Algebraic Foundations

Michael Hemmer

Cgal is targeting towards exact computation with non-linear objects,
in particular objects defined on algebraic curves and surfaces.
As a consequence types representing polynomials, algebraic extensions and
finite fields play a more important role in related implementations.
This package has been introduced to stay abreast of these changes.
Since in particular polynomials must be supported by the introduced framework
the package avoids the term *number type*. Instead the package distinguishes
between the *algebraic structure* of a type and whether a type is embeddable on
the real axis, or *real embeddable* for short.
Moreover, the package introduces the notion of *interoperable* types which
allows an explicit handling of mixed operations.

The algebraic structure concepts introduced within this section are
motivated by their well known counterparts in traditional algebra,
but we also had to pay tribute to existing types and their restrictions.
To keep the interface minimal,
it was not desirable to cover all known algebraic structures,
e.g., we did not introduce concepts for such basic structures as *groups* or
exceptional structures as *skew fields*.

Figure 4.1 shows the refinement
relationship of the algebraic structure concepts.
*IntegralDomain*, *UniqueFactorizationDomain*, *EuclideanRing* and
*Field* correspond to the algebraic structures with the
same name. *FieldWithSqrt*, *FieldWithKthRoot* and
*FieldWithRootOf* are fields that in addition are closed under
the operations 'sqrt', 'k-th root' and 'real root of a polynomial',
respectively. The concept *IntegralDomainWithoutDivision* also
corresponds to integral domains in the algebraic sense, the
distinction results from the fact that some implementations of
integral domains lack the (algebraically always well defined) integral
division.
Note that *Field* refines *IntegralDomain*. This is because
most ring-theoretic notions like greatest common divisors become trivial for
*Field*s. Hence we see *Field* as a refinement of
*IntegralDomain* and not as a
refinement of one of the more advanced ring concepts.
If an algorithm wants to rely on gcd or remainder computation, it is trying
to do things it should not do with a *Field* in the first place.

The main properties of an algebraic structure are collected in the class
*Algebraic_structure_traits*.
In particular the (most refined) concept each concrete model *AS*
fulfills is encoded in the tag
*Algebraic_structure_traits<AS>::Algebraic_category*.
An algebraic structure is at least *Assignable*,
*CopyConstructible*, *DefaultConstructible* and
*EqualityComparable*. Moreover, we require that it is
constructible from *int*.
For ease of use and since their semantic is sufficiently standard to presume
their existence, the usual arithmetic and comparison operators are required
to be realized via C`++` operator overloading.
The division operator is reserved for division in fields.
All other unary (e.g., sqrt) and binary functions
(e.g., gcd, div) must be models of the well known STL-concepts
*AdaptableUnaryFunction* or *AdaptableBinaryFunction*
concept and local to the traits class
(e.g., *Algebraic_structure_traits<AS>::Sqrt()(x)*).
This design allows us to profit from all parts in the
STL and its programming style and avoids the name-lookup and
two-pass template compilation problems experienced with the old design
using overloaded functions. However, for ease of use and backward
compatibility all functionality is also
accessible through global functions defined within namespace *CGAL*,
e.g., *CGAL::sqrt(x)*. This is realized via function templates using
the according functor of the traits class. For an overview see
Section 4.8 in the reference manual.

For a type *AS*, *Algebraic_structure_traits<AS>*
provides several tags. The most important tag is the *Algebraic_category*
tag, which indicates the most refined algebraic concept the type *AS*
fulfills. The tag is one of;
*Integral_domain_without_division_tag*, *Integral_domain_tag*,
*Field_tag*, *Field_with_sqrt_tag*, *Field_with_kth_root_tag*,
*Field_with_root_of_tag*, *Unique_factorization_domain_tag*,
*Euclidean_ring_tag*, or even *Null_tag*
in case the type is not a model of an algebraic structure concept.
The tags are derived from each other such that they reflect the
hierarchy of the algebraic structure concept, e.g.,
*Field_with_sqrt_tag* is derived from *Field_tag*.

Moreover, *Algebraic_structure_traits<AS>* provides the tags *Is_exact*
and *Is_numerical_sensitive*, which are both *Boolean_tag*s.

An algebraic structure is considered *exact*,
if all operations required by its concept are computed such that a comparison
of two algebraic expressions is always correct.

An algebraic structure is considered as *numerically sensitive*,
if the performance of the type is sensitive to the condition number of an
algorithm.
Note that there is really a difference among these two notions,
e.g., the fundamental type *int* is not numerical sensitive but
considered inexact due to overflow.
Conversely, types as *leda_real* or *CORE::Expr* are exact but sensitive
to numerical issues due to the internal use of multi precision floating point
arithmetic. We expect that *Is_numerical_sensitive* is used for dispatching
of algorithms, while *Is_exact* is useful to enable assertions that can be
check for exact types only.

Tags are very useful to dispatch between alternative implementations.
The following example illustrates a dispatch for *Field*s using overloaded
functions. The example only needs two overloads since the algebraic
category tags reflect the algebraic structure hierarchy.

File:examples/Algebraic_foundations/algebraic_structure_dispatch.cpp

#include <CGAL/basic.h> #include <CGAL/IO/io.h> #include <CGAL/Algebraic_structure_traits.h> template< typename NT > NT unit_part(const NT& x); template< typename NT > NT unit_part_(const NT& x, CGAL::Field_tag); template< typename NT > NT unit_part_(const NT& x, CGAL::Integral_domain_without_division_tag); template< typename NT > NT unit_part(const NT& x){ // the unit part of 0 is defined as 1. if (x == 0 ) return NT(1); typedef CGAL::Algebraic_structure_traits<NT> AST; typedef typename AST::Algebraic_category Algebraic_category; return unit_part_(x,Algebraic_category()); } template< typename NT > NT unit_part_(const NT& x, CGAL::Integral_domain_without_division_tag){ // For many other types the only units are just -1 and +1. return NT(int(CGAL::sign(x))); } template< typename NT > NT unit_part_(const NT& x, CGAL::Field_tag){ // For Fields every x != 0 is a unit. // Therefore, every x != 0 is its own unit part. return x; } int main(){ // Function call for a model of EuclideanRing, i.e. int. std::cout<< "int: unit_part(-3 ): " << unit_part(-3 ) << std::endl; // Function call for a model of FieldWithSqrt, i.e. double std::cout<< "double: unit_part(-3.0): " << unit_part(-3.0) << std::endl; return 0; } // Note that this is just an example // This implementation for unit part won't work for some types, e.g., // types that are not RealEmbeddable or types representing structures that have // more units than just -1 and +1. (e.g. MP_Float representing Z[1/2]) // From there Algebraic_structure_traits provides the functor Unit_part.

Most number types represent some subset of the real numbers. From those types
we expect functionality to compute the sign, absolute value or double
approximations. In particular we can expect an order on such a type that
reflects the order along the real axis.
All these properties are gathered in the concept *RealComparable*.
The concept is orthogonal to the algebraic structure concepts,
i.e., it is possible
that a type is a model of *RealEmbeddable* only,
since the type may just represent values on the real axis
but does not provide any arithmetic operations.

As for algebraic structures this concept is also traits class oriented.
The main functionality related to *RealEmbeddable* is gathered in
the class *Real_embeddable_traits*. In particular, it porivdes the boolean
tag *Is_real_embeddable* indicating whether a type is a model of
*RealEmbeddable*. The comparison operators are required to be realized via
C`++` operator overloading.
All unary functions (e.g. *sign*, *to_double*) and
binary functions (e.g. *compare* ) are models of the STL-concepts
*AdaptableUnaryFunction* and *AdaptableBinaryFunction* and are local
to *Real_embeddable_traits*.

In case a type is a model of *IntegralDomainWithoutDivision* and
*RealEmbeddable* the number represented by an object of this type is
the same for arithmetic and comparison.
It follows that the ring represented by this type is a superset of the integers
and a subset of the real numbers and hence has characteristic zero.
In case the type is a model of *Field* and *RealEmbeddable* it is a
superset of the rational numbers.

The concept *FieldNumberType* combines the requirements of the
concepts *Field* and *RealEmbeddable*, while
*RingNumberType* combines *IntegralDomainWithoutDivision* and
*RealEmbeddable*. Algebraically, the real number types do not form
distinct structures and are therefore not listed in the concept
hierarchy of Figure 4.1.

This section introduces two concepts for interoperability of types,
namely *ImplicitInteroperable* and *ExplicitInteroperable*. While
*ExplicitInteroperable* is the base concept, we start with
*ImplicitInteroperable* since it is the more intuitive one.

In general mixed operations are provided by overloaded operators and
functions or just via implicit constructor calls.
This level of interoperability is reflected by the concept
*ImplicitInteroperable*. However, within template code the result type,
or so called coercion type, of a mixed arithmetic operation may be unclear.
Therefore, the package introduces *CGAL::Coercion_traits*
giving access to the coercion type via *CGAL::Coercion_traits<A,B>::Type*
for two interoperable types *A* and *B*.

Some trivial example are *int* and *double* with coercion type double
or *CGAL::Gmpz* and *CGAL::Gmpq* with coercion type *CGAL::Gmpq*.
However, the coercion type is not necessarily one of the input types,
e.g. the coercion type of a polynomial
with integer coefficients that is multiplied by a rational type
is supposed to be a polynomial with rational coefficients.

*CGAL::Coercion_traits* is also
required to provide a functor *CGAL::Coercion_traits<A,B>::Cast()*, that
converts from an input type into the coercion type. This is in fact the core
of the more basic concept *ExplicitInteroperable*.
*ExplicitInteroperable* has been introduced to cover more complex cases
for which it is hard or impossible to guarantee implicit interoperability.
Note that this functor can be useful for *ImplicitInteroperable* types
as well, since it can be used to void redundant type conversions.

In case two types *A* and *B* are *ExplicitInteroperable* with
coercion type *C* they are valid argument types for all binary functors
provided by *Algebraic_structure_traits* and *Real_embeddable_traits* of
*C*. This is also true for the according global functions.

The following example illustrates how two write code for
*ExplicitInteroperable* types.

File:examples/Algebraic_foundations/interoperable.cpp

#include <CGAL/basic.h> #include <CGAL/Coercion_traits.h> #include <CGAL/IO/io.h> // this is an implementation for ExplicitInteroperable types // the result type is determined via Coercion_traits<A,B> template <typename A, typename B> typename CGAL::Coercion_traits<A,B>::Type binary_func(const A& a , const B& b){ typedef CGAL::Coercion_traits<A,B> CT; // check for explicit interoperability BOOST_STATIC_ASSERT((CT::Are_explicit_interoperable::value)); // CT::Cast is used to to convert both types into the coercion type typename CT::Cast cast; // all operations are performed in the coercion type return cast(a)*cast(b); } int main(){ // Function call for the interoperable types std::cout<< binary_func(double(3), int(5)) << std::endl; // Note that Coercion_traits is symmetric std::cout<< binary_func(int(3), double(5)) << std::endl; return 0; }

The following example illustrates a dispatch for *ImplicitInteroperable* and
*ExplicitInteroperable* types.
The binary function (that just multiplies its two arguments) is supposed to
take two *ExplicitInteroperable* arguments. For *ImplicitInteroperable*
types a variant that avoids the explicit cast is selected.

File:examples/Algebraic_foundations/implicit_interoperable_dispatch.cpp

#include <CGAL/basic.h> #include <CGAL/Coercion_traits.h> #include <CGAL/Quotient.h> #include <CGAL/Sqrt_extension.h> #include <CGAL/IO/io.h> // this is the implementation for ExplicitInteroperable types template <typename A, typename B> typename CGAL::Coercion_traits<A,B>::Type binary_function_(const A& a , const B& b, CGAL::Tag_false){ std::cout << "Call for ExplicitInteroperable types: " << std::endl; typedef CGAL::Coercion_traits<A,B> CT; typename CT::Cast cast; return cast(a)*cast(b); } // this is the implementation for ImplicitInteroperable types template <typename A, typename B> typename CGAL::Coercion_traits<A,B>::Type binary_function_(const A& a , const B& b, CGAL::Tag_true){ std::cout << "Call for ImpicitInteroperable types: " << std::endl; return a*b; } // this function selects the correct implementation template <typename A, typename B> typename CGAL::Coercion_traits<A,B>::Type binary_func(const A& a , const B& b){ typedef CGAL::Coercion_traits<A,B> CT; typedef typename CT::Are_implicit_interoperable Are_implicit_interoperable; return binary_function_(a,b,Are_implicit_interoperable()); } int main(){ CGAL::set_pretty_mode(std::cout); // Function call for ImplicitInteroperable types std::cout<< binary_func(double(3), int(5)) << std::endl; // Function call for ExplicitInteroperable types CGAL::Quotient<int> rational(1,3); // == 1/3 CGAL::Sqrt_extension<int,int> extension(1,2,3); // == 1+2*sqrt(3) CGAL::Sqrt_extension<CGAL::Quotient<int>,int> result = binary_func(rational, extension); std::cout<< result << std::endl; return 0; }

Beyond the need for performing algebraic operations on objects as a
whole, there are also number types which one would like to decompose into
numerator and denominator. This does not only hold for rational numbers
as *Quotient*, *Gmpq*, *mpq_class* or *leda_rational*, but
also for compound objects as *Sqrt_extension* or *Polynomial*
which may decompose into a (scalar)
denominator and a compound numerator with a simpler coefficient type
(e.g. integer instead of rational). Often operations can be performed faster on
these denominator-free multiples. In case a type is a *Fraction*
the relevant functionality as well as the numerator and denominator
type are provided by *CGAL::Fraction_traits*. In particular
*CGAL::Fraction_traits* provides a tag *Is_fraction* that can be
used for dispatching.

A related class is *CGAL::Rational_traits* which has been kept for backward
compatibility reasons. However, we recommend to use *Fraction_traits* since
it is more general and offers dispatching functionality.

The following example show a simple use of *Fraction_traits*:

File:examples/Algebraic_foundations/fraction_traits.cpp

#include <CGAL/basic.h> #include <CGAL/Fraction_traits.h> #include <CGAL/IO/io.h> #ifdef CGAL_USE_GMP #include <CGAL/Gmpz.h> #include <CGAL/Gmpq.h> int main(){ typedef CGAL::Fraction_traits<CGAL::Gmpq> FT; typedef FT::Numerator_type Numerator_type; typedef FT::Denominator_type Denominator_type; BOOST_STATIC_ASSERT((boost::is_same<Numerator_type,CGAL::Gmpz>::value)); BOOST_STATIC_ASSERT((boost::is_same<Denominator_type,CGAL::Gmpz>::value)); Numerator_type numerator; Denominator_type denominator; CGAL::Gmpq fraction(4,5); FT::Decompose()(fraction,numerator,denominator); CGAL::set_pretty_mode(std::cout); std::cout << "decompose fraction: "<< std::endl; std::cout << "fraction : " << fraction << std::endl; std::cout << "numerator : " << numerator<< std::endl; std::cout << "denominator: " << denominator << std::endl; std::cout << "re-compose fraction: "<< std::endl; fraction = FT::Compose()(numerator,denominator); std::cout << "fraction : " << fraction << std::endl; } #else int main(){ std::cout << "This examples needs GMP" << std::endl; } #endif

The following example illustrates the integralization of a vector,
i.e., the coefficient vector of a polynomial. Note that for minimizing
coefficient growth *Fraction_traits<Type>::Common_factor* is used to
compute the 'least' common multiple of the denominators.

File:examples/Algebraic_foundations/integralize.cpp

#include <CGAL/basic.h> #include <CGAL/Fraction_traits.h> #include <CGAL/IO/io.h> #include <vector> template <class Fraction> std::vector<typename CGAL::Fraction_traits<Fraction>::Numerator_type > integralize( const std::vector<Fraction>& vec, typename CGAL::Fraction_traits<Fraction>::Denominator_type& d ) { typedef CGAL::Fraction_traits<Fraction> FT; typedef typename FT::Numerator_type Numerator_type; typedef typename FT::Denominator_type Denominator_type; typename FT::Decompose decompose; std::vector<Numerator_type> num(vec.size()); std::vector<Denominator_type> den(vec.size()); // decompose each coefficient into integral part and denominator for (unsigned int i = 0; i < vec.size(); i++) { decompose(vec[i], num[i], den[i]); } // compute 'least' common multiple of all denominator // We would like to use gcd, so let's think of Common_factor as gcd. typename FT::Common_factor gcd; d = 1; for (unsigned int i = 0; i < vec.size(); i++) { d *= CGAL::integral_division(den[i], gcd(d, den[i])); } // expand each (numerator, denominator) pair to common denominator for (unsigned int i = 0; i < vec.size(); i++) { // For simplicity ImplicitInteroperability is expected in this example num[i] *= CGAL::integral_division(d, den[i]); } return num; } #ifdef CGAL_USE_GMP #include <CGAL/Gmpz.h> #include <CGAL/Gmpq.h> int main(){ std::vector<CGAL::Gmpq> vec(3); vec[0]=CGAL::Gmpq(1,4); vec[1]=CGAL::Gmpq(1,6); vec[2]=CGAL::Gmpq(1,10); std::cout<< "compute an integralized vector" << std::endl; std::cout<<"input vector: [" << vec[0] << "," << vec[1] << "," << vec[2] << "]" << std::endl; CGAL::Gmpz d; std::vector<CGAL::Gmpz> integral_vec = integralize(vec,d); std::cout<<"output vector: [" << integral_vec[0] << "," << integral_vec[1] << "," << integral_vec[2] << "]" << std::endl; std::cout<<"denominator : "<< d <<std::endl; } #else int main(){ std::cout << "This examples needs GMP" << std::endl; } #endif

The package is part of Cgal since release 3.3. Of course the package is based
on the former Number type support of CGAL. This goes back to Stefan Schirra and Andreas Fabri. But on the other hand the package is to a large extend influenced
by the experience with the number type support in Exacus [BEH^{+}05],
which in the main goes back to
Lutz Kettner, Susan Hert, Arno Eigenwillig and Michael Hemmer.
However, the package abstracts from the pure support for
number types that are embedded on the real axis which allows the support of
polynomials, finite fields, and algebraic extensions as well. See also related
subsequent chapters.

Next: Reference Manual

CGAL Open Source Project.
Release 3.9.
26 September 2011.