 CGAL 5.1 - Number Types
CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > Class Template Reference

#include <CGAL/Sqrt_extension.h>

## Definition

### template<typename NT, typename ROOT, typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false> class CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >

An instance of this class represents an extension of the type NT by one square root of the type ROOT.

NT is required to be constructible from ROOT.

NT is required to be an IntegralDomainWithoutDivision.

Sqrt_extension is RealEmbeddable if NT is RealEmbeddable.

For example, let Integer be some type representing $$\Z$$, then Sqrt_extension<Integer,Integer> is able to represent $$\Z[\sqrt{\mathrm{root}}]$$ for some arbitrary Integer $$\mathrm{root}$$. $$R[a]$$ denotes the extension of a ring $$R$$ by an element $$a$$. See also: http://mathworld.wolfram.com/ExtensionRing.html The value of $$\mathrm{root}$$ is set at construction time, or set to zero if it is not specified.

Arithmetic operations among different extensions, say $$\Z[\sqrt{a}]$$ and $$\Z[\sqrt{b}]$$, are not supported. The result would be in $$\Z[\sqrt{a},\sqrt{b}]$$, which is not representable by Sqrt_extension<Integer,Integer>.

Attention
The user is responsible to check that arithmetic operations are carried out for elements from the same extensions only.

This is not tested by Sqrt_extension for efficiency reasons. A violation of the precondition leads to undefined behavior. Be aware that for efficiency reasons the given $$\mathrm{root}$$ is stored as it is given to the constructor. In particular, an extension by a square root of a square is considered as an extension.

Since elements of Sqrt_extension that lie in different extensions are not interoperable with respect to any arithmetic operations, the full value range of Sqrt_extension does not represent an algebraic structure. However, each subset of the value range that represents the extension of NT by a particular square root is a valid algebraic structure, since this subset is closed under all provided arithmetic operations. From there, Sqrt_extension can be used as if it were a model of an algebraic structure concept, with the following correspondence:

 NT Sqrt_extension IntegralDomainWithoutDivision IntegralDomainWithoutDivision IntegralDomain IntegralDomain UniqueFactorizationDomain IntegralDomain EuclideanRing IntegralDomain Field Field

The extension of a UniqueFactorizationDomain or EuclideanRing is just an IntegralDomain, since the extension in general destroys the unique factorization property. For instance consider $$\Z[\sqrt{10}]$$, the extension of $$\Z$$ by $$\sqrt{10}$$: in $$\Z[\sqrt{10}]$$ the element 10 has two different factorizations $$\sqrt{10} \cdot \sqrt{10}$$ and $$2 \cdot 5$$. In particular, the factorization is not unique.

If NT is a model of RealEmbeddable the type Sqrt_extension is also considered as RealEmbeddable. However, by default it is not allowed to compare values from different extensions for efficiency reasons. In case such a comparison becomes necessary, use the member function compare with the according Boolean flag. If such a comparison is a very frequent case, override the default of DifferentExtensionComparable by giving CGAL::Tag_true as third template parameter. This effects the behavior of compare functions as well as the compare operators.

The fourth template argument, FilterPredicates, triggers an internal filter that may speed up comparisons and sign computations. In case FilterPredicates is set to CGAL::Tag_true the type first computes a double interval containing the represented number and tries to perform the comparison or sign computation using this interval. Once computed, this interval is stored by the corresponding Sqrt_extension object for further usage. Note that this internal filter is switched off by default, since it may conflict with other filtering methods, such as Lazy_exact_nt<Sqrt_extension>.

In case NT is not RealEmbeddable, DifferentExtensionComparable as well as FilterPredicates have no effect.

Is Model Of:

Assignable

CopyConstructible

DefaultConstructible

EqualityComparable

ImplicitInteroperable with int

ImplicitInteroperable with NT

Fraction if NT is a Fraction

RootOf_2

IntegralDomainWithoutDivision
IntegralDomain
Field
RealEmbeddable
CGAL::Tag_true
CGAL::Tag_false

## Related Functions

(Note that these are not member functions.)

Sqrt_extension operator+ (const Sqrt_extension &a, const Sqrt_extension &b)

Sqrt_extension operator- (const Sqrt_extension &a, const Sqrt_extension &b)

Sqrt_extension operator* (const Sqrt_extension &a, const Sqrt_extension &b)

Sqrt_extension operator/ (const Sqrt_extension &a, const Sqrt_extension &b)

bool operator== (const Sqrt_extension &a, const Sqrt_extension &b)

bool operator!= (const Sqrt_extension &a, const Sqrt_extension &b)

bool operator< (const Sqrt_extension &a, const Sqrt_extension &b)

bool operator<= (const Sqrt_extension &a, const Sqrt_extension &b)

bool operator> (const Sqrt_extension &a, const Sqrt_extension &b)

bool operator>= (const Sqrt_extension &a, const Sqrt_extension &b)

std::ostream & operator<< (std::ostream &os, const Sqrt_extension< NT, ROOT > &ext)
writes ext to ostream os. More...

std::istream & operator>> (std::istream &is, const Sqrt_extension< NT, ROOT > &ext)
reads ext from istream is in format EXT[a0,a1,root], the output format in mode CGAL::IO::ASCII More...

## Creation

Sqrt_extension ()
Introduces a variable ext initialized with 0.

Sqrt_extension (const Sqrt_extension &x)
Copy constructor.

Sqrt_extension (const int &i)
Introduces a variable ext initialized with $$i$$.

Sqrt_extension (const NT &x)
Introduces a variable ext initialized with $$x$$.

template<class NTX >
Sqrt_extension (const NTX &x)
Introduces a variable ext initialized with $$x$$. More...

Sqrt_extension (int a0, int a1, int r)
Constructor from int: ext $$= a0 +a1 \cdot sqrt(r)$$. More...

Sqrt_extension (NT a0, NT a1, ROOT r)
General constructor: ext $$= a0 + a1 \cdot sqrt(r)$$. More...

## Operations

An object of type Sqrt_extension represent an expression of the form: $$a0 + a1 \sqrt(\mathrm{root})$$.

const NT & a0 () const
Const access operator for a0.

const NT & a1 () const
Const access operator for a1.

const ROOT & root () const
Const access operator for root.

bool is_extended () const
Returns true in case root of ext is not zero. More...

void simplify ()
Simplifies the representation, in particular $$\mathrm{root}$$ is set to zero if $$a1$$ is zero, that is, ext becomes not extended. More...

bool is_zero () const
returns true if ext represents the value zero.

CGAL::Sign sign () const
Determines the sign of ext by (repeated) squaring. More...

Sqrt_extension abs () const
returns the absolute value of ext. More...

CGAL::Comparison_result compare (const Sqrt_extension &y, bool in_same_extension=!DifferentExtensionComparable::value) const
Compares ext with y. More...

Sqrt_extensionoperator+= (const Sqrt_extension &a)

Sqrt_extensionoperator-= (const Sqrt_extension &a)

Sqrt_extensionoperator*= (const Sqrt_extension &a)

Sqrt_extensionoperator/= (const Sqrt_extension &a)

## ◆ Sqrt_extension() [1/3]

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
template<class NTX >
 CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::Sqrt_extension ( const NTX & x )
explicit

Introduces a variable ext initialized with $$x$$.

Precondition
NT must constructible from NTX

## ◆ Sqrt_extension() [2/3]

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::Sqrt_extension ( int a0, int a1, int r )

Constructor from int: ext $$= a0 +a1 \cdot sqrt(r)$$.

Precondition
$$r \neq0$$

## ◆ Sqrt_extension() [3/3]

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::Sqrt_extension ( NT a0, NT a1, ROOT r )

General constructor: ext $$= a0 + a1 \cdot sqrt(r)$$.

Precondition
$$r \neq0$$

## ◆ abs()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::abs ( ) const

returns the absolute value of ext.

Precondition
Sqrt_extension is RealEmbeddable.

## ◆ compare()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 CGAL::Comparison_result CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::compare ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & y, bool in_same_extension = !DifferentExtensionComparable::value ) const

Compares ext with y.

The optional bool in_same_extension indicates whether ext and $$y$$ are in the same extension of NT.

## ◆ is_extended()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 bool CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::is_extended ( ) const

Returns true in case root of ext is not zero.

Note that $$a1 == 0$$ does not imply $$\mathrm{root} == 0$$.

## ◆ operator*=()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension& CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::operator*= ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a )
Precondition
(this->root()==0 or a.root()==0 or this->root() == a.root())

## ◆ operator+=()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension& CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::operator+= ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a )
Precondition
(this->root()==0 or a.root()==0 or this->root() == a.root())

## ◆ operator-=()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension& CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::operator-= ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a )
Precondition
(this->root()==0 or a.root()==0 or this->root() == a.root())

## ◆ operator/=()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension& CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::operator/= ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a )
Precondition
(this->root()==0 or a.root()==0 or this->root() == a.root())

In case NT is only an IntegralDomain operator/ implements integral division. In case NT is a Field operator/ implements the field division.

## ◆ sign()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 CGAL::Sign CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::sign ( ) const

Determines the sign of ext by (repeated) squaring.

Precondition
Sqrt_extension is RealEmbeddable.

## ◆ simplify()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 void CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >::simplify ( )

Simplifies the representation, in particular $$\mathrm{root}$$ is set to zero if $$a1$$ is zero, that is, ext becomes not extended.

Moreover, it propagates the simplify command to members of ext. see also: AlgebraicStructureTraits::Simplify.

## ◆ operator!=()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 bool operator!= ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())

## ◆ operator*()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension operator* ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())

## ◆ operator+()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension operator+ ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())

## ◆ operator-()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension operator- ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())

## ◆ operator/()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 Sqrt_extension operator/ ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())

In case NT is only an IntegralDomain operator/ implements integral division. In case NT is a Field operator/ implements the field division.

## ◆ operator==()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 bool operator== ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())

## ◆ operator>()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 bool operator> ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())
Attention
Only exists when Sqrt_extension is RealEmbeddable.

## ◆ operator>=()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 bool operator>= ( const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & a, const Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > & b )
related
Precondition
(a.root()==0 or b.root()==0 or a.root() == b.root())
Attention
Only exists when Sqrt_extension is RealEmbeddable.

## ◆ operator>>()

template<typename NT , typename ROOT , typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
 std::istream & operator>> ( std::istream & is, const Sqrt_extension< NT, ROOT > & ext )
related

reads ext from istream is in format EXT[a0,a1,root], the output format in mode CGAL::IO::ASCII

Attention
operator<< must be defined exist for ROOT and NT.