CGAL 6.1 - Algebraic Kernel

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User Manual

Authors
  Eric Berberich, Michael Hemmer, Michael Kerber, Sylvain Lazard, Luis Peñaranda, and Monique Teillaud

# Introduction

Real solving of polynomials is a fundamental problem with a wide application range. This package provides black-box implementations of algorithms to determine, compare, and approximate real roots of univariate polynomials and bivariate polynomial systems. Such a black-box is called an Algebraic Kernel. Since this package is aimed at providing more than one implementation, the interface of the algebraic kernels is expressed in concepts. The main concepts provided by this package are the `AlgebraicKernel_d_1` for univariate polynomial systems and `AlgebraicKernel_d_2` for bivariate polynomial systems, the latter being a refinement of the first.

# Algebraic Kernel Concepts

## Univariate Algebraic Kernel

## Major Types

First of all, the univariate algebraic kernel provides construction, comparison and approximation of real roots of univariate polynomials. Thus, the major public types the `AlgebraicKernel_d_1` provides are:

`AlgebraicKernel_d_1::Polynomial_1` - the type representing univariate polynomials,

`AlgebraicKernel_d_1::Coefficient` - the coefficient type of these polynomials,

`AlgebraicKernel_d_1::Algebraic_real_1` - the type representing real roots,

`AlgebraicKernel_d_1::Bound` - the type which is used to approximate these algebraic reals, in particular, it is used to represent the boundaries of isolating intervals.

## Construction of Algebraic Real Numbers

The kernel provides two different function objects to construct an `AlgebraicKernel_d_1::Algebraic_real_1`. The most general way is to use `AlgebraicKernel_d_1::Isolate_1`; The function object takes a univariate polynomial and writes all real roots into a given output iterator. It is also possible to retrieve the multiplicity of each root. The second option is to construct one particular algebraic real using `AlgebraicKernel_d_1::Construct_algebraic_real_1`. This function object provides construction from the native int type, the coefficient type as well as the bound type. Moreover, it is possible to construct an algebraic real by giving a polynomial and either an isolating interval or the index of the root. A related function object is `AlgebraicKernel_d_1::Number_of_solutions_1` computing the number of real roots of a polynomial.

## Comparison and Approximation of Algebraic Real Numbers

An `AlgebraicKernel_d_1::Algebraic_real_1` is model of `RealEmbeddable`, for instance, it is possible to compare two algebraic reals, to determine the sign of an algebraic real or to ask for its double approximation, see also section secRealEmbeddable. Moreover, `AlgebraicKernel_d_1::Compare_1` provides comparison with int, the coefficient type and the bound type.

There are several ways to approximate an `AlgebraicKernel_d_1::Algebraic_real_1`:

`AlgebraicKernel_d_1::Approximate_absolute_1` - provides an approximation that is better than the passed absolute error bound,

`AlgebraicKernel_d_1::Approximate_relative_1` - provides an approximation that is better than the passed relative error bound,

`AlgebraicKernel_d_1::Isolate_1` - returns an isolating interval with respect to a given univariate polynomial,

A related function object is `AlgebraicKernel_d_1::Bound_between_1`, which computes a number that isolates two algebraic real numbers.

## Interplay with Polynomials

It is also possible to retrieve a representing polynomial from an algebraic real using `AlgebraicKernel_d_1::Compute_polynomial_1`, which guarantees that the algebraic real is a root of the returned polynomial. As the name already indicates, this operation may be very costly since the polynomial may not be computed yet. Moreover, it is not guaranteed that the returned polynomial is the minimal polynomial of the number. Together with `AlgebraicKernel_d_1::Isolate_1`, it is possible to retrieve the traditional representation of an algebraic real as a square free polynomial and an isolating interval.

Though the `AlgebraicKernel_d_1` does not provide arithmetic operations on `AlgebraicKernel_d_1::Algebraic_real_1`, it is possible to compute the sign of a polynomial at a given algebraic real using `AlgebraicKernel_d_1::Sign_at_1`. Or alternatively, just compute whether the polynomial is zero at an algebraic real number using `AlgebraicKernel_d_1::Is_zero_at_1`. Note that this operation can be significantly less expensive, in particular if the polynomial is not zero at the given algebraic real.

## Auxiliary Functionality for Polynomials

First of all the type `AlgebraicKernel_d_1::Polynomial_1` is required to be a model of the concept `Polynomial_1`, which is defined in the Polynomial package (see chapter Chapter_Polynomial). This implies that all essential functionality is provided via `Polynomial_traits_d`. However, the algebraic kernel also provides several function objects to handle polynomials:

`AlgebraicKernel_d_1::Is_square_free_1` - determines whether a polynomial is square free,

`AlgebraicKernel_d_1::Make_square_free_1` - computes the square free part of a polynomial,

`AlgebraicKernel_d_1::Square_free_factorize_1` - computes a square free factorization of a polynomial,

`AlgebraicKernel_d_1::Is_coprime_1` - computes whether a pair of polynomials is square free,

`AlgebraicKernel_d_1::Make_coprime_1` - decomposes two polynomials into the coprime factors and their common factor.

Though the polynomial package provides similar functionality we suggest to use the function objects provided by the kernel, since the design of the algebraic kernel allows for instance internal caching by the kernel.

Also note that `AlgebraicKernel_d_1::Square_free_factorize_1` only computes the square free factorization up to a constant factor. This is a slight modification with respect to its counterpart in `Polynomial_traits_d`. In this way it was possible that the concepts just require the coefficient type to be a model of `IntegralDomain`, instead of `Field` or `UniqueFactorizationDomain`. For more details see also:

- PolynomialTraits_d::SquareFreeFactorize
- PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor

## Design Rationale

Most implementations of an `AlgebraicKernel_d_1` will represent an algebraic real number by the root of a square free polynomial and an isolating interval, that is, the number is defined as the only root of the polynomial within the interval. Usually, one will refrain from computing the minimal polynomial since the computation of the minimal polynomial is much more expensive and does not pay of. However, besides the representation by a polynomial and an isolating interval one can also imagine the representation by a polynomial and the index of the root, e.g., as the \( i\)th real root when enumerated from minus to plus infinity. Moreover, it may very well be that the kernel just computes an approximation of the number, whereas the representing polynomial is not computed yet. This is in particular relevant in relation to the `AlgebraicKernel_d_2`, where `AlgebraicKernel_d_1::Algebraic_real_1` is used to represent coordinates of solutions of bivariate systems. Hence, the design does not allow direct access to any, seemingly obvious, members of an `AlgebraicKernel_d_1::Algebraic_real_1`. Instead, there is, e.g., `AlgebraicKernel_d_1::Compute_polynomial_1` which emphasizes that the requested polynomial may not be computed yet. Similarly, there is no way to directly ask for the refinement of the current isolating interval since this would impose a state to every object of an `AlgebraicKernel_d_1::Algebraic_real_1`.

## Bivariate Algebraic Kernel

The concept `AlgebraicKernel_d_2` is a refinement of `AlgebraicKernel_d_1`, that is, a model of `AlgebraicKernel_d_2` is also a model of `AlgebraicKernel_d_1`. Hence, the `AlgebraicKernel_d_2` concept is designed such that occurring names and functionalities are as similar as possible to those in the `AlgebraicKernel_d_1` concept. The following are a direct generalization of their univariate counterparts:

`AlgebraicKernel_d_2::Polynomial_2`,

`AlgebraicKernel_d_2::Algebraic_real_2`,

`AlgebraicKernel_d_2::Construct_algebraic_real_2`,

`AlgebraicKernel_d_2::Isolate_2`,

`AlgebraicKernel_d_2::Is_square_free_2`,

`AlgebraicKernel_d_2::Make_square_free_2`,

`AlgebraicKernel_d_2::Square_free_factorize_2`,

`AlgebraicKernel_d_2::Is_coprime_2`,

`AlgebraicKernel_d_2::Make_coprime_2`,

`AlgebraicKernel_d_2::Solve_2`,

`AlgebraicKernel_d_2::Number_of_solutions_2`,

`AlgebraicKernel_d_2::Compare_xy_2`,

`AlgebraicKernel_d_2::Sign_at_2`,

`AlgebraicKernel_d_2::Is_zero_at_2`.

For instance, `AlgebraicKernel_d_2::Solve_2` provides the solution for a bivariate polynomial system. However, it is also possible to obtain the coordinates of these solutions with the additional functors:

`AlgebraicKernel_d_2::Compute_x_2`,

`AlgebraicKernel_d_2::Compute_y_2`.

In principal this would be sufficient generalization, since functions such as isolating, approximating algebraic real numbers could be implemented using these access functions ant the corresponding functionalities in the univariate algebraic kernel. However, one should be aware that an `AlgebraicKernel_d_2::Algebraic_real_2` is not necessarily represented as a pair of univariate solutions, that is, using `AlgebraicKernel_d_2::Compute_y_2` may entail considerable computations. Therefore, the concept also requires the following additional functors that may allow a model to bypass this issue:

`AlgebraicKernel_d_2::Compute_polynomial_x_2`,

`AlgebraicKernel_d_2::Compute_polynomial_y_2`,

`AlgebraicKernel_d_2::Isolate_x_2`,

`AlgebraicKernel_d_2::Isolate_y_2`,

`AlgebraicKernel_d_2::Compare_x_2`,

`AlgebraicKernel_d_2::Compare_y_2`,

`AlgebraicKernel_d_2::Approximate_absolute_x_2`,

`AlgebraicKernel_d_2::Approximate_relative_x_2`,

`AlgebraicKernel_d_2::Approximate_absolute_y_2`,

`AlgebraicKernel_d_2::Approximate_relative_y_2`,

`AlgebraicKernel_d_2::Bound_between_x_2`,

`AlgebraicKernel_d_2::Bound_between_y_2`.

# Models

## Generic Algebraic Kernels

The package provides generic models of the univariate and bivariate algebraic kernel, namely `Algebraic_kernel_d_1<Coeff>` and `Algebraic_kernel_d_2<Coeff>`, respectively. Both kernels support a large set of number types as their template argument, which defines the supported coefficient type. The supported types are, for instance, `Gmpz` and `Gmpq` as well as the corresponding types of LEDA and CORE.

The `Algebraic_kernel_d_1<Coeff>` represents an algebraic real root by a square free polynomial and an isolating interval that uniquely defines the root. The current method to isolate roots is the Bitstream Descartes method [6]. The used method to refine the approximation of an algebraic real root is a slightly modified (filtered) version of the one presented in [1]. The method has quadratic convergence.

`Algebraic_kernel_d_2<Coeff>` is based on an algorithm computing a geometric-topological analysis of a single curve [5] and of a pair of curves [4]. The main idea behind both analyses is to compute the critical x-coordinates of curves and curve pairs by projection (resultants), and compute additional information about the critical fibers using subresultants and Sturm-Habicht sequences [7]. With that information, the fiber at critical x-coordinates is computed by a variant of the Bitstream Descartes method. See also [8] for a comprehensive description of these techniques. Almost all functors in the class that take a `Polynomial_2` object as argument trigger such an analysis as a main computation step. For efficiency, these analyses (of single curves and curve pairs) are therefore cached internally for efficiency. For instance, computing the pairwise solutions of 10 `Polynomial_2` objects requires 10 curve analyses and 45 curve pair analyses to be computed internally.

A point \( p\) of type `Algebraic_real_2` is represented by its \( x\)-coordinate \( x_0\) (as described in the `Algebraic_kernel_d_1` paragraph above), an algebraic curve where \( p\) lies on, and an integer \( i\), denoting that \( p\) is the \( i\)th point in the fiber at \( x_0\), counted from the bottom (ignoring a possible vertical line at \( x_0\)). Note that this determines the point uniquely, but the \( y\)-coordinate is not stored internally in terms of an `Algebraic_real_1` object. Querying such a representation by calling `Compute_y_2` is a time-consuming step, and should be avoided for efficiency reasons if possible.

# Examples

## Construction of Algebraic Real Numbers

The following example illustrates the construction of `AlgebraicKernel_d_1::Algebraic_real_1` using `AlgebraicKernel_d_1::Construct_algebraic_real_1`:
File Algebraic_kernel_d/Construct_algebraic_real_1.cpp

// $URL$
// $Id$
#include <CGAL/config.h>
#ifdef CGAL_USE_MPFI
#include <CGAL/Algebraic_kernel_d_1.h>
#include <CGAL/Gmpz.h>
#include <vector>
#include <iostream>
typedef CGAL::Algebraic_kernel_d_1<CGAL::Gmpz> AK;
typedef AK::Polynomial_1 Polynomial_1;
typedef AK::Algebraic_real_1 Algebraic_real_1;
typedef AK::Coefficient Coefficient;
typedef AK::Bound Bound;
typedef AK::Multiplicity_type Multiplicity_type;
int main(){
AK ak; // an object of
AK::Construct_algebraic_real_1 construct_algreal_1 = ak.construct_algebraic_real_1_object();
std::cout << "Construct from int : " << construct_algreal_1(int(2)) << "\n";
std::cout << "Construct from Coefficient : " << construct_algreal_1(Coefficient(2)) << "\n";
std::cout << "Construct from Bound : " << construct_algreal_1(Bound(2)) << "\n\n";
Polynomial_1 x = CGAL::shift(AK::Polynomial_1(1),1); // the monomial x
std::cout << "Construct by index : "
<< construct_algreal_1(x*x-2,1) << "\n"
<< to_double(construct_algreal_1(x*x-2,1)) << "\n";
std::cout << "Construct by isolating interval : "
<< construct_algreal_1(x*x-2,Bound(0),Bound(2)) << "\n"
<< to_double(construct_algreal_1(x*x-2,Bound(0),Bound(2))) << "\n\n";
return 0;
}
#else
int main(){
std::cout << "This example requires CGAL to be configured with library MPFI." << std::endl;
return 0;
}
#endif
Gmpz.h
CGAL::Algebraic_kernel_d_1
The class represents an algebraic real root by a square free polynomial and an isolating interval tha...
Definition: Algebraic_kernel_d_1.h:38
CGAL::to_double
double to_double(const NT &x)
CGAL::shift
Polynomial_traits_d< Polynomial_d >::Shift::result_type shift(const Polynomial_d &p, int i, int index=Polynomial_traits_d< Polynomial_d >::d-1)

## Solving Univariate Polynomials

The following example illustrates the construction of `AlgebraicKernel_d_1::Algebraic_real_1` using `AlgebraicKernel_d_1::Solve_1`:
File Algebraic_kernel_d/Solve_1.cpp

// $URL$
// $Id$
#include <CGAL/config.h>
#ifdef CGAL_USE_MPFI
#include <CGAL/Algebraic_kernel_d_1.h>
#include <CGAL/Gmpz.h>
#include <vector>
typedef CGAL::Algebraic_kernel_d_1<CGAL::Gmpz> AK;
typedef AK::Polynomial_1 Polynomial_1;
typedef AK::Algebraic_real_1 Algebraic_real_1;
typedef AK::Bound Bound;
typedef AK::Multiplicity_type Multiplicity_type;
int main(){
AK ak; // an object of
AK::Solve_1 solve_1 = ak.solve_1_object();
Polynomial_1 x = CGAL::shift(AK::Polynomial_1(1),1); // the monomial x
// variant using a bool indicating a square free polynomial
// multiplicities are not computed
std::vector<Algebraic_real_1> roots;
solve_1(x*x-2,true, std::back_inserter(roots));
std::cout << "Number of roots is : " << roots.size() << "\n";
std::cout << "First root should be -sqrt(2): " << CGAL::to_double(roots[0]) << "\n";
std::cout << "Second root should be sqrt(2): " << CGAL::to_double(roots[1]) << "\n\n";
roots.clear();
// variant for roots in a given range of a square free polynomial
solve_1((x*x-2)*(x*x-3),true, Bound(0),Bound(10),std::back_inserter(roots));
std::cout << "Number of roots is : " << roots.size() << "\n";
std::cout << "First root should be sqrt(2): " << CGAL::to_double(roots[0]) << "\n";
std::cout << "Second root should be sqrt(3): " << CGAL::to_double(roots[1]) << "\n\n";
roots.clear();
// variant computing all roots with multiplicities
std::vector<std::pair<Algebraic_real_1,Multiplicity_type> > mroots;
solve_1((x*x-2), std::back_inserter(mroots));
std::cout << "Number of roots is : " << mroots.size() << "\n";
std::cout << "First root should be -sqrt(2): " << CGAL::to_double(mroots[0].first) << ""
<< " with multiplicity " << mroots[0].second << "\n";
std::cout << "Second root should be sqrt(2): " << CGAL::to_double(mroots[1].first) << ""
<< " with multiplicity " << mroots[1].second << "\n\n";
mroots.clear();
// variant computing roots with multiplicities for a range
solve_1((x*x-2)*(x*x-3),Bound(0),Bound(10),std::back_inserter(mroots));
std::cout << "Number of roots is : " << mroots.size() << "\n";
std::cout << "First root should be sqrt(2): " << CGAL::to_double(mroots[0].first) << ""
<< " with multiplicity " << mroots[0].second << "\n";
std::cout << "Second root should be sqrt(3): " << CGAL::to_double(mroots[1].first) << ""
<< " with multiplicity " << mroots[1].second << "\n\n";
return 0;
}
#else
int main(){
std::cout << "This example requires CGAL to be configured with library MPFI." << std::endl;
return 0;
}
#endif

## Comparison and Approximation of Algebraic Real Numbers

The following example illustrates the comparison of `AlgebraicKernel_d_1::Algebraic_real_1` numbers:
File Algebraic_kernel_d/Compare_1.cpp

// $URL$
// $Id$
#include <CGAL/config.h>
#ifdef CGAL_USE_MPFI
#include <CGAL/Algebraic_kernel_d_1.h>
#include <CGAL/Gmpz.h>
#include <vector>
typedef CGAL::Algebraic_kernel_d_1<CGAL::Gmpz> AK;
typedef AK::Coefficient Coefficient;
typedef AK::Polynomial_1 Polynomial_1;
typedef AK::Algebraic_real_1 Algebraic_real_1;
typedef AK::Bound Bound;
typedef std::pair<Bound,Bound> Interval;
int main(){
AK ak;
AK::Construct_algebraic_real_1 construct_algebraic_real_1 = ak.construct_algebraic_real_1_object();
Polynomial_1 x = CGAL::shift(AK::Polynomial_1(1),1); // the monomial x
Algebraic_real_1 a = construct_algebraic_real_1(x*x-2,1); // sqrt(2)
Algebraic_real_1 b = construct_algebraic_real_1(x*x-3,1); // sqrt(3)
// Algebraic_real_1 is RealEmbeddable (just some functions:)
std::cout << "sign of a is : " << CGAL::sign(a) << "\n";
std::cout << "double approximation of a is : " << CGAL::to_double(a) << "\n";
std::cout << "double approximation of b is : " << CGAL::to_double(b) << "\n";
std::cout << "double lower bound of a : " << CGAL::to_interval(a).first << "\n";
std::cout << "double upper bound of a : " << CGAL::to_interval(a).second << "\n";
std::cout << "LessThanComparable (a<b) : " << (a<b) << "\n\n";
// use compare_1 with int, Bound, Coefficient, Algebraic_real_1
AK::Compare_1 compare_1 = ak.compare_1_object();
std::cout << " compare with an int : " << compare_1(a ,int(2)) << "\n";
std::cout << " compare with an Coefficient : " << compare_1(a ,Coefficient(2)) << "\n";
std::cout << " compare with an Bound : " << compare_1(a ,Bound(2)) << "\n";
std::cout << " compare with another Algebraic_real_1: " << compare_1(a ,b) << "\n\n";
// get a value between two roots
AK::Bound_between_1 bound_between_1 = ak.bound_between_1_object();
std::cout << " value between sqrt(2) and sqrt(3) " << bound_between_1(a,b) << "\n";
std::cout << " is larger than sqrt(2) " << compare_1(bound_between_1(a,b),a) << "\n";
std::cout << " is less than sqrt(3) " << compare_1(bound_between_1(a,b),b) << "\n\n";
// approximate with relative precision
AK::Approximate_relative_1 approx_r = ak.approximate_relative_1_object();
std::cout << " lower bound of a with at least 100 bits: "<< approx_r(a,100).first << "\n";
std::cout << " upper bound of a with at least 100 bits: "<< approx_r(a,100).second << "\n\n";
// approximate with absolute error
AK::Approximate_absolute_1 approx_a = ak.approximate_absolute_1_object();
std::cout << " lower bound of b with error less than 2^-100: "<< approx_a(b,100).first << "\n";
std::cout << " upper bound of b with error less than 2^-100: "<< approx_a(b,100).second << "\n\n";
return 0;
}
#else
int main(){
std::cout << "This example requires CGAL to be configured with library MPFI." << std::endl;
return 0;
}
#endif
CGAL::to_interval
std::pair< double, double > to_interval(const NT &x)
CGAL::sign
result_type sign(const NT &x)

## Isolation of Algebraic Real Numbers with respect to roots of other polynomials

The following example illustrates the isolation of `AlgebraicKernel_d_1::Algebraic_real_1` numbers:
File Algebraic_kernel_d/Isolate_1.cpp

// $URL$
// $Id$
#include <CGAL/config.h>
#ifdef CGAL_USE_MPFI
#include <CGAL/Algebraic_kernel_d_1.h>
#include <CGAL/Gmpz.h>
#include <vector>
typedef CGAL::Algebraic_kernel_d_1<CGAL::Gmpz> AK;
typedef AK::Polynomial_1 Polynomial_1;
typedef AK::Algebraic_real_1 Algebraic_real_1;
typedef AK::Coefficient Coefficient;
typedef AK::Bound Bound;
typedef AK::Multiplicity_type Multiplicity_type;
int main(){
AK ak; // an object of
AK::Construct_algebraic_real_1 construct_algreal_1 = ak.construct_algebraic_real_1_object();
AK::Isolate_1 isolate_1 = ak.isolate_1_object();
AK::Compute_polynomial_1 compute_polynomial_1 = ak.compute_polynomial_1_object();
// construct an algebraic number from an integer
Algebraic_real_1 frominteger=construct_algreal_1(int(2));
std::cout << "Construct from int: " << frominteger << "\n";
// the constructed algebraic number is root of a polynomial
Polynomial_1 pol=compute_polynomial_1(frominteger);
std::cout << "The constructed number is root of: " << pol << "\n";
// construct an algebraic number from a polynomial and an isolating interval
Polynomial_1 x = CGAL::shift(AK::Polynomial_1(1),1); // the monomial x
Algebraic_real_1 frominterval=construct_algreal_1(x*x-2,Bound(0),Bound(2));
std::cout << "Construct from isolating interval: " << frominterval << "\n";
// isolate the second algebraic number from the first: this is to say,
// isolating the second algebraic number with respect to the polynomial
// of which the first constructed number is root
std::pair<Bound,Bound> isolation1 = isolate_1(frominterval,pol);
std::cout << "Isolating the second algebraic number gives: ["
<< isolation1.first << "," << isolation1.second << "]\n";
// isolate again the same algebraic number, this time with respect to
// the polynomial 10*x-14 (which has root 1.4, close to this algebraic
// number)
std::pair<Bound,Bound> isolation2 = isolate_1(frominterval,10*x-14);
std::cout << "Isolating again the second algebraic number gives: ["
<< isolation2.first << "," << isolation2.second << "]\n";
return 0;
}
#else
int main(){
std::cout << "This example requires CGAL to be configured with library MPFI." << std::endl;
return 0;
}
#endif

## Interplay with Polynomials

The following example illustrates the sign evaluation of `AlgebraicKernel_d_1::Algebraic_real_1` numbers in polynomials:
File Algebraic_kernel_d/Sign_at_1.cpp

// $URL$
// $Id$
#include <CGAL/config.h>
#ifdef CGAL_USE_MPFI
#include <CGAL/Algebraic_kernel_d_1.h>
#include <CGAL/Gmpz.h>
#include <vector>
typedef CGAL::Algebraic_kernel_d_1<CGAL::Gmpz> AK;
typedef AK::Polynomial_1 Polynomial_1;
typedef AK::Algebraic_real_1 Algebraic_real_1;
typedef AK::Coefficient Coefficient;
typedef AK::Bound Bound;
typedef AK::Multiplicity_type Multiplicity_type;
int main(){
AK ak;
AK::Solve_1 solve_1 = ak.solve_1_object();
AK::Sign_at_1 sign_at_1 = ak.sign_at_1_object();
AK::Is_zero_at_1 is_zero_at_1 = ak.is_zero_at_1_object();
// construct the polynomials p=x^2-5 and q=x-2
Polynomial_1 x = CGAL::shift(AK::Polynomial_1(1),1); // the monomial x
Polynomial_1 p = x*x-5;
std::cout << "Polynomial p: " << p << "\n";
Polynomial_1 q = x-2;
std::cout << "Polynomial q: " << q << "\n";
// find the roots of p (it has two roots) and q (one root)
std::vector<Algebraic_real_1> roots_p,roots_q;
solve_1(p,true, std::back_inserter(roots_p));
solve_1(q,true, std::back_inserter(roots_q));
// evaluate the second root of p in q
std::cout << "Sign of the evaluation of root 2 of p in q: "
<< sign_at_1(q,roots_p[1]) << "\n";
// evaluate the root of q in p
std::cout << "Sign of the evaluation of root 1 of q in p: "
<< sign_at_1(p,roots_q[0]) << "\n";
// check whether the evaluation of the first root of p in p is zero
std::cout << "Is zero the evaluation of root 1 of p in p? "
<< is_zero_at_1(p,roots_p[0]) << "\n";
return 0;
}
#else
int main(){
std::cout << "This example requires CGAL to be configured with library MPFI." << std::endl;
return 0;
}
#endif

# Design and Implementation History

This package is clearly split into a univariate and bivariate kernel. However, with respect to its history the package splits into a design part and an implementation part.

The concepts, which make up the design part, were written by Eric Berberich, Michael Hemmer, and Monique Teillaud. The design history of the package is fairly old and several ideas that influenced this package can already be found in [3]. Since then, the initial design underwent considerable changes. For instance, it was decided that the algebraic numbers should be under the control of the algebraic kernel. On the other hand the initial support for polynomials was extended to a separate and independent package that is not restricted to a certain number of variables. Thus, the authors want to thank for all the useful feedback and ideas that was brought to them throughout the last years. In particular, they want to thank Menelaos Karavelas and Elias Tsigaridas for their initial contributions.

The two generic models where initially developed as part of the Exacus [2] project. However, the models are now fully integrated into the CGAL library, since also the relevant layers of Exacus are now part of CGAL. The main authors for `Algebraic_kernel_d_1<Coeff>` and `Algebraic_kernel_d_2<Coeff>` are Michael Hemmer and Michael Kerber, respectively. Notwithstanding, the authors also want to emphasize the contribution of all authors of the Exacus project, particularly the contribution of Arno Eigenwillig, Sebastian Limbach and Pavel Emeliyanenko.

In 2010, two univariate kernels that interface the library RS [10] were written by Luis Peñaranda and Sylvain Lazard [9]. Both models interface the library RS [10] by Fabrice Rouillier. The authors want to thank Fabrice Rouillier and Elias Tsigaridas for strong support and many useful discussions that lead to the integration of RS. Due to lack of maintenance, these kernels have been removed in 2024.