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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> |
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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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