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.

The first step is the inclusion of the appropriate headers. Because the function will handle floating-point bounds, the easiest solution is:

#include <boost/numeric/interval.hpp>

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.

int sign_polynomial(double x, double P[], int sz) { using namespace boost::numeric; using namespace interval_lib;

Then we can define the interval type. Since no special behavior is required, the default policies are enough:

typedef interval<double> I;

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
`y`

:

I y = P[sz - 1]; for(int i = sz - 2; i >= 0; i--) y = y * x + P[i];

The last step is the computation of the sign of `y`

. It is
done by choosing an appropriate comparison scheme and then doing the
comparison with the usual operators:

using namespace compare::certain; if (y > 0.) return 1; if (y < 0.) return -1; return 0; }

The answer `0`

does not mean the polynomial is zero at this
point. It only means the answer is not known since `y`

contains
zero and thus does not have a precise sign.

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:

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; }

The difference between this code and the previous is the use of another
interval type. This new type `I`

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.

In `libs/numeric/interval/test/`

and
`libs/numeric/interval/examples/`

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 mini-guide.

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
`bjam`

(except for interval_test.cpp).

**add.cpp** 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.

**cmp.cpp**, **cmp_lex.cpp**, **cmp_set.cpp**, and
**cmp_tribool.cpp** test if the operators `<`

`>`

`<=`

`>=`

`==`

`!=`

behave correctly for the default, lexicographic, set, and
tristate comparisons. **cmp_exp.cpp** tests the explicit comparison
functions `cer..`

and `pos..`

behave correctly.
**cmp_exn.cpp** 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).

**det.cpp** tests if the `_std`

and `_opp`

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 `interval<float>`

and
`interval<double>`

.

**fmod.cpp** defines a minimalistic version of
`interval<int>`

and uses it in order to test
`fmod`

on some specific interval values.

**mul.cpp** exercises the multiplication, the finite division, the
square and the square root with some integer intervals leading to exact
results.

**pi.cpp** tests if the interval value of π (for `int`

,
`float`

and `double`

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).

**pow.cpp** tests if the `pow`

function behaves correctly
on some simple test cases.

**test_float.cpp** exercises the arithmetic operations of the library
for floating point base types.

**interval_test.cpp** tests if the interval library respects the
inclusion property of interval arithmetic by computing some functions and
operations for both `double`

and
`interval<double>`

.

**filter.cpp** contains filters for computational geometry able to
find the sign of a determinant. This example is inspired by the article
*Interval arithmetic yields efficient dynamic filters for computational
geometry* by Brönnimann, Burnikel and Pion, 2001.

**findroot_demo.cpp** 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.

**horner.cpp** 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).

**io.cpp** 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.

**newton-raphson.cpp** 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.

**transc.cpp** implements the transcendental part of the rounding
policy for `double`

by using an external library (the MPFR
subset of GMP in this case).

Revised 2006-12-24

*Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé
Brönnimann, Polytechnic University
Copyright © 2003 Guillaume Melquiond*

*Distributed under the Boost Software License, Version 1.0. (See
accompanying file LICENSE_1_0.txt
or copy at http://www.boost.org/LICENSE_1_0.txt)*