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- <h1>Numbers Requirements</h1>
- <p>What we call "number" is the base type of the <code>interval</code>
- class. The interval library expect a lot of properties from this base type
- in order to respect the inclusion property. All these properties are
- already detailed in the other sections of this documentation; but we will
- try to summarize them here.</p>
- <h3>Ordering</h3>
- <p>The numbers need to be supplied with an ordering. This ordering
- expresses itself by the operators <code>< <= => > == !=</code>.
- It must be a total order (reflexivity, antisymmetry, transitivity, and each
- pair of numbers is ordered). So <code>complex<T></code> will not be a
- good candidate for the base type; if you need the inclusion property of
- interval property, you should use <code>complex< interval<T>
- ></code> in place of <code>interval< complex<T> ></code>
- (but unfortunately, <code>complex</code> only provides specialization).</p>
- <p>Please note that invalid numbers are not concerned by the order; it can
- even be conceptually better if a comparison with these invalid numbers is
- always <code>false</code> (except for <code>!=</code>). If your checking
- policy uses <code>interval_lib::checking_base</code> and your base type
- contains invalid numbers, then this property is needed:
- <code>nan!=nan</code> (here <code>nan</code> is an invalid number). If this
- property is not present, then you should not use <code>checking_base</code>
- directly.</p>
- <p>Interval arithmetic involves a lot of comparison to zero. By default,
- they are done by comparing the numbers to
- <code>static_cast<T>(0)</code>. However, if the format of the numbers
- allows some faster comparisons when dealing with zero, the template
- functions in the <code>interval_lib::user</code> namespace can be
- specialized:</p>
- <pre>
- namespace user {
- template<class T> inline bool is_zero(T const &v) { return v == static_cast<T>(0); }
- template<class T> inline bool is_neg (T const &v) { return v < static_cast<T>(0); }
- template<class T> inline bool is_pos (T const &v) { return v > static_cast<T>(0); }
- }
- </pre>
- <h3>Numeric limits</h3>
- <p>Another remark about the checking policy. It normally is powerful enough
- to handle the exceptional behavior that the basic type could induce; in
- particular infinite and invalid numbers (thanks to the four functions
- <code>pos_inf</code>, <code>neg_inf</code>, <code>nan</code> and
- <code>is_nan</code>). However, if you use
- <code>interval_lib::checking_base</code> (and the default checking policy
- uses it), your base type should have a correctly specialized
- <code>std::numeric_limits<T></code>. In particular, the values
- <code>has_infinity</code> and <code>has_quiet_NaN</code>, and the functions
- <code>infinity</code> and <code>quiet_NaN</code> should be accordingly
- defined.</p>
- <p>So, to summarize, if you do not rely on the default policy and do not
- use <code>interval_lib::checking_base</code>, it is not necessary to have a
- specialization of the numeric limits for your base type.</p>
- <h3>Mathematical properties</h3>
- <p>Ensuring the numbers are correctly ordered is not enough. The basic
- operators should also respect some properties depending on the order. Here
- they are:</p>
- <ul>
- <li>0 ≤ <i>x</i> ⇒ -<i>x</i> ≤ 0</li>
- <li><i>x</i> ≤ <i>y</i> ⇒ -<i>y</i> ≤ -<i>x</i></li>
- <li><i>x</i> ≤ <i>y</i> ⇒ <i>x</i>+<i>z</i> ≤
- <i>y</i>+<i>z</i></li>
- <li><i>x</i> ≤ <i>y</i> and <i>z</i> ≥ 0 ⇒
- <i>x</i>×<i>z</i> ≤ <i>y</i>×<i>z</i></li>
- <li>0 < <i>x</i> ≤ <i>y</i> ⇒ 0 < 1/<i>y</i> ≤
- 1/<i>x</i></li>
- </ul>
- <p>The previous properties are also used (and enough) for <code>abs</code>,
- <code>square</code> and <code>pow</code>. For all the transcendental
- functions (including <code>sqrt</code>), other properties are needed. These
- functions should have the same properties than the corresponding real
- functions. For example, the expected properties for <code>cos</code>
- are:</p>
- <ul>
- <li><code>cos</code> is defined for all the valid numbers;</li>
- <li>it is 2π-periodic;</li>
- <li><code>cos</code>(2π-<i>x</i>) is equal to
- <code>cos</code>(<i>x</i>);</li>
- <li><code>cos</code> is a decreasing function on [0,2π].</li>
- </ul>
- <h3>Rounding</h3>
- <p>If you work with a base type and no inexact result is ever computed, you
- can skip the rest of this paragraph. You can also skip it if you are not
- interested in the inclusion property (if approximate results are enough).
- If you are still reading, it is probably because you want to know the basic
- properties the rounding policy should validate.</p>
- <p>Whichever operation or function you consider, the following property
- should be respected: <code>f_down(x,y) <= f(x,y) <= f_up(x,y)</code>.
- Here, <code>f</code> denotes the infinitely precise function computed and
- <code>f_down</code> and <code>f_up</code> are functions which return
- possibly inexact values but of the correct type (the base type). If
- possible, they should try to return the nearest representable value, but it
- is not always easy.</p>
- <h3>Constants</h3>
- <p>In order for the trigonometric functions to correctly work, the library
- need to know the value of the π constant (and also π/2 and 2π).
- Since these constants may not be representable in the base type, the
- library does not have to know the exact value: a lower bound and an upper
- bound are enough. If these values are not provided by the user, the default
- values will be used: they are integer values (so π is bounded by 3 and
- 4).</p>
- <h3>Operators and conversions</h3>
- <p>As explained at the beginning, the comparison operators should be
- defined for the base type. The rounding policy defines a lot of functions
- used by the interval library. So the arithmetic operators do not need to be
- defined for the base type (unless required by one of the predefined
- classes). However, there is an exception: the unary minus need to be
- defined. Moreover, this operator should only provide exact results; it is
- the reason why the rounding policy does not provide some negation
- functions.</p>
- <p>The conversion from <code>int</code> to the base type needs to be
- defined (only a few values need to be available: -1, 0, 1, 2). The
- conversion the other way around is provided by the rounding policy
- (<code>int_down</code> and <code>int_up</code> members); and no other
- conversion is strictly needed. However, it may be valuable to provide as
- much conversions as possible in the rounding policy (<code>conv_down</code>
- and <code>conv_up</code> members) in order to benefit from interval
- conversions.</p>
- <hr>
- <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
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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 © 2004 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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