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| 9 | |
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| 10 | <title>Boost Interval Arithmetic Library</title> |
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| 11 | </head> |
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| 12 | |
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| 13 | <body lang="en"> |
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| 14 | <h1><img src="../../../../boost.png" alt="boost.png (6897 bytes)" align= |
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| 15 | "middle"> Interval Arithmetic Library</h1> |
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| 16 | |
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| 17 | <center> |
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| 18 | <table width="80%" summary=""> |
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| 19 | <tbody> |
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| 20 | <tr> |
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| 21 | <td><b>Contents of this page:</b><br> |
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| 22 | <a href="#intro">Introduction</a><br> |
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| 23 | <a href="#synopsis">Synopsis</a><br> |
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| 24 | <a href="#interval">Template class <code>interval</code></a><br> |
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| 25 | <a href="#opers">Operations and functions</a><br> |
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| 26 | <a href="#interval_lib">Interval support library</a><br> |
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| 27 | <!--<a href="#compil">Compilation notes</a><br>--> |
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| 28 | <a href="#dangers">Common pitfalls and dangers</a><br> |
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| 29 | <a href="#rationale">Rationale</a><br> |
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| 30 | <a href="#acks">History and Acknowledgments</a></td> |
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| 31 | |
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| 32 | <td><b>Other pages associated with this page:</b><br> |
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| 33 | <a href="rounding.htm">Rounding policies</a><br> |
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| 34 | <a href="checking.htm">Checking policies</a><br> |
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| 35 | <a href="policies.htm">Policies manipulation</a><br> |
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| 36 | <a href="comparisons.htm">Comparisons</a><br> |
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| 37 | <a href="numbers.htm">Base number type requirements</a><br> |
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| 38 | <a href="guide.htm">Choosing your own interval type</a><br> |
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| 39 | <a href="examples.htm">Test and example programs</a><br> |
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| 40 | <a href="includes.htm">Headers inclusion</a><br> |
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| 41 | <a href="todo.htm">Some items on the todo list</a></td> |
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| 42 | </tr> |
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| 43 | </tbody> |
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| 44 | </table> |
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| 45 | </center> |
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| 46 | |
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| 47 | <h2 id="intro">Introduction and Overview</h2> |
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| 48 | |
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| 49 | <p>As implied by its name, this library is intended to help manipulating |
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| 50 | mathematical intervals. It consists of a single header <<a href= |
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| 51 | "../../../../boost/numeric/interval.hpp">boost/numeric/interval.hpp</a>> |
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| 52 | and principally a type which can be used as <code>interval<T></code>. |
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| 53 | In fact, this interval template is declared as |
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| 54 | <code>interval<T,Policies></code> where <code>Policies</code> is a |
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| 55 | policy class that controls the various behaviours of the interval class; |
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| 56 | <code>interval<T></code> just happens to pick the default policies |
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| 57 | for the type <code>T</code>.</p> |
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| 58 | |
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| 59 | <p><span style="color: #FF0000; font-weight: bold">Warning!</span> |
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| 60 | Guaranteed interval arithmetic for native floating-point format is not |
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| 61 | supported on every combination of processor, operating system, and |
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| 62 | compiler. This is a list of systems known to work correctly when using |
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| 63 | <code>interval<float></code> and <code>interval<double></code> |
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| 64 | with basic arithmetic operators.</p> |
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| 65 | |
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| 66 | <ul> |
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| 67 | <li>x86-like hardware is supported by the library with GCC, Visual C++ |
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| 68 | ≥ 7.1, Intel compiler (≥ 8 on Windows), CodeWarrior (≥ 9), as |
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| 69 | long as the traditional x87 floating-point unit is used for |
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| 70 | floating-point computations (no <code>-mfpmath=sse2</code> support).</li> |
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| 71 | |
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| 72 | <li>Sparc hardware is supported with GCC and Sun compiler.</li> |
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| 73 | |
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| 74 | <li>PowerPC hardware is supported with GCC and CodeWarrior, when |
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| 75 | floating-point computations are not done with the Altivec unit.</li> |
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| 76 | |
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| 77 | <li>Alpha hardware is supported with GCC, except maybe for the square |
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| 78 | root. The options <code>-mfp-rounding-mode=d -mieee</code> have to be |
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| 79 | used.</li> |
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| 80 | </ul> |
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| 81 | |
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| 82 | <p>The previous list is not exhaustive. And even if a system does not |
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| 83 | provide guaranteed computations for hardware floating-point types, the |
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| 84 | interval library is still usable with user-defined types and for doing box |
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| 85 | arithmetic.</p> |
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| 86 | |
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| 87 | <h3>Interval Arithmetic</h3> |
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| 88 | |
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| 89 | <p>An interval is a pair of numbers which represents all the numbers |
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| 90 | between these two. (Intervals are considered close so the bounds are |
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| 91 | included.) The purpose of this library is to extend the usual arithmetic |
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| 92 | functions to intervals. These intervals will be written [<i>a</i>,<i>b</i>] |
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| 93 | to represent all the numbers between <i>a</i> and <i>b</i> (included). |
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| 94 | <i>a</i> and <i>b</i> can be infinite (but they can not be the same |
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| 95 | infinite) and <i>a</i> ≤ <i>b</i>.</p> |
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| 96 | |
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| 97 | <p>The fundamental property of interval arithmetic is the |
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| 98 | <em><strong>inclusion property</strong></em>:</p> |
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| 99 | |
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| 100 | <dl> |
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| 101 | <dd>``if <i>f</i> is a function on a set of numbers, <i>f</i> can be |
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| 102 | extended to a new function defined on intervals. This new function |
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| 103 | <i>f</i> takes one interval argument and returns an interval result such |
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| 104 | as: ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>], <i>f</i>(<i>x</i>) |
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| 105 | ∈ <i>f</i>([<i>a</i>,<i>b</i>]).''</dd> |
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| 106 | </dl> |
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| 107 | |
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| 108 | <p>Such a property is not limited to functions with only one argument. |
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| 109 | Whenever possible, the interval result should be the smallest one able to |
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| 110 | satisfy the property (it is not really useful if the new functions always |
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| 111 | answer [-∞,+∞]).</p> |
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| 112 | |
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| 113 | <p>There are at least two reasons a user would like to use this library. |
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| 114 | The obvious one is when the user has to compute with intervals. One example |
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| 115 | is when input data have some builtin imprecision: instead of a number, an |
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| 116 | input variable can be passed as an interval. Another example application is |
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| 117 | to solve equations, by bisecting an interval until the interval width is |
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| 118 | small enough. A third example application is in computer graphics, where |
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| 119 | computations with boxes, segments or rays can be reduced to computations |
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| 120 | with points via intervals.</p> |
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| 121 | |
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| 122 | <p>Another common reason to use interval arithmetic is when the computer |
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| 123 | doesn't produce exact results: by using intervals, it is possible to |
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| 124 | quantify the propagation of rounding errors. This approach is used often in |
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| 125 | numerical computation. For example, let's assume the computer stores |
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| 126 | numbers with ten decimal significant digits. To the question 1 + 1E-100 - |
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| 127 | 1, the computer will answer 0 although the correct answer would be 1E-100. |
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| 128 | With the help of interval arithmetic, the computer will answer [0,1E-9]. |
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| 129 | This is quite a huge interval for such a little result, but the precision |
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| 130 | is now known, without having to compute error propagation.</p> |
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| 131 | |
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| 132 | <h3>Numbers, rounding, and exceptional behavior</h3> |
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| 133 | |
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| 134 | <p>The <em><strong>base number type</strong></em> is the type that holds |
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| 135 | the bounds of the interval. In order to successfully use interval |
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| 136 | arithmetic, the base number type must present some <a href= |
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| 137 | "rounding.htm">characteristics</a>. Firstly, due to the definition of an |
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| 138 | interval, the base numbers have to be totally ordered so, for instance, |
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| 139 | <code>complex<T></code> is not usable as base number type for |
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| 140 | intervals. The mathematical functions for the base number type should also |
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| 141 | be compatible with the total order (for instance if x>y and z>t, then |
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| 142 | it should also hold that x+z > y+t), so modulo types are not usable |
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| 143 | either.</p> |
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| 144 | |
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| 145 | <p>Secondly, the computations must be exact or provide some rounding |
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| 146 | methods (for instance, toward minus or plus infinity) if we want to |
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| 147 | guarantee the inclusion property. Note that we also may explicitely specify |
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| 148 | no rounding, for instance if the base number type is exact, i.e. the result |
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| 149 | of a mathematic operation is always computed and represented without loss |
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| 150 | of precision. If the number type is not exact, we may still explicitely |
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| 151 | specify no rounding, with the obvious consequence that the inclusion |
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| 152 | property is no longuer guaranteed.</p> |
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| 153 | |
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| 154 | <p>Finally, because heavy loss of precision is always possible, some |
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| 155 | numbers have to represent infinities or an exceptional behavior must be |
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| 156 | defined. The same situation also occurs for NaN (<i>Not a Number</i>).</p> |
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| 157 | |
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| 158 | <p>Given all this, one may want to limit the template argument T of the |
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| 159 | class template <code>interval</code> to the floating point types |
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| 160 | <code>float</code>, <code>double</code>, and <code>long double</code>, as |
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| 161 | defined by the IEEE-754 Standard. Indeed, if the interval arithmetic is |
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| 162 | intended to replace the arithmetic provided by the floating point unit of a |
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| 163 | processor, these types are the best choice. Unlike |
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| 164 | <code>std::complex</code>, however, we don't want to limit <code>T</code> |
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| 165 | to these types. This is why we allow the rounding and exceptional behaviors |
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| 166 | to be given by the two policies (rounding and checking). We do nevertheless |
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| 167 | provide highly optimized rounding and checking class specializations for |
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| 168 | the above-mentioned floating point types.</p> |
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| 169 | |
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| 170 | <h3>Operations and functions</h3> |
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| 171 | |
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| 172 | <p>It is straightforward to define the elementary arithmetic operations on |
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| 173 | intervals, being guided by the inclusion property. For instance, if [a,b] |
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| 174 | and [c,d] are intervals, [a,b]+[c,d] can be computed by taking the smallest |
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| 175 | interval that contains all the numbers x+y for x in [a,b] and y in [c,d]; |
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| 176 | in this case, rounding a+c down and b+d up will suffice. Other operators |
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| 177 | and functions are similarly defined (see their definitions below).</p> |
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| 178 | |
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| 179 | <h3>Comparisons</h3> |
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| 180 | |
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| 181 | <p>It is also possible to define some comparison operators. Given two |
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| 182 | intervals, the result is a tri-state boolean type |
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| 183 | {<i>false</i>,<i>true,indeterminate</i>}. The answers <i>false</i> and |
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| 184 | <i>true</i> are easy to manipulate since they can directly be mapped on the |
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| 185 | boolean <i>true</i> and <i>false</i>. But it is not the case for the answer |
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| 186 | <em>indeterminate</em> since comparison operators are supposed to be |
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| 187 | boolean functions. So, what to do in order to obtain boolean answers?</p> |
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| 188 | |
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| 189 | <p>One solution consists of deciding to adopt an exceptional behavior, such |
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| 190 | as a failed assertion or raising an exception. In this case, the |
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| 191 | exceptional behavior will be triggered when the result is |
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| 192 | indeterminate.</p> |
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| 193 | |
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| 194 | <p>Another solution is to map <em>indeterminate</em> always to |
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| 195 | <i>false,</i> or always to <i>true</i>. If <i>false</i> is chosen, the |
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| 196 | comparison will be called "<i>certain</i>;" indeed, the result of |
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| 197 | [<i>a</i>,<i>b</i>] < [<i>c</i>,<i>d</i>] will be <i>true</i> if and |
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| 198 | only if: ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i> |
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| 199 | ∈ [<i>c</i>,<i>d</i>], <i>x</i> < <i>y</i>. If <i>true</i> is |
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| 200 | chosen, the comparison will be called "<i>possible</i>;" indeed, the result |
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| 201 | of [<i>a</i>,<i>b</i>] < [<i>c</i>,<i>d</i>] will be <i>true</i> if and |
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| 202 | only if: ∃ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∃ <i>y</i> |
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| 203 | ∈ [<i>c</i>,<i>d</i>], <i>x</i> < <i>y</i>.</p> |
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| 204 | |
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| 205 | <p>Since any of these solution has a clearly defined semantics, it is not |
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| 206 | clear that we should enforce either of them. For this reason, the default |
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| 207 | behavior consists to mimic the real comparisons by throwing an exception in |
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| 208 | the indeterminate case. Other behaviors can be selected bu using specific |
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| 209 | comparison namespace. There is also a bunch of explicitely named comparison |
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| 210 | functions. See <a href="comparisons.htm">comparisons</a> pages for further |
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| 211 | details.</p> |
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| 212 | |
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| 213 | <h3>Overview of the library, and usage</h3> |
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| 214 | |
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| 215 | <p>This library provides two quite distinct levels of usage. One is to use |
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| 216 | the basic class template <code>interval<T></code> without specifying |
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| 217 | the policy. This only requires to know and understand the concepts |
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| 218 | developed above and the content of the namespace boost. In addition to the |
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| 219 | class <code>interval<T></code>, this level of usage provides |
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| 220 | arithmetic operators (<code>+</code>, <code>-</code>, <code>*</code>, |
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| 221 | <code>/</code>), algebraic and piecewise-algebraic functions |
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| 222 | (<code>abs</code>, <code>square</code>, <code>sqrt</code>, |
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| 223 | <code>pow</code>), transcendental and trigonometric functions |
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| 224 | (<code>exp</code>, <code>log</code>, <code>sin</code>, <code>cos</code>, |
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| 225 | <code>tan</code>, <code>asin</code>, <code>acos</code>, <code>atan</code>, |
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| 226 | <code>sinh</code>, <code>cosh</code>, <code>tanh</code>, |
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| 227 | <code>asinh</code>, <code>acosh</code>, <code>atanh</code>), and the |
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| 228 | standard comparison operators (<code><</code>, <code><=</code>, |
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| 229 | <code>></code>, <code>>=</code>, <code>==</code>, <code>!=</code>), |
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| 230 | as well as several interval-specific functions (<code>min</code>, |
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| 231 | <code>max</code>, which have a different meaning than <code>std::min</code> |
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| 232 | and <code>std::max</code>; <code>lower</code>, <code>upper</code>, |
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| 233 | <code>width</code>, <code>median</code>, <code>empty</code>, |
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| 234 | <code>singleton</code>, <code>equal</code>, <code>in</code>, |
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| 235 | <code>zero_in</code>, <code>subset</code>, <code>proper_subset</code>, |
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| 236 | <code>overlap</code>, <code>intersection</code>, <code>hull</code>, |
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| 237 | <code>bisect</code>).</p> |
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| 238 | |
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| 239 | <p>For some functions which take several parameters of type |
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| 240 | <code>interval<T></code>, all combinations of argument types |
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| 241 | <code>T</code> and <code>interval<T></code> which contain at least |
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| 242 | one <code>interval<T></code>, are considered in order to avoid a |
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| 243 | conversion from the arguments of type <code>T</code> to a singleton of type |
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| 244 | <code>interval<T></code>. This is done for efficiency reasons (the |
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| 245 | fact that an argument is a singleton sometimes renders some tests |
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| 246 | unnecessary).</p> |
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| 247 | |
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| 248 | <p>A somewhat more advanced usage of this library is to hand-pick the |
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| 249 | policies <code>Rounding</code> and <code>Checking</code> and pass them to |
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| 250 | <code>interval<T, Policies></code> through the use of <code>Policies |
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| 251 | := boost::numeric::interval_lib::policies<Rounding,Checking></code>. |
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| 252 | Appropriate policies can be fabricated by using the various classes |
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| 253 | provided in the namespace <code>boost::numeric::interval_lib</code> as |
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| 254 | detailed in section <a href="#interval_lib">Interval Support Library</a>. |
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| 255 | It is also possible to choose the comparison scheme by overloading |
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| 256 | operators through namespaces.</p> |
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| 257 | |
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| 258 | <h2><a name="synopsis" id="synopsis"></a>Synopsis</h2> |
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| 259 | <pre> |
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| 260 | namespace boost { |
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| 261 | namespace numeric { |
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| 262 | |
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| 263 | namespace interval_lib { |
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| 264 | |
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| 265 | /* this declaration is necessary for the declaration of interval */ |
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| 266 | template <class T> struct default_policies; |
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| 267 | |
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| 268 | /* ... ; the full synopsis of namespace interval_lib can be found <a href= |
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| 269 | "#interval_lib">here</a> */ |
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| 270 | |
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| 271 | } // namespace interval_lib |
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| 272 | |
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| 273 | |
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| 274 | /* template interval_policies; class definition can be found <a href= |
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| 275 | "policies.htm">here</a> */ |
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| 276 | template<class Rounding, class Checking> |
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| 277 | struct interval_policies; |
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| 278 | |
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| 279 | /* template class interval; class definition can be found <a href= |
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| 280 | "#interval">here</a> */ |
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| 281 | template<class T, class Policies = typename interval_lib::default_policies<T>::type > class interval; |
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| 282 | |
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| 283 | /* arithmetic operators involving intervals */ |
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| 284 | template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x); |
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| 285 | template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x); |
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| 286 | |
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| 287 | template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 288 | template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x, const T& y); |
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| 289 | template <class T, class Policies> interval<T, Policies> operator+(const T& x, const interval<T, Policies>& y); |
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| 290 | |
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| 291 | template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 292 | template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x, const T& y); |
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| 293 | template <class T, class Policies> interval<T, Policies> operator-(const T& x, const interval<T, Policies>& y); |
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| 294 | |
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| 295 | template <class T, class Policies> interval<T, Policies> operator*(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 296 | template <class T, class Policies> interval<T, Policies> operator*(const interval<T, Policies>& x, const T& y); |
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| 297 | template <class T, class Policies> interval<T, Policies> operator*(const T& x, const interval<T, Policies>& y); |
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| 298 | |
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| 299 | template <class T, class Policies> interval<T, Policies> operator/(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 300 | template <class T, class Policies> interval<T, Policies> operator/(const interval<T, Policies>& x, const T& y); |
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| 301 | template <class T, class Policies> interval<T, Policies> operator/(const T& r, const interval<T, Policies>& x); |
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| 302 | |
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| 303 | /* algebraic functions: sqrt, abs, square, pow, root */ |
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| 304 | template <class T, class Policies> interval<T, Policies> abs(const interval<T, Policies>& x); |
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| 305 | template <class T, class Policies> interval<T, Policies> sqrt(const interval<T, Policies>& x); |
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| 306 | template <class T, class Policies> interval<T, Policies> square(const interval<T, Policies>& x); |
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| 307 | template <class T, class Policies> interval<T, Policies> pow(const interval<T, Policies>& x, int y); |
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| 308 | template <class T, class Policies> interval<T, Policies> root(const interval<T, Policies>& x, int y); |
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| 309 | |
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| 310 | /* transcendental functions: exp, log */ |
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| 311 | template <class T, class Policies> interval<T, Policies> exp(const interval<T, Policies>& x); |
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| 312 | template <class T, class Policies> interval<T, Policies> log(const interval<T, Policies>& x); |
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| 313 | |
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| 314 | /* fmod, for trigonometric function argument reduction (see below) */ |
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| 315 | template <class T, class Policies> interval<T, Policies> fmod(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 316 | template <class T, class Policies> interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y); |
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| 317 | template <class T, class Policies> interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y); |
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| 318 | |
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| 319 | /* trigonometric functions */ |
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| 320 | template <class T, class Policies> interval<T, Policies> sin(const interval<T, Policies>& x); |
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| 321 | template <class T, class Policies> interval<T, Policies> cos(const interval<T, Policies>& x); |
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| 322 | template <class T, class Policies> interval<T, Policies> tan(const interval<T, Policies>& x); |
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| 323 | template <class T, class Policies> interval<T, Policies> asin(const interval<T, Policies>& x); |
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| 324 | template <class T, class Policies> interval<T, Policies> acos(const interval<T, Policies>& x); |
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| 325 | template <class T, class Policies> interval<T, Policies> atan(const interval<T, Policies>& x); |
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| 326 | |
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| 327 | /* hyperbolic trigonometric functions */ |
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| 328 | template <class T, class Policies> interval<T, Policies> sinh(const interval<T, Policies>& x); |
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| 329 | template <class T, class Policies> interval<T, Policies> cosh(const interval<T, Policies>& x); |
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| 330 | template <class T, class Policies> interval<T, Policies> tanh(const interval<T, Policies>& x); |
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| 331 | template <class T, class Policies> interval<T, Policies> asinh(const interval<T, Policies>& x); |
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| 332 | template <class T, class Policies> interval<T, Policies> acosh(const interval<T, Policies>& x); |
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| 333 | template <class T, class Policies> interval<T, Policies> atanh(const interval<T, Policies>& x); |
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| 334 | |
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| 335 | /* min, max external functions (NOT std::min/max, see below) */ |
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| 336 | template <class T, class Policies> interval<T, Policies> max(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 337 | template <class T, class Policies> interval<T, Policies> max(const interval<T, Policies>& x, const T& y); |
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| 338 | template <class T, class Policies> interval<T, Policies> max(const T& x, const interval<T, Policies>& y); |
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| 339 | template <class T, class Policies> interval<T, Policies> min(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 340 | template <class T, class Policies> interval<T, Policies> min(const interval<T, Policies>& x, const T& y); |
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| 341 | template <class T, class Policies> interval<T, Policies> min(const T& x, const interval<T, Policies>& y); |
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| 342 | |
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| 343 | /* bounds-related interval functions */ |
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| 344 | template <class T, class Policies> T lower(const interval<T, Policies>& x); |
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| 345 | template <class T, class Policies> T upper(const interval<T, Policies>& x); |
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| 346 | template <class T, class Policies> T width(const interval<T, Policies>& x); |
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| 347 | template <class T, class Policies> T median(const interval<T, Policies>& x); |
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| 348 | template <class T, class Policies> T norm(const interval<T, Policies>& x); |
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| 349 | |
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| 350 | /* bounds-related interval functions */ |
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| 351 | template <class T, class Policies> bool empty(const interval<T, Policies>& b); |
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| 352 | template <class T, class Policies> bool singleton(const interval<T, Policies>& x); |
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| 353 | template <class T, class Policies> bool equal(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 354 | template <class T, class Policies> bool in(const T& r, const interval<T, Policies>& b); |
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| 355 | template <class T, class Policies> bool zero_in(const interval<T, Policies>& b); |
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| 356 | template <class T, class Policies> bool subset(const interval<T, Policies>& a, const interval<T, Policies>& b); |
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| 357 | template <class T, class Policies> bool proper_subset(const interval<T, Policies>& a, const interval<T, Policies>& b); |
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| 358 | template <class T, class Policies> bool overlap(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 359 | |
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| 360 | /* set manipulation interval functions */ |
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| 361 | template <class T, class Policies> interval<T, Policies> intersection(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 362 | template <class T, class Policies> interval<T, Policies> hull(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 363 | template <class T, class Policies> interval<T, Policies> hull(const interval<T, Policies>& x, const T& y); |
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| 364 | template <class T, class Policies> interval<T, Policies> hull(const T& x, const interval<T, Policies>& y); |
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| 365 | template <class T, class Policies> interval<T, Policies> hull(const T& x, const T& y); |
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| 366 | template <class T, class Policies> std::pair<interval<T, Policies>, interval<T, Policies> > bisect(const interval<T, Policies>& x); |
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| 367 | |
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| 368 | /* interval comparison operators */ |
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| 369 | template<class T, class Policies> bool operator<(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 370 | template<class T, class Policies> bool operator<(const interval<T, Policies>& x, const T& y); |
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| 371 | template<class T, class Policies> bool operator<(const T& x, const interval<T, Policies>& y); |
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| 372 | |
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| 373 | template<class T, class Policies> bool operator<=(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 374 | template<class T, class Policies> bool operator<=(const interval<T, Policies>& x, const T& y); |
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| 375 | template<class T, class Policies> bool operator<=(const T& x, const interval<T, Policies>& y); |
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| 376 | |
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| 377 | template<class T, class Policies> bool operator>(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 378 | template<class T, class Policies> bool operator>(const interval<T, Policies>& x, const T& y); |
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| 379 | template<class T, class Policies> bool operator>(const T& x, const interval<T, Policies>& y); |
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| 380 | |
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| 381 | template<class T, class Policies> bool operator>=(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 382 | template<class T, class Policies> bool operator>=(const interval<T, Policies>& x, const T& y); |
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| 383 | template<class T, class Policies> bool operator>=(const T& x, const interval<T, Policies>& y); |
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| 384 | </pre> |
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| 385 | <pre> |
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| 386 | template<class T, class Policies> bool operator==(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 387 | template<class T, class Policies> bool operator==(const interval<T, Policies>& x, const T& y); |
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| 388 | template<class T, class Policies> bool operator==(const T& x, const interval<T, Policies>& y); |
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| 389 | |
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| 390 | template<class T, class Policies> bool operator!=(const interval<T, Policies>& x, const interval<T, Policies>& y); |
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| 391 | template<class T, class Policies> bool operator!=(const interval<T, Policies>& x, const T& y); |
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| 392 | template<class T, class Policies> bool operator!=(const T& x, const interval<T, Policies>& y); |
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| 393 | |
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| 394 | namespace interval_lib { |
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| 395 | |
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| 396 | template<class T, class Policies> interval<T, Policies> division_part1(const interval<T, Policies>& x, const interval<T, Policies& y, bool& b); |
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| 397 | template<class T, class Policies> interval<T, Policies> division_part2(const interval<T, Policies>& x, const interval<T, Policies& y, bool b = true); |
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| 398 | template<class T, class Policies> interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x); |
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| 399 | |
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| 400 | template<class I> I add(const typename I::base_type& x, const typename I::base_type& y); |
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| 401 | template<class I> I sub(const typename I::base_type& x, const typename I::base_type& y); |
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| 402 | template<class I> I mul(const typename I::base_type& x, const typename I::base_type& y); |
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| 403 | template<class I> I div(const typename I::base_type& x, const typename I::base_type& y); |
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| 404 | |
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| 405 | } // namespace interval_lib |
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| 406 | |
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| 407 | } // namespace numeric |
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| 408 | } // namespace boost |
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| 409 | </pre> |
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| 410 | |
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| 411 | <h2><a name="interval" id="interval"></a>Template class |
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| 412 | <code>interval</code></h2>The public interface of the template class |
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| 413 | interval itself is kept at a simplest minimum: |
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| 414 | <pre> |
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| 415 | template <class T, class Policies = typename interval_lib::default_policies<T>::type> |
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| 416 | class interval |
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| 417 | { |
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| 418 | public: |
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| 419 | typedef T base_type; |
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| 420 | typedef Policies traits_type; |
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| 421 | |
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| 422 | interval(); |
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| 423 | interval(T const &v); |
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| 424 | template<class T1> interval(T1 const &v); |
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| 425 | interval(T const &l, T const &u); |
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| 426 | template<class T1, class T2> interval(T1 const &l, T2 const &u); |
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| 427 | interval(interval<T, Policies> const &r); |
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| 428 | template<class Policies1> interval(interval<T, Policies1> const &r); |
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| 429 | template<class T1, class Policies1> interval(interval<T1, Policies1> const &r); |
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| 430 | |
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| 431 | interval &operator=(T const &v); |
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| 432 | template<class T1> interval &operator=(T1 const &v); |
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| 433 | interval &operator=(interval<T, Policies> const &r); |
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| 434 | template<class Policies1> interval &operator=(interval<T, Policies1> const &r); |
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| 435 | template<class T1, class Policies1> interval &operator=(interval<T1, Policies1> const &r); |
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| 436 | |
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| 437 | void assign(T const &l, T const &u); |
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| 438 | |
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| 439 | T const &lower() const; |
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| 440 | T const &upper() const; |
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| 441 | |
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| 442 | static interval empty(); |
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| 443 | static interval whole(); |
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| 444 | static interval hull(T const &x, T const &y); |
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| 445 | |
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| 446 | interval& operator+= (T const &r); |
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| 447 | interval& operator-= (T const &r); |
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| 448 | interval& operator*= (T const &r); |
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| 449 | interval& operator/= (T const &r); |
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| 450 | interval& operator+= (interval const &r); |
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| 451 | interval& operator-= (interval const &r); |
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| 452 | interval& operator*= (interval const &r); |
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| 453 | interval& operator/= (interval const &r); |
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| 454 | }; |
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| 455 | </pre> |
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| 456 | |
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| 457 | <p>The constructors create an interval enclosing their arguments. If there |
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| 458 | are two arguments, the first one is assumed to be the left bound and the |
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| 459 | second one is the right bound. Consequently, the arguments need to be |
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| 460 | ordered. If the property !(l <= u) is not respected, the checking policy |
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| 461 | will be used to create an empty interval. If no argument is given, the |
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| 462 | created interval is the singleton zero.</p> |
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| 463 | |
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| 464 | <p>If the type of the arguments is the same as the base number type, the |
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| 465 | values are directly used for the bounds. If it is not the same type, the |
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| 466 | library will use the rounding policy in order to convert the arguments |
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| 467 | (<code>conv_down</code> and <code>conv_up</code>) and create an enclosing |
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| 468 | interval. When the argument is an interval with a different policy, the |
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| 469 | input interval is checked in order to correctly propagate its emptiness (if |
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| 470 | empty).</p> |
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| 471 | |
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| 472 | <p>The assignment operators behave similarly, except they obviously take |
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| 473 | one argument only. There is also an <code>assign</code> function in order |
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| 474 | to directly change the bounds of an interval. It behaves like the |
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| 475 | two-arguments constructors if the bounds are not ordered. There is no |
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| 476 | assign function that directly takes an interval or only one number as a |
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| 477 | parameter; just use the assignment operators in such a case.</p> |
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| 478 | |
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| 479 | <p>The type of the bounds and the policies of the interval type define the |
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| 480 | set of values the intervals contain. E.g. with the default policies, |
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| 481 | intervals are subsets of the set of real numbers. The static functions |
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| 482 | <code>empty</code> and <code>whole</code> produce the intervals/subsets |
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| 483 | that are repectively the empty subset and the whole set. They are static |
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| 484 | member functions rather than global functions because they cannot guess |
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| 485 | their return types. Likewise for <code>hull</code>. <code>empty</code> and |
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| 486 | <code>whole</code> involve the checking policy in order to get the bounds |
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| 487 | of the resulting intervals.</p> |
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| 488 | |
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| 489 | <h2><a name="opers" id="opers"></a>Operations and Functions</h2> |
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| 490 | |
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| 491 | <p>Some of the following functions expect <code>min</code> and |
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| 492 | <code>max</code> to be defined for the base type. Those are the only |
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| 493 | requirements for the <code>interval</code> class (but the policies can have |
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| 494 | other requirements).</p> |
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| 495 | |
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| 496 | <h4>Operators <code>+</code> <code>-</code> <code>*</code> <code>/</code> |
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| 497 | <code>+=</code> <code>-=</code> <code>*=</code> <code>/=</code></h4> |
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| 498 | |
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| 499 | <p>The basic operations are the unary minus and the binary <code>+</code> |
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| 500 | <code>-</code> <code>*</code> <code>/</code>. The unary minus takes an |
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| 501 | interval and returns an interval. The binary operations take two intervals, |
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| 502 | or one interval and a number, and return an interval. If an argument is a |
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| 503 | number instead of an interval, you can expect the result to be the same as |
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| 504 | if the number was first converted to an interval. This property will be |
|---|
| 505 | true for all the following functions and operators.</p> |
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| 506 | |
|---|
| 507 | <p>There are also some assignment operators <code>+=</code> <code>-=</code> |
|---|
| 508 | <code>*=</code> <code>/=</code>. There is not much to say: <code>x op= |
|---|
| 509 | y</code> is equivalent to <code>x = x op y</code>. If an exception is |
|---|
| 510 | thrown during the computations, the l-value is not modified (but it may be |
|---|
| 511 | corrupt if an exception is thrown by the base type during an |
|---|
| 512 | assignment).</p> |
|---|
| 513 | |
|---|
| 514 | <p>The operators <code>/</code> and <code>/=</code> will try to produce an |
|---|
| 515 | empty interval if the denominator is exactly zero. If the denominator |
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| 516 | contains zero (but not only zero), the result will be the smallest interval |
|---|
| 517 | containing the set of division results; so one of its bound will be |
|---|
| 518 | infinite, but it may not be the whole interval.</p> |
|---|
| 519 | |
|---|
| 520 | <h4><code>lower</code> <code>upper</code> <code>median</code> |
|---|
| 521 | <code>width</code> <code>norm</code></h4> |
|---|
| 522 | |
|---|
| 523 | <p><code>lower</code>, <code>upper</code>, <code>median</code> respectively |
|---|
| 524 | compute the lower bound, the upper bound, and the median number of an |
|---|
| 525 | interval (<code>(lower+upper)/2</code> rounded to nearest). |
|---|
| 526 | <code>width</code> computes the width of an interval |
|---|
| 527 | (<code>upper-lower</code> rounded toward plus infinity). <code>norm</code> |
|---|
| 528 | computes an upper bound of the interval in absolute value; it is a |
|---|
| 529 | mathematical norm (hence the name) similar to the absolute value for real |
|---|
| 530 | numbers.</p> |
|---|
| 531 | |
|---|
| 532 | <h4><code>min</code> <code>max</code> <code>abs</code> <code>square</code> |
|---|
| 533 | <code>pow</code> <code>root</code> <code>division_part?</code> |
|---|
| 534 | <code>multiplicative_inverse</code></h4> |
|---|
| 535 | |
|---|
| 536 | <p>The functions <code>min</code>, <code>max</code> and <code>abs</code> |
|---|
| 537 | are also defined. Please do not mistake them for the functions defined in |
|---|
| 538 | the standard library (aka <code>a<b?a:b</code>, <code>a>b?a:b</code>, |
|---|
| 539 | <code>a<0?-a:a</code>). These functions are compatible with the |
|---|
| 540 | elementary property of interval arithmetic. For example, |
|---|
| 541 | max([<i>a</i>,<i>b</i>], [<i>c</i>,<i>d</i>]) = {max(<i>x</i>,<i>y</i>) |
|---|
| 542 | such that <i>x</i> in [<i>a</i>,<i>b</i>] and <i>y</i> in |
|---|
| 543 | [<i>c</i>,<i>d</i>]}. They are not defined in the <code>std</code> |
|---|
| 544 | namespace but in the boost namespace in order to avoid conflict with the |
|---|
| 545 | other definitions.</p> |
|---|
| 546 | |
|---|
| 547 | <p>The <code>square</code> function is quite particular. As you can expect |
|---|
| 548 | from its name, it computes the square of its argument. The reason this |
|---|
| 549 | function is provided is: <code>square(x)</code> is not <code>x*x</code> but |
|---|
| 550 | only a subset when <code>x</code> contains zero. For example, [-2,2]*[-2,2] |
|---|
| 551 | = [-4,4] but [-2,2]² = [0,4]; the result is a smaller interval. |
|---|
| 552 | Consequently, <code>square(x)</code> should be used instead of |
|---|
| 553 | <code>x*x</code> because of its better accuracy and a small performance |
|---|
| 554 | improvement.</p> |
|---|
| 555 | |
|---|
| 556 | <p>As for <code>square</code>, <code>pow</code> provides an efficient and |
|---|
| 557 | more accurate way to compute the integer power of an interval. Please note: |
|---|
| 558 | when the power is 0 and the interval is not empty, the result is 1, even if |
|---|
| 559 | the input interval contains 0. <code>root</code> computes the integer root |
|---|
| 560 | of an interval (<code>root(pow(x,k),k)</code> encloses <code>x</code> or |
|---|
| 561 | <code>abs(x)</code> depending on whether <code>k</code> is odd or even). |
|---|
| 562 | The behavior of <code>root</code> is not defined if the integer argument is |
|---|
| 563 | not positive. <code>multiplicative_inverse</code> computes |
|---|
| 564 | <code>1/x</code>.</p> |
|---|
| 565 | |
|---|
| 566 | <p>The functions <code>division_part1</code> and |
|---|
| 567 | <code>division_part2</code> are useful when the user expects the division |
|---|
| 568 | to return disjoint intervals if necessary. For example, the narrowest |
|---|
| 569 | closed set containg [2,3] / [-2,1] is not ]-∞,∞[ but the union |
|---|
| 570 | of ]-∞,-1] and [2,∞[. When the result of the division is |
|---|
| 571 | representable by only one interval, <code>division_part1</code> returns |
|---|
| 572 | this interval and sets the boolean reference to <code>false</code>. |
|---|
| 573 | However, if the result needs two intervals, <code>division_part1</code> |
|---|
| 574 | returns the negative part and sets the boolean reference to |
|---|
| 575 | <code>true</code>; a call to <code>division_part2</code> is now needed to |
|---|
| 576 | get the positive part. This second function can take the boolean returned |
|---|
| 577 | by the first function as last argument. If this bool is not given, its |
|---|
| 578 | value is assumed to be true and the behavior of the function is then |
|---|
| 579 | undetermined if the division does not produce a second interval.</p> |
|---|
| 580 | |
|---|
| 581 | <h4><code>intersect</code> <code>hull</code> <code>overlap</code> |
|---|
| 582 | <code>in</code> <code>zero_in</code> <code>subset</code> |
|---|
| 583 | <code>proper_subset</code> <code>empty</code> <code>singleton</code> |
|---|
| 584 | <code>equal</code></h4> |
|---|
| 585 | |
|---|
| 586 | <p><code>intersect</code> computes the set intersection of two closed sets, |
|---|
| 587 | <code>hull</code> computes the smallest interval which contains the two |
|---|
| 588 | parameters; those parameters can be numbers or intervals. If one of the |
|---|
| 589 | arguments is an invalid number or an empty interval, the function will only |
|---|
| 590 | use the other argument to compute the resulting interval (if allowed by the |
|---|
| 591 | checking policy).</p> |
|---|
| 592 | |
|---|
| 593 | <p>There is no union function since the union of two intervals is not an |
|---|
| 594 | interval if they do not overlap. If they overlap, the <code>hull</code> |
|---|
| 595 | function computes the union.</p> |
|---|
| 596 | |
|---|
| 597 | <p>The function <code>overlap</code> tests if two intervals have some |
|---|
| 598 | common subset. <code>in</code> tests if a number is in an interval; |
|---|
| 599 | <code>zero_in</code> is a variant which tests if zero is in the interval. |
|---|
| 600 | <code>subset</code> tests if the first interval is a subset of the second; |
|---|
| 601 | and <code>proper_subset</code> tests if it is a proper subset. |
|---|
| 602 | <code>empty</code> and <code>singleton</code> test if an interval is empty |
|---|
| 603 | or is a singleton. Finally, <code>equal</code> tests if two intervals are |
|---|
| 604 | equal.</p> |
|---|
| 605 | |
|---|
| 606 | <h4><code>sqrt</code> <code>log</code> <code>exp</code> <code>sin</code> |
|---|
| 607 | <code>cos</code> <code>tan</code> <code>asin</code> <code>acos</code> |
|---|
| 608 | <code>atan</code> <code>sinh</code> <code>cosh</code> <code>tanh</code> |
|---|
| 609 | <code>asinh</code> <code>acosh</code> <code>atanh</code> |
|---|
| 610 | <code>fmod</code></h4> |
|---|
| 611 | |
|---|
| 612 | <p>The functions <code>sqrt</code>, <code>log</code>, <code>exp</code>, |
|---|
| 613 | <code>sin</code>, <code>cos</code>, <code>tan</code>, <code>asin</code>, |
|---|
| 614 | <code>acos</code>, <code>atan</code>, <code>sinh</code>, <code>cosh</code>, |
|---|
| 615 | <code>tanh</code>, <code>asinh</code>, <code>acosh</code>, |
|---|
| 616 | <code>atanh</code> are also defined. There is not much to say; these |
|---|
| 617 | functions extend the traditional functions to the intervals and respect the |
|---|
| 618 | basic property of interval arithmetic. They use the <a href= |
|---|
| 619 | "checking.htm">checking</a> policy to produce empty intervals when the |
|---|
| 620 | input interval is strictly outside of the domain of the function.</p> |
|---|
| 621 | |
|---|
| 622 | <p>The function <code>fmod(interval x, interval y)</code> expects the lower |
|---|
| 623 | bound of <code>y</code> to be strictly positive and returns an interval |
|---|
| 624 | <code>z</code> such as <code>0 <= z.lower() < y.upper()</code> and |
|---|
| 625 | such as <code>z</code> is a superset of <code>x-n*y</code> (with |
|---|
| 626 | <code>n</code> being an integer). So, if the two arguments are positive |
|---|
| 627 | singletons, this function <code>fmod(interval, interval)</code> will behave |
|---|
| 628 | like the traditional function <code>fmod(double, double)</code>.</p> |
|---|
| 629 | |
|---|
| 630 | <p>Please note that <code>fmod</code> does not respect the inclusion |
|---|
| 631 | property of arithmetic interval. For example, the result of |
|---|
| 632 | <code>fmod</code>([13,17],[7,8]) should be [0,8] (since it must contain |
|---|
| 633 | [0,3] and [5,8]). But this answer is not really useful when the purpose is |
|---|
| 634 | to restrict an interval in order to compute a periodic function. It is the |
|---|
| 635 | reason why <code>fmod</code> will answer [5,10].</p> |
|---|
| 636 | |
|---|
| 637 | <h4><code>add</code> <code>sub</code> <code>mul</code> |
|---|
| 638 | <code>div</code></h4> |
|---|
| 639 | |
|---|
| 640 | <p>These four functions take two numbers and return the enclosing interval |
|---|
| 641 | for the operations. It avoids converting a number to an interval before an |
|---|
| 642 | operation, it can result in a better code with poor optimizers.</p> |
|---|
| 643 | |
|---|
| 644 | <h3>Constants</h3> |
|---|
| 645 | |
|---|
| 646 | <p>Some constants are hidden in the |
|---|
| 647 | <code>boost::numeric::interval_lib</code> namespace. They need to be |
|---|
| 648 | explicitely templated by the interval type. The functions are |
|---|
| 649 | <code>pi<I>()</code>, <code>pi_half<I>()</code> and |
|---|
| 650 | <code>pi_twice<I>()</code>, and they return an object of interval |
|---|
| 651 | type <code>I</code>. Their respective values are π, π/2 and |
|---|
| 652 | 2π.</p> |
|---|
| 653 | |
|---|
| 654 | <h3>Exception throwing</h3> |
|---|
| 655 | |
|---|
| 656 | <p>The interval class and all the functions defined around this class never |
|---|
| 657 | throw any exceptions by themselves. However, it does not mean that an |
|---|
| 658 | operation will never throw an exception. For example, let's consider the |
|---|
| 659 | copy constructor. As explained before, it is the default copy constructor |
|---|
| 660 | generated by the compiler. So it will not throw an exception if the copy |
|---|
| 661 | constructor of the base type does not throw an exception.</p> |
|---|
| 662 | |
|---|
| 663 | <p>The same situation applies to all the functions: exceptions will only be |
|---|
| 664 | thrown if the base type or one of the two policies throws an exception.</p> |
|---|
| 665 | |
|---|
| 666 | <h2 id="interval_lib">Interval Support Library</h2> |
|---|
| 667 | |
|---|
| 668 | <p>The interval support library consists of a collection of classes that |
|---|
| 669 | can be used and combined to fabricate almost various commonly-needed |
|---|
| 670 | interval policies. In contrast to the basic classes and functions which are |
|---|
| 671 | used in conjunction with <code>interval<T></code> (and the default |
|---|
| 672 | policies as the implicit second template parameter in this type), which |
|---|
| 673 | belong simply to the namespace <code>boost</code>, these components belong |
|---|
| 674 | to the namespace <code>boost::numeric::interval_lib</code>.</p> |
|---|
| 675 | |
|---|
| 676 | <p>We merely give the synopsis here and defer each section to a separate |
|---|
| 677 | web page since it is only intended for the advanced user. This allows to |
|---|
| 678 | expand on each topic with examples, without unduly stretching the limits of |
|---|
| 679 | this document.</p> |
|---|
| 680 | |
|---|
| 681 | <h4>Synopsis</h4> |
|---|
| 682 | <pre> |
|---|
| 683 | namespace boost { |
|---|
| 684 | namespace numeric { |
|---|
| 685 | namespace interval_lib { |
|---|
| 686 | |
|---|
| 687 | <span style= |
|---|
| 688 | "color: #FF0000">/* built-in rounding policy and its specializations */</span> |
|---|
| 689 | template <class T> struct rounded_math; |
|---|
| 690 | template <> struct rounded_math<float>; |
|---|
| 691 | template <> struct rounded_math<double>; |
|---|
| 692 | template <> struct rounded_math<long double>; |
|---|
| 693 | |
|---|
| 694 | <span style= |
|---|
| 695 | "color: #FF0000">/* built-in rounding construction blocks */</span> |
|---|
| 696 | template <class T> struct rounding_control; |
|---|
| 697 | |
|---|
| 698 | template <class T, class Rounding = rounding_control<T> > struct rounded_arith_exact; |
|---|
| 699 | template <class T, class Rounding = rounding_control<T> > struct rounded_arith_std; |
|---|
| 700 | template <class T, class Rounding = rounding_control<T> > struct rounded_arith_opp; |
|---|
| 701 | |
|---|
| 702 | template <class T, class Rounding> struct rounded_transc_dummy; |
|---|
| 703 | template <class T, class Rounding = rounded_arith_exact<T> > struct rounded_transc_exact; |
|---|
| 704 | template <class T, class Rounding = rounded_arith_std <T> > struct rounded_transc_std; |
|---|
| 705 | template <class T, class Rounding = rounded_arith_opp <T> > struct rounded_transc_opp; |
|---|
| 706 | |
|---|
| 707 | template <class Rounding> struct save_state; |
|---|
| 708 | template <class Rounding> struct save_state_nothing; |
|---|
| 709 | |
|---|
| 710 | <span style="color: #FF0000">/* built-in checking policies */</span> |
|---|
| 711 | template <class T> struct checking_base; |
|---|
| 712 | template <class T, class Checking = checking_base<T>, class Exception = exception_create_empty> struct checking_no_empty; |
|---|
| 713 | template <class T, class Checking = checking_base<T> > struct checking_no_nan; |
|---|
| 714 | template <class T, class Checking = checking_base<T>, class Exception = exception_invalid_number> struct checking_catch_nan; |
|---|
| 715 | template <class T> struct checking_strict; |
|---|
| 716 | |
|---|
| 717 | <span style= |
|---|
| 718 | "color: #FF0000">/* some metaprogramming to manipulate interval policies */</span> |
|---|
| 719 | template <class Rounding, class Checking> struct policies; |
|---|
| 720 | template <class OldInterval, class NewRounding> struct change_rounding; |
|---|
| 721 | template <class OldInterval, class NewChecking> struct change_checking; |
|---|
| 722 | template <class OldInterval> struct unprotect; |
|---|
| 723 | |
|---|
| 724 | <span style= |
|---|
| 725 | "color: #FF0000">/* constants, need to be explicitly templated */</span> |
|---|
| 726 | template<class I> I pi(); |
|---|
| 727 | template<class I> I pi_half(); |
|---|
| 728 | template<class I> I pi_twice(); |
|---|
| 729 | |
|---|
| 730 | <span style="color: #FF0000">/* interval explicit comparison functions: |
|---|
| 731 | * the mode can be cer=certainly or pos=possibly, |
|---|
| 732 | * the function lt=less_than, gt=greater_than, le=less_than_or_equal_to, ge=greater_than_or_equal_to |
|---|
| 733 | * eq=equal_to, ne= not_equal_to */</span> |
|---|
| 734 | template <class T, class Policies> bool cerlt(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 735 | template <class T, class Policies> bool cerlt(const interval<T, Policies>& x, const T& y); |
|---|
| 736 | template <class T, class Policies> bool cerlt(const T& x, const interval<T, Policies>& y); |
|---|
| 737 | |
|---|
| 738 | template <class T, class Policies> bool cerle(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 739 | template <class T, class Policies> bool cerle(const interval<T, Policies>& x, const T& y); |
|---|
| 740 | template <class T, class Policies> bool cerle(const T& x, const interval<T, Policies>& y); |
|---|
| 741 | |
|---|
| 742 | template <class T, class Policies> bool cergt(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 743 | template <class T, class Policies> bool cergt(const interval<T, Policies>& x, const T& y); |
|---|
| 744 | template <class T, class Policies> bool cergt(const T& x, const interval<T, Policies>& y); |
|---|
| 745 | |
|---|
| 746 | template <class T, class Policies> bool cerge(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 747 | template <class T, class Policies> bool cerge(const interval<T, Policies>& x, const T& y); |
|---|
| 748 | template <class T, class Policies> bool cerge(const T& x, const interval<T, Policies>& y); |
|---|
| 749 | |
|---|
| 750 | template <class T, class Policies> bool cereq(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 751 | template <class T, class Policies> bool cereq(const interval<T, Policies>& x, const T& y); |
|---|
| 752 | template <class T, class Policies> bool cereq(const T& x, const interval<T, Policies>& y); |
|---|
| 753 | |
|---|
| 754 | template <class T, class Policies> bool cerne(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 755 | template <class T, class Policies> bool cerne(const interval<T, Policies>& x, const T& y); |
|---|
| 756 | template <class T, class Policies> bool cerne(const T& x, const interval<T, Policies>& y); |
|---|
| 757 | |
|---|
| 758 | template <class T, class Policies> bool poslt(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 759 | template <class T, class Policies> bool poslt(const interval<T, Policies>& x, const T& y); |
|---|
| 760 | template <class T, class Policies> bool poslt(const T& x, const interval<T, Policies>& y); |
|---|
| 761 | |
|---|
| 762 | template <class T, class Policies> bool posle(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 763 | template <class T, class Policies> bool posle(const interval<T, Policies>& x, const T& y); |
|---|
| 764 | template <class T, class Policies> bool posle(const T& x, const interval<T, Policies>& y); |
|---|
| 765 | |
|---|
| 766 | template <class T, class Policies> bool posgt(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 767 | template <class T, class Policies> bool posgt(const interval<T, Policies>& x, const T& y); |
|---|
| 768 | template <class T, class Policies> bool posgt(const T& x, const interval<T, Policies> & y); |
|---|
| 769 | |
|---|
| 770 | template <class T, class Policies> bool posge(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 771 | template <class T, class Policies> bool posge(const interval<T, Policies>& x, const T& y); |
|---|
| 772 | template <class T, class Policies> bool posge(const T& x, const interval<T, Policies>& y); |
|---|
| 773 | |
|---|
| 774 | template <class T, class Policies> bool poseq(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 775 | template <class T, class Policies> bool poseq(const interval<T, Policies>& x, const T& y); |
|---|
| 776 | template <class T, class Policies> bool poseq(const T& x, const interval<T, Policies>& y); |
|---|
| 777 | |
|---|
| 778 | template <class T, class Policies> bool posne(const interval<T, Policies>& x, const interval<T, Policies>& y); |
|---|
| 779 | template <class T, class Policies> bool posne(const interval<T, Policies>& x, const T& y); |
|---|
| 780 | template <class T, class Policies> bool posne(const T& x, const interval<T, Policies>& y); |
|---|
| 781 | |
|---|
| 782 | <span style="color: #FF0000">/* comparison namespaces */</span> |
|---|
| 783 | namespace compare { |
|---|
| 784 | namespace certain; |
|---|
| 785 | namespace possible; |
|---|
| 786 | namespace lexicographic; |
|---|
| 787 | namespace set; |
|---|
| 788 | namespace tribool; |
|---|
| 789 | } // namespace compare |
|---|
| 790 | |
|---|
| 791 | } // namespace interval_lib |
|---|
| 792 | } // namespace numeric |
|---|
| 793 | } // namespace boost |
|---|
| 794 | </pre> |
|---|
| 795 | |
|---|
| 796 | <p>Each component of the interval support library is detailed in its own |
|---|
| 797 | page.</p> |
|---|
| 798 | |
|---|
| 799 | <ul> |
|---|
| 800 | <li><a href="comparisons.htm">Comparisons</a></li> |
|---|
| 801 | |
|---|
| 802 | <li><a href="rounding.htm">Rounding</a></li> |
|---|
| 803 | |
|---|
| 804 | <li><a href="checking.htm">Checking</a></li> |
|---|
| 805 | </ul> |
|---|
| 806 | |
|---|
| 807 | <h2 id="dangers">Common Pitfalls and Dangers</h2> |
|---|
| 808 | |
|---|
| 809 | <h4>Comparisons</h4> |
|---|
| 810 | |
|---|
| 811 | <p>One of the biggest problems is problably the correct use of the |
|---|
| 812 | comparison functions and operators. First, functions and operators do not |
|---|
| 813 | try to know if two intervals are the same mathematical object. So, if the |
|---|
| 814 | comparison used is "certain", then <code>x != x</code> is always true |
|---|
| 815 | unless <code>x</code> is a singleton interval; and the same problem arises |
|---|
| 816 | with <code>cereq</code> and <code>cerne</code>.</p> |
|---|
| 817 | |
|---|
| 818 | <p>Another misleading interpretation of the comparison is: you cannot |
|---|
| 819 | always expect [a,b] < [c,d] to be !([a,b] >= [c,d]) since the |
|---|
| 820 | comparison is not necessarily total. Equality and less comparison should be |
|---|
| 821 | seen as two distincts relational operators. However the default comparison |
|---|
| 822 | operators do respect this property since they throw an exception whenever |
|---|
| 823 | [a,b] and [c,d] overlap.</p> |
|---|
| 824 | |
|---|
| 825 | <h4>Interval values and references</h4> |
|---|
| 826 | |
|---|
| 827 | <p>This problem is a corollary of the previous problem with <code>x != |
|---|
| 828 | x</code>. All the functions of the library only consider the value of an |
|---|
| 829 | interval and not the reference of an interval. In particular, you should |
|---|
| 830 | not expect (unless <code>x</code> is a singleton) the following values to |
|---|
| 831 | be equal: <code>x/x</code> and 1, <code>x*x</code> and |
|---|
| 832 | <code>square(x)</code>, <code>x-x</code> and 0, etc. So the main cause of |
|---|
| 833 | wide intervals is that interval arithmetic does not identify different |
|---|
| 834 | occurences of the same variable. So, whenever possible, the user has to |
|---|
| 835 | rewrite the formulas to eliminate multiple occurences of the same variable. |
|---|
| 836 | For example, <code>square(x)-2*x</code> is far less precise than |
|---|
| 837 | <code>square(x-1)-1</code>.</p> |
|---|
| 838 | |
|---|
| 839 | <h4>Unprotected rounding</h4> |
|---|
| 840 | |
|---|
| 841 | <p>As explained in <a href="rounding.htm#perf">this section</a>, a good way |
|---|
| 842 | to speed up computations when the base type is a basic floating-point type |
|---|
| 843 | is to unprotect the intervals at the hot spots of the algorithm. This |
|---|
| 844 | method is safe and really an improvement for interval computations. But |
|---|
| 845 | please remember that any basic floating-point operation executed inside the |
|---|
| 846 | unprotection blocks will probably have an undefined behavior (but only for |
|---|
| 847 | the current thread). And do not forget to create a rounding object as |
|---|
| 848 | explained in the <a href="rounding.htm#perfexp">example</a>.</p> |
|---|
| 849 | |
|---|
| 850 | <h2 id="rationale">Rationale</h2> |
|---|
| 851 | |
|---|
| 852 | <p>The purpose of this library is to provide an efficient and generalized |
|---|
| 853 | way to deal with interval arithmetic through the use of a templatized class |
|---|
| 854 | <code>boost::interval</code>. The big contention for which we provide a |
|---|
| 855 | rationale is the format of this class template.</p> |
|---|
| 856 | |
|---|
| 857 | <p>It would have been easier to provide a class interval whose base type is |
|---|
| 858 | double. Or to follow <code>std::complex</code> and allow only |
|---|
| 859 | specializations for <code>float</code>, <code>double</code>, and <code>long |
|---|
| 860 | double</code>. We decided not to do this to allow intervals on custom |
|---|
| 861 | types, e.g. fixed-precision bigfloat library types (MPFR, etc), rational |
|---|
| 862 | numbers, and so on.</p> |
|---|
| 863 | |
|---|
| 864 | <p><strong>Policy design.</strong> Although it was tempting to make it a |
|---|
| 865 | class template with only one template argument, the diversity of uses for |
|---|
| 866 | an interval arithmetic practically forced us to use policies. The behavior |
|---|
| 867 | of this class can be fixed by two policies. These policies are packaged |
|---|
| 868 | into a single policy class, rather than making <code>interval</code> with |
|---|
| 869 | three template parameters. This is both for ease of use (the policy class |
|---|
| 870 | can be picked by default) and for readability.</p> |
|---|
| 871 | |
|---|
| 872 | <p>The first policy provides all the mathematical functions on the base |
|---|
| 873 | type needed to define the functions on the interval type. The second one |
|---|
| 874 | sets the way exceptional cases encountered during computations are |
|---|
| 875 | handled.</p> |
|---|
| 876 | |
|---|
| 877 | <p>We could foresee situations where any combination of these policies |
|---|
| 878 | would be appropriate. Moreover, we wanted to enable the user of the library |
|---|
| 879 | to reuse the <code>interval</code> class template while at the same time |
|---|
| 880 | choosing his own behavior. See this <a href="guide.htm">page</a> for some |
|---|
| 881 | examples.</p> |
|---|
| 882 | |
|---|
| 883 | <p><strong>Rounding policy.</strong> The library provides specialized |
|---|
| 884 | implementations of the rounding policy for the primitive types float and |
|---|
| 885 | double. In order for these implementations to be correct and fast, the |
|---|
| 886 | library needs to work a lot with rounding modes. Some processors are |
|---|
| 887 | directly dealt with and some mecanisms are provided in order to speed up |
|---|
| 888 | the computations. It seems to be heavy and hazardous optimizations for a |
|---|
| 889 | gain of only a few computer cycles; but in reality, the speed-up factor can |
|---|
| 890 | easily go past 2 or 3 depending on the computer. Moreover, these |
|---|
| 891 | optimizations do not impact the interface in any major way (with the design |
|---|
| 892 | we have chosen, everything can be added by specialization or by passing |
|---|
| 893 | different template parameters).</p> |
|---|
| 894 | |
|---|
| 895 | <p><strong>Pred/succ.</strong> In a previous version, two functions |
|---|
| 896 | <code>pred</code> and <code>succ</code>, with various corollaries like |
|---|
| 897 | <code>widen</code>, were supplied. The intent was to enlarge the interval |
|---|
| 898 | by one ulp (as little as possible), e.g. to ensure the inclusion property. |
|---|
| 899 | Since making interval a template of T, we could not define <i>ulp</i> for a |
|---|
| 900 | random parameter. In turn, rounding policies let us eliminate entirely the |
|---|
| 901 | use of ulp while making the intervals tighter (if a result is a |
|---|
| 902 | representable singleton, there is no use to widen the interval). We decided |
|---|
| 903 | to drop those functions.</p> |
|---|
| 904 | |
|---|
| 905 | <p><strong>Specialization of <code>std::less</code>.</strong> Since the |
|---|
| 906 | operator <code><</code> depends on the comparison namespace locally |
|---|
| 907 | chosen by the user, it is not possible to correctly specialize |
|---|
| 908 | <code>std::less</code>. So you have to explicitely provide such a class to |
|---|
| 909 | all the algorithms and templates that could require it (for example, |
|---|
| 910 | <code>std::map</code>).</p> |
|---|
| 911 | |
|---|
| 912 | <p><strong>Input/output.</strong> The interval library does not include I/O |
|---|
| 913 | operators. Printing an interval value allows a lot of customization: some |
|---|
| 914 | people may want to output the bounds, others may want to display the median |
|---|
| 915 | and the width of intervals, and so on. The example file io.cpp shows some |
|---|
| 916 | possibilities and may serve as a foundation in order for the user to define |
|---|
| 917 | her own operators.</p> |
|---|
| 918 | |
|---|
| 919 | <p><strong>Mixed operations with integers.</strong> When using and reusing |
|---|
| 920 | template codes, it is common there are operations like <code>2*x</code>. |
|---|
| 921 | However, the library does not provide them by default because the |
|---|
| 922 | conversion from <code>int</code> to the base number type is not always |
|---|
| 923 | correct (think about the conversion from a 32bit integer to a single |
|---|
| 924 | precision floating-point number). So the functions have been put in a |
|---|
| 925 | separate header and the user needs to include them explicitely if she wants |
|---|
| 926 | to benefit from these mixed operators. Another point, there is no mixed |
|---|
| 927 | comparison operators due to the technical way they are defined.</p> |
|---|
| 928 | |
|---|
| 929 | <p><strong>Interval-aware functions.</strong> All the functions defined by |
|---|
| 930 | the library are obviously aware they manipulate intervals and they do it |
|---|
| 931 | accordingly to general interval arithmetic principles. Consequently they |
|---|
| 932 | may have a different behavior than the one commonly encountered with |
|---|
| 933 | functions not interval-aware. For example, <code>max</code> is defined by |
|---|
| 934 | canonical set extension and the result is not always one of the two |
|---|
| 935 | arguments (if the intervals do not overlap, then the result is one of the |
|---|
| 936 | two intervals).</p> |
|---|
| 937 | |
|---|
| 938 | <p>This behavior is different from <code>std::max</code> which returns a |
|---|
| 939 | reference on one of its arguments. So if the user expects a reference to be |
|---|
| 940 | returned, she should use <code>std::max</code> since it is exactly what |
|---|
| 941 | this function does. Please note that <code>std::max</code> will throw an |
|---|
| 942 | exception when the intervals overlap. This behavior does not predate the |
|---|
| 943 | one described by the C++ standard since the arguments are not "equivalent" |
|---|
| 944 | and it allows to have an equivalence between <code>a <= b</code> and |
|---|
| 945 | <code>&b == &std::max(a,b)</code>(some particular cases may be |
|---|
| 946 | implementation-defined). However it is different from the one described by |
|---|
| 947 | SGI since it does not return the first argument even if "neither is greater |
|---|
| 948 | than the other".</p> |
|---|
| 949 | |
|---|
| 950 | <h2 id="acks">History and Acknowledgments</h2> |
|---|
| 951 | |
|---|
| 952 | <p>This library was mostly inspired by previous work from Jens Maurer. Some |
|---|
| 953 | discussions about his work are reproduced <a href= |
|---|
| 954 | "http://www.mscs.mu.edu/%7Egeorgec/IFAQ/maurer1.html">here</a>. Jeremy Siek |
|---|
| 955 | and Maarten Keijzer provided some rounding control for MSVC and Sparc |
|---|
| 956 | platforms.</p> |
|---|
| 957 | |
|---|
| 958 | <p>Guillaume Melquiond, Hervé Brönnimann and Sylvain Pion |
|---|
| 959 | started from the library left by Jens and added the policy design. |
|---|
| 960 | Guillaume and Sylvain worked hard on the code, especially the porting and |
|---|
| 961 | mostly tuning of the rounding modes to the different architectures. |
|---|
| 962 | Guillaume did most of the coding, while Sylvain and Hervé have |
|---|
| 963 | provided some useful comments in order for this library to be written. |
|---|
| 964 | Hervé reorganized and wrote chapters of the documentation based on |
|---|
| 965 | Guillaume's great starting point.</p> |
|---|
| 966 | |
|---|
| 967 | <p>This material is partly based upon work supported by the National |
|---|
| 968 | Science Foundation under NSF CAREER Grant CCR-0133599. Any opinions, |
|---|
| 969 | findings and conclusions or recommendations expressed in this material are |
|---|
| 970 | those of the author(s) and do not necessarily reflect the views of the |
|---|
| 971 | National Science Foundation (NSF).</p> |
|---|
| 972 | <hr> |
|---|
| 973 | |
|---|
| 974 | <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src= |
|---|
| 975 | "http://www.w3.org/Icons/valid-html401" alt="Valid HTML 4.01 Transitional" |
|---|
| 976 | height="31" width="88"></a></p> |
|---|
| 977 | |
|---|
| 978 | <p>Revised |
|---|
| 979 | <!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-25<!--webbot bot="Timestamp" endspan i-checksum="12174" --></p> |
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| 980 | |
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| 981 | <p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé |
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| 982 | Brönnimann, Polytechnic University<br> |
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| 983 | Copyright © 2003-2006 Guillaume Melquiond, ENS Lyon</i></p> |
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| 984 | |
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| 985 | <p><i>Distributed under the Boost Software License, Version 1.0. (See |
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| 986 | accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> |
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| 987 | or copy at <a href= |
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| 988 | "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p> |
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| 989 | </body> |
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| 990 | </html> |
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