Relates Algebraic Extensions

Classes
struct	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

compute_roots_of_2()

template<typename RT , typename OutputIterator >

OutputIterator CGAL::compute_roots_of_2	(	const RT &	a,
		const RT &	b,
		const RT &	c,
		OutputIterator	oit
	)

#include <CGAL/Root_of_traits.h>

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>

make_root_of_2() [1/2]

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
	)

#include <CGAL/Root_of_traits.h>

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>

make_root_of_2() [2/2]

template<typename RT >

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

#include <CGAL/Root_of_traits.h>

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>

make_sqrt()

template<typename RT >

Root_of_traits<RT>::Root_of_2 CGAL::make_sqrt

(

const RT &

x

)

#include <CGAL/Root_of_traits.h>

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>