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205 lines
8.3 KiB
HTML
205 lines
8.3 KiB
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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"http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<head>
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<meta http-equiv="Content-Language" content="en-us">
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<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
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<link rel="stylesheet" type="text/css" href="../../../../boost.css">
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<title>Comparisons</title>
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</head>
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<body>
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<h1>Comparisons</h1>
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<p>As was said before, the definition of the comparison operators induces a
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slight problem. There are many ways to define them, depending of the return
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type or the expected order. It is the reason why the meaning of the
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operators is not fixed once and for all.</p>
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<p>The way the operators are defined could have been influenced by a
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policy, as it is already the case for the rounding and the checking.
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However, comparisons are more an external property of the the class rather
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than an internal one. They are meant to be locally modified, independantly
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of the type of the intervals.</p>
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<p>The operators <code><</code>, <code><=</code>, <code>></code>,
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<code>>=</code>, <code>==</code>, <code>!=</code> are defined each time;
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and like the arithmetic operators they can take an argument of the base
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type. However, due to technical limitations, this base type can only be the
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second argument; so the operators are unfortunately not fully symmetric.
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The return type is not always <code>bool</code>, since some interesting
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results can be achieved by using a tri-state return type. So here is the
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common signatures of the operators:</p>
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<pre>
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template<class T, class Policies1, class Policies2>
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return_type operator== (const interval<T, Policies1>&, const interval<T, Policies2>&);
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template<class T, class Policies>
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return_type operator== (const interval<T, Policies>&, const T&);
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</pre>
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<h2>vided comparisons</h2>
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<h3>Default comparison</h3>
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<p>If nothing is specified, the meaning of the comparison operators are an
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extension of the operator on the base type. More precisely, if one of the
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argument is invalid or empty, an exception is thrown. If the arguments are
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valid, the following rules are applied to determine the result of
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[<i>a</i>,<i>b</i>] <code>op</code> [<i>c</i>,<i>d</i>] (just consider
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<i>c</i> <code>==</code> <i>d</i> if the second argument is of type
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<code>T</code>):</p>
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<ul>
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<li>if ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i>
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∈ [<i>c</i>,<i>d</i>] <code>(</code><i>x</i> <code>op</code>
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y<code>)</code>, then <code>true</code></li>
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<li>if ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i>
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∈ [<i>c</i>,<i>d</i>] <code>!(</code><i>x</i> <code>op</code>
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y<code>)</code>, then <code>false</code></li>
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<li>otherwise throw an exception.</li>
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</ul>
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<p>This comparison allows to replace base types by interval types without
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changing the meaning of a program. Indeed, if no exception is thrown, the
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result is the same as before; and if an exception is thrown, the previous
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comparison was unsure and should have been rewritten.</p>
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<h3>Other comparisons</h3>
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<p>The other comparisons are selected by using a namespace. These
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namespaces are located under
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<code>boost::numeric::interval_lib::compare</code> and are invoked by:</p>
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<pre>
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using namespace boost::numeric::interval_lib::compare::the_comparison_to_select;
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</pre>
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<p>After this line, the default meaning of the operators will have been
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replaced by the meaning located in the namespace. Please note that because
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of C++ lookup rules, it is not possible to use two namespaces one after
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another and they must be used in different block hierarchies. Otherwise the
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compiler will complain about ambiguous operators. To summarize:</p>
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<pre>
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// example 1: BAD
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using namespace compare1;
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...
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using namespace compare2;
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...
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// example 2: GOOD
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{ using namespace compare1;
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... }
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{ using namespace compare2;
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... }
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// example 3: BAD
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using namespace compare1;
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...
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{ using namespace compare2;
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... }
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</pre>
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<p>Now comes the list of the provided comparisons. They all are located in
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their respective header files under
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<code><boost/numeric/interval/compare/...></code>. And as for the
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default comparison, the operators will generally complain by throwing an
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exception if feed by invalid values.</p>
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<ul>
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<li><code>certain</code>: this comparison is equivalent to the default
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scheme with the exceptional case mapped to <code>false</code>. So these
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operators answer <code>true</code> only when the comparison is verified
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for all pairs of elements.</li>
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<li><code>possible</code>: this time, the exceptional case is mapped to
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<code>true</code>. The operators answer <code>true</code> as soon as the
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comparison is verified for a pair of elements.<br></li>
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<li><code>lexicographic</code>: the lexicographic order (the lower bounds
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are first compared, and if it is not enough to know the result, the upper
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bounds are then compared). This order does not have a meaning in interval
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arithmetic. However, since it is the natural total order on pair of
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(totally ordered) numbers, it may be handy in some cases.</li>
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<li><code>set</code>: the set inclusion partial order. This time, an
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empty interval is not considered to be invalid (but an invalid number is
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still invalid). <code><=</code> and <code><</code> are the subset
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and proper subset relations; and <code>>=</code> and <code>></code>
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are the superset and proper superset relations.</li>
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<li><code>tribool</code>: this comparison relies on the Boost tristate
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boolean library and changes the default operators so that an explicit
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indeterminate value is returned in the third case instead of throwing an
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exception.</li>
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</ul>
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<h3>Exception</h3>
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<pre>
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namespace boost {
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namespace numeric {
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namespace interval_lib {
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class comparison_error: std::runtime_error; // "boost::interval: uncertain comparison"
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} // namespace interval_lib
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} // namespace numeric
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} // namespace boost
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</pre>
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<h2>Explicit comparison functions</h2>
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<p>In some situation, you may want to perform direct comparisons on the
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bounds and avoid the indeterminate case that appears with default
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operators. Some functions are provided for this purpose. They expect their
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arguments to be valid and return a result after only one comparison. Their
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names are composed by <code>cer</code> (for "certain", if the default
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comparison is true, the result is true) or <code>pos</code> (for
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"possible", if the default comparison is false, the result is false)
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followed by <code>lt</code>, <code>le</code>, <code>gt</code>,
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<code>ge</code>, <code>eq</code> or <code>ne</code>. They are located in
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<code><boost/numeric/interval/compare/explicit.hpp></code>. Each of
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these functions takes two parameters and returns a boolean; the parameters
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are expected to be valid, undefined behavior may result otherwise. For
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example, the definition of the "certainly less than" comparison is:</p>
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<pre>
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namespace boost {
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namespace numeric {
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namespace interval_lib {
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template<class T, class Policies1, class Policies2>
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bool cerlt(const interval<T, Policies1>& x, const interval<T, Policies2>& y);
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template<class T, class Policies>
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bool cerlt(const interval<T, Policies>& x, const T& y);
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template<class T, class Policies>
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bool cerlt(const T& x, const interval<T, Policies>& y);
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} // namespace interval_lib
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} // namespace numeric
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} // namespace boost
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</pre>
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<hr>
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<p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
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"../../../../doc/images/valid-html401.png" alt="Valid HTML 4.01 Transitional"
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height="31" width="88"></a></p>
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<p>Revised
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<!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-24<!--webbot bot="Timestamp" endspan i-checksum="12172" --></p>
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<p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé
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Brönnimann, Polytechnic University<br>
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Copyright © 2003 Guillaume Melquiond</i></p>
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<p><i>Distributed under the Boost Software License, Version 1.0. (See
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accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a>
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or copy at <a href=
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"http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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</body>
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</html>
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