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184 lines
7.9 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-Type" content="text/html; charset=iso-8859-1">
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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 a
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value and returns the sign of this polynomial at this point. This function is
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a filter: if the answer is not guaranteed, the functions says so. The reason
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of using a filter rather than a simple evaluation function is: computations
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with floating-point numbers will incur approximations and it can be enough to
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change the sign of the polynomial. So, in order to validate the result, the
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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>#include <boost/numeric/interval.hpp></pre>
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<p>Now, let's begin the function. The polynomial is given by the array of its
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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>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;</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> typedef interval<double> I;</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 <code>y</code>:</p>
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<pre> 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];</pre>
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<p>The last step is the computation of the sign of <code>y</code>. It is done
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by choosing an appropriate comparison scheme and then doing the comparison
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with the usual operators:</p>
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<pre> 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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}</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 can
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be useful to say to optimize the code; or rather, to let the library optimize
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it. The main condition for this optimization is that the interval code should
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not be mixed with floating-point code. In this example, it is the case, since
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all the operations done in the functions involve the library. So the code can
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be rewritten:</p>
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<pre>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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}</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 all
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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 object
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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
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href="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 of
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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 is
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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 exceptional
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cases. All these tests use some simple intervals ([1,2] and [3,4], [1,3] and
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[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 Gauss
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scheme is used on an unstable matrix (in order to exercise rounding). The
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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
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<code>int</code>, <code>float</code> and <code>double</code> base types)
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contains the number π (defined with 21 decimal digits) and if it is a
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subset of [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 on
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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 find
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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 dichotomy
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and even produces gnuplot data for one of them. The processor has to
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correctly handle elementary functions for this example to properly 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 by
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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 implement
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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 of
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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 and
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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 subset
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of GMP in this case).</p>
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<hr>
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<p>Revised: 2003-08-16<br>
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Copyright (c) Guillaume Melquiond, Sylvain Pion, Hervé Brönnimann, 2002.
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Polytechnic University.<br>
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Copyright (c) Guillaume Melquiond, 2003.</p>
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</body>
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</html>
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