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This commit is contained in:
Andreas Fabri 2012-10-30 13:11:33 +00:00
parent 7736798599
commit c4572e8cb0
1 changed files with 13 additions and 13 deletions
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26
Polynomial/doc/Polynomial/Polynomial.txt
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@ -78,7 +78,7 @@ structure, that is, the ring operations \f$ \{+, -, \cdot\}\f$ are
provided due to the fact that `Polynomial_d` refines at least the
concept `IntegralDomainWithoutDivision`.
However, a model of `Polynomial_d` has to be accompanied by a traits class
`CGAL::Polynomial_traits_d<Polynomial_d>` being a model of
`Polynomial_traits_d<Polynomial_d>` being a model of
`PolynomialTraits_d`.
This traits class provides all further functionalities on polynomials.
@ -97,12 +97,12 @@ the polynomial as an element of \f$ R [x_0,\dots,x_{d-1}]\f$.
According to these two different views the traits class is required to provide
two different coefficient types:
<UL>
<LI>`CGAL::Polynomial_traits_d::Coefficient_type` representing \f$ R[x_0,\dots,x_{d-2}]\f$.
<LI>`CGAL::Polynomial_traits_d::Innermost_coefficient_type` representing the base ring \f$ R\f$.
<LI>`Polynomial_traits_d::Coefficient_type` representing \f$ R[x_0,\dots,x_{d-2}]\f$.
<LI>`Polynomial_traits_d::Innermost_coefficient_type` representing the base ring \f$ R\f$.
</UL>
Another important type which is introduced by this package is
`CGAL::Exponent_vector`.
`Exponent_vector`.
It is derived from `std::vector<int>` and used to identify multivariate
monomials. For instance the exponent vector containing the sequence
\f$ [3,2,4]\f$ corresponds to the trivariate monomial \f$ x_0^3x_1^2x_2^4\f$.
@ -120,15 +120,15 @@ refines `IntegralDomainWithoutDivision` which in turn refines
constant polynomials.
In general a polynomial is constructed using the functor
`CGAL::Polynomial_traits_d::Construct_polynomial` a model of
`Polynomial_traits_d::Construct_polynomial` a model of
`PolynomialTraits_d::ConstructPolynomial`. Basically there are two options:
<UL>
<LI>The polynomial is constructed from an iterator range with value type
`CGAL::Polynomial_traits_d::Coefficient_type`,
`Polynomial_traits_d::Coefficient_type`,
where the `begin` iterator refers to the constant term
(constant with respect to the outermost variable).
<LI>The polynomial is constructed from an iterator range with value type
`std::pair< CGAL::Exponent_vector, CGAL::Polynomial_traits_d::Innermost_coefficient_type>`,
`std::pair<Exponent_vector, Polynomial_traits_d::Innermost_coefficient_type>`,
where each pair defines the coefficient for the monomial defined by
the exponent vector.
</UL>
@ -137,7 +137,7 @@ However, in some cases it might be more convenient to just construct
the polynomials representing the different variables and to obtain the
final polynomial using
algebraic expressions. The most elegant way to construct a certain variable is
`CGAL::Polynomial_traits_d::Shift` being a model of
`Polynomial_traits_d::Shift` being a model of
`PolynomialTraits_d::Shift`.
## Example ##
@ -236,7 +236,7 @@ structure of the innermost coefficient, for instance, a gcd is available
if and only if the innermost coefficient is a `Field` or a
`UniqueFactorizationDomain`. Hence, we can not provide a \f$ gcd\f$ if the
innermost coefficient is just an `IntegralDomain` since it is simply
not well defined\footnote{An example for such a number type is the template `CGAL::Sqrt_extension<NT,ROOT>` representing an algebraic extension of degree two. This is just an `IntegralDomain` if NT is not a `Field`. }.
not well defined\footnote{An example for such a number type is the template `Sqrt_extension<NT,ROOT>` representing an algebraic extension of degree two. This is just an `IntegralDomain` if NT is not a `Field`. }.
However, if we would consider the polynomial over the quotient field of the
integral domain the \f$ gcd\f$ would be well defined. The only problem is
that the result can not be represented over the ring since it contains
@ -365,7 +365,7 @@ reference pages.
The principal Sturm-Habicht sequence allows to count the number of
real roots of a polynomial using the function
<UL>
<LI>\ref ::CGAL::number_of_real_roots.
<LI>number_of_real_roots().
</UL>
As input, this function requires an iterator range that represents
the principal Sturm-Habicht coefficients.
@ -379,7 +379,7 @@ and its (principal) Sturm-Habicht coefficients
Sturm-Habicht sequence for \f$ f_{t_0}\f$. The same holds for (principal)
subresultants. Thus, it is enough in such situations to compute
the sequence once for the parameter \f$ t\f$, and call
`CGAL::number_of_real_roots` for each specialized parameter
`number_of_real_roots()` for each specialized parameter
value.
We finally remark that computing subresultants and Sturm-Habicht sequences
@ -402,7 +402,7 @@ of a polynomial using its principal Sturm-Habicht coefficients.
This package is the result of the integration process of the NumeriX library
of <span class="textsc">Exacus</span> \cite beh-eeeafcs-05 into \cgal.
The class `CGAL::Polynomial<Coeff>` had been started by Michael Seel within
The class `Polynomial<Coeff>` had been started by Michael Seel within
CGAL as part of the Nef_2 package. As part of the <span class="textsc">Exacus</span> project
it got significantly improved by Arno Eigenwillig and Michael Hemmer.
@ -410,7 +410,7 @@ However, due to the recursive definition the class was rather restricted to the
univariate view. Moreover, it is clear that depending on the context
other classes that are symmetric in all variables or dedicated
for sparse polynomials may be more efficient. As a consequence this package
introduced the `CGAL::Polynomial_traits_d<Polynomial_d>` giving also
introduced the `Polynomial_traits_d<Polynomial_d>` giving also
the symmetric view on polynomials and the opportunity to introduce and use
other classes representing polynomials within \cgal.