LegendreGaussLobattoQuadrature

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

Interface

܀ Interface

INTERFACE
  MODULE SUBROUTINE LegendreGaussLobattoQuadrature(n, pt, wt)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of Legendre polynomials
    REAL(DFP), INTENT(OUT) :: pt(:)
    !! n+2 quad points indexed from 1 to n+2
    REAL(DFP), OPTIONAL, INTENT(OUT) :: wt(:)
    !! n+2 weights, index from 1 to n+2
  END SUBROUTINE LegendreGaussLobattoQuadrature
END INTERFACE

n

Order of Legendre polynomial. The order of accuracy of this rule is $2n+1$.

pt

$n+2$ quadrature points

wt

$n+2$ weights.

️܀ See example

This example shows the usage of LegendreGaussLobattoQuadrature method.

This routine returns the quadrature points for Legendre weights.

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( : )
  type(string) :: msg, astr
  n = 2
  call reallocate( pt, n+2, wt, n+2 )
  call LegendreGaussLobattoQuadrature( n=n, pt=pt, wt=wt )
  msg="Legendre Gauss Lobatto points, n+2 = " &
      & // tostring( n+2 )
  call display(msg%chars())
  astr = MdEncode( pt .COLCONCAT. wt )
  call display( astr%chars(), "" )
end program main

Legendre Gauss Lobatto points, n+2 = 4

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

↢

Some LegendreGaussLobattoQuadrature

n+2 = 3

Point	Weight
-1	0.33333
-2.66578E-17	1.3333
1	0.33333

n+2 = 4

Point	Weight
-1	0.16667
-0.44721	0.83333
0.44721	0.83333
1	0.16667

n+2 = 5

Point	Weight
-1	0.1
-0.65465	0.54444
4.1375E-17	0.71111
0.65465	0.54444
1	0.1

n+2 = 6

Point	Weight
-1	6.66667E-02
-0.76506	0.37847
-0.28523	0.55486
0.28523	0.55486
0.76506	0.37847
1	6.66667E-02

n+2 = 7

Point	Weight
-1	4.7619E-02
-0.83022	0.27683
-0.46885	0.43175
2.39125E-16	0.48762
0.46885	0.43175
0.83022	0.27683
1	4.7619E-02

n+2 = 8

Point	Weight
-1	3.57143E-02
-0.87174	0.2107
-0.5917	0.34112
-0.2093	0.41246
0.2093	0.41246
0.5917	0.34112
0.87174	0.2107
1	3.57143E-02

n+2 = 9

Point	Weight
-1	2.77778E-02
-0.89976	0.1655
-0.67719	0.27454
-0.36312	0.34643
-2.81541E-16	0.37152
0.36312	0.34643
0.67719	0.27454
0.89976	0.1655
1	2.77778E-02

n+2 = 10

Point	Weight
-1	2.22222E-02
-0.91953	0.13331
-0.73877	0.22489
-0.47792	0.29204
-0.16528	0.32754
0.16528	0.32754
0.47792	0.29204
0.73877	0.22489
0.91953	0.13331
1	2.22222E-02