CGAL 4.7 - Polynomial: PolynomialTraits_d::SturmHabichtSequenceWithCofactors Concept Reference

\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.7 - Polynomial

All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
List of all members
PolynomialTraits_d::SturmHabichtSequenceWithCofactors Concept Reference
Concepts

## Definition

Note: This functor is optional!

Computes the Sturm-Habicht polynomials of a polynomial \( f\) of degree \( n\), as defined in the documentation of `PolynomialTraits_d::SturmHabichtSequence`. Moreover, for \( \mathrm{Stha}_i(f)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq n-i-2\) and \( \deg v_i\leq n-i-1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i f + v_i f'\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Stha}_i(f)\).

The result is written in three output ranges, each of length \( \min\{n,m\}+1\), starting with the \( 0\)-th Sturm-Habicht polynomial \( \mathrm{Stha_0(f)}\) and the corresponding cofactors.

Refines:

  `AdaptableBinaryFunction`

  `CopyConstructible`

  `DefaultConstructible`

See Also
  `Polynomial_d`
  `PolynomialTraits_d`
  `PolynomialTraits_d::Resultant`
  `PolynomialTraits_d::SturmHabichtSequence`
  `PolynomialTraits_d::PrincipalSturmHabichtSequence`
  `PolynomialTraits_d::PolynomialSubresultantsWithCofactors`

## Operations

template<typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >
OutputIterator1 | operator() (Polynomial_d f, OutputIterator1 stha, OutputIterator2 co_f, OutputIterator3 co_fx)
| computes the Sturm-Habicht sequence of \( f\), and the cofactors, with respect to the outermost variable. More...
template<typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >
OutputIterator1 | operator() (Polynomial_d f, OutputIterator1 stha, OutputIterator2 co_f, OutputIterator3 co_fx, int i)
| computes the Sturm-Habicht sequence of \( f\), and the cofactors, with respect to \( x_i\). More...

## Member Function Documentation

template<typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >

OutputIterator1 PolynomialTraits_d::SturmHabichtSequenceWithCofactors::operator() | ( | Polynomial_d | f,
|  | OutputIterator1 | stha,
|  | OutputIterator2 | co_f,
|  | OutputIterator3 | co_fx
| ) |  |

computes the Sturm-Habicht sequence of \( f\), and the cofactors, with respect to the outermost variable.

Each element is of type `PolynomialTraits_d::Polynomial_d`.

template<typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >

OutputIterator1 PolynomialTraits_d::SturmHabichtSequenceWithCofactors::operator() | ( | Polynomial_d | f,
|  | OutputIterator1 | stha,
|  | OutputIterator2 | co_f,
|  | OutputIterator3 | co_fx,
|  | int | i
| ) |  |

computes the Sturm-Habicht sequence of \( f\), and the cofactors, with respect to \( x_i\).

Each element is of type `PolynomialTraits_d::Polynomial_d`.