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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> |
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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<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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