LegendreGaussQuadrature

This routine computes the n Gauss-Quadrature points.

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

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