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| <title>Boost Interval Arithmetic Library</title> |
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| <h1><img src="../../../../boost.png" alt="boost.png (6897 bytes)" align= |
| "middle"> Interval Arithmetic Library</h1> |
| |
| <center> |
| <table width="80%" summary=""> |
| <tbody> |
| <tr> |
| <td><b>Contents of this page:</b><br> |
| <a href="#intro">Introduction</a><br> |
| <a href="#synopsis">Synopsis</a><br> |
| <a href="#interval">Template class <code>interval</code></a><br> |
| <a href="#opers">Operations and functions</a><br> |
| <a href="#interval_lib">Interval support library</a><br> |
| <!--<a href="#compil">Compilation notes</a><br>--> |
| <a href="#dangers">Common pitfalls and dangers</a><br> |
| <a href="#rationale">Rationale</a><br> |
| <a href="#acks">History and Acknowledgments</a></td> |
| |
| <td><b>Other pages associated with this page:</b><br> |
| <a href="rounding.htm">Rounding policies</a><br> |
| <a href="checking.htm">Checking policies</a><br> |
| <a href="policies.htm">Policies manipulation</a><br> |
| <a href="comparisons.htm">Comparisons</a><br> |
| <a href="numbers.htm">Base number type requirements</a><br> |
| <a href="guide.htm">Choosing your own interval type</a><br> |
| <a href="examples.htm">Test and example programs</a><br> |
| <a href="includes.htm">Headers inclusion</a><br> |
| <a href="todo.htm">Some items on the todo list</a></td> |
| </tr> |
| </tbody> |
| </table> |
| </center> |
| |
| <h2 id="intro">Introduction and Overview</h2> |
| |
| <p>As implied by its name, this library is intended to help manipulating |
| mathematical intervals. It consists of a single header <<a href= |
| "../../../../boost/numeric/interval.hpp">boost/numeric/interval.hpp</a>> |
| and principally a type which can be used as <code>interval<T></code>. |
| In fact, this interval template is declared as |
| <code>interval<T,Policies></code> where <code>Policies</code> is a |
| policy class that controls the various behaviours of the interval class; |
| <code>interval<T></code> just happens to pick the default policies |
| for the type <code>T</code>.</p> |
| |
| <p><span style="color: #FF0000; font-weight: bold">Warning!</span> |
| Guaranteed interval arithmetic for native floating-point format is not |
| supported on every combination of processor, operating system, and |
| compiler. This is a list of systems known to work correctly when using |
| <code>interval<float></code> and <code>interval<double></code> |
| with basic arithmetic operators.</p> |
| |
| <ul> |
| <li>x86-like hardware is supported by the library with GCC, Visual C++ |
| ≥ 7.1, Intel compiler (≥ 8 on Windows), CodeWarrior (≥ 9), as |
| long as the traditional x87 floating-point unit is used for |
| floating-point computations (no <code>-mfpmath=sse2</code> support).</li> |
| |
| <li>Sparc hardware is supported with GCC and Sun compiler.</li> |
| |
| <li>PowerPC hardware is supported with GCC and CodeWarrior, when |
| floating-point computations are not done with the Altivec unit.</li> |
| |
| <li>Alpha hardware is supported with GCC, except maybe for the square |
| root. The options <code>-mfp-rounding-mode=d -mieee</code> have to be |
| used.</li> |
| </ul> |
| |
| <p>The previous list is not exhaustive. And even if a system does not |
| provide guaranteed computations for hardware floating-point types, the |
| interval library is still usable with user-defined types and for doing box |
| arithmetic.</p> |
| |
| <h3>Interval Arithmetic</h3> |
| |
| <p>An interval is a pair of numbers which represents all the numbers |
| between these two. (Intervals are considered closed so the bounds are |
| included.) The purpose of this library is to extend the usual arithmetic |
| functions to intervals. These intervals will be written [<i>a</i>,<i>b</i>] |
| to represent all the numbers between <i>a</i> and <i>b</i> (included). |
| <i>a</i> and <i>b</i> can be infinite (but they can not be the same |
| infinite) and <i>a</i> ≤ <i>b</i>.</p> |
| |
| <p>The fundamental property of interval arithmetic is the |
| <em><strong>inclusion property</strong></em>:</p> |
| |
| <dl> |
| <dd>``if <i>f</i> is a function on a set of numbers, <i>f</i> can be |
| extended to a new function defined on intervals. This new function |
| <i>f</i> takes one interval argument and returns an interval result such |
| as: ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>], <i>f</i>(<i>x</i>) |
| ∈ <i>f</i>([<i>a</i>,<i>b</i>]).''</dd> |
| </dl> |
| |
| <p>Such a property is not limited to functions with only one argument. |
| Whenever possible, the interval result should be the smallest one able to |
| satisfy the property (it is not really useful if the new functions always |
| answer [-∞,+∞]).</p> |
| |
| <p>There are at least two reasons a user would like to use this library. |
| The obvious one is when the user has to compute with intervals. One example |
| is when input data have some builtin imprecision: instead of a number, an |
| input variable can be passed as an interval. Another example application is |
| to solve equations, by bisecting an interval until the interval width is |
| small enough. A third example application is in computer graphics, where |
| computations with boxes, segments or rays can be reduced to computations |
| with points via intervals.</p> |
| |
| <p>Another common reason to use interval arithmetic is when the computer |
| doesn't produce exact results: by using intervals, it is possible to |
| quantify the propagation of rounding errors. This approach is used often in |
| numerical computation. For example, let's assume the computer stores |
| numbers with ten decimal significant digits. To the question 1 + 1E-100 - |
| 1, the computer will answer 0 although the correct answer would be 1E-100. |
| With the help of interval arithmetic, the computer will answer [0,1E-9]. |
| This is quite a huge interval for such a little result, but the precision |
| is now known, without having to compute error propagation.</p> |
| |
| <h3>Numbers, rounding, and exceptional behavior</h3> |
| |
| <p>The <em><strong>base number type</strong></em> is the type that holds |
| the bounds of the interval. In order to successfully use interval |
| arithmetic, the base number type must present some <a href= |
| "rounding.htm">characteristics</a>. Firstly, due to the definition of an |
| interval, the base numbers have to be totally ordered so, for instance, |
| <code>complex<T></code> is not usable as base number type for |
| intervals. The mathematical functions for the base number type should also |
| be compatible with the total order (for instance if x>y and z>t, then |
| it should also hold that x+z > y+t), so modulo types are not usable |
| either.</p> |
| |
| <p>Secondly, the computations must be exact or provide some rounding |
| methods (for instance, toward minus or plus infinity) if we want to |
| guarantee the inclusion property. Note that we also may explicitely specify |
| no rounding, for instance if the base number type is exact, i.e. the result |
| of a mathematic operation is always computed and represented without loss |
| of precision. If the number type is not exact, we may still explicitely |
| specify no rounding, with the obvious consequence that the inclusion |
| property is no longuer guaranteed.</p> |
| |
| <p>Finally, because heavy loss of precision is always possible, some |
| numbers have to represent infinities or an exceptional behavior must be |
| defined. The same situation also occurs for NaN (<i>Not a Number</i>).</p> |
| |
| <p>Given all this, one may want to limit the template argument T of the |
| class template <code>interval</code> to the floating point types |
| <code>float</code>, <code>double</code>, and <code>long double</code>, as |
| defined by the IEEE-754 Standard. Indeed, if the interval arithmetic is |
| intended to replace the arithmetic provided by the floating point unit of a |
| processor, these types are the best choice. Unlike |
| <code>std::complex</code>, however, we don't want to limit <code>T</code> |
| to these types. This is why we allow the rounding and exceptional behaviors |
| to be given by the two policies (rounding and checking). We do nevertheless |
| provide highly optimized rounding and checking class specializations for |
| the above-mentioned floating point types.</p> |
| |
| <h3>Operations and functions</h3> |
| |
| <p>It is straightforward to define the elementary arithmetic operations on |
| intervals, being guided by the inclusion property. For instance, if [a,b] |
| and [c,d] are intervals, [a,b]+[c,d] can be computed by taking the smallest |
| interval that contains all the numbers x+y for x in [a,b] and y in [c,d]; |
| in this case, rounding a+c down and b+d up will suffice. Other operators |
| and functions are similarly defined (see their definitions below).</p> |
| |
| <h3>Comparisons</h3> |
| |
| <p>It is also possible to define some comparison operators. Given two |
| intervals, the result is a tri-state boolean type |
| {<i>false</i>,<i>true,indeterminate</i>}. The answers <i>false</i> and |
| <i>true</i> are easy to manipulate since they can directly be mapped on the |
| boolean <i>true</i> and <i>false</i>. But it is not the case for the answer |
| <em>indeterminate</em> since comparison operators are supposed to be |
| boolean functions. So, what to do in order to obtain boolean answers?</p> |
| |
| <p>One solution consists of deciding to adopt an exceptional behavior, such |
| as a failed assertion or raising an exception. In this case, the |
| exceptional behavior will be triggered when the result is |
| indeterminate.</p> |
| |
| <p>Another solution is to map <em>indeterminate</em> always to |
| <i>false,</i> or always to <i>true</i>. If <i>false</i> is chosen, the |
| comparison will be called "<i>certain</i>;" indeed, the result of |
| [<i>a</i>,<i>b</i>] < [<i>c</i>,<i>d</i>] will be <i>true</i> if and |
| only if: ∀ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∀ <i>y</i> |
| ∈ [<i>c</i>,<i>d</i>], <i>x</i> < <i>y</i>. If <i>true</i> is |
| chosen, the comparison will be called "<i>possible</i>;" indeed, the result |
| of [<i>a</i>,<i>b</i>] < [<i>c</i>,<i>d</i>] will be <i>true</i> if and |
| only if: ∃ <i>x</i> ∈ [<i>a</i>,<i>b</i>] ∃ <i>y</i> |
| ∈ [<i>c</i>,<i>d</i>], <i>x</i> < <i>y</i>.</p> |
| |
| <p>Since any of these solution has a clearly defined semantics, it is not |
| clear that we should enforce either of them. For this reason, the default |
| behavior consists to mimic the real comparisons by throwing an exception in |
| the indeterminate case. Other behaviors can be selected bu using specific |
| comparison namespace. There is also a bunch of explicitely named comparison |
| functions. See <a href="comparisons.htm">comparisons</a> pages for further |
| details.</p> |
| |
| <h3>Overview of the library, and usage</h3> |
| |
| <p>This library provides two quite distinct levels of usage. One is to use |
| the basic class template <code>interval<T></code> without specifying |
| the policy. This only requires to know and understand the concepts |
| developed above and the content of the namespace boost. In addition to the |
| class <code>interval<T></code>, this level of usage provides |
| arithmetic operators (<code>+</code>, <code>-</code>, <code>*</code>, |
| <code>/</code>), algebraic and piecewise-algebraic functions |
| (<code>abs</code>, <code>square</code>, <code>sqrt</code>, |
| <code>pow</code>), transcendental and trigonometric functions |
| (<code>exp</code>, <code>log</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>), and the |
| standard comparison operators (<code><</code>, <code><=</code>, |
| <code>></code>, <code>>=</code>, <code>==</code>, <code>!=</code>), |
| as well as several interval-specific functions (<code>min</code>, |
| <code>max</code>, which have a different meaning than <code>std::min</code> |
| and <code>std::max</code>; <code>lower</code>, <code>upper</code>, |
| <code>width</code>, <code>median</code>, <code>empty</code>, |
| <code>singleton</code>, <code>equal</code>, <code>in</code>, |
| <code>zero_in</code>, <code>subset</code>, <code>proper_subset</code>, |
| <code>overlap</code>, <code>intersection</code>, <code>hull</code>, |
| <code>bisect</code>).</p> |
| |
| <p>For some functions which take several parameters of type |
| <code>interval<T></code>, all combinations of argument types |
| <code>T</code> and <code>interval<T></code> which contain at least |
| one <code>interval<T></code>, are considered in order to avoid a |
| conversion from the arguments of type <code>T</code> to a singleton of type |
| <code>interval<T></code>. This is done for efficiency reasons (the |
| fact that an argument is a singleton sometimes renders some tests |
| unnecessary).</p> |
| |
| <p>A somewhat more advanced usage of this library is to hand-pick the |
| policies <code>Rounding</code> and <code>Checking</code> and pass them to |
| <code>interval<T, Policies></code> through the use of <code>Policies |
| := boost::numeric::interval_lib::policies<Rounding,Checking></code>. |
| Appropriate policies can be fabricated by using the various classes |
| provided in the namespace <code>boost::numeric::interval_lib</code> as |
| detailed in section <a href="#interval_lib">Interval Support Library</a>. |
| It is also possible to choose the comparison scheme by overloading |
| operators through namespaces.</p> |
| |
| <h2><a name="synopsis" id="synopsis"></a>Synopsis</h2> |
| <pre> |
| namespace boost { |
| namespace numeric { |
| |
| namespace interval_lib { |
| |
| /* this declaration is necessary for the declaration of interval */ |
| template <class T> struct default_policies; |
| |
| /* ... ; the full synopsis of namespace interval_lib can be found <a href= |
| "#interval_lib">here</a> */ |
| |
| } // namespace interval_lib |
| |
| |
| /* template interval_policies; class definition can be found <a href= |
| "policies.htm">here</a> */ |
| template<class Rounding, class Checking> |
| struct interval_policies; |
| |
| /* template class interval; class definition can be found <a href= |
| "#interval">here</a> */ |
| template<class T, class Policies = typename interval_lib::default_policies<T>::type > class interval; |
| |
| /* arithmetic operators involving intervals */ |
| template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x); |
| template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x); |
| |
| 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& 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& 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& 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> intersection(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 repectively 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 containg [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 problably 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 |
| occurences 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 mecanisms 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> |