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