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