Reference documentation for deal.II version GIT d20b776824 2022-09-30 19:00:01+00:00
Polynomials Namespace Reference

internal

## Classes

class  Polynomial

class  Monomial

class  LagrangeEquidistant

class  Legendre

class  Lobatto

class  Hierarchical

class  HermiteInterpolation

class  HermiteLikeInterpolation

class  PiecewisePolynomial

## Functions

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)

std::vector< PiecewisePolynomial< double > > generate_complete_linear_basis_on_subdivisions (const std::vector< Point< 1 >> &points)

## Detailed Description

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

## ◆ 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 702 of file polynomial.cc.

## ◆ 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 1038 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 1079 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 191 of file polynomials_piecewise.cc.

## ◆ generate_complete_linear_basis_on_subdivisions()

 std::vector< PiecewisePolynomial< double > > Polynomials::generate_complete_linear_basis_on_subdivisions ( const std::vector< Point< 1 >> & points )

Generates a complete linear basis on a subdivision of the unit interval in smaller intervals for a given vector of points.

Definition at line 216 of file polynomials_piecewise.cc.