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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 &lt;<a href=
  "../../../../boost/numeric/interval.hpp">boost/numeric/interval.hpp</a>&gt;
  and principally a type which can be used as <code>interval&lt;T&gt;</code>.
  In fact, this interval template is declared as
  <code>interval&lt;T,Policies&gt;</code> where <code>Policies</code> is a
  policy class that controls the various behaviours of the interval class;
  <code>interval&lt;T&gt;</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&lt;float&gt;</code> and <code>interval&lt;double&gt;</code>
  with basic arithmetic operators.</p>

  <ul>
    <li>x86-like hardware is supported by the library with GCC, Visual C++
    &ge; 7.1, Intel compiler (&ge; 8 on Windows), CodeWarrior (&ge; 9), as
    long as the traditional x87 floating-point unit is used for
    floating-point computations (no <code>-mfpmath=sse2</code> support).
      <ul>
        <li>clang (though v8) does not have proper
          <a href="http://lists.llvm.org/pipermail/llvm-dev/2018-May/123529.html">rounding support</a>.</li>
        <li>gcc requires <tt>-frounding-math</tt> to be specified.</li>
        <li>msvc requires <tt>/fp:strict</tt> to be specified, which means
            MSVC 10.0 and earlier will not work properly.</li>
      </ul>
    </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>

    <li>valgrind cannot be used as it lacks necessary rounding support.</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> &le; <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: &forall; <i>x</i> &isin; [<i>a</i>,<i>b</i>], <i>f</i>(<i>x</i>)
    &isin; <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 [-&infin;,+&infin;]).</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&lt;T&gt;</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&gt;y and z&gt;t, then
  it should also hold that x+z &gt; 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 mathematical 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 longer 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>] &lt; [<i>c</i>,<i>d</i>] will be <i>true</i> if and
  only if: &forall; <i>x</i> &isin; [<i>a</i>,<i>b</i>] &forall; <i>y</i>
  &isin; [<i>c</i>,<i>d</i>], <i>x</i> &lt; <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>] &lt; [<i>c</i>,<i>d</i>] will be <i>true</i> if and
  only if: &exist; <i>x</i> &isin; [<i>a</i>,<i>b</i>] &exist; <i>y</i>
  &isin; [<i>c</i>,<i>d</i>], <i>x</i> &lt; <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&lt;T&gt;</code> without specifying
  the policy. This only requires one to know and understand the concepts
  developed above and the content of the namespace boost. In addition to the
  class <code>interval&lt;T&gt;</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>&lt;</code>, <code>&lt;=</code>,
  <code>&gt;</code>, <code>&gt;=</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>intersect</code>, <code>hull</code>,
  <code>bisect</code>).</p>

  <p>For some functions which take several parameters of type
  <code>interval&lt;T&gt;</code>, all combinations of argument types
  <code>T</code> and <code>interval&lt;T&gt;</code> which contain at least
  one <code>interval&lt;T&gt;</code>, are considered in order to avoid a
  conversion from the arguments of type <code>T</code> to a singleton of type
  <code>interval&lt;T&gt;</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&lt;T, Policies&gt;</code> through the use of <code>Policies
  := boost::numeric::interval_lib::policies&lt;Rounding,Checking&gt;</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 &lt;class T&gt; 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&lt;class Rounding, class Checking&gt;
struct interval_policies;

/* template class interval; class definition can be found <a href=
"#interval">here</a> */
template&lt;class T, class Policies = typename interval_lib::default_policies&lt;T&gt;::type &gt; class interval;

/* arithmetic operators involving intervals */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator+(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator-(const interval&lt;T, Policies&gt;&amp; x);

template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator+(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator+(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator+(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator-(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator-(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator-(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator*(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator*(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator*(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator/(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator/(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; operator/(const T&amp; r, const interval&lt;T, Policies&gt;&amp; x);

/* algebraic functions: sqrt, abs, square, pow, nth_root */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; abs(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; sqrt(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; square(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; pow(const interval&lt;T, Policies&gt;&amp; x, int y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; nth_root(const interval&lt;T, Policies&gt;&amp; x, int y);

/* transcendental functions: exp, log */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; exp(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; log(const interval&lt;T, Policies&gt;&amp; x);

/* fmod, for trigonometric function argument reduction (see below) */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; fmod(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; fmod(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; fmod(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

/* trigonometric functions */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; sin(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; cos(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; tan(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; asin(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; acos(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; atan(const interval&lt;T, Policies&gt;&amp; x);

/* hyperbolic trigonometric functions */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; sinh(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; cosh(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; tanh(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; asinh(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; acosh(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; atanh(const interval&lt;T, Policies&gt;&amp; x);

/* min, max external functions (NOT std::min/max, see below) */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; max(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; max(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; max(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; min(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; min(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; min(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

/* bounds-related interval functions */
template &lt;class T, class Policies&gt;  T lower(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  T upper(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  T width(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  T median(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  T norm(const interval&lt;T, Policies&gt;&amp; x);

/* bounds-related interval functions */
template &lt;class T, class Policies&gt;  bool empty(const interval&lt;T, Policies&gt;&amp; b);
template &lt;class T, class Policies&gt;  bool singleton(const interval&lt;T, Policies&gt;&amp; x);
template &lt;class T, class Policies&gt;  bool equal(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool in(const T&amp; r, const interval&lt;T, Policies&gt;&amp; b);
template &lt;class T, class Policies&gt;  bool zero_in(const interval&lt;T, Policies&gt;&amp; b);
template &lt;class T, class Policies&gt;  bool subset(const interval&lt;T, Policies&gt;&amp; a, const interval&lt;T, Policies&gt;&amp; b);
template &lt;class T, class Policies&gt;  bool proper_subset(const interval&lt;T, Policies&gt;&amp; a, const interval&lt;T, Policies&gt;&amp; b);
template &lt;class T, class Policies&gt;  bool overlap(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);

/* set manipulation interval functions */
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; intersect(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; hull(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; hull(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; hull(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; hull(const T&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  std::pair&lt;interval&lt;T, Policies&gt;, interval&lt;T, Policies&gt; &gt; bisect(const interval&lt;T, Policies&gt;&amp; x);

/* interval comparison operators */
template&lt;class T, class Policies&gt;  bool operator&lt;(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template&lt;class T, class Policies&gt;  bool operator&lt;(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template&lt;class T, class Policies&gt;  bool operator&lt;(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template&lt;class T, class Policies&gt;  bool operator&lt;=(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template&lt;class T, class Policies&gt;  bool operator&lt;=(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template&lt;class T, class Policies&gt;  bool operator&lt;=(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template&lt;class T, class Policies&gt;  bool operator&gt;(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template&lt;class T, class Policies&gt;  bool operator&gt;(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template&lt;class T, class Policies&gt;  bool operator&gt;(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template&lt;class T, class Policies&gt;  bool operator&gt;=(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template&lt;class T, class Policies&gt;  bool operator&gt;=(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template&lt;class T, class Policies&gt;  bool operator&gt;=(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);
</pre>
  <pre>
template&lt;class T, class Policies&gt;  bool operator==(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template&lt;class T, class Policies&gt;  bool operator==(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template&lt;class T, class Policies&gt;  bool operator==(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template&lt;class T, class Policies&gt;  bool operator!=(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template&lt;class T, class Policies&gt;  bool operator!=(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template&lt;class T, class Policies&gt;  bool operator!=(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

namespace interval_lib {

template&lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; division_part1(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&amp; y, bool&amp; b);
template&lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; division_part2(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&amp; y, bool b = true);
template&lt;class T, class Policies&gt;  interval&lt;T, Policies&gt; multiplicative_inverse(const interval&lt;T, Policies&gt;&amp; x);

template&lt;class I&gt;  I add(const typename I::base_type&amp; x, const typename I::base_type&amp; y);
template&lt;class I&gt;  I sub(const typename I::base_type&amp; x, const typename I::base_type&amp; y);
template&lt;class I&gt;  I mul(const typename I::base_type&amp; x, const typename I::base_type&amp; y);
template&lt;class I&gt;  I div(const typename I::base_type&amp; x, const typename I::base_type&amp; 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 &lt;class T, class Policies = typename interval_lib::default_policies&lt;T&gt;::type&gt;
class interval
{
  public:
  typedef T base_type;
  typedef Policies traits_type;

  interval();
  interval(T const &amp;v);
  template&lt;class T1&gt; interval(T1 const &amp;v);
  interval(T const &amp;l, T const &amp;u);
  template&lt;class T1, class T2&gt; interval(T1 const &amp;l, T2 const &amp;u);
  interval(interval&lt;T, Policies&gt; const &amp;r);
  template&lt;class Policies1&gt; interval(interval&lt;T, Policies1&gt; const &amp;r);
  template&lt;class T1, class Policies1&gt; interval(interval&lt;T1, Policies1&gt; const &amp;r);

  interval &amp;operator=(T const &amp;v);
  template&lt;class T1&gt; interval &amp;operator=(T1 const &amp;v);
  interval &amp;operator=(interval&lt;T, Policies&gt; const &amp;r);
  template&lt;class Policies1&gt; interval &amp;operator=(interval&lt;T, Policies1&gt; const &amp;r);
  template&lt;class T1, class Policies1&gt; interval &amp;operator=(interval&lt;T1, Policies1&gt; const &amp;r);

  void assign(T const &amp;l, T const &amp;u);

  T const &amp;lower() const;
  T const &amp;upper() const;

  static interval empty();
  static interval whole();
  static interval hull(T const &amp;x, T const &amp;y);

  interval&amp; operator+= (T const &amp;r);
  interval&amp; operator-= (T const &amp;r);
  interval&amp; operator*= (T const &amp;r);
  interval&amp; operator/= (T const &amp;r);
  interval&amp; operator+= (interval const &amp;r);
  interval&amp; operator-= (interval const &amp;r);
  interval&amp; operator*= (interval const &amp;r);
  interval&amp; operator/= (interval const &amp;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 &lt;= 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&lt;b?a:b</code>, <code>a&gt;b?a:b</code>,
  <code>a&lt;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]&sup2; = [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 ]-&infin;,&infin;[ but the union
  of ]-&infin;,-1] and [2,&infin;[. 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 &lt;= z.lower() &lt; 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&lt;I&gt;()</code>, <code>pi_half&lt;I&gt;()</code> and
  <code>pi_twice&lt;I&gt;()</code>, and they return an object of interval
  type <code>I</code>. Their respective values are &pi;, &pi;/2 and
  2&pi;.</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&lt;T&gt;</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 &lt;class T&gt;  struct rounded_math;
template &lt;&gt;         struct rounded_math&lt;float&gt;;
template &lt;&gt;         struct rounded_math&lt;double&gt;;
template &lt;&gt;         struct rounded_math&lt;long double&gt;;

<span style=
"color: #FF0000">/* built-in rounding construction blocks */</span>
template &lt;class T&gt;  struct rounding_control;

template &lt;class T, class Rounding = rounding_control&lt;T&gt; &gt;  struct rounded_arith_exact;
template &lt;class T, class Rounding = rounding_control&lt;T&gt; &gt;  struct rounded_arith_std;
template &lt;class T, class Rounding = rounding_control&lt;T&gt; &gt;  struct rounded_arith_opp;

template &lt;class T, class Rounding&gt;  struct rounded_transc_dummy;
template &lt;class T, class Rounding = rounded_arith_exact&lt;T&gt; &gt;  struct rounded_transc_exact;
template &lt;class T, class Rounding = rounded_arith_std  &lt;T&gt; &gt;  struct rounded_transc_std;
template &lt;class T, class Rounding = rounded_arith_opp  &lt;T&gt; &gt;  struct rounded_transc_opp;

template &lt;class Rounding&gt; struct save_state;
template &lt;class Rounding&gt; struct save_state_nothing;

<span style="color: #FF0000">/* built-in checking policies */</span>
template &lt;class T&gt; struct checking_base;
template &lt;class T, class Checking = checking_base&lt;T&gt;, class Exception = exception_create_empty&gt;   struct checking_no_empty;
template &lt;class T, class Checking = checking_base&lt;T&gt; &gt;                                            struct checking_no_nan;
template &lt;class T, class Checking = checking_base&lt;T&gt;, class Exception = exception_invalid_number&gt; struct checking_catch_nan;
template &lt;class T&gt; struct checking_strict;

<span style=
"color: #FF0000">/* some metaprogramming to manipulate interval policies */</span>
template &lt;class Rounding, class Checking&gt; struct policies;
template &lt;class OldInterval, class NewRounding&gt; struct change_rounding;
template &lt;class OldInterval, class NewChecking&gt; struct change_checking;
template &lt;class OldInterval&gt; struct unprotect;

<span style=
"color: #FF0000">/* constants, need to be explicitly templated */</span>
template&lt;class I&gt; I pi();
template&lt;class I&gt; I pi_half();
template&lt;class I&gt; 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 &lt;class T, class Policies&gt;  bool cerlt(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool cerlt(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool cerlt(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool cerle(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool cerle(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool cerle(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool cergt(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool cergt(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool cergt(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool cerge(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool cerge(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool cerge(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool cereq(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool cereq(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool cereq(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool cerne(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool cerne(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool cerne(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool poslt(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool poslt(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool poslt(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool posle(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool posle(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool posle(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool posgt(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool posgt(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool posgt(const T&amp; x, const interval&lt;T, Policies&gt; &amp; y);

template &lt;class T, class Policies&gt;  bool posge(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool posge(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool posge(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool poseq(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool poseq(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool poseq(const T&amp; x, const interval&lt;T, Policies&gt;&amp; y);

template &lt;class T, class Policies&gt;  bool posne(const interval&lt;T, Policies&gt;&amp; x, const interval&lt;T, Policies&gt;&amp; y);
template &lt;class T, class Policies&gt;  bool posne(const interval&lt;T, Policies&gt;&amp; x, const T&amp; y);
template &lt;class T, class Policies&gt;  bool posne(const T&amp; x, const interval&lt;T, Policies&gt;&amp; 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] &lt; [c,d] to be !([a,b] &gt;= [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>&lt;</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 &lt;= b</code> and
  <code>&amp;b == &amp;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, Herv&eacute; Br&ouml;nnimann 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 Herv&eacute; have
  provided some useful comments in order for this library to be written.
  Herv&eacute; 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=
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  <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 &copy; 2002 Guillaume Melquiond, Sylvain Pion, Herv&eacute;
  Br&ouml;nnimann, Polytechnic University<br>
  Copyright &copy; 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>
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