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  <h1>Comparisons</h1>

  <p>As was said before, the definition of the comparison operators induces a
  slight problem. There are many ways to define them, depending of the return
  type or the expected order. It is the reason why the meaning of the
  operators is not fixed once and for all.</p>

  <p>The way the operators are defined could have been influenced by a
  policy, as it is already the case for the rounding and the checking.
  However, comparisons are more an external property of the the class rather
  than an internal one. They are meant to be locally modified, independantly
  of the type of the intervals.</p>

  <p>The operators <code>&lt;</code>, <code>&lt;=</code>, <code>&gt;</code>,
  <code>&gt;=</code>, <code>==</code>, <code>!=</code> are defined each time;
  and like the arithmetic operators they can take an argument of the base
  type. However, due to technical limitations, this base type can only be the
  second argument; so the operators are unfortunately not fully symmetric.
  The return type is not always <code>bool</code>, since some interesting
  results can be achieved by using a tri-state return type. So here is the
  common signatures of the operators:</p>
  <pre>
template&lt;class T, class Policies1, class Policies2&gt;
return_type operator== (const interval&lt;T, Policies1&gt;&amp;, const interval&lt;T, Policies2&gt;&amp;);

template&lt;class T, class Policies&gt;
return_type operator== (const interval&lt;T, Policies&gt;&amp;, const T&amp;);
</pre>

  <h2>vided comparisons</h2>

  <h3>Default comparison</h3>

  <p>If nothing is specified, the meaning of the comparison operators are an
  extension of the operator on the base type. More precisely, if one of the
  argument is invalid or empty, an exception is thrown. If the arguments are
  valid, the following rules are applied to determine the result of
  [<i>a</i>,<i>b</i>] <code>op</code> [<i>c</i>,<i>d</i>] (just consider
  <i>c</i> <code>==</code> <i>d</i> if the second argument is of type
  <code>T</code>):</p>

  <ul>
    <li>if &forall; <i>x</i> &isin; [<i>a</i>,<i>b</i>] &forall; <i>y</i>
    &isin; [<i>c</i>,<i>d</i>] <code>(</code><i>x</i> <code>op</code>
    y<code>)</code>, then <code>true</code></li>

    <li>if &forall; <i>x</i> &isin; [<i>a</i>,<i>b</i>] &forall; <i>y</i>
    &isin; [<i>c</i>,<i>d</i>] <code>!(</code><i>x</i> <code>op</code>
    y<code>)</code>, then <code>false</code></li>

    <li>otherwise throw an exception.</li>
  </ul>

  <p>This comparison allows to replace base types by interval types without
  changing the meaning of a program. Indeed, if no exception is thrown, the
  result is the same as before; and if an exception is thrown, the previous
  comparison was unsure and should have been rewritten.</p>

  <h3>Other comparisons</h3>

  <p>The other comparisons are selected by using a namespace. These
  namespaces are located under
  <code>boost::numeric::interval_lib::compare</code> and are invoked by:</p>
  <pre>
using namespace boost::numeric::interval_lib::compare::the_comparison_to_select;
</pre>

  <p>After this line, the default meaning of the operators will have been
  replaced by the meaning located in the namespace. Please note that because
  of C++ lookup rules, it is not possible to use two namespaces one after
  another and they must be used in different block hierarchies. Otherwise the
  compiler will complain about ambiguous operators. To summarize:</p>
  <pre>
// example 1: BAD
using namespace compare1;
...
using namespace compare2;
...

// example 2: GOOD
{ using namespace compare1;
  ...                       }
{ using namespace compare2;
  ...                       }

// example 3: BAD
using namespace compare1;
...
{ using namespace compare2;
  ...                       }
</pre>

  <p>Now comes the list of the provided comparisons. They all are located in
  their respective header files under
  <code>&lt;boost/numeric/interval/compare/...&gt;</code>. And as for the
  default comparison, the operators will generally complain by throwing an
  exception if feed by invalid values.</p>

  <ul>
    <li><code>certain</code>: this comparison is equivalent to the default
    scheme with the exceptional case mapped to <code>false</code>. So these
    operators answer <code>true</code> only when the comparison is verified
    for all pairs of elements.</li>

    <li><code>possible</code>: this time, the exceptional case is mapped to
    <code>true</code>. The operators answer <code>true</code> as soon as the
    comparison is verified for a pair of elements.<br></li>

    <li><code>lexicographic</code>: the lexicographic order (the lower bounds
    are first compared, and if it is not enough to know the result, the upper
    bounds are then compared). This order does not have a meaning in interval
    arithmetic. However, since it is the natural total order on pair of
    (totally ordered) numbers, it may be handy in some cases.</li>

    <li><code>set</code>: the set inclusion partial order. This time, an
    empty interval is not considered to be invalid (but an invalid number is
    still invalid). <code>&lt;=</code> and <code>&lt;</code> are the subset
    and proper subset relations; and <code>&gt;=</code> and <code>&gt;</code>
    are the superset and proper superset relations.</li>

    <li><code>tribool</code>: this comparison relies on the Boost tristate
    boolean library and changes the default operators so that an explicit
    indeterminate value is returned in the third case instead of throwing an
    exception.</li>
  </ul>

  <h3>Exception</h3>
  <pre>
namespace boost {
namespace numeric {
namespace interval_lib {

class comparison_error: std::runtime_error; // "boost::interval: uncertain comparison"

} // namespace interval_lib
} // namespace numeric
} // namespace boost
</pre>

  <h2>Explicit comparison functions</h2>

  <p>In some situation, you may want to perform direct comparisons on the
  bounds and avoid the indeterminate case that appears with default
  operators. Some functions are provided for this purpose. They expect their
  arguments to be valid and return a result after only one comparison. Their
  names are composed by <code>cer</code> (for "certain", if the default
  comparison is true, the result is true) or <code>pos</code> (for
  "possible", if the default comparison is false, the result is false)
  followed by <code>lt</code>, <code>le</code>, <code>gt</code>,
  <code>ge</code>, <code>eq</code> or <code>ne</code>. They are located in
  <code>&lt;boost/numeric/interval/compare/explicit.hpp&gt;</code>. Each of
  these functions takes two parameters and returns a boolean; the parameters
  are expected to be valid, undefined behavior may result otherwise. For
  example, the definition of the "certainly less than" comparison is:</p>
  <pre>
namespace boost {
namespace numeric {
namespace interval_lib {

template&lt;class T, class Policies1, class Policies2&gt;
bool cerlt(const interval&lt;T, Policies1&gt;&amp; x, const interval&lt;T, Policies2&gt;&amp; y);

template&lt;class T, class Policies&gt;
bool cerlt(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);

template&lt;class T, class Policies&gt;
bool cerlt(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

} // namespace interval_lib
} // namespace numeric
} // namespace boost
</pre>
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  <p>Revised 
  <!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-24<!--webbot bot="Timestamp" endspan i-checksum="12172" --></p>

  <p><i>Copyright &copy; 2002 Guillaume Melquiond, Sylvain Pion, Herv&eacute;
  Br&ouml;nnimann, Polytechnic University<br>
  Copyright &copy; 2003 Guillaume Melquiond</i></p>

  <p><i>Distributed under the Boost Software License, Version 1.0. (See
  accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a>
  or copy at <a href=
  "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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