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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>&lt; &lt;= =&gt; &gt; == !=</code>.
  It must be a total order (reflexivity, antisymmetry, transitivity, and each
  pair of numbers is ordered). So <code>complex&lt;T&gt;</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&lt; interval&lt;T&gt;
  &gt;</code> in place of <code>interval&lt; complex&lt;T&gt; &gt;</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&lt;T&gt;(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&lt;class T&gt; inline bool is_zero(T const &amp;v) { return v == static_cast&lt;T&gt;(0); }
template&lt;class T&gt; inline bool is_neg (T const &amp;v) { return v &lt;  static_cast&lt;T&gt;(0); }
template&lt;class T&gt; inline bool is_pos (T const &amp;v) { return v &gt;  static_cast&lt;T&gt;(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&lt;T&gt;</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 &le; <i>x</i> &rArr; -<i>x</i> &le; 0</li>

    <li><i>x</i> &le; <i>y</i> &rArr; -<i>y</i> &le; -<i>x</i></li>

    <li><i>x</i> &le; <i>y</i> &rArr; <i>x</i>+<i>z</i> &le;
    <i>y</i>+<i>z</i></li>

    <li><i>x</i> &le; <i>y</i> and <i>z</i> &ge; 0 &rArr;
    <i>x</i>&times;<i>z</i> &le; <i>y</i>&times;<i>z</i></li>

    <li>0 &lt; <i>x</i> &le; <i>y</i> &rArr; 0 &lt; 1/<i>y</i> &le;
    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 2&pi;-periodic;</li>

    <li><code>cos</code>(2&pi;-<i>x</i>) is equal to
    <code>cos</code>(<i>x</i>);</li>

    <li><code>cos</code> is a decreasing function on [0,2&pi;].</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) &lt;= f(x,y) &lt;= 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 &pi; constant (and also &pi;/2 and 2&pi;).
  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 &pi; 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>

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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 &copy; 2002 Guillaume Melquiond, Sylvain Pion, Herv&eacute;
  Br&ouml;nnimann, Polytechnic University<br>
  Copyright &copy; 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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