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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><</code>, <code><=</code>, <code>></code>,
<code>>=</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<class T, class Policies1, class Policies2>
return_type operator== (const interval<T, Policies1>&, const interval<T, Policies2>&);
template<class T, class Policies>
return_type operator== (const interval<T, Policies>&, const T&);
</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 ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i>
∈ [<i>c</i>,<i>d</i>] <code>(</code><i>x</i> <code>op</code>
y<code>)</code>, then <code>true</code></li>
<li>if ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i>
∈ [<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><boost/numeric/interval/compare/...></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><=</code> and <code><</code> are the subset
and proper subset relations; and <code>>=</code> and <code>></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><boost/numeric/interval/compare/explicit.hpp></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<class T, class Policies1, class Policies2>
bool cerlt(const interval<T, Policies1>& x, const interval<T, Policies2>& y);
template<class T, class Policies>
bool cerlt(const interval<T, Policies>& x, const T& y);
template<class T, class Policies>
bool cerlt(const T& x, const interval<T, Policies>& 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 © 2002 Guillaume Melquiond, Sylvain Pion, Hervé
Brönnimann, Polytechnic University<br>
Copyright © 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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