mirror of https://github.com/CGAL/cgal
Updated Manual wrt to new functionality
This commit is contained in:
Michael Kerber
parent
b0be205def
commit
ba2bc3dddb
|
|
@ -2864,13 +2864,14 @@ Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_IsZeroAtHomogeneous.tex -te
|
|||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_MonomialRepresentation.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Move.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Permute.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Polynomial_subresultants.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Principal_sturm_habicht_sequence.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Principal_subresultants.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_PolynomialSubresultants.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_PolynomialSubresultantsWithCofactors.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_PrincipalSturmHabichtSequence.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_PrincipalSubresultants.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_SignAt.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_SignAtHomogeneous.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Sturm_habicht_sequence.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Sturm_habicht_sequence_with_cofactors.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_SturmHabichtSequence.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_SturmHabichtSequenceWithCofactors.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_Substitute.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/PolynomialTraits_d_SubstituteHomogeneous.tex -text
|
||||
Polynomial/doc_tex/Polynomial_ref/Polynomial_1.tex -text
|
||||
|
|
|
|||
|
|
@ -188,17 +188,18 @@ is not a model of \ccc{UniqueFactorizationDomain}, this is of type \ccc{CGAL::Nu
|
|||
%resultant
|
||||
\ccNestedType{Resultant}{ A model of \ccc{PolynomialTraits_d::Resultant}.}
|
||||
|
||||
% This was added by Michael Kerber, no review so far
|
||||
%\ccNestedType{Polynomial_subresultants}
|
||||
%{ A model of \ccc{PolynomialTraits_d::PolynomialSubresultant}.}
|
||||
%\ccNestedType{Principal_subresultants}
|
||||
%{ A model of \ccc{PolynomialTraits_d::PrincipalSubresultant}.}
|
||||
%\ccNestedType{Sturm_habicht_sequence}
|
||||
%{ A model of \ccc{PolynomialTraits_d::SturmHabichtSequence}.}
|
||||
%\ccNestedType{Sturm_habicht_sequence_with_cofactors}
|
||||
%{ A model of \ccc{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}.}
|
||||
%\ccNestedType{Principal_sturm_habicht_sequence}
|
||||
%{ A model of \ccc{PolynomialTraits_d::PrincipalSturmHabichtSequence}.}
|
||||
\ccNestedType{Polynomial_subresultants}
|
||||
{ Either \ccc{CGAL::Null_functor} or a model of \ccc{PolynomialTraits_d::PolynomialSubresultants}.}
|
||||
\ccNestedType{Polynomial_subresultants_with_cofactors}
|
||||
{ Either \ccc{CGAL::Null_functor} or a model of \ccc{PolynomialTraits_d::PolynomialSubresultants_with_cofactors}.}
|
||||
\ccNestedType{Principal_subresultants}
|
||||
{ Either \ccc{CGAL::Null_functor} or a model of \ccc{PolynomialTraits_d::PrincipalSubresultants}.}
|
||||
\ccNestedType{Sturm_habicht_sequence}
|
||||
{ Either \ccc{CGAL::Null_functor} or a model of \ccc{PolynomialTraits_d::SturmHabichtSequence}.}
|
||||
\ccNestedType{Sturm_habicht_sequence_with_cofactors}
|
||||
{ Either \ccc{CGAL::Null_functor} or a model of \ccc{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}.}
|
||||
\ccNestedType{Principal_sturm_habicht_sequence}
|
||||
{ Either \ccc{CGAL::Null_functor} or a model of \ccc{PolynomialTraits_d::PrincipalSturmHabichtSequence}.}
|
||||
|
||||
|
||||
\ccSeeAlso
|
||||
|
|
@ -209,4 +210,4 @@ is not a model of \ccc{UniqueFactorizationDomain}, this is of type \ccc{CGAL::Nu
|
|||
|
||||
\ccRefIdfierPage{CGAL::Polynomial_traits_d<Polynomial_d>}
|
||||
|
||||
\end{ccRefConcept}
|
||||
\end{ccRefConcept}
|
||||
|
|
|
|||
|
|
@ -0,0 +1,55 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::PolynomialSubresultants}
|
||||
\ccDefinition
|
||||
|
||||
Computes the polynomial subresultant of two polynomials $p$ and $q$ of
|
||||
type \ccc{PolynomialTraits_d::Polynomial_d} with respect to outermost variable.
|
||||
Let $p=\sum_{i=0}^n p_i t^i$ and $q\sum_{i=0}^m q_i t^i$, where $t$
|
||||
is the outermost variable.
|
||||
The $i$th subresultant (with $i=0,\ldots,\min\{n,m\}$) is defined by
|
||||
\begin{eqnarray*}
|
||||
\mathrm{Sres}_i(p,q)&=&\det \left(\begin{array}{cccccc}
|
||||
p_n & \ldots &\ldots& p_{2i-m+2}&t^{m-i-1}p \\
|
||||
&\ddots&&\vdots&\vdots\\
|
||||
&p_n&\ldots&p_{i+1}&p\\
|
||||
q_m & \ldots &\ldots & q_{2i-n+2}&t^{n-i-1}q \\
|
||||
&\ddots&&\vdots&\vdots\\
|
||||
&q_m&\ldots&q_{i+1}&q
|
||||
\end{array}\right)
|
||||
\end{eqnarray*}
|
||||
In the exceptional case that $n=m$, $\mathrm{Sres_n}$ is set to $q$.
|
||||
|
||||
The result is written in an output range, starting with the $0$th subresultant
|
||||
$\mathrm{Sres}_0(p,q)$
|
||||
(aka as the resultant of $p$ and $q$).
|
||||
|
||||
A default implementation for the case that
|
||||
\ccc{Polynomial_traits_d::Coefficient_type}
|
||||
is a model of \ccc{CGAL::Integral_domain_without_division}
|
||||
is provided by the function \ccc{CGAL::polynomial_subresultants}.
|
||||
|
||||
\ccOperations
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d p,
|
||||
Polynomial_d q,
|
||||
OutputIterator out);}
|
||||
{ computes the polynomial subresultants of $p$ and $q$,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d p,
|
||||
Polynomial_d q,
|
||||
OutputIterator out,
|
||||
int i);}
|
||||
{ computes the polynomial subresultants of $p$ and $q$,
|
||||
with respect to the variable $x_i$.}
|
||||
|
||||
%\ccHasModels
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
\ccRefIdfierPage{CGAL::polynomial_subresultants}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,56 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::PolynomialSubresultantsWithCofactors}
|
||||
\ccDefinition
|
||||
|
||||
Computes the polynomial subresultant of two polynomials $p$ and $q$ of degree
|
||||
$n$ and $m$, respectively,
|
||||
as defined in the documentation of \ccc{PolynomialTraits_d::PolynomialSubresultants}.
|
||||
Moreover, for $\mathrm{Sres}_i(p,q)$, polynomials $u_i$ and $v_i$
|
||||
with $\deg u_i\leq m-i-1$ and $\deg v_i\leq n-i-1$ are computed
|
||||
such that $\mathrm{Sres}_i(p,q)=u_i p + v_i q$. $u_i$ and $v_i$ are called
|
||||
the \emph{cofactors} of $\mathrm{Sres}_i(p,q)$.
|
||||
|
||||
The result is written in three output ranges, each of length $\min\{n,m\}+1$,
|
||||
starting with the $0$th subresultant and the corresponding cofactors.
|
||||
|
||||
A default implementation for the case that
|
||||
\ccc{Polynomial_traits_d::Coefficient_type}
|
||||
is a model of \ccc{CGAL::Integral_division}
|
||||
is provided by the function \ccc{CGAL::polynomial_subresultants_with_cofactors}.
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
\ccOperations
|
||||
\ccMethod{template< typename OutputIterator1,
|
||||
typename OutputIterator2,
|
||||
typename OutputIterator3 >
|
||||
OutputIterator1 operator()(Polynomial_d p,
|
||||
Polynomial_d q,
|
||||
OutputIterator1 sres,
|
||||
OutputIterator2 co_p,
|
||||
OutputIterator3 co_q);}
|
||||
{ computes the subresultants of $p$ and $q$, and the cofactors,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
|
||||
\ccMethod{template< typename OutputIterator1,
|
||||
typename OutputIterator2,
|
||||
typename OutputIterator3 >
|
||||
OutputIterator1 operator()(Polynomial_d p,
|
||||
Polynomial_d q,
|
||||
OutputIterator1 sres,
|
||||
OutputIterator2 co_p,
|
||||
OutputIterator3 co_q,
|
||||
int i);}
|
||||
{ computes the subresultants of $p$ and $q$, and the cofactors,
|
||||
with respect to $x_i$. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
|
||||
%\ccHasModels
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d::PolynomialSubresultants}\\
|
||||
\ccRefIdfierPage{CGAL::polynomial_subresultants_with_cofactors}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -1,33 +0,0 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::Polynomial_subresultants}
|
||||
\ccDefinition
|
||||
|
||||
Computes the polynomial subresultant of two polynomials $f$ and $g$ of
|
||||
type \ccc{PolynomialTraits_d::Polynomial_d} with respect a certain variable.
|
||||
The result is written in an output range, starting with the $0$th subresultant
|
||||
(aka as the resultant of $f$ and $g$).
|
||||
|
||||
\ccOperations
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d f,
|
||||
Polynomial_d g,
|
||||
OutputIterator out);}
|
||||
{ computes the polynomial subresultants of $f$ and $g$,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d f,
|
||||
Polynomial_d g,
|
||||
OutputIterator out,
|
||||
int i);}
|
||||
{ computes the polynomial subresultants of $f$ and $g$,
|
||||
with respect to the variable $x_i$.}
|
||||
|
||||
%\ccHasModels
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -1,7 +1,7 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::Principal_sturm_habicht_sequence}
|
||||
\begin{ccRefConcept}{PolynomialTraits_d::PrincipalSturmHabichtSequence}
|
||||
\ccDefinition
|
||||
|
||||
Computes the principal coefficients of the Sturm-Habicht sequence
|
||||
Computes the principal leading coefficients of the Sturm-Habicht sequence
|
||||
of a polynomials $f$ of type \ccc{PolynomialTraits_d::Polynomial_d}
|
||||
with respect a certain variable $x_i$.
|
||||
This means that for the $j$th Sturm-Habicht polynomial, this methods returns
|
||||
|
|
@ -11,6 +11,15 @@ Note that the degree of the $j$th Sturm-Habicht polynomial is at most $j$,
|
|||
but the principal coefficient might be zero, thus, this functor does not
|
||||
necessarily give the leading coefficient of the Sturm-Habicht polynomials
|
||||
|
||||
In case that \ccc{PolynomialTraits_d::Coefficient_type} is \ccc{RealEmbeddable}, the function \ccc{CGAL::stha_count_number_of_real_roots} can be used
|
||||
on the resulting sequence to count the number of distinct real roots of
|
||||
the polynomial~$f$.
|
||||
|
||||
A default implementation for the case that
|
||||
\ccc{Polynomial_traits_d::Coefficient_type}
|
||||
is a model of \ccc{CGAL::Integral_domain_without_division}
|
||||
is provided by the function \ccc{CGAL::principal_sturm_habicht_sequence}.
|
||||
|
||||
\ccOperations
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d f,
|
||||
|
|
@ -34,5 +43,7 @@ necessarily give the leading coefficient of the Sturm-Habicht polynomials
|
|||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
\ccRefIdfierPage{CGAL::principal_sturm_habicht_sequence}\\
|
||||
\ccRefIdfierPage{CGAL::stha_count_number_of_real_roots}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,48 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::PrincipalSubresultants}
|
||||
\ccDefinition
|
||||
|
||||
Computes the principal subresultant of two polynomials $p$ and $q$ of
|
||||
type \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
with respect to the outermost variable.
|
||||
The $i$th principal subresultant, $\mathrm{sres}_i(p,q)$,
|
||||
is defined as the coefficient at $t^i$ of the $i$th polynomial
|
||||
subresultant $\mathrm{Sres}_i(p,q)$. Thus, it is either the leading
|
||||
coefficient of $\mathrm{Sres}_i$, or zero in the case where its degree is
|
||||
below $i$.
|
||||
|
||||
The result is written in an output range, starting with the $0$th subresultant
|
||||
$\mathrm{sres}_0(p,q)$
|
||||
,aka as the resultant of $p$ and $q$.
|
||||
(Note that $\mathrm{sres}_0(p,q)=\mathrm{Sres}_0(p,q)$ by definition)
|
||||
|
||||
A default implementation for the case that
|
||||
\ccc{Polynomial_traits_d::Coefficient_type}
|
||||
is a model of \ccc{CGAL::Integral_domain_without_division}
|
||||
is provided by the function \ccc{CGAL::principal_subresultants}.
|
||||
|
||||
\ccOperations
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d p,
|
||||
Polynomial_d q,
|
||||
OutputIterator out);}
|
||||
{ computes the principal subresultants of $p$ and $q$,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Coefficient_type}.}
|
||||
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d p,
|
||||
Polynomial_d q,
|
||||
OutputIterator out,
|
||||
int i);}
|
||||
{ computes the principal subresultants of $p$ and $q$,
|
||||
with respect to the variable $x_i$.}
|
||||
|
||||
%\ccHasModels
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
\ccRefIdfierPage{CGAL::principal_subresultants}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -1,43 +0,0 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::Principal_subresultants}
|
||||
\ccDefinition
|
||||
|
||||
Computes the principal subresultant of two polynomials $f$ and $g$ of
|
||||
type \ccc{PolynomialTraits_d::Polynomial_d}
|
||||
with respect a certain variable $x_i$.
|
||||
The principal subresultants are also known as {\it scalar} subresultants.
|
||||
|
||||
The $j$th such principal subresultant is defined to be the coefficient
|
||||
of $x_i^j$ in the $j$-th subresultant polynomial of $f$ and $g$.
|
||||
Since the degree of the $j$-th subresultant polynomial is at most $j$,
|
||||
this principal coefficients are sometimes called the
|
||||
{\tt formal leading coefficients} (``formal'' because they might vanish).
|
||||
|
||||
The result is written in an output range,
|
||||
starting with the $0$th principal subresultant
|
||||
(aka as the resultant of $f$ and $g$).
|
||||
|
||||
\ccOperations
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d f,
|
||||
Polynomial_d g,
|
||||
OutputIterator out);}
|
||||
{ computes the principal subresultants of $f$ and $g$,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Coefficient_type}.}
|
||||
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
OutputIterator operator()(Polynomial_d f,
|
||||
Polynomial_d g,
|
||||
OutputIterator out,
|
||||
int i);}
|
||||
{ computes the principal subresultants of $f$ and $g$,
|
||||
with respect to the variable $x_i$.}
|
||||
|
||||
%\ccHasModels
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -1,18 +1,34 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::Sturm_habicht_sequence}
|
||||
\begin{ccRefConcept}{PolynomialTraits_d::SturmHabichtSequence}
|
||||
\ccDefinition
|
||||
|
||||
Computes the Sturm-Habicht sequence of a polynomials $f$ of type
|
||||
Computes the Sturm-Habicht sequence
|
||||
(aka the signed subresultant sequence)
|
||||
of a polynomials $f$ of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d} with respect a certain variable $x_i$.
|
||||
The Sturm-Habicht sequence is similar to the polynomial subresultant sequence
|
||||
of $f$ and its derivative with respect to $x_i$, but some entries
|
||||
are multiplied by $-1$. With that, it becomes possible to count
|
||||
the number of real roots of (univariate) polynomials.
|
||||
Sturm-Habicht sequences are also sometimes
|
||||
called signed subresultant sequences in the literature.
|
||||
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 {\it $k$th Sturm-Habicht polynomial}
|
||||
of $f$ is defined as:
|
||||
$$\mathrm{Stha}_k(f)=\left\{\begin{array}{cc}
|
||||
f & \textrm{if\ } k=n\\
|
||||
f' & \textrm{if\ } k=n-1\\
|
||||
\delta_{n-k-1}\mathrm{Sres}_k(f,f') &\textrm{if\ } 0\leq k\leq n-2
|
||||
\end{array}\color{white}\right\}\color{black}$$
|
||||
where $\mathrm{Sres}_k(f,f')$ is defined
|
||||
as in the concept \ccc{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)
|
||||
|
||||
A default implementation for the case that
|
||||
\ccc{Polynomial_traits_d::Coefficient_type}
|
||||
is a model of \ccc{CGAL::Integral_domain_without_division}
|
||||
is provided by the function \ccc{CGAL::sturm_habicht_sequence}.
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
\ccOperations
|
||||
\ccMethod{template<typename OutputIterator>
|
||||
|
|
@ -35,5 +51,6 @@ the discriminant of $f$ up to a multiple of the leading coefficient)
|
|||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
\ccRefIdfierPage{CGAL::sturm_habicht_sequence}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -1,6 +1,22 @@
|
|||
\begin{ccRefConcept}{PolynomialTraits_d::Sturm_habicht_sequence_with_cofactors}
|
||||
\begin{ccRefConcept}{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}
|
||||
\ccDefinition
|
||||
|
||||
Computes the Sturm-Habicht polynomials of a polynomial $f$ of degree $n$,
|
||||
as defined in the documentation of \ccc{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 \emph{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.
|
||||
|
||||
A default implementation for the case that
|
||||
\ccc{Polynomial_traits_d::Coefficient_type}
|
||||
is a model of \ccc{CGAL::Integral_division}
|
||||
is provided by the function \ccc{CGAL::sturm_habicht_sequence_with_cofactors}.
|
||||
|
||||
Computes the Sturm-Habicht sequence of a polynomials $f$ of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d} with respect a certain variable $x_i$.
|
||||
Additionally, it computes two ranges of cofactors, {\tt co\_f} and {\tt co\_fx}
|
||||
|
|
@ -37,5 +53,7 @@ with the property that {\tt stha[i] == co\_f[i] f + co\_fx[i] f'}.
|
|||
|
||||
\ccRefIdfierPage{Polynomial_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d}\\
|
||||
\ccRefIdfierPage{PolynomialTraits_d::SturmHabichtSequence}\\
|
||||
\ccRefIdfierPage{CGAL::sturm_habicht_sequence_with_cofactors}\\
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -80,11 +80,12 @@
|
|||
%\ccRefConceptPage{PolynomialTraits_d::ScaleDown}\\
|
||||
|
||||
\ccRefConceptPage{PolynomialTraits_d::Resultant}\\
|
||||
%\ccRefConceptPage{PolynomialTraits_d::Polynomial_subresultants}\\
|
||||
%\ccRefConceptPage{PolynomialTraits_d::Principal_subresultants}\\
|
||||
%\ccRefConceptPage{PolynomialTraits_d::Sturm_habicht_sequence}\\
|
||||
%\ccRefConceptPage{PolynomialTraits_d::Sturm_habicht_sequence_with_cofactors}\\
|
||||
%\ccRefConceptPage{PolynomialTraits_d::Principal_sturm_habicht_sequence}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PolynomialSubresultants}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PolynomialSubresultantsWithCofactors}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PrincipalSubresultants}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::SturmHabichtSequence}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PrincipalSturmHabichtSequence}\\
|
||||
|
||||
|
||||
\subsubsection*{Classes}
|
||||
|
|
@ -152,8 +153,10 @@
|
|||
%\ccRefIdfierPage{CGAL::ScaleDown}\\
|
||||
|
||||
\ccRefIdfierPage{CGAL::resultant}\\
|
||||
%\ccRefIdfierPage{CGAL::Polynomial_subresultants}\\
|
||||
%\ccRefIdfierPage{CGAL::Principal_subresultants}\\
|
||||
%\ccRefIdfierPage{CGAL::Sturm_habicht_sequence}\\
|
||||
%\ccRefIdfierPage{CGAL::Sturm_habicht_sequence_with_cofactors}\\
|
||||
%\ccRefIdfierPage{CGAL::Principal_sturm_habicht_sequence}\\
|
||||
\ccRefIdfierPage{CGAL::polynomial_subresultants}\\
|
||||
\ccRefIdfierPage{CGAL::polynomial_subresultants_with_cofactors}\\
|
||||
\ccRefIdfierPage{CGAL::principal_subresultants}\\
|
||||
\ccRefIdfierPage{CGAL::sturm_habicht_sequence}\\
|
||||
\ccRefIdfierPage{CGAL::sturm_habicht_sequence_with_cofactors}\\
|
||||
\ccRefIdfierPage{CGAL::principal_sturm_habicht_sequence}\\
|
||||
\ccRefIdfierPage{CGAL::stha_count_number_of_real_roots}\\
|
||||
|
|
|
|||
|
|
@ -71,12 +71,12 @@
|
|||
|
||||
\input{Polynomial_ref/PolynomialTraits_d_Resultant.tex}
|
||||
|
||||
%// This was added by Michael Kerber, missing review
|
||||
%%\input{Polynomial_ref/PolynomialTraits_d_Polynomial_subresultants.tex}
|
||||
%%\input{Polynomial_ref/PolynomialTraits_d_Principal_subresultants.tex}
|
||||
%%\input{Polynomial_ref/PolynomialTraits_d_Sturm_habicht_sequence.tex}
|
||||
%%\input{Polynomial_ref/PolynomialTraits_d_Sturm_habicht_sequence_with_cofactors.tex}
|
||||
%%\input{Polynomial_ref/PolynomialTraits_d_Principal_sturm_habicht_sequence.tex}
|
||||
\input{Polynomial_ref/PolynomialTraits_d_PolynomialSubresultants.tex}
|
||||
\input{Polynomial_ref/PolynomialTraits_d_PolynomialSubresultantsWithCofactors.tex}
|
||||
\input{Polynomial_ref/PolynomialTraits_d_PrincipalSubresultants.tex}
|
||||
\input{Polynomial_ref/PolynomialTraits_d_SturmHabichtSequence.tex}
|
||||
\input{Polynomial_ref/PolynomialTraits_d_SturmHabichtSequenceWithCofactors.tex}
|
||||
\input{Polynomial_ref/PolynomialTraits_d_PrincipalSturmHabichtSequence.tex}
|
||||
|
||||
\input{Polynomial_ref/Polynomial.tex}
|
||||
\input{Polynomial_ref/Polynomial_traits_d.tex}
|
||||
|
|
|
|||
|
|
@ -802,3 +802,202 @@ Adapts \ccc{Polynomial_traits_d<Polynomial_d>::Resultant}.
|
|||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::Resultant}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
% polynomial_subresultants
|
||||
\begin{ccRefFunction}{polynomial_subresultants}
|
||||
\ccDefinition
|
||||
|
||||
Computes the polynomial subresultants
|
||||
of two polynomials, defined as in the concept
|
||||
\ccc{Polynomial_traits_d::PolynomialSubresultants},
|
||||
if \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
is at least a model of \ccc{CGAL::Integral_domain_without_division}.
|
||||
|
||||
The computation method depends on the algebraic category of
|
||||
\ccc{PolynomialTraits_d::Coefficient_type}.
|
||||
if it models \ccc{CGAL::Integral_division},
|
||||
the subresultants are computed
|
||||
by a pseudo-division approach~\cite{ducos-optimizations}.
|
||||
If they only model a
|
||||
\ccc{CGAL::Integral_domain_without_division}, the subresultants
|
||||
are computed by evaluating the determinants directly, avoiding divisions
|
||||
completely~\cite{kerber-division}.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{template<typename Polynomial_traits_d,typename OutputIterator>
|
||||
OutputIterator polynomial_subresultants
|
||||
(typename Polynomial_traits_d::Polynomial_d p,
|
||||
typename Polynomial_traits_d::Polynomial_d q,
|
||||
OutputIterator out);}
|
||||
{ computes the polynomial subresultants of $p$ and $q$,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PolynomialSubresultants}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
% polynomial_subresultants_with_cofactors
|
||||
\begin{ccRefFunction}{polynomial_subresultants_with_cofactors}
|
||||
\ccDefinition
|
||||
|
||||
Computes the polynomial subresultants
|
||||
of two polynomials and their cofactors, defined as in the concept
|
||||
\ccc{Polynomial_traits_d::PolynomialSubresultantsWithCofactors},
|
||||
if \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
is at least a model of \ccc{CGAL::Integral_division}.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{template<typename Polynomial_traits_d,
|
||||
typename OutputIterator1,
|
||||
typename OutputIterator2,
|
||||
typename OutputIterator3>
|
||||
OutputIterator1 polynomial_subresultants_with_cofactors
|
||||
(typename Polynomial_traits_d::Polynomial_d p,
|
||||
typename Polynomial_traits_d::Polynomial_d q,
|
||||
OutputIterator1 sres_out,
|
||||
OutputIterator2 coP_out,
|
||||
OutputIterator3 coQ_out);}
|
||||
{ computes the polynomial subresultants of $p$ and $q$,
|
||||
\ccc{sres_out}, with respect to the outermost variable, and
|
||||
the cofactors for $P$, \ccc{coP_out} and $Q$, \ccc{coQ_out}.
|
||||
The elements of each output range are of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PolynomialSubresultantsWithCofactors}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
|
||||
% principal_subresultants
|
||||
\begin{ccRefFunction}{principal_subresultants}
|
||||
\ccDefinition
|
||||
|
||||
Computes the principal subresultants
|
||||
of two polynomials, defined as in the concept
|
||||
\ccc{Polynomial_traits_d::PrincipalSubresultants},
|
||||
if \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
is at least a model of \ccc{CGAL::Integral_domain_without_division}.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{template<typename Polynomial_traits_d,typename OutputIterator>
|
||||
OutputIterator principal_subresultants
|
||||
(typename Polynomial_traits_d::Polynomial_d p,
|
||||
typename Polynomial_traits_d::Polynomial_d q,
|
||||
OutputIterator out);}
|
||||
{ computes the principal subresultants of $p$ and $q$,
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Coefficient_type}.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PrincipalSubresultants}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
% sturm_habicht_sequence
|
||||
\begin{ccRefFunction}{sturm_habicht_sequence}
|
||||
\ccDefinition
|
||||
|
||||
Computes the Sturm-Habicht-polynomials
|
||||
of a polynomial, defined as in the concept
|
||||
\ccc{Polynomial_traits_d::SturmHabichtSequence},
|
||||
if \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
is at least a model of \ccc{CGAL::Integral_domain_without_division}.
|
||||
|
||||
The computation works simply by calling \ccc{CGAL::polynomial_subresultants},
|
||||
and adjusting the signs.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{template<typename Polynomial_traits_d,typename OutputIterator> OutputIterator
|
||||
sturm_habicht_sequence(typename Polynomial_traits_d::Polynomial_d f,
|
||||
OutputIterator out);}
|
||||
{ computes the Sturm-Habicht-sequence of $f$
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::SturmHabichtSequence}\\
|
||||
\ccRefIdfierPage{CGAL::polynomial_subresultants}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
% sturm_habicht_sequence_with_cofactors
|
||||
\begin{ccRefFunction}{sturm_habicht_sequence_with_cofactors}
|
||||
\ccDefinition
|
||||
|
||||
Computes the Sturm-Habicht-sequence of a polynomial
|
||||
and its cofactors, defined as in the concept
|
||||
\ccc{Polynomial_traits_d::SturmHabichtSequenceWithCofactors},
|
||||
if \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
is at least a model of \ccc{CGAL::Integral_division}.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{template<typename Polynomial_traits_d,
|
||||
typename OutputIterator1,
|
||||
typename OutputIterator2,
|
||||
typename OutputIterator3>
|
||||
OutputIterator1
|
||||
sturm_habicht_sequence_with_cofactors
|
||||
(typename Polynomial_traits_d::Polynomial_d f,
|
||||
OutputIterator1 stha_out,
|
||||
OutputIterator2 cof_out,
|
||||
OutputIterator3 cofx_out);}
|
||||
{ computes the Sturm-Habicht sequence of $f$
|
||||
\ccc{stha_out}, with respect to the outermost variable, and
|
||||
the cofactors for $f$, \ccc{cof_out} and $f'$, \ccc{cofx_out}.
|
||||
The elements of each output range are of type
|
||||
\ccc{PolynomialTraits_d::Polynomial_d}.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
|
||||
% principal_subresultants
|
||||
\begin{ccRefFunction}{principal_sturm_habicht_sequence}
|
||||
\ccDefinition
|
||||
|
||||
Computes the principal Sturm-Habicht coefficients
|
||||
of a polynomial, defined as in the concept
|
||||
\ccc{Polynomial_traits_d::PrincipalSturmHabichtSequence},
|
||||
if \ccc{PolynomialTraits_d::Coefficient_type}
|
||||
is at least a model of \ccc{CGAL::Integral_domain_without_division}.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{ template <typename Polynomial_traits_d,typename OutputIterator> inline
|
||||
OutputIterator
|
||||
principal_sturm_habicht_sequence
|
||||
(typename Polynomial_traits_d::Polynomial_d A,
|
||||
OutputIterator out);}
|
||||
{ computes the principal Sturm-Habicht coefficients of $f$
|
||||
with respect to the outermost variable. Each element is of type
|
||||
\ccc{PolynomialTraits_d::Coefficient_type}.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PrincipalSturmHabichtSequence}\\
|
||||
\end{ccRefFunction}
|
||||
|
||||
% stha_count_number_of_real_roots
|
||||
\begin{ccRefFunction}{stha_count_number_of_real_roots}
|
||||
\ccDefinition
|
||||
|
||||
Given a range that is interpreted as the principal Sturm-Habicht coefficients
|
||||
of a polynomial $f$, the function computes
|
||||
$$m:=\# \{\alpha\in\mathbb{R}\mid f(\alpha)=0\}$$
|
||||
that is, the number of distinct real roots of $f$.
|
||||
|
||||
The value type of the iterator range must be a model of \ccc{RealEmbeddable}.
|
||||
|
||||
\ccOperations
|
||||
\ccFunction{ template<typename InputIterator>
|
||||
int stha_count_number_of_real_roots(InputIterator start,InputIterator end);}
|
||||
{ computes the number of distinct real roots of $f$ whose principal Sturm-Habicht coefficients are passed by the iterator range.}
|
||||
\ccSeeAlso
|
||||
\ccRefConceptPage{Polynomial_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d}\\
|
||||
\ccRefConceptPage{PolynomialTraits_d::PrincipalSturmHabichtSequence}\\
|
||||
\end{ccRefFunction}
|
||||
|
|
|
|||
Loading…
Reference in New Issue