namespace CGAL {
/*!

\mainpage User Manual
\anchor Chapter_Algebraic_Kernel
\anchor chapteralgebraickerneld
\cgalAutoToc
\authors Eric Berberich, Michael Hemmer, Michael Kerber, Sylvain Lazard, Luis Pe&ntilde;aranda, and Monique Teillaud

\section Algebraic_kernel_dIntroduction 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.

\section Algebraic_kernel_dAlgebraic Algebraic Kernel Concepts

\subsection Algebraic_kernel_dUnivariateAlgebraicKernel Univariate Algebraic Kernel

\subsection Algebraic_kernel_dMajorTypes 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.

\subsection Algebraic_kernel_dConstructionofAlgebraic 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.

\subsection Algebraic_kernel_dComparisonandApproximation 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 \ref 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.

\subsection Algebraic_kernel_dInterplaywithPolynomials 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.

\subsection Algebraic_kernel_dAuxiliaryFunctionality 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 \ref 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:

- \ref PolynomialTraits_d::SquareFreeFactorize
- \ref PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor

\subsection Algebraic_kernel_dDesignRationale 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 \f$ i\f$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`.

\subsection Algebraic_kernel_dBivariateAlgebraicKernel 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`.

\section Algebraic_kernel_dModels Models

\subsection Algebraic_kernel_dGenericAlgebraicKernels 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 \cgalCite{eigenwillig-phd-08}.
The used method to refine the approximation of an algebraic real root is a
slightly modified (filtered) version of the one presented in \cgalCite{abbott-qir-06}.
The method has quadratic convergence.

`Algebraic_kernel_d_2<Coeff>` is based on an algorithm computing a
geometric-topological analysis of a single curve \cgalCite{ekw-fast-07} and of a
pair of curves \cgalCite{ek-exact-08}.
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 \cgalCite{grlr-sturm-habicht-98}.
With that information, the fiber at
critical x-coordinates is computed by a variant of the Bitstream
Descartes method.
See also \cgalCite{kerber-phd-09} 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 \f$ p\f$ of type `Algebraic_real_2` is represented
by its \f$ x\f$-coordinate \f$ x_0\f$ (as described in the `Algebraic_kernel_d_1`
paragraph above), an algebraic curve where \f$ p\f$ lies on, and an
integer \f$ i\f$, denoting that \f$ p\f$ is the \f$ i\f$th point in the fiber at \f$ x_0\f$,
counted from the bottom (ignoring a possible vertical line at \f$ x_0\f$).
Note that this determines the point uniquely, but the \f$ y\f$-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.


\section Algebraic_kernel_dExamples Examples

\subsection CGALAK1ConstructionExamples 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`:
\cgalExample{Algebraic_kernel_d/Construct_algebraic_real_1.cpp}

\subsection CGALAK1Solving Solving Univariate Polynomials

The following example illustrates the construction of `AlgebraicKernel_d_1::Algebraic_real_1`
using `AlgebraicKernel_d_1::Solve_1`: \cgalExample{Algebraic_kernel_d/Solve_1.cpp}

\subsection CGALAK1EGCompare_1 Comparison and Approximation of Algebraic Real Numbers

The following example illustrates the comparison of `AlgebraicKernel_d_1::Algebraic_real_1` numbers:
\cgalExample{Algebraic_kernel_d/Compare_1.cpp}

\subsection CGALAK1EGIsolate_1 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:
\cgalExample{Algebraic_kernel_d/Isolate_1.cpp}

\subsection CGALAK1EGSign_at_1 Interplay with Polynomials

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

\section Algebraic_kernel_dDesign 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 \cgalCite{cgal:bhkt-risak-07}. 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 \cgalCite{beh-eeeafcs-05} 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 \cgalCite{cgal:r-rs} were
written by Luis Peñaranda and Sylvain Lazard \cgalCite{cgal:2009-ESA}.
Both models interface the library RS \cgalCite{cgal:r-rs} 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.


*/
} /* namespace CGAL */
