CGAL 6.1.1 - Polynomial

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PolynomialTraits_d::SturmHabichtSequence Concept Reference
Reference Manual » Concepts

## Definition

Computes the Sturm-Habicht sequence (aka the signed subresultant sequence) of a polynomial \( f\) of type `PolynomialTraits_d::Polynomial_d` with respect to a certain variable \( x_i\). The Sturm-Habicht sequence is similar to the polynomial subresultant sequence of \( f\) and its derivative \( f':=\frac{\partial f}{\partial x_i}\) with respect to \( x_i\). The implementation is based on the following definition:

Let \( n:=\deg f\) and \( \delta_k:=(-1)^{k(k+1)/2}\). For \( k\in\{0,\ldots,n\}\), the \( k\)-th Sturm-Habicht polynomial of \( f\) is defined as:

\[ \mathrm{Stha}_{k}(f) = \left \{ \begin{array}{cll} f & \text{if} & k = n \\ f' & \text{if} & k = n - 1 \\ \delta_{n - k - 1}\mathrm{Sres}_{k}(f,f') & \text{if} & 0 \leq k \leq n - 2 \end{array} \right . \]

where \( \mathrm{Sres}_k(f,f')\) is defined as in the concept `PolynomialTraits_d::PolynomialSubresultants`.

The result is written in an output range, starting with the \( 0\)-th Sturm-Habicht polynomial (which is equal to the discriminant of \( f\) up to a multiple of the leading coefficient).

Note
  This functor is optional.

Refines
  `AdaptableBinaryFunction`
  `CopyConstructible`
  `DefaultConstructible`

See also
  `Polynomial_d`
  `PolynomialTraits_d`
  `PolynomialTraits_d::Resultant`
  `PolynomialTraits_d::PrincipalSturmHabichtSequence`
  `PolynomialTraits_d::SturmHabichtSequenceWithCofactors`
  `PolynomialTraits_d::PolynomialSubresultants`

## Operations
template<typename OutputIterator >
OutputIterator | operator() (Polynomial_d f, OutputIterator out)
| computes the Sturm-Habicht sequence of \( f\), with respect to the outermost variable.
template<typename OutputIterator >
OutputIterator | operator() (Polynomial_d f, OutputIterator out, int i)
| computes the Sturm-Habicht sequence of \( f\) with respect to the variable \( x_i\).

## Member Function Documentation

## ◆ operator()()

template<typename OutputIterator >

OutputIterator PolynomialTraits_d::SturmHabichtSequence::operator() | ( | Polynomial_d | f,
|  | OutputIterator | out
| ) |  |

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

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