JacobiGaussLobattoQuadrature

This routine returns the $n+2$ Quadrature points and weights.

The Gauss-Lobatto quadrature points consists both $\pm 1$ as quadrature points.

• The first quadrature point is $-1$
• The second quadrature point is $+1$

The remaining $n$ points are internal to $(-1, +1)$, and they are n-zeros of Jacobi polynomial of order n with respect to the following weight.

$$(1-x)^{\alpha} (1+x)^{\beta} (x+1)(1-x)$$

Here n is the order of Jacobi polynomial.

INTERFACE
  MODULE SUBROUTINE JacobiGaussLobattoQuadrature(n, alpha, beta, pt, wt)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: alpha
    REAL(DFP), INTENT(IN) :: beta
    REAL(DFP), INTENT(OUT) :: pt(:)
    !! n+2 quad points indexed from 1 to n+2
    REAL(DFP), INTENT(OUT) :: wt(:)
    !! n+2 weights, index from 1 to n+2
  END SUBROUTINE JacobiGaussLobattoQuadrature
END INTERFACE

Examples

️܀ See example

• This example shows the usage of JacobiGaussLobattoQuadrature method.
• This routine returns the quadrature points for Jacobi weights.

In this example $\alpha=\beta=0.0$ (that is, Legendre polynomial) Note that this routine returns n+2 points, the first point is -1 and last point is +1

program main
  use easifembase
  implicit none
  integer( i4b ) :: n
  real( dfp ), allocatable :: pt( : ), wt( : )
  real( dfp ), parameter :: alpha=0.0, beta=0.0
  type(string) :: msg, astr

order=0

  n = 0; call callme

See results

Jacobi Gauss Lobatto points, n+2 = 3 alpha=0 beta=0

pt	wt
-1	0.33333
-4.48639E-17	1.3333
1	0.33333

order=1

  n = 1; call callme

See results

Jacobi Gauss Lobatto points, n+2 = 3 alpha=0 beta=0

pt	wt
-1	0.33333
-4.48639E-17	1.3333
1	0.33333

order=2.

  n = 2; call callme

See results

Jacobi Gauss Lobatto points, n+2 = 4 alpha=0 beta=0

pt	wt
-1	0.16667
-0.44721	0.83333
0.44721	0.83333
1	0.16667

order=4.

  n = 4; call callme

See results

Jacobi Gauss Lobatto points, n+2 = 4 alpha=0 beta=0

pt	wt
-1	6.66667E-02
-0.76506	0.37847
-0.28523	0.55486
0.28523	0.55486
0.76506	0.37847
1	6.66667E-02

  contains
    subroutine callme
    call reallocate( pt, n+2, wt, n+2 )
    call JacobiGaussLobattoQuadrature( n=n, alpha=alpha, &
      & beta=beta, pt=pt, wt=wt )
    msg="Jacobi Gauss Lobatto points, n+2 = " &
        & // tostring( n+2 ) // " alpha="//tostring( alpha ) // &
        & " beta="//tostring( beta )
    call display(msg%chars())
    astr = MdEncode( pt .COLCONCAT. wt )
    call display( astr%chars(), "" )
  end subroutine callme

end program main

↢