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| <title>Choosing Your Own Interval Type</title> |
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| <h1>Choosing Your Own Interval Type</h1> |
| |
| <p>First of all, you need to select your base type. In order to obtain an |
| useful interval type, the numbers should respect some requirements. Please |
| refer to <a href="numbers.htm">this page</a> in order to see them. When |
| your base type is robust enough, you can go to the next step: the choice of |
| the policies.</p> |
| |
| <p>As you should already know if you did not come to this page by accident, |
| the <code>interval</code> class expect a policies argument describing the |
| <a href="rounding.htm">rounding</a> and <a href="checking.htm">checking</a> |
| policies. The first thing to do is to verify if the default policies are or |
| are not adapted to your case. If your base type is not <code>float</code>, |
| <code>double</code>, or <code>long double</code>, the default rounding |
| policy is probably not adapted. However, by specializing |
| <code>interval_lib::rounded_math</code> to your base type, the default |
| rounding policy will be suitable.</p> |
| |
| <p>The default policies define an interval type that performs precise |
| computations (for <code>float</code>, <code>double</code>, <code>long |
| double</code>), detects invalid numbers and throws exception each times an |
| empty interval is created. This is a brief description and you should refer |
| to the corresponding sections for a more precise description of the default |
| policies. Unless you need some special behavior, this default type is |
| usable in a lot of situations.</p> |
| |
| <p>After having completely defined the interval type (and its policies), |
| the only thing left to do is to verify that the constants are defined and |
| <code>std::numeric_limits</code> is correct (if needed). Now you can use |
| your brand new interval type.</p> |
| |
| <h2>Some Examples</h2> |
| |
| <h3>Solving systems</h3> |
| |
| <p>If you use the interval library in order to solve equation and |
| inequation systems by bisection, something like |
| <code>boost::interval<double></code> is probably what you need. The |
| computations are precise, and they may be fast if enclosed in a protected |
| rounding mode block (see the <a href="rounding.htm#perf">performance</a> |
| section). The comparison are "certain"; it is probably the most used type |
| of comparison, and the other comparisons are still accessible by the |
| explicit comparison functions. The checking forbid empty interval; they are |
| not needed since there would be an empty interval at end of the computation |
| if an empty interval is created during the computation, and no root would |
| be inside. The checking also forbid invalid numbers (NaN for floating-point |
| numbers). It can be a minor performance hit if you only use exact |
| floating-point constants (which are clearly not NaNs); however, if |
| performance really does matter, you will probably use a good compiler which |
| knows how to inline functions and all these annoying little tests will |
| magically disappear (if not, it is time to upgrade your compiler).</p> |
| |
| <h3>Manipulating wide intervals</h3> |
| |
| <p>You may want to use the library on intervals with imprecise bounds or on |
| inexact numbers. In particular, it may be an existing algorithm that you |
| want to rewrite and simplify by using the library. In that case, you are |
| not really interested by the inclusion property; you are only interested by |
| the computation algorithms the library provides. So you do not need to use |
| any rounding; the checking also may not be useful. Use an "exact |
| computation" rounding (you are allowed to think the name stangely applies |
| to the situation) and a checking that never tests for any invalid numbers |
| or empty intervals. By doing that, you will obtain library functions |
| reduced to their minimum (an addition of two intervals will only be two |
| additions of numbers).</p> |
| |
| <h3>Computing ranges</h3> |
| |
| <p>The inputs of your program may be empty intervals or invalid values (for |
| example, a database can allow undefined values in some field) and the core |
| of your program could also do some non-arithmetic computations that do not |
| always propagate empty intervals. For example, in the library, the |
| <code>hull</code> function can happily receive an empty interval but not |
| generate an empty interval if the other input is valid. The |
| <code>intersect</code> function is also able to produce empty intervals if |
| the intervals do not overlap. In that case, it is not really interesting if |
| an exception is thrown each time an empty interval is produced or an |
| invalid value is used; it would be better to generate and propagate empty |
| intervals. So you need to change the checking policy to something like |
| <code>interval_lib::checking_base<T></code>.</p> |
| |
| <h3>Switching interval types</h3> |
| |
| <p>This example does not deal with a full case, but with a situation that |
| can occur often. Sometimes, it can be useful to change the policies of an |
| interval by converting it to another type. For example, this happens when |
| you use an unprotected version of the interval type in order to speed up |
| the computations; it is a change of the rounding policy. It also happens |
| when you want to temporarily allow empty intervals to be created; it is a |
| change of the checking policy. These changes should not be prohibited: they |
| can greatly enhance a program (lisibility, interest, performance).</p> |
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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 © 2002 Guillaume Melquiond, Sylvain Pion, Hervé |
| Brönnimann, Polytechnic University</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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