels. That said, abscissa-weight pairs for new levels
  57         are computed only when a new level is actually required (see thread safety),
  58         none the less, you should avoid setting the maximum arbitrarily high "just
  59         in case" as the time and space requirements for a large number of levels
  60         can quickly grow out of control.
  61       </p>
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  66       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
  67       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
  68       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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