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ROL
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#include <iostream>#include <iomanip>#include <cmath>#include "ROL_StdVector.hpp"#include "Teuchos_LAPACK.hpp"#include "LinearAlgebra.hpp"Go to the source code of this file.
Functions | |
| template<class Real > | |
| void | rec_jacobi (ROL::Ptr< Teuchos::LAPACK< int, Real > > lapack, const double alpha, const double beta, std::vector< Real > &a, std::vector< Real > &b) |
| Generate the Jacobi polynomial recursion coeffcients \(a_k,b_k\). More... | |
| template<class Real > | |
| void | vandermonde (const std::vector< Real > &a, const std::vector< Real > &b, const std::vector< Real > &x, std::vector< Real > &V) |
| Construct the generalized Vandermonde matrix (in column stacked form) based upon the recurrence coefficients (a,b) on the grid x. More... | |
| template<class Real > | |
| void | gauss (ROL::Ptr< Teuchos::LAPACK< int, Real > > lapack, const std::vector< Real > &a, const std::vector< Real > &b, std::vector< Real > &x, std::vector< Real > &w) |
| Compute the Gauss quadrature nodes and weights for the polynomials generated by the recurrence coefficients. More... | |
| template<class Real > | |
| void | rec_lobatto (ROL::Ptr< Teuchos::LAPACK< int, Real > > const lapack, const double xl1, const double xl2, std::vector< Real > &a, std::vector< Real > &b) |
| Modify the given recurrence coefficients so that the set of zeros of the maximal order polynomial include the two prescribed points. More... | |
| void rec_jacobi | ( | ROL::Ptr< Teuchos::LAPACK< int, Real > > | lapack, |
| const double | alpha, | ||
| const double | beta, | ||
| std::vector< Real > & | a, | ||
| std::vector< Real > & | b | ||
| ) |
Generate the Jacobi polynomial recursion coeffcients \(a_k,b_k\).
The Jacobi polynomials satisfy the recurrence relation
\[ P^{(\alpha,\beta)}_{k+1}(x) = (x-a_k)P_{k}^{(\alpha,\beta)}(x) - b_k P_{k-1}^{(\alpha,\beta)}(x) \]
and form an orthogonal basis on \([-1,1]\) with respect to the weight function \(w(x)=(1-x)^\alpha(1+x)^\beta\).
| [in] | lapack | is a pointer to the Teuchos::LAPACK interface |
| [in] | alpha | is a parameter that defines the weight function |
| [in] | beta | is a parameter that defines the weight function |
| [out] | ap | is a vector of recursion coefficients |
| [out] | bp | is a vector of recursion coefficients |
Adapted from the MATLAB code by Dirk Laurie and Walter Gautschi http://www.cs.purdue.edu/archives/2002/wxg/codes/r_jacobi.m
Definition at line 38 of file OrthogonalPolynomials.hpp.
Referenced by NodalBasis< Real >::NodalBasis().
| void vandermonde | ( | const std::vector< Real > & | a, |
| const std::vector< Real > & | b, | ||
| const std::vector< Real > & | x, | ||
| std::vector< Real > & | V | ||
| ) |
Construct the generalized Vandermonde matrix (in column stacked form) based upon the recurrence coefficients (a,b) on the grid x.
| [in] | a | vector of recursion coefficients |
| [in] | b | vector of recursion coefficients |
| [in] | x | vector of quadrature nodes |
| [in] | V | column-stacked Vandermonde matrix |
Definition at line 81 of file OrthogonalPolynomials.hpp.
| void gauss | ( | ROL::Ptr< Teuchos::LAPACK< int, Real > > | lapack, |
| const std::vector< Real > & | a, | ||
| const std::vector< Real > & | b, | ||
| std::vector< Real > & | x, | ||
| std::vector< Real > & | w | ||
| ) |
Compute the Gauss quadrature nodes and weights for the polynomials generated by the recurrence coefficients.
| [in] | lapack | pointer to the Teuchos::LAPACK interface |
| [in] | a | vector of recursion coefficients |
| [in] | b | vector of recursion coefficients |
| [out] | x | vector of quadrature nodes |
| [out] | w | vector of quadrature weights |
Adapted from the MATLAB code by Walter Gautschi http://www.cs.purdue.edu/archives/2002/wxg/codes/gauss.m
Definition at line 113 of file OrthogonalPolynomials.hpp.
Referenced by NodalBasis< Real >::NodalBasis().
| void rec_lobatto | ( | ROL::Ptr< Teuchos::LAPACK< int, Real > > const | lapack, |
| const double | xl1, | ||
| const double | xl2, | ||
| std::vector< Real > & | a, | ||
| std::vector< Real > & | b | ||
| ) |
Modify the given recurrence coefficients so that the set of zeros of the maximal order polynomial include the two prescribed points.
| [in] | lapack | pointer to the Teuchos::LAPACK interface |
| [in] | xl1 | location of one pre-assigned node |
| [in] | xl2 | location of another pre-assigned node |
| in/out] | ap pointer to vector of recursion coefficients | |
| in/out] | bp pointer to vector of recursion coefficients |
Based on the section 7 of the paper "Some modified matrix eigenvalue problems" by Gene Golub, SIAM Review Vol 15, No. 2, April 1973, pp.318–334
Definition at line 160 of file OrthogonalPolynomials.hpp.
References trisolve().
Referenced by NodalBasis< Real >::NodalBasis().
1.8.5