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991 lines
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HTML
991 lines
58 KiB
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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"http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<head>
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<meta http-equiv="Content-Language" content="en-us">
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<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
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<link rel="stylesheet" type="text/css" href="../../../../boost.css">
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<title>Boost Interval Arithmetic Library</title>
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</head>
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<body lang="en">
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<h1><img src="../../../../boost.png" alt="boost.png (6897 bytes)" align=
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"middle"> Interval Arithmetic Library</h1>
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<center>
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<table width="80%" summary="">
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<tbody>
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<tr>
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<td><b>Contents of this page:</b><br>
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<a href="#intro">Introduction</a><br>
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<a href="#synopsis">Synopsis</a><br>
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<a href="#interval">Template class <code>interval</code></a><br>
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<a href="#opers">Operations and functions</a><br>
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<a href="#interval_lib">Interval support library</a><br>
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<!--<a href="#compil">Compilation notes</a><br>-->
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<a href="#dangers">Common pitfalls and dangers</a><br>
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<a href="#rationale">Rationale</a><br>
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<a href="#acks">History and Acknowledgments</a></td>
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<td><b>Other pages associated with this page:</b><br>
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<a href="rounding.htm">Rounding policies</a><br>
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<a href="checking.htm">Checking policies</a><br>
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<a href="policies.htm">Policies manipulation</a><br>
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<a href="comparisons.htm">Comparisons</a><br>
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<a href="numbers.htm">Base number type requirements</a><br>
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<a href="guide.htm">Choosing your own interval type</a><br>
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<a href="examples.htm">Test and example programs</a><br>
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<a href="includes.htm">Headers inclusion</a><br>
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<a href="todo.htm">Some items on the todo list</a></td>
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</tr>
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</tbody>
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</table>
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</center>
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<h2 id="intro">Introduction and Overview</h2>
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<p>As implied by its name, this library is intended to help manipulating
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mathematical intervals. It consists of a single header <<a href=
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"../../../../boost/numeric/interval.hpp">boost/numeric/interval.hpp</a>>
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and principally a type which can be used as <code>interval<T></code>.
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In fact, this interval template is declared as
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<code>interval<T,Policies></code> where <code>Policies</code> is a
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policy class that controls the various behaviours of the interval class;
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<code>interval<T></code> just happens to pick the default policies
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for the type <code>T</code>.</p>
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<p><span style="color: #FF0000; font-weight: bold">Warning!</span>
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Guaranteed interval arithmetic for native floating-point format is not
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supported on every combination of processor, operating system, and
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compiler. This is a list of systems known to work correctly when using
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<code>interval<float></code> and <code>interval<double></code>
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with basic arithmetic operators.</p>
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<ul>
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<li>x86-like hardware is supported by the library with GCC, Visual C++
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≥ 7.1, Intel compiler (≥ 8 on Windows), CodeWarrior (≥ 9), as
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long as the traditional x87 floating-point unit is used for
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floating-point computations (no <code>-mfpmath=sse2</code> support).</li>
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<li>Sparc hardware is supported with GCC and Sun compiler.</li>
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<li>PowerPC hardware is supported with GCC and CodeWarrior, when
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floating-point computations are not done with the Altivec unit.</li>
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<li>Alpha hardware is supported with GCC, except maybe for the square
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root. The options <code>-mfp-rounding-mode=d -mieee</code> have to be
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used.</li>
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</ul>
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<p>The previous list is not exhaustive. And even if a system does not
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provide guaranteed computations for hardware floating-point types, the
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interval library is still usable with user-defined types and for doing box
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arithmetic.</p>
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<h3>Interval Arithmetic</h3>
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<p>An interval is a pair of numbers which represents all the numbers
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between these two. (Intervals are considered closed so the bounds are
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included.) The purpose of this library is to extend the usual arithmetic
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functions to intervals. These intervals will be written [<i>a</i>,<i>b</i>]
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to represent all the numbers between <i>a</i> and <i>b</i> (included).
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<i>a</i> and <i>b</i> can be infinite (but they can not be the same
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infinite) and <i>a</i> ≤ <i>b</i>.</p>
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<p>The fundamental property of interval arithmetic is the
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<em><strong>inclusion property</strong></em>:</p>
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<dl>
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<dd>``if <i>f</i> is a function on a set of numbers, <i>f</i> can be
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extended to a new function defined on intervals. This new function
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<i>f</i> takes one interval argument and returns an interval result such
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as: ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>], <i>f</i>(<i>x</i>)
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∈ <i>f</i>([<i>a</i>,<i>b</i>]).''</dd>
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</dl>
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<p>Such a property is not limited to functions with only one argument.
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Whenever possible, the interval result should be the smallest one able to
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satisfy the property (it is not really useful if the new functions always
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answer [-∞,+∞]).</p>
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<p>There are at least two reasons a user would like to use this library.
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The obvious one is when the user has to compute with intervals. One example
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is when input data have some builtin imprecision: instead of a number, an
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input variable can be passed as an interval. Another example application is
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to solve equations, by bisecting an interval until the interval width is
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small enough. A third example application is in computer graphics, where
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computations with boxes, segments or rays can be reduced to computations
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with points via intervals.</p>
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<p>Another common reason to use interval arithmetic is when the computer
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doesn't produce exact results: by using intervals, it is possible to
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quantify the propagation of rounding errors. This approach is used often in
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numerical computation. For example, let's assume the computer stores
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numbers with ten decimal significant digits. To the question 1 + 1E-100 -
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1, the computer will answer 0 although the correct answer would be 1E-100.
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With the help of interval arithmetic, the computer will answer [0,1E-9].
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This is quite a huge interval for such a little result, but the precision
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is now known, without having to compute error propagation.</p>
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<h3>Numbers, rounding, and exceptional behavior</h3>
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<p>The <em><strong>base number type</strong></em> is the type that holds
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the bounds of the interval. In order to successfully use interval
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arithmetic, the base number type must present some <a href=
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"rounding.htm">characteristics</a>. Firstly, due to the definition of an
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interval, the base numbers have to be totally ordered so, for instance,
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<code>complex<T></code> is not usable as base number type for
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intervals. The mathematical functions for the base number type should also
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be compatible with the total order (for instance if x>y and z>t, then
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it should also hold that x+z > y+t), so modulo types are not usable
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either.</p>
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<p>Secondly, the computations must be exact or provide some rounding
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methods (for instance, toward minus or plus infinity) if we want to
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guarantee the inclusion property. Note that we also may explicitely specify
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no rounding, for instance if the base number type is exact, i.e. the result
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of a mathematical operation is always computed and represented without loss
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of precision. If the number type is not exact, we may still explicitely
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specify no rounding, with the obvious consequence that the inclusion
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property is no longer guaranteed.</p>
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<p>Finally, because heavy loss of precision is always possible, some
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numbers have to represent infinities or an exceptional behavior must be
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defined. The same situation also occurs for NaN (<i>Not a Number</i>).</p>
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<p>Given all this, one may want to limit the template argument T of the
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class template <code>interval</code> to the floating point types
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<code>float</code>, <code>double</code>, and <code>long double</code>, as
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defined by the IEEE-754 Standard. Indeed, if the interval arithmetic is
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intended to replace the arithmetic provided by the floating point unit of a
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processor, these types are the best choice. Unlike
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<code>std::complex</code>, however, we don't want to limit <code>T</code>
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to these types. This is why we allow the rounding and exceptional behaviors
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to be given by the two policies (rounding and checking). We do nevertheless
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provide highly optimized rounding and checking class specializations for
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the above-mentioned floating point types.</p>
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<h3>Operations and functions</h3>
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<p>It is straightforward to define the elementary arithmetic operations on
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intervals, being guided by the inclusion property. For instance, if [a,b]
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and [c,d] are intervals, [a,b]+[c,d] can be computed by taking the smallest
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interval that contains all the numbers x+y for x in [a,b] and y in [c,d];
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in this case, rounding a+c down and b+d up will suffice. Other operators
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and functions are similarly defined (see their definitions below).</p>
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<h3>Comparisons</h3>
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<p>It is also possible to define some comparison operators. Given two
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intervals, the result is a tri-state boolean type
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{<i>false</i>,<i>true,indeterminate</i>}. The answers <i>false</i> and
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<i>true</i> are easy to manipulate since they can directly be mapped on the
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boolean <i>true</i> and <i>false</i>. But it is not the case for the answer
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<em>indeterminate</em> since comparison operators are supposed to be
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boolean functions. So, what to do in order to obtain boolean answers?</p>
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<p>One solution consists of deciding to adopt an exceptional behavior, such
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as a failed assertion or raising an exception. In this case, the
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exceptional behavior will be triggered when the result is
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indeterminate.</p>
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<p>Another solution is to map <em>indeterminate</em> always to
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<i>false,</i> or always to <i>true</i>. If <i>false</i> is chosen, the
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comparison will be called "<i>certain</i>;" indeed, the result of
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[<i>a</i>,<i>b</i>] < [<i>c</i>,<i>d</i>] will be <i>true</i> if and
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only if: ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i>
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∈ [<i>c</i>,<i>d</i>], <i>x</i> < <i>y</i>. If <i>true</i> is
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chosen, the comparison will be called "<i>possible</i>;" indeed, the result
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of [<i>a</i>,<i>b</i>] < [<i>c</i>,<i>d</i>] will be <i>true</i> if and
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only if: ∃ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∃ <i>y</i>
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∈ [<i>c</i>,<i>d</i>], <i>x</i> < <i>y</i>.</p>
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<p>Since any of these solution has a clearly defined semantics, it is not
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clear that we should enforce either of them. For this reason, the default
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behavior consists to mimic the real comparisons by throwing an exception in
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the indeterminate case. Other behaviors can be selected bu using specific
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comparison namespace. There is also a bunch of explicitely named comparison
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functions. See <a href="comparisons.htm">comparisons</a> pages for further
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details.</p>
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<h3>Overview of the library, and usage</h3>
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<p>This library provides two quite distinct levels of usage. One is to use
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the basic class template <code>interval<T></code> without specifying
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the policy. This only requires one to know and understand the concepts
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developed above and the content of the namespace boost. In addition to the
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class <code>interval<T></code>, this level of usage provides
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arithmetic operators (<code>+</code>, <code>-</code>, <code>*</code>,
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<code>/</code>), algebraic and piecewise-algebraic functions
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(<code>abs</code>, <code>square</code>, <code>sqrt</code>,
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<code>pow</code>), transcendental and trigonometric functions
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(<code>exp</code>, <code>log</code>, <code>sin</code>, <code>cos</code>,
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<code>tan</code>, <code>asin</code>, <code>acos</code>, <code>atan</code>,
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<code>sinh</code>, <code>cosh</code>, <code>tanh</code>,
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<code>asinh</code>, <code>acosh</code>, <code>atanh</code>), and the
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standard comparison operators (<code><</code>, <code><=</code>,
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<code>></code>, <code>>=</code>, <code>==</code>, <code>!=</code>),
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as well as several interval-specific functions (<code>min</code>,
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<code>max</code>, which have a different meaning than <code>std::min</code>
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and <code>std::max</code>; <code>lower</code>, <code>upper</code>,
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<code>width</code>, <code>median</code>, <code>empty</code>,
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<code>singleton</code>, <code>equal</code>, <code>in</code>,
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<code>zero_in</code>, <code>subset</code>, <code>proper_subset</code>,
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<code>overlap</code>, <code>intersect</code>, <code>hull</code>,
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<code>bisect</code>).</p>
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<p>For some functions which take several parameters of type
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<code>interval<T></code>, all combinations of argument types
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<code>T</code> and <code>interval<T></code> which contain at least
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one <code>interval<T></code>, are considered in order to avoid a
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conversion from the arguments of type <code>T</code> to a singleton of type
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<code>interval<T></code>. This is done for efficiency reasons (the
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fact that an argument is a singleton sometimes renders some tests
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unnecessary).</p>
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<p>A somewhat more advanced usage of this library is to hand-pick the
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policies <code>Rounding</code> and <code>Checking</code> and pass them to
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<code>interval<T, Policies></code> through the use of <code>Policies
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:= boost::numeric::interval_lib::policies<Rounding,Checking></code>.
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Appropriate policies can be fabricated by using the various classes
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provided in the namespace <code>boost::numeric::interval_lib</code> as
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detailed in section <a href="#interval_lib">Interval Support Library</a>.
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It is also possible to choose the comparison scheme by overloading
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operators through namespaces.</p>
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<h2><a name="synopsis" id="synopsis"></a>Synopsis</h2>
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<pre>
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namespace boost {
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namespace numeric {
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namespace interval_lib {
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/* this declaration is necessary for the declaration of interval */
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template <class T> struct default_policies;
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/* ... ; the full synopsis of namespace interval_lib can be found <a href=
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"#interval_lib">here</a> */
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} // namespace interval_lib
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/* template interval_policies; class definition can be found <a href=
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"policies.htm">here</a> */
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template<class Rounding, class Checking>
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struct interval_policies;
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/* template class interval; class definition can be found <a href=
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"#interval">here</a> */
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template<class T, class Policies = typename interval_lib::default_policies<T>::type > class interval;
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/* arithmetic operators involving intervals */
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template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x);
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template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x);
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template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x, const interval<T, Policies>& y);
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template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x, const T& y);
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template <class T, class Policies> interval<T, Policies> operator+(const T& x, const interval<T, Policies>& y);
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template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x, const interval<T, Policies>& y);
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template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x, const T& y);
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template <class T, class Policies> interval<T, Policies> operator-(const T& x, const interval<T, Policies>& y);
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template <class T, class Policies> interval<T, Policies> operator*(const interval<T, Policies>& x, const interval<T, Policies>& y);
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template <class T, class Policies> interval<T, Policies> operator*(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> operator*(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> interval<T, Policies> operator/(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> operator/(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> operator/(const T& r, const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
/* algebraic functions: sqrt, abs, square, pow, nth_root */
|
||
|
|
template <class T, class Policies> interval<T, Policies> abs(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> sqrt(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> square(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> pow(const interval<T, Policies>& x, int y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> nth_root(const interval<T, Policies>& x, int y);
|
||
|
|
|
||
|
|
/* transcendental functions: exp, log */
|
||
|
|
template <class T, class Policies> interval<T, Policies> exp(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> log(const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
/* fmod, for trigonometric function argument reduction (see below) */
|
||
|
|
template <class T, class Policies> interval<T, Policies> fmod(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
/* trigonometric functions */
|
||
|
|
template <class T, class Policies> interval<T, Policies> sin(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> cos(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> tan(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> asin(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> acos(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> atan(const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
/* hyperbolic trigonometric functions */
|
||
|
|
template <class T, class Policies> interval<T, Policies> sinh(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> cosh(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> tanh(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> asinh(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> acosh(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> interval<T, Policies> atanh(const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
/* min, max external functions (NOT std::min/max, see below) */
|
||
|
|
template <class T, class Policies> interval<T, Policies> max(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> max(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> max(const T& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> min(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> min(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> min(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
/* bounds-related interval functions */
|
||
|
|
template <class T, class Policies> T lower(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> T upper(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> T width(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> T median(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> T norm(const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
/* bounds-related interval functions */
|
||
|
|
template <class T, class Policies> bool empty(const interval<T, Policies>& b);
|
||
|
|
template <class T, class Policies> bool singleton(const interval<T, Policies>& x);
|
||
|
|
template <class T, class Policies> bool equal(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool in(const T& r, const interval<T, Policies>& b);
|
||
|
|
template <class T, class Policies> bool zero_in(const interval<T, Policies>& b);
|
||
|
|
template <class T, class Policies> bool subset(const interval<T, Policies>& a, const interval<T, Policies>& b);
|
||
|
|
template <class T, class Policies> bool proper_subset(const interval<T, Policies>& a, const interval<T, Policies>& b);
|
||
|
|
template <class T, class Policies> bool overlap(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
/* set manipulation interval functions */
|
||
|
|
template <class T, class Policies> interval<T, Policies> intersect(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> hull(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> hull(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> hull(const T& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> interval<T, Policies> hull(const T& x, const T& y);
|
||
|
|
template <class T, class Policies> std::pair<interval<T, Policies>, interval<T, Policies> > bisect(const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
/* interval comparison operators */
|
||
|
|
template<class T, class Policies> bool operator<(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template<class T, class Policies> bool operator<(const interval<T, Policies>& x, const T& y);
|
||
|
|
template<class T, class Policies> bool operator<(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template<class T, class Policies> bool operator<=(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template<class T, class Policies> bool operator<=(const interval<T, Policies>& x, const T& y);
|
||
|
|
template<class T, class Policies> bool operator<=(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template<class T, class Policies> bool operator>(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template<class T, class Policies> bool operator>(const interval<T, Policies>& x, const T& y);
|
||
|
|
template<class T, class Policies> bool operator>(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template<class T, class Policies> bool operator>=(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template<class T, class Policies> bool operator>=(const interval<T, Policies>& x, const T& y);
|
||
|
|
template<class T, class Policies> bool operator>=(const T& x, const interval<T, Policies>& y);
|
||
|
|
</pre>
|
||
|
|
<pre>
|
||
|
|
template<class T, class Policies> bool operator==(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template<class T, class Policies> bool operator==(const interval<T, Policies>& x, const T& y);
|
||
|
|
template<class T, class Policies> bool operator==(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template<class T, class Policies> bool operator!=(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template<class T, class Policies> bool operator!=(const interval<T, Policies>& x, const T& y);
|
||
|
|
template<class T, class Policies> bool operator!=(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
namespace interval_lib {
|
||
|
|
|
||
|
|
template<class T, class Policies> interval<T, Policies> division_part1(const interval<T, Policies>& x, const interval<T, Policies& y, bool& b);
|
||
|
|
template<class T, class Policies> interval<T, Policies> division_part2(const interval<T, Policies>& x, const interval<T, Policies& y, bool b = true);
|
||
|
|
template<class T, class Policies> interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x);
|
||
|
|
|
||
|
|
template<class I> I add(const typename I::base_type& x, const typename I::base_type& y);
|
||
|
|
template<class I> I sub(const typename I::base_type& x, const typename I::base_type& y);
|
||
|
|
template<class I> I mul(const typename I::base_type& x, const typename I::base_type& y);
|
||
|
|
template<class I> I div(const typename I::base_type& x, const typename I::base_type& y);
|
||
|
|
|
||
|
|
} // namespace interval_lib
|
||
|
|
|
||
|
|
} // namespace numeric
|
||
|
|
} // namespace boost
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
<h2><a name="interval" id="interval"></a>Template class
|
||
|
|
<code>interval</code></h2>The public interface of the template class
|
||
|
|
interval itself is kept at a simplest minimum:
|
||
|
|
<pre>
|
||
|
|
template <class T, class Policies = typename interval_lib::default_policies<T>::type>
|
||
|
|
class interval
|
||
|
|
{
|
||
|
|
public:
|
||
|
|
typedef T base_type;
|
||
|
|
typedef Policies traits_type;
|
||
|
|
|
||
|
|
interval();
|
||
|
|
interval(T const &v);
|
||
|
|
template<class T1> interval(T1 const &v);
|
||
|
|
interval(T const &l, T const &u);
|
||
|
|
template<class T1, class T2> interval(T1 const &l, T2 const &u);
|
||
|
|
interval(interval<T, Policies> const &r);
|
||
|
|
template<class Policies1> interval(interval<T, Policies1> const &r);
|
||
|
|
template<class T1, class Policies1> interval(interval<T1, Policies1> const &r);
|
||
|
|
|
||
|
|
interval &operator=(T const &v);
|
||
|
|
template<class T1> interval &operator=(T1 const &v);
|
||
|
|
interval &operator=(interval<T, Policies> const &r);
|
||
|
|
template<class Policies1> interval &operator=(interval<T, Policies1> const &r);
|
||
|
|
template<class T1, class Policies1> interval &operator=(interval<T1, Policies1> const &r);
|
||
|
|
|
||
|
|
void assign(T const &l, T const &u);
|
||
|
|
|
||
|
|
T const &lower() const;
|
||
|
|
T const &upper() const;
|
||
|
|
|
||
|
|
static interval empty();
|
||
|
|
static interval whole();
|
||
|
|
static interval hull(T const &x, T const &y);
|
||
|
|
|
||
|
|
interval& operator+= (T const &r);
|
||
|
|
interval& operator-= (T const &r);
|
||
|
|
interval& operator*= (T const &r);
|
||
|
|
interval& operator/= (T const &r);
|
||
|
|
interval& operator+= (interval const &r);
|
||
|
|
interval& operator-= (interval const &r);
|
||
|
|
interval& operator*= (interval const &r);
|
||
|
|
interval& operator/= (interval const &r);
|
||
|
|
};
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
<p>The constructors create an interval enclosing their arguments. If there
|
||
|
|
are two arguments, the first one is assumed to be the left bound and the
|
||
|
|
second one is the right bound. Consequently, the arguments need to be
|
||
|
|
ordered. If the property !(l <= u) is not respected, the checking policy
|
||
|
|
will be used to create an empty interval. If no argument is given, the
|
||
|
|
created interval is the singleton zero.</p>
|
||
|
|
|
||
|
|
<p>If the type of the arguments is the same as the base number type, the
|
||
|
|
values are directly used for the bounds. If it is not the same type, the
|
||
|
|
library will use the rounding policy in order to convert the arguments
|
||
|
|
(<code>conv_down</code> and <code>conv_up</code>) and create an enclosing
|
||
|
|
interval. When the argument is an interval with a different policy, the
|
||
|
|
input interval is checked in order to correctly propagate its emptiness (if
|
||
|
|
empty).</p>
|
||
|
|
|
||
|
|
<p>The assignment operators behave similarly, except they obviously take
|
||
|
|
one argument only. There is also an <code>assign</code> function in order
|
||
|
|
to directly change the bounds of an interval. It behaves like the
|
||
|
|
two-arguments constructors if the bounds are not ordered. There is no
|
||
|
|
assign function that directly takes an interval or only one number as a
|
||
|
|
parameter; just use the assignment operators in such a case.</p>
|
||
|
|
|
||
|
|
<p>The type of the bounds and the policies of the interval type define the
|
||
|
|
set of values the intervals contain. E.g. with the default policies,
|
||
|
|
intervals are subsets of the set of real numbers. The static functions
|
||
|
|
<code>empty</code> and <code>whole</code> produce the intervals/subsets
|
||
|
|
that are respectively the empty subset and the whole set. They are static
|
||
|
|
member functions rather than global functions because they cannot guess
|
||
|
|
their return types. Likewise for <code>hull</code>. <code>empty</code> and
|
||
|
|
<code>whole</code> involve the checking policy in order to get the bounds
|
||
|
|
of the resulting intervals.</p>
|
||
|
|
|
||
|
|
<h2><a name="opers" id="opers"></a>Operations and Functions</h2>
|
||
|
|
|
||
|
|
<p>Some of the following functions expect <code>min</code> and
|
||
|
|
<code>max</code> to be defined for the base type. Those are the only
|
||
|
|
requirements for the <code>interval</code> class (but the policies can have
|
||
|
|
other requirements).</p>
|
||
|
|
|
||
|
|
<h4>Operators <code>+</code> <code>-</code> <code>*</code> <code>/</code>
|
||
|
|
<code>+=</code> <code>-=</code> <code>*=</code> <code>/=</code></h4>
|
||
|
|
|
||
|
|
<p>The basic operations are the unary minus and the binary <code>+</code>
|
||
|
|
<code>-</code> <code>*</code> <code>/</code>. The unary minus takes an
|
||
|
|
interval and returns an interval. The binary operations take two intervals,
|
||
|
|
or one interval and a number, and return an interval. If an argument is a
|
||
|
|
number instead of an interval, you can expect the result to be the same as
|
||
|
|
if the number was first converted to an interval. This property will be
|
||
|
|
true for all the following functions and operators.</p>
|
||
|
|
|
||
|
|
<p>There are also some assignment operators <code>+=</code> <code>-=</code>
|
||
|
|
<code>*=</code> <code>/=</code>. There is not much to say: <code>x op=
|
||
|
|
y</code> is equivalent to <code>x = x op y</code>. If an exception is
|
||
|
|
thrown during the computations, the l-value is not modified (but it may be
|
||
|
|
corrupt if an exception is thrown by the base type during an
|
||
|
|
assignment).</p>
|
||
|
|
|
||
|
|
<p>The operators <code>/</code> and <code>/=</code> will try to produce an
|
||
|
|
empty interval if the denominator is exactly zero. If the denominator
|
||
|
|
contains zero (but not only zero), the result will be the smallest interval
|
||
|
|
containing the set of division results; so one of its bound will be
|
||
|
|
infinite, but it may not be the whole interval.</p>
|
||
|
|
|
||
|
|
<h4><code>lower</code> <code>upper</code> <code>median</code>
|
||
|
|
<code>width</code> <code>norm</code></h4>
|
||
|
|
|
||
|
|
<p><code>lower</code>, <code>upper</code>, <code>median</code> respectively
|
||
|
|
compute the lower bound, the upper bound, and the median number of an
|
||
|
|
interval (<code>(lower+upper)/2</code> rounded to nearest).
|
||
|
|
<code>width</code> computes the width of an interval
|
||
|
|
(<code>upper-lower</code> rounded toward plus infinity). <code>norm</code>
|
||
|
|
computes an upper bound of the interval in absolute value; it is a
|
||
|
|
mathematical norm (hence the name) similar to the absolute value for real
|
||
|
|
numbers.</p>
|
||
|
|
|
||
|
|
<h4><code>min</code> <code>max</code> <code>abs</code> <code>square</code>
|
||
|
|
<code>pow</code> <code>nth_root</code> <code>division_part?</code>
|
||
|
|
<code>multiplicative_inverse</code></h4>
|
||
|
|
|
||
|
|
<p>The functions <code>min</code>, <code>max</code> and <code>abs</code>
|
||
|
|
are also defined. Please do not mistake them for the functions defined in
|
||
|
|
the standard library (aka <code>a<b?a:b</code>, <code>a>b?a:b</code>,
|
||
|
|
<code>a<0?-a:a</code>). These functions are compatible with the
|
||
|
|
elementary property of interval arithmetic. For example,
|
||
|
|
max([<i>a</i>,<i>b</i>], [<i>c</i>,<i>d</i>]) = {max(<i>x</i>,<i>y</i>)
|
||
|
|
such that <i>x</i> in [<i>a</i>,<i>b</i>] and <i>y</i> in
|
||
|
|
[<i>c</i>,<i>d</i>]}. They are not defined in the <code>std</code>
|
||
|
|
namespace but in the boost namespace in order to avoid conflict with the
|
||
|
|
other definitions.</p>
|
||
|
|
|
||
|
|
<p>The <code>square</code> function is quite particular. As you can expect
|
||
|
|
from its name, it computes the square of its argument. The reason this
|
||
|
|
function is provided is: <code>square(x)</code> is not <code>x*x</code> but
|
||
|
|
only a subset when <code>x</code> contains zero. For example, [-2,2]*[-2,2]
|
||
|
|
= [-4,4] but [-2,2]² = [0,4]; the result is a smaller interval.
|
||
|
|
Consequently, <code>square(x)</code> should be used instead of
|
||
|
|
<code>x*x</code> because of its better accuracy and a small performance
|
||
|
|
improvement.</p>
|
||
|
|
|
||
|
|
<p>As for <code>square</code>, <code>pow</code> provides an efficient and
|
||
|
|
more accurate way to compute the integer power of an interval. Please note:
|
||
|
|
when the power is 0 and the interval is not empty, the result is 1, even if
|
||
|
|
the input interval contains 0. <code>nth_root</code> computes the integer root
|
||
|
|
of an interval (<code>nth_root(pow(x,k),k)</code> encloses <code>x</code> or
|
||
|
|
<code>abs(x)</code> depending on whether <code>k</code> is odd or even).
|
||
|
|
The behavior of <code>nth_root</code> is not defined if the integer argument is
|
||
|
|
not positive. <code>multiplicative_inverse</code> computes
|
||
|
|
<code>1/x</code>.</p>
|
||
|
|
|
||
|
|
<p>The functions <code>division_part1</code> and
|
||
|
|
<code>division_part2</code> are useful when the user expects the division
|
||
|
|
to return disjoint intervals if necessary. For example, the narrowest
|
||
|
|
closed set containing [2,3] / [-2,1] is not ]-∞,∞[ but the union
|
||
|
|
of ]-∞,-1] and [2,∞[. When the result of the division is
|
||
|
|
representable by only one interval, <code>division_part1</code> returns
|
||
|
|
this interval and sets the boolean reference to <code>false</code>.
|
||
|
|
However, if the result needs two intervals, <code>division_part1</code>
|
||
|
|
returns the negative part and sets the boolean reference to
|
||
|
|
<code>true</code>; a call to <code>division_part2</code> is now needed to
|
||
|
|
get the positive part. This second function can take the boolean returned
|
||
|
|
by the first function as last argument. If this bool is not given, its
|
||
|
|
value is assumed to be true and the behavior of the function is then
|
||
|
|
undetermined if the division does not produce a second interval.</p>
|
||
|
|
|
||
|
|
<h4><code>intersect</code> <code>hull</code> <code>overlap</code>
|
||
|
|
<code>in</code> <code>zero_in</code> <code>subset</code>
|
||
|
|
<code>proper_subset</code> <code>empty</code> <code>singleton</code>
|
||
|
|
<code>equal</code></h4>
|
||
|
|
|
||
|
|
<p><code>intersect</code> computes the set intersection of two closed sets,
|
||
|
|
<code>hull</code> computes the smallest interval which contains the two
|
||
|
|
parameters; those parameters can be numbers or intervals. If one of the
|
||
|
|
arguments is an invalid number or an empty interval, the function will only
|
||
|
|
use the other argument to compute the resulting interval (if allowed by the
|
||
|
|
checking policy).</p>
|
||
|
|
|
||
|
|
<p>There is no union function since the union of two intervals is not an
|
||
|
|
interval if they do not overlap. If they overlap, the <code>hull</code>
|
||
|
|
function computes the union.</p>
|
||
|
|
|
||
|
|
<p>The function <code>overlap</code> tests if two intervals have some
|
||
|
|
common subset. <code>in</code> tests if a number is in an interval;
|
||
|
|
<code>zero_in</code> is a variant which tests if zero is in the interval.
|
||
|
|
<code>subset</code> tests if the first interval is a subset of the second;
|
||
|
|
and <code>proper_subset</code> tests if it is a proper subset.
|
||
|
|
<code>empty</code> and <code>singleton</code> test if an interval is empty
|
||
|
|
or is a singleton. Finally, <code>equal</code> tests if two intervals are
|
||
|
|
equal.</p>
|
||
|
|
|
||
|
|
<h4><code>sqrt</code> <code>log</code> <code>exp</code> <code>sin</code>
|
||
|
|
<code>cos</code> <code>tan</code> <code>asin</code> <code>acos</code>
|
||
|
|
<code>atan</code> <code>sinh</code> <code>cosh</code> <code>tanh</code>
|
||
|
|
<code>asinh</code> <code>acosh</code> <code>atanh</code>
|
||
|
|
<code>fmod</code></h4>
|
||
|
|
|
||
|
|
<p>The functions <code>sqrt</code>, <code>log</code>, <code>exp</code>,
|
||
|
|
<code>sin</code>, <code>cos</code>, <code>tan</code>, <code>asin</code>,
|
||
|
|
<code>acos</code>, <code>atan</code>, <code>sinh</code>, <code>cosh</code>,
|
||
|
|
<code>tanh</code>, <code>asinh</code>, <code>acosh</code>,
|
||
|
|
<code>atanh</code> are also defined. There is not much to say; these
|
||
|
|
functions extend the traditional functions to the intervals and respect the
|
||
|
|
basic property of interval arithmetic. They use the <a href=
|
||
|
|
"checking.htm">checking</a> policy to produce empty intervals when the
|
||
|
|
input interval is strictly outside of the domain of the function.</p>
|
||
|
|
|
||
|
|
<p>The function <code>fmod(interval x, interval y)</code> expects the lower
|
||
|
|
bound of <code>y</code> to be strictly positive and returns an interval
|
||
|
|
<code>z</code> such as <code>0 <= z.lower() < y.upper()</code> and
|
||
|
|
such as <code>z</code> is a superset of <code>x-n*y</code> (with
|
||
|
|
<code>n</code> being an integer). So, if the two arguments are positive
|
||
|
|
singletons, this function <code>fmod(interval, interval)</code> will behave
|
||
|
|
like the traditional function <code>fmod(double, double)</code>.</p>
|
||
|
|
|
||
|
|
<p>Please note that <code>fmod</code> does not respect the inclusion
|
||
|
|
property of arithmetic interval. For example, the result of
|
||
|
|
<code>fmod</code>([13,17],[7,8]) should be [0,8] (since it must contain
|
||
|
|
[0,3] and [5,8]). But this answer is not really useful when the purpose is
|
||
|
|
to restrict an interval in order to compute a periodic function. It is the
|
||
|
|
reason why <code>fmod</code> will answer [5,10].</p>
|
||
|
|
|
||
|
|
<h4><code>add</code> <code>sub</code> <code>mul</code>
|
||
|
|
<code>div</code></h4>
|
||
|
|
|
||
|
|
<p>These four functions take two numbers and return the enclosing interval
|
||
|
|
for the operations. It avoids converting a number to an interval before an
|
||
|
|
operation, it can result in a better code with poor optimizers.</p>
|
||
|
|
|
||
|
|
<h3>Constants</h3>
|
||
|
|
|
||
|
|
<p>Some constants are hidden in the
|
||
|
|
<code>boost::numeric::interval_lib</code> namespace. They need to be
|
||
|
|
explicitely templated by the interval type. The functions are
|
||
|
|
<code>pi<I>()</code>, <code>pi_half<I>()</code> and
|
||
|
|
<code>pi_twice<I>()</code>, and they return an object of interval
|
||
|
|
type <code>I</code>. Their respective values are π, π/2 and
|
||
|
|
2π.</p>
|
||
|
|
|
||
|
|
<h3>Exception throwing</h3>
|
||
|
|
|
||
|
|
<p>The interval class and all the functions defined around this class never
|
||
|
|
throw any exceptions by themselves. However, it does not mean that an
|
||
|
|
operation will never throw an exception. For example, let's consider the
|
||
|
|
copy constructor. As explained before, it is the default copy constructor
|
||
|
|
generated by the compiler. So it will not throw an exception if the copy
|
||
|
|
constructor of the base type does not throw an exception.</p>
|
||
|
|
|
||
|
|
<p>The same situation applies to all the functions: exceptions will only be
|
||
|
|
thrown if the base type or one of the two policies throws an exception.</p>
|
||
|
|
|
||
|
|
<h2 id="interval_lib">Interval Support Library</h2>
|
||
|
|
|
||
|
|
<p>The interval support library consists of a collection of classes that
|
||
|
|
can be used and combined to fabricate almost various commonly-needed
|
||
|
|
interval policies. In contrast to the basic classes and functions which are
|
||
|
|
used in conjunction with <code>interval<T></code> (and the default
|
||
|
|
policies as the implicit second template parameter in this type), which
|
||
|
|
belong simply to the namespace <code>boost</code>, these components belong
|
||
|
|
to the namespace <code>boost::numeric::interval_lib</code>.</p>
|
||
|
|
|
||
|
|
<p>We merely give the synopsis here and defer each section to a separate
|
||
|
|
web page since it is only intended for the advanced user. This allows to
|
||
|
|
expand on each topic with examples, without unduly stretching the limits of
|
||
|
|
this document.</p>
|
||
|
|
|
||
|
|
<h4>Synopsis</h4>
|
||
|
|
<pre>
|
||
|
|
namespace boost {
|
||
|
|
namespace numeric {
|
||
|
|
namespace interval_lib {
|
||
|
|
|
||
|
|
<span style=
|
||
|
|
"color: #FF0000">/* built-in rounding policy and its specializations */</span>
|
||
|
|
template <class T> struct rounded_math;
|
||
|
|
template <> struct rounded_math<float>;
|
||
|
|
template <> struct rounded_math<double>;
|
||
|
|
template <> struct rounded_math<long double>;
|
||
|
|
|
||
|
|
<span style=
|
||
|
|
"color: #FF0000">/* built-in rounding construction blocks */</span>
|
||
|
|
template <class T> struct rounding_control;
|
||
|
|
|
||
|
|
template <class T, class Rounding = rounding_control<T> > struct rounded_arith_exact;
|
||
|
|
template <class T, class Rounding = rounding_control<T> > struct rounded_arith_std;
|
||
|
|
template <class T, class Rounding = rounding_control<T> > struct rounded_arith_opp;
|
||
|
|
|
||
|
|
template <class T, class Rounding> struct rounded_transc_dummy;
|
||
|
|
template <class T, class Rounding = rounded_arith_exact<T> > struct rounded_transc_exact;
|
||
|
|
template <class T, class Rounding = rounded_arith_std <T> > struct rounded_transc_std;
|
||
|
|
template <class T, class Rounding = rounded_arith_opp <T> > struct rounded_transc_opp;
|
||
|
|
|
||
|
|
template <class Rounding> struct save_state;
|
||
|
|
template <class Rounding> struct save_state_nothing;
|
||
|
|
|
||
|
|
<span style="color: #FF0000">/* built-in checking policies */</span>
|
||
|
|
template <class T> struct checking_base;
|
||
|
|
template <class T, class Checking = checking_base<T>, class Exception = exception_create_empty> struct checking_no_empty;
|
||
|
|
template <class T, class Checking = checking_base<T> > struct checking_no_nan;
|
||
|
|
template <class T, class Checking = checking_base<T>, class Exception = exception_invalid_number> struct checking_catch_nan;
|
||
|
|
template <class T> struct checking_strict;
|
||
|
|
|
||
|
|
<span style=
|
||
|
|
"color: #FF0000">/* some metaprogramming to manipulate interval policies */</span>
|
||
|
|
template <class Rounding, class Checking> struct policies;
|
||
|
|
template <class OldInterval, class NewRounding> struct change_rounding;
|
||
|
|
template <class OldInterval, class NewChecking> struct change_checking;
|
||
|
|
template <class OldInterval> struct unprotect;
|
||
|
|
|
||
|
|
<span style=
|
||
|
|
"color: #FF0000">/* constants, need to be explicitly templated */</span>
|
||
|
|
template<class I> I pi();
|
||
|
|
template<class I> I pi_half();
|
||
|
|
template<class I> I pi_twice();
|
||
|
|
|
||
|
|
<span style="color: #FF0000">/* interval explicit comparison functions:
|
||
|
|
* the mode can be cer=certainly or pos=possibly,
|
||
|
|
* the function lt=less_than, gt=greater_than, le=less_than_or_equal_to, ge=greater_than_or_equal_to
|
||
|
|
* eq=equal_to, ne= not_equal_to */</span>
|
||
|
|
template <class T, class Policies> bool cerlt(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool cerlt(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool cerlt(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool cerle(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool cerle(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool cerle(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool cergt(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool cergt(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool cergt(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool cerge(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool cerge(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool cerge(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool cereq(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool cereq(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool cereq(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool cerne(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool cerne(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool cerne(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool poslt(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool poslt(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool poslt(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool posle(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool posle(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool posle(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool posgt(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool posgt(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool posgt(const T& x, const interval<T, Policies> & y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool posge(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool posge(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool posge(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool poseq(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool poseq(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool poseq(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
template <class T, class Policies> bool posne(const interval<T, Policies>& x, const interval<T, Policies>& y);
|
||
|
|
template <class T, class Policies> bool posne(const interval<T, Policies>& x, const T& y);
|
||
|
|
template <class T, class Policies> bool posne(const T& x, const interval<T, Policies>& y);
|
||
|
|
|
||
|
|
<span style="color: #FF0000">/* comparison namespaces */</span>
|
||
|
|
namespace compare {
|
||
|
|
namespace certain;
|
||
|
|
namespace possible;
|
||
|
|
namespace lexicographic;
|
||
|
|
namespace set;
|
||
|
|
namespace tribool;
|
||
|
|
} // namespace compare
|
||
|
|
|
||
|
|
} // namespace interval_lib
|
||
|
|
} // namespace numeric
|
||
|
|
} // namespace boost
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
<p>Each component of the interval support library is detailed in its own
|
||
|
|
page.</p>
|
||
|
|
|
||
|
|
<ul>
|
||
|
|
<li><a href="comparisons.htm">Comparisons</a></li>
|
||
|
|
|
||
|
|
<li><a href="rounding.htm">Rounding</a></li>
|
||
|
|
|
||
|
|
<li><a href="checking.htm">Checking</a></li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<h2 id="dangers">Common Pitfalls and Dangers</h2>
|
||
|
|
|
||
|
|
<h4>Comparisons</h4>
|
||
|
|
|
||
|
|
<p>One of the biggest problems is probably the correct use of the
|
||
|
|
comparison functions and operators. First, functions and operators do not
|
||
|
|
try to know if two intervals are the same mathematical object. So, if the
|
||
|
|
comparison used is "certain", then <code>x != x</code> is always true
|
||
|
|
unless <code>x</code> is a singleton interval; and the same problem arises
|
||
|
|
with <code>cereq</code> and <code>cerne</code>.</p>
|
||
|
|
|
||
|
|
<p>Another misleading interpretation of the comparison is: you cannot
|
||
|
|
always expect [a,b] < [c,d] to be !([a,b] >= [c,d]) since the
|
||
|
|
comparison is not necessarily total. Equality and less comparison should be
|
||
|
|
seen as two distincts relational operators. However the default comparison
|
||
|
|
operators do respect this property since they throw an exception whenever
|
||
|
|
[a,b] and [c,d] overlap.</p>
|
||
|
|
|
||
|
|
<h4>Interval values and references</h4>
|
||
|
|
|
||
|
|
<p>This problem is a corollary of the previous problem with <code>x !=
|
||
|
|
x</code>. All the functions of the library only consider the value of an
|
||
|
|
interval and not the reference of an interval. In particular, you should
|
||
|
|
not expect (unless <code>x</code> is a singleton) the following values to
|
||
|
|
be equal: <code>x/x</code> and 1, <code>x*x</code> and
|
||
|
|
<code>square(x)</code>, <code>x-x</code> and 0, etc. So the main cause of
|
||
|
|
wide intervals is that interval arithmetic does not identify different
|
||
|
|
occurrences of the same variable. So, whenever possible, the user has to
|
||
|
|
rewrite the formulas to eliminate multiple occurences of the same variable.
|
||
|
|
For example, <code>square(x)-2*x</code> is far less precise than
|
||
|
|
<code>square(x-1)-1</code>.</p>
|
||
|
|
|
||
|
|
<h4>Unprotected rounding</h4>
|
||
|
|
|
||
|
|
<p>As explained in <a href="rounding.htm#perf">this section</a>, a good way
|
||
|
|
to speed up computations when the base type is a basic floating-point type
|
||
|
|
is to unprotect the intervals at the hot spots of the algorithm. This
|
||
|
|
method is safe and really an improvement for interval computations. But
|
||
|
|
please remember that any basic floating-point operation executed inside the
|
||
|
|
unprotection blocks will probably have an undefined behavior (but only for
|
||
|
|
the current thread). And do not forget to create a rounding object as
|
||
|
|
explained in the <a href="rounding.htm#perfexp">example</a>.</p>
|
||
|
|
|
||
|
|
<h2 id="rationale">Rationale</h2>
|
||
|
|
|
||
|
|
<p>The purpose of this library is to provide an efficient and generalized
|
||
|
|
way to deal with interval arithmetic through the use of a templatized class
|
||
|
|
<code>boost::numeric::interval</code>. The big contention for which we provide a
|
||
|
|
rationale is the format of this class template.</p>
|
||
|
|
|
||
|
|
<p>It would have been easier to provide a class interval whose base type is
|
||
|
|
double. Or to follow <code>std::complex</code> and allow only
|
||
|
|
specializations for <code>float</code>, <code>double</code>, and <code>long
|
||
|
|
double</code>. We decided not to do this to allow intervals on custom
|
||
|
|
types, e.g. fixed-precision bigfloat library types (MPFR, etc), rational
|
||
|
|
numbers, and so on.</p>
|
||
|
|
|
||
|
|
<p><strong>Policy design.</strong> Although it was tempting to make it a
|
||
|
|
class template with only one template argument, the diversity of uses for
|
||
|
|
an interval arithmetic practically forced us to use policies. The behavior
|
||
|
|
of this class can be fixed by two policies. These policies are packaged
|
||
|
|
into a single policy class, rather than making <code>interval</code> with
|
||
|
|
three template parameters. This is both for ease of use (the policy class
|
||
|
|
can be picked by default) and for readability.</p>
|
||
|
|
|
||
|
|
<p>The first policy provides all the mathematical functions on the base
|
||
|
|
type needed to define the functions on the interval type. The second one
|
||
|
|
sets the way exceptional cases encountered during computations are
|
||
|
|
handled.</p>
|
||
|
|
|
||
|
|
<p>We could foresee situations where any combination of these policies
|
||
|
|
would be appropriate. Moreover, we wanted to enable the user of the library
|
||
|
|
to reuse the <code>interval</code> class template while at the same time
|
||
|
|
choosing his own behavior. See this <a href="guide.htm">page</a> for some
|
||
|
|
examples.</p>
|
||
|
|
|
||
|
|
<p><strong>Rounding policy.</strong> The library provides specialized
|
||
|
|
implementations of the rounding policy for the primitive types float and
|
||
|
|
double. In order for these implementations to be correct and fast, the
|
||
|
|
library needs to work a lot with rounding modes. Some processors are
|
||
|
|
directly dealt with and some mechanisms are provided in order to speed up
|
||
|
|
the computations. It seems to be heavy and hazardous optimizations for a
|
||
|
|
gain of only a few computer cycles; but in reality, the speed-up factor can
|
||
|
|
easily go past 2 or 3 depending on the computer. Moreover, these
|
||
|
|
optimizations do not impact the interface in any major way (with the design
|
||
|
|
we have chosen, everything can be added by specialization or by passing
|
||
|
|
different template parameters).</p>
|
||
|
|
|
||
|
|
<p><strong>Pred/succ.</strong> In a previous version, two functions
|
||
|
|
<code>pred</code> and <code>succ</code>, with various corollaries like
|
||
|
|
<code>widen</code>, were supplied. The intent was to enlarge the interval
|
||
|
|
by one ulp (as little as possible), e.g. to ensure the inclusion property.
|
||
|
|
Since making interval a template of T, we could not define <i>ulp</i> for a
|
||
|
|
random parameter. In turn, rounding policies let us eliminate entirely the
|
||
|
|
use of ulp while making the intervals tighter (if a result is a
|
||
|
|
representable singleton, there is no use to widen the interval). We decided
|
||
|
|
to drop those functions.</p>
|
||
|
|
|
||
|
|
<p><strong>Specialization of <code>std::less</code>.</strong> Since the
|
||
|
|
operator <code><</code> depends on the comparison namespace locally
|
||
|
|
chosen by the user, it is not possible to correctly specialize
|
||
|
|
<code>std::less</code>. So you have to explicitely provide such a class to
|
||
|
|
all the algorithms and templates that could require it (for example,
|
||
|
|
<code>std::map</code>).</p>
|
||
|
|
|
||
|
|
<p><strong>Input/output.</strong> The interval library does not include I/O
|
||
|
|
operators. Printing an interval value allows a lot of customization: some
|
||
|
|
people may want to output the bounds, others may want to display the median
|
||
|
|
and the width of intervals, and so on. The example file io.cpp shows some
|
||
|
|
possibilities and may serve as a foundation in order for the user to define
|
||
|
|
her own operators.</p>
|
||
|
|
|
||
|
|
<p><strong>Mixed operations with integers.</strong> When using and reusing
|
||
|
|
template codes, it is common there are operations like <code>2*x</code>.
|
||
|
|
However, the library does not provide them by default because the
|
||
|
|
conversion from <code>int</code> to the base number type is not always
|
||
|
|
correct (think about the conversion from a 32bit integer to a single
|
||
|
|
precision floating-point number). So the functions have been put in a
|
||
|
|
separate header and the user needs to include them explicitely if she wants
|
||
|
|
to benefit from these mixed operators. Another point, there is no mixed
|
||
|
|
comparison operators due to the technical way they are defined.</p>
|
||
|
|
|
||
|
|
<p><strong>Interval-aware functions.</strong> All the functions defined by
|
||
|
|
the library are obviously aware they manipulate intervals and they do it
|
||
|
|
accordingly to general interval arithmetic principles. Consequently they
|
||
|
|
may have a different behavior than the one commonly encountered with
|
||
|
|
functions not interval-aware. For example, <code>max</code> is defined by
|
||
|
|
canonical set extension and the result is not always one of the two
|
||
|
|
arguments (if the intervals do not overlap, then the result is one of the
|
||
|
|
two intervals).</p>
|
||
|
|
|
||
|
|
<p>This behavior is different from <code>std::max</code> which returns a
|
||
|
|
reference on one of its arguments. So if the user expects a reference to be
|
||
|
|
returned, she should use <code>std::max</code> since it is exactly what
|
||
|
|
this function does. Please note that <code>std::max</code> will throw an
|
||
|
|
exception when the intervals overlap. This behavior does not predate the
|
||
|
|
one described by the C++ standard since the arguments are not "equivalent"
|
||
|
|
and it allows to have an equivalence between <code>a <= b</code> and
|
||
|
|
<code>&b == &std::max(a,b)</code>(some particular cases may be
|
||
|
|
implementation-defined). However it is different from the one described by
|
||
|
|
SGI since it does not return the first argument even if "neither is greater
|
||
|
|
than the other".</p>
|
||
|
|
|
||
|
|
<h2 id="acks">History and Acknowledgments</h2>
|
||
|
|
|
||
|
|
<p>This library was mostly inspired by previous work from Jens Maurer. Some
|
||
|
|
discussions about his work are reproduced <a href=
|
||
|
|
"http://www.mscs.mu.edu/%7Egeorgec/IFAQ/maurer1.html">here</a>. Jeremy Siek
|
||
|
|
and Maarten Keijzer provided some rounding control for MSVC and Sparc
|
||
|
|
platforms.</p>
|
||
|
|
|
||
|
|
<p>Guillaume Melquiond, Hervé Brönnimann and Sylvain Pion
|
||
|
|
started from the library left by Jens and added the policy design.
|
||
|
|
Guillaume and Sylvain worked hard on the code, especially the porting and
|
||
|
|
mostly tuning of the rounding modes to the different architectures.
|
||
|
|
Guillaume did most of the coding, while Sylvain and Hervé have
|
||
|
|
provided some useful comments in order for this library to be written.
|
||
|
|
Hervé reorganized and wrote chapters of the documentation based on
|
||
|
|
Guillaume's great starting point.</p>
|
||
|
|
|
||
|
|
<p>This material is partly based upon work supported by the National
|
||
|
|
Science Foundation under NSF CAREER Grant CCR-0133599. Any opinions,
|
||
|
|
findings and conclusions or recommendations expressed in this material are
|
||
|
|
those of the author(s) and do not necessarily reflect the views of the
|
||
|
|
National Science Foundation (NSF).</p>
|
||
|
|
<hr>
|
||
|
|
|
||
|
|
<p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
|
||
|
|
"../../../../doc/images/valid-html401.png" alt="Valid HTML 4.01 Transitional"
|
||
|
|
height="31" width="88"></a></p>
|
||
|
|
|
||
|
|
<p>Revised
|
||
|
|
<!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-25<!--webbot bot="Timestamp" endspan i-checksum="12174" --></p>
|
||
|
|
|
||
|
|
<p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé
|
||
|
|
Brönnimann, Polytechnic University<br>
|
||
|
|
Copyright © 2003-2006 Guillaume Melquiond, ENS Lyon</i></p>
|
||
|
|
|
||
|
|
<p><i>Distributed under the Boost Software License, Version 1.0. (See
|
||
|
|
accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a>
|
||
|
|
or copy at <a href=
|
||
|
|
"http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
|
||
|
|
</body>
|
||
|
|
</html>
|