\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.5 - Number Types
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Relates Algebraic Extensions

Classes

class  CGAL::Root_of_traits< RT >
 For a RealEmbeddable IntegralDomain RT, the class template Root_of_traits<RT> associates a type Root_of_2, which represents algebraic numbers of degree 2 over RT. More...
 
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. More...
 

Functions

template<typename RT , typename OutputIterator >
OutputIterator CGAL::compute_roots_of_2 (const RT &a, const RT &b, const RT &c, OutputIterator oit)
 The function compute_roots_of_2() solves a univariate polynomial as it is defined by the coefficients given to the function. More...
 
template<typename RT >
Root_of_traits< RT >::Root_of_2 CGAL::make_root_of_2 (const RT &a, const RT &b, const RT &c, bool s)
 The function make_root_of_2() constructs an algebraic number of degree 2 over a ring number type. More...
 
template<typename RT >
Root_of_traits< RT >::Root_of_2 CGAL::make_root_of_2 (RT alpha, RT beta, RT gamma)
 The function make_root_of_2() constructs an algebraic number of degree 2 over a ring number type. More...
 
template<typename RT >
Root_of_traits< RT >::Root_of_2 CGAL::make_sqrt (const RT &x)
 The function make_sqrt() constructs a square root of a given value of type \( RT\). More...
 

Function Documentation

template<typename RT , typename OutputIterator >
OutputIterator CGAL::compute_roots_of_2 ( const RT &  a,
const RT &  b,
const RT &  c,
OutputIterator  oit 
)

The function compute_roots_of_2() solves a univariate polynomial as it is defined by the coefficients given to the function.

The solutions are written into the given OutputIterator. Writes the real roots of the polynomial \( aX^2+bX+c\) into \( oit\) in ascending order.

OutputIterator is required to accept Root_of_traits<RT>::Root_of_2.

Multiplicities are not reported.

Precondition
RT is an IntegralDomainWithoutDivision.
\( a\neq0\) or \( b\neq0\).
See Also
RootOf_2
CGAL::Root_of_traits<RT>
CGAL::make_root_of_2()
CGAL::make_sqrt()
CGAL::Sqrt_extension<NT,ROOT>

#include <CGAL/Root_of_traits.h>

template<typename RT >
Root_of_traits<RT>::Root_of_2 CGAL::make_root_of_2 ( const RT &  a,
const RT &  b,
const RT &  c,
bool  s 
)

The function make_root_of_2() constructs an algebraic number of degree 2 over a ring number type.

Returns the smallest real root of the polynomial \( aX^2+bX+c\) if \( s\) is true, and the largest root is \( s\) is false.

Precondition
RT is an IntegralDomainWithoutDivision.
The polynomial has at least one real root.
See Also
RootOf_2
CGAL::Root_of_traits<RT>
CGAL::make_sqrt()
CGAL::compute_roots_of_2()
CGAL::Sqrt_extension<NT,ROOT>

#include <CGAL/Root_of_traits.h>

template<typename RT >
Root_of_traits<RT>::Root_of_2 CGAL::make_root_of_2 ( RT  alpha,
RT  beta,
RT  gamma 
)

The function make_root_of_2() constructs an algebraic number of degree 2 over a ring number type.

Constructs the number \( \alpha+ \beta\sqrt{\gamma}\).

Precondition
RT is an IntegralDomainWithoutDivision.
\( \gamma\geq0\)
See Also
RootOf_2
CGAL::Root_of_traits<RT>
CGAL::make_sqrt()
CGAL::compute_roots_of_2()
CGAL::Sqrt_extension<NT,ROOT>

#include <CGAL/Root_of_traits.h>

template<typename RT >
Root_of_traits<RT>::Root_of_2 CGAL::make_sqrt ( const RT &  x)

The function make_sqrt() constructs a square root of a given value of type \( RT\).

Depending on the type \( RT\) the square root may be returned in a new type that can represent algebraic extensions of degree \( 2\).

Returns
\( \sqrt{x}.\)
Precondition
RT is a RealEmbeddable IntegralDomain.
\( x \leq0 \)
See Also
RootOf_2
CGAL::make_root_of_2()
CGAL::Root_of_traits<RT>

#include <CGAL/Root_of_traits.h>