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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.roots_noderiv.root_termination"></a><a class="link" href="root_termination.html" title="Termination Condition Functors">Termination
  28       Condition Functors</a>
  29 </h3></div></div></div>
  30 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  31 <span class="keyword">struct</span> <span class="identifier">eps_tolerance</span>
  32 <span class="special">{</span>
  33    <span class="identifier">eps_tolerance</span><span class="special">();</span>
  34    <span class="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
  35    <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  36 <span class="special">};</span>
  37 </pre>
  38 <p>
  39         <code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual
  40         termination condition used with these root finding functions. Its <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code>
  41         will return true when the relative distance between <span class="emphasis"><em>a</em></span>
  42         and <span class="emphasis"><em>b</em></span> is less than four times the machine epsilon for
  43         T, or 2<sup>1-bits</sup>, whichever is the larger. In other words, you set <span class="emphasis"><em>bits</em></span>
  44         to the number of bits of precision you want in the result. The minimal tolerance
  45         of <span class="emphasis"><em>four times the machine epsilon of type T</em></span> is required
  46         to ensure that we get back a bracketing interval, since this must clearly
  47         be at greater than one epsilon in size. While in theory a maximum distance
  48         of twice machine epsilon is possible to achieve, in practice this results
  49         in a great deal of "thrashing" given that the function whose root
  50         is being found can only ever be accurate to 1 epsilon at best.
  51       </p>
  52 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span>
  53 <span class="special">{</span>
  54    <span class="identifier">equal_floor</span><span class="special">();</span>
  55    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  56 <span class="special">};</span>
  57 </pre>
  58 <p>
  59         This termination condition is used when you want to find an integer result
  60         that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate
  61         as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>.
  62       </p>
  63 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span>
  64 <span class="special">{</span>
  65    <span class="identifier">equal_ceil</span><span class="special">();</span>
  66    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  67 <span class="special">};</span>
  68 </pre>
  69 <p>
  70         This termination condition is used when you want to find an integer result
  71         that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate
  72         as soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>.
  73       </p>
  74 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span>
  75 <span class="special">{</span>
  76    <span class="identifier">equal_nearest_integer</span><span class="special">();</span>
  77    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  78 <span class="special">};</span>
  79 </pre>
  80 <p>
  81         This termination condition is used when you want to find an integer result
  82         that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate
  83         as soon as both ends of the interval round to the same nearest integer.
  84       </p>
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