| 1 | <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" |
|---|
| 2 | "http://www.w3.org/TR/html4/loose.dtd"> |
|---|
| 3 | |
|---|
| 4 | <html> |
|---|
| 5 | <head> |
|---|
| 6 | <meta http-equiv="Content-Language" content="en-us"> |
|---|
| 7 | <meta http-equiv="Content-Type" content="text/html; charset=us-ascii"> |
|---|
| 8 | <link rel="stylesheet" type="text/css" href="../../../../boost.css"> |
|---|
| 9 | |
|---|
| 10 | <title>Numbers Requirements</title> |
|---|
| 11 | </head> |
|---|
| 12 | |
|---|
| 13 | <body lang="en"> |
|---|
| 14 | <h1>Numbers Requirements</h1> |
|---|
| 15 | |
|---|
| 16 | <p>What we call "number" is the base type of the <code>interval</code> |
|---|
| 17 | class. The interval library expect a lot of properties from this base type |
|---|
| 18 | in order to respect the inclusion property. All these properties are |
|---|
| 19 | already detailed in the other sections of this documentation; but we will |
|---|
| 20 | try to summarize them here.</p> |
|---|
| 21 | |
|---|
| 22 | <h3>Ordering</h3> |
|---|
| 23 | |
|---|
| 24 | <p>The numbers need to be supplied with an ordering. This ordering |
|---|
| 25 | expresses itself by the operators <code>< <= => > == !=</code>. |
|---|
| 26 | It must be a total order (reflexivity, antisymmetry, transitivity, and each |
|---|
| 27 | pair of numbers is ordered). So <code>complex<T></code> will not be a |
|---|
| 28 | good candidate for the base type; if you need the inclusion property of |
|---|
| 29 | interval property, you should use <code>complex< interval<T> |
|---|
| 30 | ></code> in place of <code>interval< complex<T> ></code> |
|---|
| 31 | (but unfortunately, <code>complex</code> only provides specialization).</p> |
|---|
| 32 | |
|---|
| 33 | <p>Please note that invalid numbers are not concerned by the order; it can |
|---|
| 34 | even be conceptually better if a comparison with these invalid numbers is |
|---|
| 35 | always <code>false</code> (except for <code>!=</code>). If your checking |
|---|
| 36 | policy uses <code>interval_lib::checking_base</code> and your base type |
|---|
| 37 | contains invalid numbers, then this property is needed: |
|---|
| 38 | <code>nan!=nan</code> (here <code>nan</code> is an invalid number). If this |
|---|
| 39 | property is not present, then you should not use <code>checking_base</code> |
|---|
| 40 | directly.</p> |
|---|
| 41 | |
|---|
| 42 | <p>Interval arithmetic involves a lot of comparison to zero. By default, |
|---|
| 43 | they are done by comparing the numbers to |
|---|
| 44 | <code>static_cast<T>(0)</code>. However, if the format of the numbers |
|---|
| 45 | allows some faster comparisons when dealing with zero, the template |
|---|
| 46 | functions in the <code>interval_lib::user</code> namespace can be |
|---|
| 47 | specialized:</p> |
|---|
| 48 | <pre> |
|---|
| 49 | namespace user { |
|---|
| 50 | template<class T> inline bool is_zero(T const &v) { return v == static_cast<T>(0); } |
|---|
| 51 | template<class T> inline bool is_neg (T const &v) { return v < static_cast<T>(0); } |
|---|
| 52 | template<class T> inline bool is_pos (T const &v) { return v > static_cast<T>(0); } |
|---|
| 53 | } |
|---|
| 54 | </pre> |
|---|
| 55 | |
|---|
| 56 | <h3>Numeric limits</h3> |
|---|
| 57 | |
|---|
| 58 | <p>Another remark about the checking policy. It normally is powerful enough |
|---|
| 59 | to handle the exceptional behavior that the basic type could induce; in |
|---|
| 60 | particular infinite and invalid numbers (thanks to the four functions |
|---|
| 61 | <code>pos_inf</code>, <code>neg_inf</code>, <code>nan</code> and |
|---|
| 62 | <code>is_nan</code>). However, if you use |
|---|
| 63 | <code>interval_lib::checking_base</code> (and the default checking policy |
|---|
| 64 | uses it), your base type should have a correctly specialized |
|---|
| 65 | <code>std::numeric_limits<T></code>. In particular, the values |
|---|
| 66 | <code>has_infinity</code> and <code>has_quiet_NaN</code>, and the functions |
|---|
| 67 | <code>infinity</code> and <code>quiet_NaN</code> should be accordingly |
|---|
| 68 | defined.</p> |
|---|
| 69 | |
|---|
| 70 | <p>So, to summarize, if you do not rely on the default policy and do not |
|---|
| 71 | use <code>interval_lib::checking_base</code>, it is not necessary to have a |
|---|
| 72 | specialization of the numeric limits for your base type.</p> |
|---|
| 73 | |
|---|
| 74 | <h3>Mathematical properties</h3> |
|---|
| 75 | |
|---|
| 76 | <p>Ensuring the numbers are correctly ordered is not enough. The basic |
|---|
| 77 | operators should also respect some properties depending on the order. Here |
|---|
| 78 | they are:</p> |
|---|
| 79 | |
|---|
| 80 | <ul> |
|---|
| 81 | <li>0 ≤ <i>x</i> ⇒ -<i>x</i> ≤ 0</li> |
|---|
| 82 | |
|---|
| 83 | <li><i>x</i> ≤ <i>y</i> ⇒ -<i>y</i> ≤ -<i>x</i></li> |
|---|
| 84 | |
|---|
| 85 | <li><i>x</i> ≤ <i>y</i> ⇒ <i>x</i>+<i>z</i> ≤ |
|---|
| 86 | <i>y</i>+<i>z</i></li> |
|---|
| 87 | |
|---|
| 88 | <li><i>x</i> ≤ <i>y</i> and <i>z</i> ≥ 0 ⇒ |
|---|
| 89 | <i>x</i>×<i>z</i> ≤ <i>y</i>×<i>z</i></li> |
|---|
| 90 | |
|---|
| 91 | <li>0 < <i>x</i> ≤ <i>y</i> ⇒ 0 < 1/<i>y</i> ≤ |
|---|
| 92 | 1/<i>x</i></li> |
|---|
| 93 | </ul> |
|---|
| 94 | |
|---|
| 95 | <p>The previous properties are also used (and enough) for <code>abs</code>, |
|---|
| 96 | <code>square</code> and <code>pow</code>. For all the transcendental |
|---|
| 97 | functions (including <code>sqrt</code>), other properties are needed. These |
|---|
| 98 | functions should have the same properties than the corresponding real |
|---|
| 99 | functions. For example, the expected properties for <code>cos</code> |
|---|
| 100 | are:</p> |
|---|
| 101 | |
|---|
| 102 | <ul> |
|---|
| 103 | <li><code>cos</code> is defined for all the valid numbers;</li> |
|---|
| 104 | |
|---|
| 105 | <li>it is 2π-periodic;</li> |
|---|
| 106 | |
|---|
| 107 | <li><code>cos</code>(2π-<i>x</i>) is equal to |
|---|
| 108 | <code>cos</code>(<i>x</i>);</li> |
|---|
| 109 | |
|---|
| 110 | <li><code>cos</code> is a decreasing function on [0,2π].</li> |
|---|
| 111 | </ul> |
|---|
| 112 | |
|---|
| 113 | <h3>Rounding</h3> |
|---|
| 114 | |
|---|
| 115 | <p>If you work with a base type and no inexact result is ever computed, you |
|---|
| 116 | can skip the rest of this paragraph. You can also skip it if you are not |
|---|
| 117 | interested in the inclusion property (if approximate results are enough). |
|---|
| 118 | If you are still reading, it is probably because you want to know the basic |
|---|
| 119 | properties the rounding policy should validate.</p> |
|---|
| 120 | |
|---|
| 121 | <p>Whichever operation or function you consider, the following property |
|---|
| 122 | should be respected: <code>f_down(x,y) <= f(x,y) <= f_up(x,y)</code>. |
|---|
| 123 | Here, <code>f</code> denotes the infinitely precise function computed and |
|---|
| 124 | <code>f_down</code> and <code>f_up</code> are functions which return |
|---|
| 125 | possibly inexact values but of the correct type (the base type). If |
|---|
| 126 | possible, they should try to return the nearest representable value, but it |
|---|
| 127 | is not always easy.</p> |
|---|
| 128 | |
|---|
| 129 | <h3>Constants</h3> |
|---|
| 130 | |
|---|
| 131 | <p>In order for the trigonometric functions to correctly work, the library |
|---|
| 132 | need to know the value of the π constant (and also π/2 and 2π). |
|---|
| 133 | Since these constants may not be representable in the base type, the |
|---|
| 134 | library does not have to know the exact value: a lower bound and an upper |
|---|
| 135 | bound are enough. If these values are not provided by the user, the default |
|---|
| 136 | values will be used: they are integer values (so π is bounded by 3 and |
|---|
| 137 | 4).</p> |
|---|
| 138 | |
|---|
| 139 | <h3>Operators and conversions</h3> |
|---|
| 140 | |
|---|
| 141 | <p>As explained at the beginning, the comparison operators should be |
|---|
| 142 | defined for the base type. The rounding policy defines a lot of functions |
|---|
| 143 | used by the interval library. So the arithmetic operators do not need to be |
|---|
| 144 | defined for the base type (unless required by one of the predefined |
|---|
| 145 | classes). However, there is an exception: the unary minus need to be |
|---|
| 146 | defined. Moreover, this operator should only provide exact results; it is |
|---|
| 147 | the reason why the rounding policy does not provide some negation |
|---|
| 148 | functions.</p> |
|---|
| 149 | |
|---|
| 150 | <p>The conversion from <code>int</code> to the base type needs to be |
|---|
| 151 | defined (only a few values need to be available: -1, 0, 1, 2). The |
|---|
| 152 | conversion the other way around is provided by the rounding policy |
|---|
| 153 | (<code>int_down</code> and <code>int_up</code> members); and no other |
|---|
| 154 | conversion is strictly needed. However, it may be valuable to provide as |
|---|
| 155 | much conversions as possible in the rounding policy (<code>conv_down</code> |
|---|
| 156 | and <code>conv_up</code> members) in order to benefit from interval |
|---|
| 157 | conversions.</p> |
|---|
| 158 | <hr> |
|---|
| 159 | |
|---|
| 160 | <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src= |
|---|
| 161 | "http://www.w3.org/Icons/valid-html401" alt="Valid HTML 4.01 Transitional" |
|---|
| 162 | height="31" width="88"></a></p> |
|---|
| 163 | |
|---|
| 164 | <p>Revised |
|---|
| 165 | <!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-24<!--webbot bot="Timestamp" endspan i-checksum="12172" --></p> |
|---|
| 166 | |
|---|
| 167 | <p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé |
|---|
| 168 | Brönnimann, Polytechnic University<br> |
|---|
| 169 | Copyright © 2004 Guillaume Melquiond</i></p> |
|---|
| 170 | |
|---|
| 171 | <p><i>Distributed under the Boost Software License, Version 1.0. (See |
|---|
| 172 | accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> |
|---|
| 173 | or copy at <a href= |
|---|
| 174 | "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p> |
|---|
| 175 | </body> |
|---|
| 176 | </html> |
|---|
