LegendreGaussQuadrature

This routine computes the n Gauss-Quadrature points.

All Gauss-Quadrature points are inside $(-1, 1)$

Interface

܀ Interface

INTERFACE
  MODULE SUBROUTINE LegendreGaussQuadrature(n, pt, wt)
    INTEGER(I4B), INTENT(IN) :: n
    !! It represents the order of Legendre polynomial
    REAL(DFP), INTENT(OUT) :: pt(:)
    !! the size is 1 to n
    REAL(DFP), OPTIONAL, INTENT(OUT) :: wt(:)
    !! the size is 1 to n
  END SUBROUTINE LegendreGaussQuadrature
END INTERFACE

️܀ See example

This example shows the usage of LegendreGaussQuadrature method.

This routine returns the quadrature points for Legendre polynom.

program main
  use easifembase
  implicit none
  integer( i4b ) :: n
  real( dfp ), allocatable :: pt( : ), wt( : )
  type(string) :: msg, astr
  n = 5
  call reallocate( pt, n, wt, n )
  call LegendreGaussQuadrature( n=n, pt=pt, wt=wt )
  msg = "Legendre Gauss Quadrature n = " // tostring( n )
  call display(msg%chars())
  astr = MdEncode( pt .COLCONCAT. wt )
  call display( astr%chars(), "" )
end program main

Zeros of J(x), n = 5 alpha=0 beta=0

pt	wt
-0.90618	0.23693
-0.53847	0.47863
-1.56541E-16	0.56889
0.53847	0.47863
0.90618	0.23693

↢

Some Legendre Gauss Quadrature points

n = 1

Point	Weight
0	2

n = 2

Point	Weight
-0.57735	1
0.57735	1

n = 3

Point	Weight
-0.7746	0.55556
3.71231E-16	0.88889
0.7746	0.55556

n = 4

Point	Weight
-0.86114	0.34785
-0.33998	0.65215
0.33998	0.65215
0.86114	0.34785

n = 5

Point	Weight
-0.90618	0.23693
-0.53847	0.47863
2.66893E-17	0.56889
0.53847	0.47863
0.90618	0.23693

n = 6

Point	Weight
-0.93247	0.17132
-0.66121	0.36076
-0.23862	0.46791
0.23862	0.46791
0.66121	0.36076
0.93247	0.17132

n = 7

Point	Weight
-0.94911	0.12948
-0.74153	0.27971
-0.40585	0.38183
1.88509E-16	0.41796
0.40585	0.38183
0.74153	0.27971
0.94911	0.12948

n = 8

Point	Weight
-0.96029	0.10123
-0.79667	0.22238
-0.52553	0.31371
-0.18343	0.36268
0.18343	0.36268
0.52553	0.31371
0.79667	0.22238
0.96029	0.10123

n = 9

Point	Weight
-0.96816	8.12744E-02
-0.83603	0.18065
-0.61337	0.26061
-0.32425	0.31235
2.76366E-17	0.33024
0.32425	0.31235
0.61337	0.26061
0.83603	0.18065
0.96816	8.12744E-02

n = 10

Point	Weight
-0.97391	6.66713E-02
-0.86506	0.14945
-0.67941	0.21909
-0.4334	0.26927
-0.14887	0.29552
0.14887	0.29552
0.4334	0.26927
0.67941	0.21909
0.86506	0.14945
0.97391	6.66713E-02