212 lines
8.8 KiB
HTML
212 lines
8.8 KiB
HTML
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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"http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<head>
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<meta http-equiv="Content-Language" content="en-us">
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<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
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<link rel="stylesheet" type="text/css" href="../../../../boost.css">
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<title>Tests and Examples</title>
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</head>
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<body lang="en">
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<h1>Tests and Examples</h1>
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<h2>A first example</h2>
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<p>This example shows how to design a function which takes a polynomial and
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a value and returns the sign of this polynomial at this point. This
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function is a filter: if the answer is not guaranteed, the functions says
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so. The reason of using a filter rather than a simple evaluation function
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is: computations with floating-point numbers will incur approximations and
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it can be enough to change the sign of the polynomial. So, in order to
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validate the result, the function will use interval arithmetic.</p>
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<p>The first step is the inclusion of the appropriate headers. Because the
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function will handle floating-point bounds, the easiest solution is:</p>
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<pre>
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#include <boost/numeric/interval.hpp>
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</pre>
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<p>Now, let's begin the function. The polynomial is given by the array of
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its coefficients and its size (strictly greater to its degree). In order to
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simplify the code, two namespaces of the library are included.</p>
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<pre>
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int sign_polynomial(double x, double P[], int sz) {
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using namespace boost::numeric;
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using namespace interval_lib;
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</pre>
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<p>Then we can define the interval type. Since no special behavior is
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required, the default policies are enough:</p>
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<pre>
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typedef interval<double> I;
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</pre>
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<p>For the evaluation, let's just use the Horner scheme with interval
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arithmetic. The library overloads all the arithmetic operators and provides
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mixed operations, so the only difference between the code with and without
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interval arithmetic lies in the type of the iterated value
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<code>y</code>:</p>
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<pre>
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I y = P[sz - 1];
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for(int i = sz - 2; i >= 0; i--)
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y = y * x + P[i];
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</pre>
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<p>The last step is the computation of the sign of <code>y</code>. It is
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done by choosing an appropriate comparison scheme and then doing the
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comparison with the usual operators:</p>
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<pre>
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using namespace compare::certain;
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if (y > 0.) return 1;
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if (y < 0.) return -1;
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return 0;
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}
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</pre>
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<p>The answer <code>0</code> does not mean the polynomial is zero at this
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point. It only means the answer is not known since <code>y</code> contains
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zero and thus does not have a precise sign.</p>
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<p>Now we have the expected function. However, due to the poor
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implementations of floating-point rounding in most of the processors, it
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can be useful to say to optimize the code; or rather, to let the library
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optimize it. The main condition for this optimization is that the interval
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code should not be mixed with floating-point code. In this example, it is
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the case, since all the operations done in the functions involve the
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library. So the code can be rewritten:</p>
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<pre>
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int sign_polynomial(double x, double P[], int sz) {
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using namespace boost::numeric;
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using namespace interval_lib;
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typedef interval<double> I_aux;
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I_aux::traits_type::rounding rnd;
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typedef unprotect<I_aux>::type I;
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I y = P[sz - 1];
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for(int i = sz - 2; i >= 0; i--)
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y = y * x + P[i];
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using namespace compare::certain;
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if (y > 0.) return 1;
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if (y < 0.) return -1;
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return 0;
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}
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</pre>
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<p>The difference between this code and the previous is the use of another
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interval type. This new type <code>I</code> indicates to the library that
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all the computations can be done without caring for the rounding mode. And
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because of that, it is up to the function to care about it: a rounding
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object need to be alive whenever the optimized type is used.</p>
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<h2>Other tests and examples</h2>
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<p>In <code>libs/numeric/interval/test/</code> and
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<code>libs/numeric/interval/examples/</code> are some test and example
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programs.. The examples illustrate a few uses of intervals. For a general
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description and considerations on using this library, and some potential
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domains of application, please read this <a href=
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"guide.htm">mini-guide</a>.</p>
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<h3>Tests</h3>
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<p>The test programs are as follows. Please note that they require the use
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of the Boost.test library and can be automatically tested by using
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<code>bjam</code> (except for interval_test.cpp).</p>
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<p><b>add.cpp</b> tests if the additive and subtractive operators and the
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respective _std and _opp rounding functions are correctly implemented. It
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is done by using symbolic expressions as a base type.</p>
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<p><b>cmp.cpp</b>, <b>cmp_lex.cpp</b>, <b>cmp_set.cpp</b>, and
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<b>cmp_tribool.cpp</b> test if the operators <code><</code>
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<code>></code> <code><=</code> <code>>=</code> <code>==</code>
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<code>!=</code> behave correctly for the default, lexicographic, set, and
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tristate comparisons. <b>cmp_exp.cpp</b> tests the explicit comparison
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functions <code>cer..</code> and <code>pos..</code> behave correctly.
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<b>cmp_exn.cpp</b> tests if the various policies correctly detect
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exceptional cases. All these tests use some simple intervals ([1,2] and
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[3,4], [1,3] and [2,4], [1,2] and [2,3], etc).</p>
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<p><b>det.cpp</b> tests if the <code>_std</code> and <code>_opp</code>
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versions in protected and unprotected mode produce the same result when
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Gauss scheme is used on an unstable matrix (in order to exercise rounding).
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The tests are done for <code>interval<float></code> and
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<code>interval<double></code>.</p>
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<p><b>fmod.cpp</b> defines a minimalistic version of
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<code>interval<int></code> and uses it in order to test
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<code>fmod</code> on some specific interval values.</p>
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<p><b>mul.cpp</b> exercises the multiplication, the finite division, the
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square and the square root with some integer intervals leading to exact
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results.</p>
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<p><b>pi.cpp</b> tests if the interval value of π (for <code>int</code>,
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<code>float</code> and <code>double</code> base types) contains the number
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π (defined with 21 decimal digits) and if it is a subset of
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[π±1ulp] (in order to ensure some precision).</p>
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<p><b>pow.cpp</b> tests if the <code>pow</code> function behaves correctly
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on some simple test cases.</p>
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<p><b>test_float.cpp</b> exercises the arithmetic operations of the library
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for floating point base types.</p>
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<p><b>interval_test.cpp</b> tests if the interval library respects the
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inclusion property of interval arithmetic by computing some functions and
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operations for both <code>double</code> and
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<code>interval<double></code>.</p>
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<h2>Examples</h2>
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<p><b>filter.cpp</b> contains filters for computational geometry able to
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find the sign of a determinant. This example is inspired by the article
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<em>Interval arithmetic yields efficient dynamic filters for computational
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geometry</em> by Brönnimann, Burnikel and Pion, 2001.</p>
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<p><b>findroot_demo.cpp</b> finds zeros of some functions by using
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dichotomy and even produces gnuplot data for one of them. The processor has
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to correctly handle elementary functions for this example to properly
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work.</p>
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<p><b>horner.cpp</b> is a really basic example of unprotecting the interval
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operations for a whole function (which computes the value of a polynomial
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by using Horner scheme).</p>
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<p><b>io.cpp</b> shows some stream input and output operators for intervals
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.The wide variety of possibilities explains why the library do not
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implement i/o operators and they are left to the user.</p>
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<p><b>newton-raphson.cpp</b> is an implementation of a specialized version
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of Newton-Raphson algorithm for finding the zeros of a function knowing its
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derivative. It exercises unprotecting, full division, some set operations
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and empty intervals.</p>
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<p><b>transc.cpp</b> implements the transcendental part of the rounding
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policy for <code>double</code> by using an external library (the MPFR
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subset of GMP in this case).</p>
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<hr>
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<p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
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"../../../../doc/images/valid-html401.png" alt="Valid HTML 4.01 Transitional"
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height="31" width="88"></a></p>
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<p>Revised
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<!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-24<!--webbot bot="Timestamp" endspan i-checksum="12172" --></p>
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<p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé
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Brönnimann, Polytechnic University<br>
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Copyright © 2003 Guillaume Melquiond</i></p>
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<p><i>Distributed under the Boost Software License, Version 1.0. (See
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accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a>
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or copy at <a href=
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"http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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</body>
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</html>
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