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| 7 | <title>Tests and Examples</title> |
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| 8 | </head> |
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| 9 | |
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| 10 | <body lang="en"> |
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| 11 | <h1>Tests and Examples</h1> |
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| 12 | |
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| 13 | <h2>A first example</h2> |
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| 14 | |
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| 15 | <p>This example shows how to design a function which takes a polynomial and a |
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| 16 | value and returns the sign of this polynomial at this point. This function is |
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| 17 | a filter: if the answer is not guaranteed, the functions says so. The reason |
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| 18 | of using a filter rather than a simple evaluation function is: computations |
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| 19 | with floating-point numbers will incur approximations and it can be enough to |
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| 20 | change the sign of the polynomial. So, in order to validate the result, the |
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| 21 | function will use interval arithmetic.</p> |
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| 22 | |
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| 23 | <p>The first step is the inclusion of the appropriate headers. Because the |
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| 24 | function will handle floating-point bounds, the easiest solution is:</p> |
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| 25 | <pre>#include <boost/numeric/interval.hpp></pre> |
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| 26 | |
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| 27 | <p>Now, let's begin the function. The polynomial is given by the array of its |
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| 28 | coefficients and its size (strictly greater to its degree). In order to |
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| 29 | simplify the code, two namespaces of the library are included.</p> |
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| 30 | <pre>int sign_polynomial(double x, double P[], int sz) { |
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| 31 | using namespace boost::numeric; |
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| 32 | using namespace interval_lib;</pre> |
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| 33 | |
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| 34 | <p>Then we can define the interval type. Since no special behavior is |
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| 35 | required, the default policies are enough:</p> |
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| 36 | <pre> typedef interval<double> I;</pre> |
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| 37 | |
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| 38 | <p>For the evaluation, let's just use the Horner scheme with interval |
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| 39 | arithmetic. The library overloads all the arithmetic operators and provides |
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| 40 | mixed operations, so the only difference between the code with and without |
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| 41 | interval arithmetic lies in the type of the iterated value <code>y</code>:</p> |
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| 42 | <pre> I y = P[sz - 1]; |
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| 43 | for(int i = sz - 2; i >= 0; i--) |
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| 44 | y = y * x + P[i];</pre> |
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| 45 | |
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| 46 | <p>The last step is the computation of the sign of <code>y</code>. It is done |
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| 47 | by choosing an appropriate comparison scheme and then doing the comparison |
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| 48 | with the usual operators:</p> |
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| 49 | <pre> using namespace compare::certain; |
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| 50 | if (y > 0.) return 1; |
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| 51 | if (y < 0.) return -1; |
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| 52 | return 0; |
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| 53 | }</pre> |
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| 54 | |
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| 55 | <p>The answer <code>0</code> does not mean the polynomial is zero at this |
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| 56 | point. It only means the answer is not known since <code>y</code> contains |
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| 57 | zero and thus does not have a precise sign.</p> |
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| 58 | |
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| 59 | <p>Now we have the expected function. However, due to the poor |
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| 60 | implementations of floating-point rounding in most of the processors, it can |
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| 61 | be useful to say to optimize the code; or rather, to let the library optimize |
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| 62 | it. The main condition for this optimization is that the interval code should |
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| 63 | not be mixed with floating-point code. In this example, it is the case, since |
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| 64 | all the operations done in the functions involve the library. So the code can |
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| 65 | be rewritten:</p> |
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| 66 | <pre>int sign_polynomial(double x, double P[], int sz) { |
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| 67 | using namespace boost::numeric; |
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| 68 | using namespace interval_lib; |
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| 69 | typedef interval<double> I_aux; |
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| 70 | |
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| 71 | I_aux::traits_type::rounding rnd; |
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| 72 | typedef unprotect<I_aux>::type I; |
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| 73 | |
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| 74 | I y = P[sz - 1]; |
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| 75 | for(int i = sz - 2; i >= 0; i--) |
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| 76 | y = y * x + P[i]; |
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| 77 | |
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| 78 | using namespace compare::certain; |
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| 79 | if (y > 0.) return 1; |
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| 80 | if (y < 0.) return -1; |
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| 81 | return 0; |
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| 82 | }</pre> |
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| 83 | |
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| 84 | <p>The difference between this code and the previous is the use of another |
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| 85 | interval type. This new type <code>I</code> indicates to the library that all |
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| 86 | the computations can be done without caring for the rounding mode. And |
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| 87 | because of that, it is up to the function to care about it: a rounding object |
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| 88 | need to be alive whenever the optimized type is used.</p> |
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| 89 | |
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| 90 | <h2>Other tests and examples</h2> |
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| 91 | |
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| 92 | <p>In <code>libs/numeric/interval/test/</code> and |
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| 93 | <code>libs/numeric/interval/examples/</code> are some test and example |
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| 94 | programs.. The examples illustrate a few uses of intervals. For a general |
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| 95 | description and considerations on using this library, and some potential |
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| 96 | domains of application, please read this <a |
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| 97 | href="guide.htm">mini-guide</a>.</p> |
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| 98 | |
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| 99 | <h3>Tests</h3> |
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| 100 | |
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| 101 | <p>The test programs are as follows. Please note that they require the use of |
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| 102 | the Boost.test library and can be automatically tested by using |
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| 103 | <code>bjam</code> (except for interval_test.cpp).</p> |
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| 104 | |
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| 105 | <p><b>add.cpp</b> tests if the additive and subtractive operators and the |
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| 106 | respective _std and _opp rounding functions are correctly implemented. It is |
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| 107 | done by using symbolic expressions as a base type.</p> |
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| 108 | |
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| 109 | <p><b>cmp.cpp</b>, <b>cmp_lex.cpp</b>, <b>cmp_set.cpp</b>, and |
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| 110 | <b>cmp_tribool.cpp</b> test if the operators <code><</code> |
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| 111 | <code>></code> <code><=</code> <code>>=</code> <code>==</code> |
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| 112 | <code>!=</code> behave correctly for the default, lexicographic, set, and |
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| 113 | tristate comparisons. <b>cmp_exp.cpp</b> tests the explicit comparison |
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| 114 | functions <code>cer..</code> and <code>pos..</code> behave correctly. |
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| 115 | <b>cmp_exn.cpp</b> tests if the various policies correctly detect exceptional |
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| 116 | cases. All these tests use some simple intervals ([1,2] and [3,4], [1,3] and |
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| 117 | [2,4], [1,2] and [2,3], etc).</p> |
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| 118 | |
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| 119 | <p><b>det.cpp</b> tests if the <code>_std</code> and <code>_opp</code> |
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| 120 | versions in protected and unprotected mode produce the same result when Gauss |
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| 121 | scheme is used on an unstable matrix (in order to exercise rounding). The |
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| 122 | tests are done for <code>interval<float></code> and |
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| 123 | <code>interval<double></code>.</p> |
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| 124 | |
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| 125 | <p><b>fmod.cpp</b> defines a minimalistic version of |
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| 126 | <code>interval<int></code> and uses it in order to test |
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| 127 | <code>fmod</code> on some specific interval values.</p> |
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| 128 | |
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| 129 | <p><b>mul.cpp</b> exercises the multiplication, the finite division, the |
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| 130 | square and the square root with some integer intervals leading to exact |
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| 131 | results.</p> |
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| 132 | |
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| 133 | <p><b>pi.cpp</b> tests if the interval value of π (for |
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| 134 | <code>int</code>, <code>float</code> and <code>double</code> base types) |
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| 135 | contains the number π (defined with 21 decimal digits) and if it is a |
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| 136 | subset of [π±1ulp] (in order to ensure some precision).</p> |
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| 137 | |
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| 138 | <p><b>pow.cpp</b> tests if the <code>pow</code> function behaves correctly on |
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| 139 | some simple test cases.</p> |
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| 140 | |
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| 141 | <p><b>test_float.cpp</b> exercises the arithmetic operations of the library |
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| 142 | for floating point base types.</p> |
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| 143 | |
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| 144 | <p><b>interval_test.cpp</b> tests if the interval library respects the |
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| 145 | inclusion property of interval arithmetic by computing some functions and |
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| 146 | operations for both <code>double</code> and |
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| 147 | <code>interval<double></code>.</p> |
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| 148 | |
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| 149 | <h2>Examples</h2> |
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| 150 | |
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| 151 | <p><b>filter.cpp</b> contains filters for computational geometry able to find |
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| 152 | the sign of a determinant. This example is inspired by the article |
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| 153 | <em>Interval arithmetic yields efficient dynamic filters for computational |
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| 154 | geometry</em> by Brönnimann, Burnikel and Pion, 2001.</p> |
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| 155 | |
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| 156 | <p><b>findroot_demo.cpp</b> finds zeros of some functions by using dichotomy |
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| 157 | and even produces gnuplot data for one of them. The processor has to |
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| 158 | correctly handle elementary functions for this example to properly work.</p> |
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| 159 | |
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| 160 | <p><b>horner.cpp</b> is a really basic example of unprotecting the interval |
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| 161 | operations for a whole function (which computes the value of a polynomial by |
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| 162 | using Horner scheme).</p> |
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| 163 | |
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| 164 | <p><b>io.cpp</b> shows some stream input and output operators for intervals |
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| 165 | .The wide variety of possibilities explains why the library do not implement |
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| 166 | i/o operators and they are left to the user.</p> |
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| 167 | |
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| 168 | <p><b>newton-raphson.cpp</b> is an implementation of a specialized version of |
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| 169 | Newton-Raphson algorithm for finding the zeros of a function knowing its |
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| 170 | derivative. It exercises unprotecting, full division, some set operations and |
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| 171 | empty intervals.</p> |
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| 172 | |
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| 173 | <p><b>transc.cpp</b> implements the transcendental part of the rounding |
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| 174 | policy for <code>double</code> by using an external library (the MPFR subset |
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| 175 | of GMP in this case).</p> |
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| 176 | <hr> |
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| 177 | |
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| 178 | <p>Revised: 2003-08-16<br> |
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| 179 | Copyright (c) Guillaume Melquiond, Sylvain Pion, Hervé Brönnimann, 2002. |
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| 180 | Polytechnic University.<br> |
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| 181 | Copyright (c) Guillaume Melquiond, 2003.</p> |
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| 182 | </body> |
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| 183 | </html> |
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