Reference documentation for deal.II version 9.3.3
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
Namespaces | Classes | Functions
Polynomials Namespace Reference


namespace  internal


class  HermiteInterpolation
class  HermiteLikeInterpolation
class  Hierarchical
class  LagrangeEquidistant
class  Legendre
class  Lobatto
class  Monomial
class  PiecewisePolynomial
class  Polynomial


std::vector< Polynomial< double > > generate_complete_Lagrange_basis (const std::vector< Point< 1 > > &points)
template<typename Number >
Number jacobi_polynomial_value (const unsigned int degree, const int alpha, const int beta, const Number x)
template<typename Number >
std::vector< Number > jacobi_polynomial_roots (const unsigned int degree, const int alpha, const int beta)
std::vector< PiecewisePolynomial< double > > generate_complete_Lagrange_basis_on_subdivisions (const unsigned int n_subdivisions, const unsigned int base_degree)

Detailed Description

A namespace in which classes relating to the description of 1d polynomial spaces are declared.

Function Documentation

◆ generate_complete_Lagrange_basis()

std::vector< Polynomial< double > > Polynomials::generate_complete_Lagrange_basis ( const std::vector< Point< 1 > > &  points)

Given a set of points along the real axis, this function returns all Lagrange polynomials for interpolation of these points. The number of polynomials is equal to the number of points and the maximum degree is one less.

Definition at line 701 of file

◆ jacobi_polynomial_value()

template<typename Number >
Number Polynomials::jacobi_polynomial_value ( const unsigned int  degree,
const int  alpha,
const int  beta,
const Number  x 

Definition at line 976 of file polynomial.h.

◆ jacobi_polynomial_roots()

template<typename Number >
std::vector< Number > Polynomials::jacobi_polynomial_roots ( const unsigned int  degree,
const int  alpha,
const int  beta 

Compute the roots of the Jacobi polynomials on the unit interval \([0, 1]\) of the given degree. These roots are used in several places inside the deal.II library, such as the Gauss-Lobatto quadrature formula or for the Hermite-like interpolation.

The algorithm uses a Newton algorithm, using the zeros of the Chebyshev polynomials as an initial guess. This code has been tested for alpha and beta equal to zero (Legendre case), one (Gauss-Lobatto case) as well as two, so be careful when using it for other values as the Newton iteration might or might not converge.

Definition at line 1017 of file polynomial.h.

◆ generate_complete_Lagrange_basis_on_subdivisions()

std::vector< PiecewisePolynomial< double > > Polynomials::generate_complete_Lagrange_basis_on_subdivisions ( const unsigned int  n_subdivisions,
const unsigned int  base_degree 

Generates a complete Lagrange basis on a subdivision of the unit interval in smaller intervals for a given degree on the subintervals and number of intervals.

Definition at line 134 of file