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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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