Intrepid2
Classes | Static Public Member Functions | List of all members
Intrepid2::Polylib::Serial Struct Reference

Classes

struct  Cubature
 Gauss-Jacobi/Gauss-Radau-Jacobi/Gauss-Lobatto zeros and weights. More...
struct  Derivative
 Compute the Derivative Matrix and its transpose associated with the Gauss-Jacobi/Gauss-Radau-Jacobi/Gauss-Lobatto-Jacobi zeros. More...
struct  LagrangianInterpolant
 Compute the value of the i th Lagrangian interpolant through the np Gauss-Jacobi/Gauss-Radau-Jacobi/Gauss-Lobatto points zgj at the arbitrary location z. More...
struct  InterpolationOperator
 Interpolation Operator from Gauss-Jacobi points to an arbitrary distribution at points zm. More...

Static Public Member Functions

template<typename zViewType, typename wViewType>
static KOKKOS_INLINE_FUNCTION void getCubature (zViewType z, wViewType w, const ordinal_type np, const double alpha, const double beta, const EPolyType poly)
template<typename DViewType, typename zViewType>
static KOKKOS_INLINE_FUNCTION void getDerivative (DViewType D, const zViewType z, const ordinal_type np, const double alpha, const double beta, const EPolyType poly)
template<typename zViewType>
static KOKKOS_INLINE_FUNCTION zViewType::value_type getLagrangianInterpolant (const ordinal_type i, const typename zViewType::value_type z, const zViewType zgj, const ordinal_type np, const double alpha, const double beta, const EPolyType poly)
template<typename imViewType, typename zgrjViewType, typename zmViewType>
static KOKKOS_INLINE_FUNCTION void getInterpolationOperator (imViewType im, const zgrjViewType zgrj, const zmViewType zm, const ordinal_type nz, const ordinal_type mz, const double alpha, const double beta, const EPolyType poly)
template<typename zViewType, typename polyiViewType, typename polydViewType>
static KOKKOS_INLINE_FUNCTION void JacobiPolynomial (const ordinal_type np, const zViewType z, polyiViewType poly_in, polydViewType polyd, const ordinal_type n, const double alpha, const double beta)
 Routine to calculate Jacobi polynomials, $P^{\alpha,\beta}_n(z) $, and their first derivative, $\frac{d}{dz} P^{\alpha,\beta}_n(z) $.
template<typename zViewType, typename polydViewType>
static KOKKOS_INLINE_FUNCTION void JacobiPolynomialDerivative (const ordinal_type np, const zViewType z, polydViewType polyd, const ordinal_type n, const double alpha, const double beta)
 Calculate the derivative of Jacobi polynomials.
template<typename zViewType, bool DeflationEnabled = false>
static KOKKOS_INLINE_FUNCTION void JacobiZeros (zViewType z, const ordinal_type n, const double alpha, const double beta)
 Calculate the n zeros, z, of the Jacobi polynomial, i.e. $ P_n^{\alpha,\beta}(z) = 0 $.
template<typename zViewType>
static KOKKOS_INLINE_FUNCTION void JacobiZerosPolyDeflation (zViewType z, const ordinal_type n, const double alpha, const double beta)
template<typename aViewType>
static KOKKOS_INLINE_FUNCTION void JacobiZerosTriDiagonal (aViewType a, const ordinal_type n, const double alpha, const double beta)
template<typename aViewType, typename bViewType>
static KOKKOS_INLINE_FUNCTION void JacobiZeros (aViewType a, bViewType b, const ordinal_type n, const double alpha, const double beta)
 Zero determination through the eigenvalues of a tridiagonal matrix from the three term recursion relationship.
template<typename dViewType, typename eViewType>
static KOKKOS_INLINE_FUNCTION void TriQL (dViewType d, eViewType e, const ordinal_type n)
 QL algorithm for symmetric tridiagonal matrix.
static KOKKOS_INLINE_FUNCTION double GammaFunction (const double x)
 Calculate the Gamma function , $ \Gamma(x)$, for integer values x and halves.

Detailed Description

Definition at line 213 of file Intrepid2_Polylib.hpp.

Member Function Documentation

◆ GammaFunction()

KOKKOS_INLINE_FUNCTION double Intrepid2::Polylib::Serial::GammaFunction ( const double x)
static

Calculate the Gamma function , $ \Gamma(x)$, for integer values x and halves.

Determine the value of $\Gamma(x)$ using:

$ \Gamma(x) = (x-1)! \mbox{ or } \Gamma(x+1/2) = (x-1/2)\Gamma(x-1/2)$

where $ \Gamma(1/2) = \sqrt{\pi}$

Definition at line 884 of file Intrepid2_PolylibDef.hpp.

Referenced by JacobiZeros().

◆ getCubature()

template<typename zViewType, typename wViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::getCubature ( zViewType z,
wViewType w,
const ordinal_type np,
const double alpha,
const double beta,
const EPolyType poly )
inlinestatic

Definition at line 253 of file Intrepid2_Polylib.hpp.

◆ getDerivative()

template<typename DViewType, typename zViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::getDerivative ( DViewType D,
const zViewType z,
const ordinal_type np,
const double alpha,
const double beta,
const EPolyType poly )
inlinestatic

Definition at line 300 of file Intrepid2_Polylib.hpp.

◆ getInterpolationOperator()

template<typename imViewType, typename zgrjViewType, typename zmViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::getInterpolationOperator ( imViewType im,
const zgrjViewType zgrj,
const zmViewType zm,
const ordinal_type nz,
const ordinal_type mz,
const double alpha,
const double beta,
const EPolyType poly )
inlinestatic

Definition at line 453 of file Intrepid2_Polylib.hpp.

◆ getLagrangianInterpolant()

template<typename zViewType>
KOKKOS_INLINE_FUNCTION zViewType::value_type Intrepid2::Polylib::Serial::getLagrangianInterpolant ( const ordinal_type i,
const typename zViewType::value_type z,
const zViewType zgj,
const ordinal_type np,
const double alpha,
const double beta,
const EPolyType poly )
inlinestatic

Definition at line 397 of file Intrepid2_Polylib.hpp.

◆ JacobiPolynomial()

template<typename zViewType, typename polyiViewType, typename polydViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::JacobiPolynomial ( const ordinal_type np,
const zViewType z,
polyiViewType poly_in,
polydViewType polyd,
const ordinal_type n,
const double alpha,
const double beta )
static

Routine to calculate Jacobi polynomials, $P^{\alpha,\beta}_n(z) $, and their first derivative, $\frac{d}{dz} P^{\alpha,\beta}_n(z) $.

  • This function returns the vectors poly_in and poly_d containing the value of the n-th order Jacobi polynomial $ P^{\alpha,\beta}_n(z) \alpha > -1, \beta > -1 $ and its derivative at the np points in z[i]
  • If poly_in = NULL then only calculate derivative
  • If polyd = NULL then only calculate polynomial
  • To calculate the polynomial this routine uses the recursion relationship (see appendix A ref [4]) : $ \begin{array}{rcl} P^{\alpha,\beta}_0(z) &=& 1 \\ P^{\alpha,\beta}_1(z) &=& \frac{1}{2} [ \alpha-\beta+(\alpha+\beta+2)z] \\ a^1_n P^{\alpha,\beta}_{n+1}(z) &=& (a^2_n + a^3_n z) P^{\alpha,\beta}_n(z) - a^4_n P^{\alpha,\beta}_{n-1}(z) \\ a^1_n &=& 2(n+1)(n+\alpha + \beta + 1)(2n + \alpha + \beta) \\ a^2_n &=& (2n + \alpha + \beta + 1)(\alpha^2 - \beta^2) \\ a^3_n &=& (2n + \alpha + \beta)(2n + \alpha + \beta + 1) (2n + \alpha + \beta + 2) \\ a^4_n &=& 2(n+\alpha)(n+\beta)(2n + \alpha + \beta + 2) \end{array} $
  • To calculate the derivative of the polynomial this routine uses the relationship (see appendix A ref [4]) : $ \begin{array}{rcl} b^1_n(z)\frac{d}{dz} P^{\alpha,\beta}_n(z)&=&b^2_n(z)P^{\alpha,\beta}_n(z) + b^3_n(z) P^{\alpha,\beta}_{n-1}(z) \hspace{2.2cm} \\ b^1_n(z) &=& (2n+\alpha + \beta)(1-z^2) \\ b^2_n(z) &=& n[\alpha - \beta - (2n+\alpha + \beta)z]\\ b^3_n(z) &=& 2(n+\alpha)(n+\beta) \end{array} $
  • Note the derivative from this routine is only valid for -1 < z < 1.

Definition at line 598 of file Intrepid2_PolylibDef.hpp.

Referenced by JacobiPolynomialDerivative(), and Intrepid2::Polynomials::legendreDerivativeValues().

◆ JacobiPolynomialDerivative()

template<typename zViewType, typename polydViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::JacobiPolynomialDerivative ( const ordinal_type np,
const zViewType z,
polydViewType polyd,
const ordinal_type n,
const double alpha,
const double beta )
static

Calculate the derivative of Jacobi polynomials.

  • Generates a vector poly of values of the derivative of the n-th order Jacobi polynomial $ P^(\alpha,\beta)_n(z)$ at the np points z.
  • To do this we have used the relation
    $ \frac{d}{dz} P^{\alpha,\beta}_n(z) = \frac{1}{2} (\alpha + \beta + n + 1) P^{\alpha,\beta}_n(z) $
  • This formulation is valid for $ -1 \leq z \leq 1 $

Definition at line 685 of file Intrepid2_PolylibDef.hpp.

References JacobiPolynomial().

◆ JacobiZeros() [1/2]

template<typename aViewType, typename bViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::JacobiZeros ( aViewType a,
bViewType b,
const ordinal_type n,
const double alpha,
const double beta )
static

Zero determination through the eigenvalues of a tridiagonal matrix from the three term recursion relationship.

Set up a symmetric tridiagonal matrix

$ \left [ \begin{array}{ccccc} a[0] & b[0] & & & \\ b[0] & a[1] & b[1] & & \\ 0 & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & b[n-2] \\ & & & b[n-2] & a[n-1] \end{array} \right ] $

Where the coefficients a[n], b[n] come from the recurrence relation

$ b_j p_j(z) = (z - a_j ) p_{j-1}(z) - b_{j-1} p_{j-2}(z) $

where $ j=n+1$ and $p_j(z)$ are the Jacobi (normalized) orthogonal polynomials $ \alpha,\beta > -1$( integer values and halves). Since the polynomials are orthonormalized, the tridiagonal matrix is guaranteed to be symmetric. The eigenvalues of this matrix are the zeros of the Jacobi polynomial.

References GammaFunction(), and TriQL().

◆ JacobiZeros() [2/2]

template<typename zViewType, bool DeflationEnabled = false>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::JacobiZeros ( zViewType z,
const ordinal_type n,
const double alpha,
const double beta )
static

Calculate the n zeros, z, of the Jacobi polynomial, i.e. $ P_n^{\alpha,\beta}(z) = 0 $.

This routine is only valid for $( \alpha > -1, \beta > -1)$ and uses polynomial deflation in a Newton iteration

Definition at line 710 of file Intrepid2_PolylibDef.hpp.

◆ JacobiZerosPolyDeflation()

template<typename zViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::JacobiZerosPolyDeflation ( zViewType z,
const ordinal_type n,
const double alpha,
const double beta )
static

Definition at line 724 of file Intrepid2_PolylibDef.hpp.

◆ JacobiZerosTriDiagonal()

template<typename aViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::JacobiZerosTriDiagonal ( aViewType a,
const ordinal_type n,
const double alpha,
const double beta )
static

Definition at line 770 of file Intrepid2_PolylibDef.hpp.

◆ TriQL()

template<typename dViewType, typename eViewType>
KOKKOS_INLINE_FUNCTION void Intrepid2::Polylib::Serial::TriQL ( dViewType d,
eViewType e,
const ordinal_type n )
static

QL algorithm for symmetric tridiagonal matrix.

This subroutine is a translation of an algol procedure, num. math. 12, 377-383(1968) by martin and wilkinson, as modified in num. math. 15, 450(1970) by dubrulle. Handbook for auto. comp., vol.ii-linear algebra, 241-248(1971). This is a modified version from numerical recipes.

This subroutine finds the eigenvalues and first components of the eigenvectors of a symmetric tridiagonal matrix by the implicit QL method.

on input:

  • n is the order of the matrix;
  • d contains the diagonal elements of the input matrix;
  • e contains the subdiagonal elements of the input matrix in its first n-1 positions. e(n) is arbitrary;

on output:

  • d contains the eigenvalues in ascending order.
  • e has been destroyed;

Definition at line 814 of file Intrepid2_PolylibDef.hpp.

Referenced by JacobiZeros().

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