mirror of https://github.com/CGAL/cgal
78 lines
2.4 KiB
TeX
78 lines
2.4 KiB
TeX
\begin{ccRefConcept}{PolynomialTraits_d::PolynomialSubresultants}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the polynomial subresultant of two polynomials $p$ and $q$ of
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type \ccc{PolynomialTraits_d::Polynomial_d} with respect to outermost variable.
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Let
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$p=\ccSum{i=0,\ldots,n}{} p_i x^i$ and
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$q=\ccSum{i=0,\ldots,m}{} q_i x^i$, where $x$
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is the outermost variable.
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The $i$-th subresultant (with $i=0,\ldots,\min\{n,m\}$) is defined by
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\begin{ccTexOnly}
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\begin{eqnarray*}
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\mathrm{Sres}_i(p,q)&=&\det \left(\begin{array}{ccccccc}
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p_n & \ldots &&\ldots& p_{2i-m+2}&x^{m-i-1}p \\
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&\ddots&&&\vdots&\vdots\\
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&&p_n&\ldots&p_{i+1}&p\\
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q_m & \ldots &&\ldots & q_{2i-n+2}&x^{n-i-1}q \\
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&\ddots&&&\vdots&\vdots\\
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&&q_m&\ldots&q_{i+1}&q
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\end{array}\right)
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\end{eqnarray*}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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<CENTER>
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<IMG BORDER=0 SRC="./subresultant_def.gif" ALIGN=middle ALT="Subresultants">
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</CENTER>
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\end{ccHtmlOnly}
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where $p_i$ and $q_i$ are set to zero if $i<0$.
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In the case that $n=m$, $\mathrm{Sres_n}$ is set to $q$.
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The result is written in an output range, starting with the $0$-th subresultant
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$\mathrm{Sres}_0(p,q)$
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(aka as the resultant of $p$ and $q$).
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\ccRefines
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\ccc{AdaptableBinaryFunction}\\
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\ccc{CopyConstructible}\\
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\ccc{DefaultConstructible}\\
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\ccCreationVariable{fo}
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\ccOperations
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\ccMethod{template<typename OutputIterator>
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OutputIterator operator()(Polynomial_d p,
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Polynomial_d q,
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OutputIterator out);}
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{ computes the polynomial subresultants of $p$ and $q$,
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with respect to the outermost variable. Each element is of type
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\ccc{PolynomialTraits_d::Polynomial_d}.}
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\ccMethod{template<typename OutputIterator>
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OutputIterator operator()(Polynomial_d p,
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Polynomial_d q,
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OutputIterator out,
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int i);}
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{ computes the polynomial subresultants of $p$ and $q$,
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with respect to the variable $x_i$.}
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%\ccHasModels
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\ccSeeAlso
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\ccRefIdfierPage{Polynomial_d}\\
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\ccRefIdfierPage{PolynomialTraits_d}\\
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\ccRefIdfierPage{PolynomialTraits_d::Resultant}\\
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\ccRefIdfierPage{PolynomialTraits_d::PrincipalSubresultants}\\
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\ccRefIdfierPage{PolynomialTraits_d::PolynomialSubresultantsWithCofactors}\\
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\ccRefIdfierPage{PolynomialTraits_d::SturmHabichtSequence}\\
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\end{ccRefConcept}
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