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  <h1>Checking policies</h1>

  <p>A checking policy controls how the <code>interval</code> class will deal
  with special cases like: empty intervals, infinite numbers, invalid
  values.</p>

  <p>For example, let's consider <code>operator+(interval, T)</code>. The
  second argument could be an invalid value (for a floating-point number, it
  is a NaN). What to do in such a case? First, we could say that the second
  argument can never be an invalid number. Second, we could also say such a
  situation can arise but is forbidden. Third, we could allow such values and
  generate an empty interval when encountered. And there is many other
  possibilities.</p>

  <p>It is the reason why such a policy is used: there is a lot of
  interesting behaviors and it would be sad to arbitrarily select one of
  these.</p>

  <h2>Requirements</h2>

  <p>The checking class should satisfy the following requirement (in the form
  of an interface):</p>
  <pre>
/* requirements for checking policy */
struct checking
{
  static T pos_inf();
  static T neg_inf();
  static T nan();
  static bool is_nan(const T&amp;);
  static T empty_lower();
  static T empty_upper();
  static bool is_empty(const T&amp;, const T&amp;);
};
</pre>

  <p>The first two functions, <code>pos_inf</code> and <code>neg_inf</code>,
  are invoked each time the library has to create the infinite bound of an
  interval. For example, <code>interval::whole</code> computes
  <code>interval(checking::neg_inf(), checking::pos_inf())</code>. If
  infinite values are allowed and
  <code>std::numeric_limits&lt;T&gt;::infinity()</code> returns a correct
  value, such a value can be used.</p>

  <p>Next comes <code>nan</code>. This function is used each time a function
  need to return a value of type <code>T</code> but is unable to compute it.
  It only happens when one of the arguments of the function is invalid. For
  example, if you ask what the median value of an empty interval is,
  <code>nan</code> will be used. But please remember: <code>lower</code> and
  <code>upper</code> directly return the value stocked in the interval; so,
  if the interval is empty, <code>lower</code> will not answer
  <code>by</code> a call to <code>checking::nan</code> (but will return the
  same value than <code>checking::empty_lower</code> could return).</p>

  <p><code>empty_lower</code> and <code>empty_upper</code> respectively
  return the lower and upper bound of the empty interval. There is no
  requirements for <code>empty_lower</code> and <code>empty_upper</code> to
  return the same value than <code>checking::nan</code>. For example, if the
  type <code>T</code> does not have any invalid value, the
  <code>empty_</code> functions can return the [1;0] interval.</p>

  <p><code>is_nan</code> is used to test if a value of type <code>T</code> is
  invalid or not. <code>is_empty</code> tests if the interval formed by the
  two arguments is empty or not. Such tests will generally be at the
  beginning of each function which involves an argument of type
  <code>T</code>. If one of the inputs is declared invalid, the the function
  will try to produce an invalid value or an input interval.</p>

  <h2>Synopsis</h2>
  <pre>
namespace boost {
namespace numeric {
namespace interval_lib {

template&lt;class T&gt;
struct checking_base;
template&lt;class T, class Checking = checking_base&lt;T&gt;, class Exception = exception_create_empty&lt;T&gt; &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&lt;T&gt; &gt;
struct checking_catch_nan;

template&lt;class T&gt; struct exception_create_empty { T operator()(); };
template&lt;class T&gt; struct exception_invalid_number { void operator()(); };

} // namespace numeric
} // namespace interval_lib
} // namespace boost
</pre>

  <h2>Predefined classes</h2>

  <p>In order to simplify the customization of the policy, some templates are
  already defined in the library.</p>

  <p>First of all, there is <code>checking_base</code>. Thanks to the
  information provided by <code>std::numeric_limits&lt;T&gt;</code>, this
  class is able to generate a base for the policy. If <code>T</code> has
  quiet NaNs (as said by <code>numeric_limits::has_quiet_NaN</code>), then
  the value is used for <code>nan</code>, <code>empty_lower</code>,
  <code>empty_upper</code>; and a basic test is used for <code>is_nan</code>
  (it is <code>x!=x</code>). If <code>T</code> does not have quiet NaNs, then
  <code>nan</code> is an <code>assert(false)</code>, the empty interval is
  [1,0], and <code>is_nan</code> always return <code>false</code>. As for
  <code>nan</code>, <code>pos_inf</code> returns
  <code>numeric_limits::infinity()</code> if possible, or is an
  <code>assert(false</code>) otherwise. <code>neg_inf</code> returns the
  opposite. Finally, <code>is_empty(T l,T u)</code> is always defined by
  <code>!(l&lt;=u)</code>.</p>

  <p>Next comes <code>checking_no_empty</code>. Using it means that each time
  an empty interval should be produced (by <code>empty_lower</code> and
  <code>empty_upper</code>), the function object given by the
  <code>Exception</code> argument of the template is invoked and the value it
  returns is propagated. So, if <code>Exception</code> is appropriately
  defined (for example it could throw an exception, hence the name of the
  argument), you can be sure no empty interval will ever be created. So
  <code>is_empty</code> will always return <code>false</code> (since there is
  no need to test for an empty interval). And as explained before, in that
  case we can also replace <code>nan</code> by an <code>assert(false)</code>;
  you will be sure no invalid number will ever be produced. If this template
  is not used, it implicitly means that all the functions can produce empty
  intervals and they correctly deal with empty interval arguments.</p>

  <p>Finally there are <code>checking_no_nan</code> and
  <code>checking_catch_nan</code>. The first one expresses the functions of
  the library will never get an invalid number as argument. So
  <code>is_nan</code> will only return <code>false</code>. The other one
  means the arguments can be an invalid number but in that case,
  <code>is_nan</code> will call the function object <code>Exception</code>
  and return <code>false</code>. Indeed, this template means invalid numbers
  should never make their way through to the body of the function. If none of
  this two templates is used, it implicitly means that all the functions can
  get invalid number arguments and they will correctly deal with them.</p>

  <p><code>exception_create_empty</code> throws
  <code>std::runtime_error</code> with the message <code>"boost::interval:
  empty interval created"</code> and <code>exception_invalid_number</code>
  throws <code>std::invalid_argument</code> with the message
  <code>"boost::interval: invalid number"</code>.</p>

  <h2>Customizing your own checking policy</h2>

  <p>In order to define a suitable policy, you need to correctly say what you
  expect from your interval class. First of all, are you interested in
  getting empty intervals at the end of a calculus? If you do not want to
  obtain empty intervals, <code>empty_lower</code> and
  <code>empty_upper</code> have to fail when invoked (they can throw an
  exception, set a flag, etc). However, if no function is able to produce an
  empty interval, it is no more necessary to do the test, so
  <code>is_empty</code> may always return <code>false</code>. In this case, a
  good compiler will do a lot of optimizations.</p>

  <p>You could also be interested in getting empty intervals at the end of
  the calculus. For example, if you need to transform an array of unsure
  values (or intervals) in a new array of intervals, you may not want to stop
  the conversion at the first encountered problem. So
  <code>empty_lower</code> and <code>empty_upper</code> need to return
  suitable values in order to define an empty interval (you can use an upper
  bound which is not greater or equal than the lower bound for example); and
  <code>is_empty</code> must be able to distinguish empty intervals from the
  valid intervals.</p>

  <p>Another important question is: is it possible that some base numbers
  (objects of type <code>T</code>) are invalid? And if it is possible, are
  they allowed or not ? If it is not possible, no test is necessary;
  <code>is_nan</code> may always return <code>false</code>. In this case too,
  a good compiler will do a lot of optimizations. If function arguments can
  hold invalid numbers, two cases must be considered according to whether
  they are allowed or not. If they are allowed, <code>is_nan</code> just has
  to test if they are invalid or not. If they are forbidden,
  <code>is_nan</code> should fail (exception, assert, etc.) when invoked on
  an invalid argument and return <code>false</code> otherwise. The value
  returned by <code>nan</code> does not have any interest since the interval
  functions are guaranteed not to produce invalid interval bounds unless the
  user passes invalid numbers to the constructors. So you can put an assert
  inside if you do not trust the library. :-)</p>

  <p>And finally, you need to decide what to do with <code>nan</code> if it
  has not already been decided at the beginning, and with
  <code>pos_inf</code> and <code>neg_inf</code>. These functions should
  return a value or start an exceptional behavior (especially if the base
  type does not have corresponding values).</p>

  <h2>Some examples</h2>

  <ul>
    <li>If you need a checking policy that allows the library to correctly
    manipulate data, even if they contain invalid numbers and empty
    intervals, then <code>checking_base&lt;T&gt;</code> is a
    possibility.</li>

    <li>If you do not want empty intervals to be created and are not sure all
    the numbers are valid, then <code>checking_catch_nan&lt;T,
    checking_no_empty&lt;T&gt; &gt;</code> can help you.</li>

    <li>If all the numbers will be valid and if no empty interval is supposed
    to be created (or if you do not want them to be created), then you can
    use <code>checking_no_nan&lt;T, checking_no_empty&lt;T&gt; &gt;</code>.
    Please note that if <code>T</code> does not have a way to represent
    invalid numbers, then this policy will behave the same way as
    <code>checking_no_empty&lt;T&gt;</code>. This is the default policy and
    it is also called <code>interval_lib::checking_strict</code>.</li>

    <li>If all numerical data are valid but the algorithm can produce and
    manipulate empty intervals, then <code>checking_no_nan&lt;T&gt;</code>
    should be used.</li>

    <li>Similarly, if invalid data have to be signaled and the algorithm can
    manipulate empty intervals, the <code>checking_catch_nan&lt;T&gt;</code>
    is a solution.</li>

    <li>If you do not mind having undefined results when an empty interval or
    an interval number is produced, your best bet is to create your own
    policy by overloading <code>checking_base</code> and modifying
    <code>is_nan</code> et <code>is_empty</code> in order for them to always
    return <code>false</code>. It is probably the fastest checking policy
    available; however, it suffers from its deficient security.</li>
  </ul>
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  <p>Revised 
  <!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-24<!--webbot bot="Timestamp" endspan i-checksum="12172" --></p>

  <p><i>Copyright &copy; 2002 Guillaume Melquiond, Sylvain Pion, Herv&eacute;
  Br&ouml;nnimann, Polytechnic University<br>
  Copyright &copy; 2003-2004 Guillaume Melquiond</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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