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| 9 | |
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| 10 | <title>Tests and Examples</title> |
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| 11 | </head> |
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| 12 | |
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| 13 | <body lang="en"> |
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| 14 | <h1>Tests and Examples</h1> |
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| 15 | |
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| 16 | <h2>A first example</h2> |
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| 17 | |
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| 18 | <p>This example shows how to design a function which takes a polynomial and |
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| 19 | a value and returns the sign of this polynomial at this point. This |
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| 20 | function is a filter: if the answer is not guaranteed, the functions says |
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| 21 | so. The reason of using a filter rather than a simple evaluation function |
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| 22 | is: computations with floating-point numbers will incur approximations and |
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| 23 | it can be enough to change the sign of the polynomial. So, in order to |
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| 24 | validate the result, the function will use interval arithmetic.</p> |
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| 25 | |
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| 26 | <p>The first step is the inclusion of the appropriate headers. Because the |
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| 27 | function will handle floating-point bounds, the easiest solution is:</p> |
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| 28 | <pre> |
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| 29 | #include <boost/numeric/interval.hpp> |
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| 30 | </pre> |
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| 31 | |
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| 32 | <p>Now, let's begin the function. The polynomial is given by the array of |
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| 33 | its coefficients and its size (strictly greater to its degree). In order to |
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| 34 | simplify the code, two namespaces of the library are included.</p> |
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| 35 | <pre> |
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| 36 | int sign_polynomial(double x, double P[], int sz) { |
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| 37 | using namespace boost::numeric; |
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| 38 | using namespace interval_lib; |
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| 39 | </pre> |
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| 40 | |
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| 41 | <p>Then we can define the interval type. Since no special behavior is |
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| 42 | required, the default policies are enough:</p> |
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| 43 | <pre> |
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| 44 | typedef interval<double> I; |
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| 45 | </pre> |
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| 46 | |
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| 47 | <p>For the evaluation, let's just use the Horner scheme with interval |
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| 48 | arithmetic. The library overloads all the arithmetic operators and provides |
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| 49 | mixed operations, so the only difference between the code with and without |
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| 50 | interval arithmetic lies in the type of the iterated value |
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| 51 | <code>y</code>:</p> |
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| 52 | <pre> |
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| 53 | I y = P[sz - 1]; |
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| 54 | for(int i = sz - 2; i >= 0; i--) |
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| 55 | y = y * x + P[i]; |
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| 56 | </pre> |
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| 57 | |
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| 58 | <p>The last step is the computation of the sign of <code>y</code>. It is |
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| 59 | done by choosing an appropriate comparison scheme and then doing the |
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| 60 | comparison with the usual operators:</p> |
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| 61 | <pre> |
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| 62 | using namespace compare::certain; |
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| 63 | if (y > 0.) return 1; |
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| 64 | if (y < 0.) return -1; |
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| 65 | return 0; |
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| 66 | } |
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| 67 | </pre> |
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| 68 | |
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| 69 | <p>The answer <code>0</code> does not mean the polynomial is zero at this |
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| 70 | point. It only means the answer is not known since <code>y</code> contains |
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| 71 | zero and thus does not have a precise sign.</p> |
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| 72 | |
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| 73 | <p>Now we have the expected function. However, due to the poor |
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| 74 | implementations of floating-point rounding in most of the processors, it |
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| 75 | can be useful to say to optimize the code; or rather, to let the library |
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| 76 | optimize it. The main condition for this optimization is that the interval |
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| 77 | code should not be mixed with floating-point code. In this example, it is |
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| 78 | the case, since all the operations done in the functions involve the |
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| 79 | library. So the code can be rewritten:</p> |
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| 80 | <pre> |
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| 81 | int sign_polynomial(double x, double P[], int sz) { |
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| 82 | using namespace boost::numeric; |
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| 83 | using namespace interval_lib; |
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| 84 | typedef interval<double> I_aux; |
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| 85 | |
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| 86 | I_aux::traits_type::rounding rnd; |
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| 87 | typedef unprotect<I_aux>::type I; |
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| 88 | |
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| 89 | I y = P[sz - 1]; |
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| 90 | for(int i = sz - 2; i >= 0; i--) |
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| 91 | y = y * x + P[i]; |
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| 92 | |
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| 93 | using namespace compare::certain; |
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| 94 | if (y > 0.) return 1; |
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| 95 | if (y < 0.) return -1; |
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| 96 | return 0; |
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| 97 | } |
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| 98 | </pre> |
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| 99 | |
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| 100 | <p>The difference between this code and the previous is the use of another |
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| 101 | interval type. This new type <code>I</code> indicates to the library that |
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| 102 | all the computations can be done without caring for the rounding mode. And |
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| 103 | because of that, it is up to the function to care about it: a rounding |
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| 104 | object need to be alive whenever the optimized type is used.</p> |
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| 105 | |
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| 106 | <h2>Other tests and examples</h2> |
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| 107 | |
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| 108 | <p>In <code>libs/numeric/interval/test/</code> and |
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| 109 | <code>libs/numeric/interval/examples/</code> are some test and example |
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| 110 | programs.. The examples illustrate a few uses of intervals. For a general |
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| 111 | description and considerations on using this library, and some potential |
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| 112 | domains of application, please read this <a href= |
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| 113 | "guide.htm">mini-guide</a>.</p> |
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| 114 | |
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| 115 | <h3>Tests</h3> |
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| 116 | |
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| 117 | <p>The test programs are as follows. Please note that they require the use |
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| 118 | of the Boost.test library and can be automatically tested by using |
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| 119 | <code>bjam</code> (except for interval_test.cpp).</p> |
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| 120 | |
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| 121 | <p><b>add.cpp</b> tests if the additive and subtractive operators and the |
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| 122 | respective _std and _opp rounding functions are correctly implemented. It |
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| 123 | is done by using symbolic expressions as a base type.</p> |
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| 124 | |
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| 125 | <p><b>cmp.cpp</b>, <b>cmp_lex.cpp</b>, <b>cmp_set.cpp</b>, and |
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| 126 | <b>cmp_tribool.cpp</b> test if the operators <code><</code> |
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| 127 | <code>></code> <code><=</code> <code>>=</code> <code>==</code> |
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| 128 | <code>!=</code> behave correctly for the default, lexicographic, set, and |
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| 129 | tristate comparisons. <b>cmp_exp.cpp</b> tests the explicit comparison |
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| 130 | functions <code>cer..</code> and <code>pos..</code> behave correctly. |
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| 131 | <b>cmp_exn.cpp</b> tests if the various policies correctly detect |
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| 132 | exceptional cases. All these tests use some simple intervals ([1,2] and |
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| 133 | [3,4], [1,3] and [2,4], [1,2] and [2,3], etc).</p> |
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| 134 | |
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| 135 | <p><b>det.cpp</b> tests if the <code>_std</code> and <code>_opp</code> |
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| 136 | versions in protected and unprotected mode produce the same result when |
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| 137 | Gauss scheme is used on an unstable matrix (in order to exercise rounding). |
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| 138 | The tests are done for <code>interval<float></code> and |
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| 139 | <code>interval<double></code>.</p> |
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| 140 | |
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| 141 | <p><b>fmod.cpp</b> defines a minimalistic version of |
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| 142 | <code>interval<int></code> and uses it in order to test |
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| 143 | <code>fmod</code> on some specific interval values.</p> |
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| 144 | |
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| 145 | <p><b>mul.cpp</b> exercises the multiplication, the finite division, the |
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| 146 | square and the square root with some integer intervals leading to exact |
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| 147 | results.</p> |
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| 148 | |
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| 149 | <p><b>pi.cpp</b> tests if the interval value of π (for <code>int</code>, |
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| 150 | <code>float</code> and <code>double</code> base types) contains the number |
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| 151 | π (defined with 21 decimal digits) and if it is a subset of |
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| 152 | [π±1ulp] (in order to ensure some precision).</p> |
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| 153 | |
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| 154 | <p><b>pow.cpp</b> tests if the <code>pow</code> function behaves correctly |
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| 155 | on some simple test cases.</p> |
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| 156 | |
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| 157 | <p><b>test_float.cpp</b> exercises the arithmetic operations of the library |
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| 158 | for floating point base types.</p> |
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| 159 | |
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| 160 | <p><b>interval_test.cpp</b> tests if the interval library respects the |
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| 161 | inclusion property of interval arithmetic by computing some functions and |
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| 162 | operations for both <code>double</code> and |
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| 163 | <code>interval<double></code>.</p> |
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| 164 | |
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| 165 | <h2>Examples</h2> |
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| 166 | |
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| 167 | <p><b>filter.cpp</b> contains filters for computational geometry able to |
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| 168 | find the sign of a determinant. This example is inspired by the article |
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| 169 | <em>Interval arithmetic yields efficient dynamic filters for computational |
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| 170 | geometry</em> by Brönnimann, Burnikel and Pion, 2001.</p> |
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| 171 | |
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| 172 | <p><b>findroot_demo.cpp</b> finds zeros of some functions by using |
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| 173 | dichotomy and even produces gnuplot data for one of them. The processor has |
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| 174 | to correctly handle elementary functions for this example to properly |
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| 175 | work.</p> |
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| 176 | |
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| 177 | <p><b>horner.cpp</b> is a really basic example of unprotecting the interval |
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| 178 | operations for a whole function (which computes the value of a polynomial |
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| 179 | by using Horner scheme).</p> |
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| 180 | |
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| 181 | <p><b>io.cpp</b> shows some stream input and output operators for intervals |
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| 182 | .The wide variety of possibilities explains why the library do not |
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| 183 | implement i/o operators and they are left to the user.</p> |
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| 184 | |
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| 185 | <p><b>newton-raphson.cpp</b> is an implementation of a specialized version |
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| 186 | of Newton-Raphson algorithm for finding the zeros of a function knowing its |
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| 187 | derivative. It exercises unprotecting, full division, some set operations |
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| 188 | and empty intervals.</p> |
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| 189 | |
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| 190 | <p><b>transc.cpp</b> implements the transcendental part of the rounding |
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| 191 | policy for <code>double</code> by using an external library (the MPFR |
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| 192 | subset of GMP in this case).</p> |
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| 193 | <hr> |
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| 194 | |
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| 195 | <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src= |
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| 197 | height="31" width="88"></a></p> |
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| 198 | |
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| 199 | <p>Revised |
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| 200 | <!--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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| 201 | |
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| 202 | <p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé |
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| 203 | Brönnimann, Polytechnic University<br> |
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| 204 | Copyright © 2003 Guillaume Melquiond</i></p> |
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| 205 | |
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| 206 | <p><i>Distributed under the Boost Software License, Version 1.0. (See |
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| 207 | accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> |
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| 208 | or copy at <a href= |
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| 209 | "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p> |
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| 210 | </body> |
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| 211 | </html> |
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