xwlg

XWLG  Computes nodes and weights of the Legendre-Gauss  quadrature formula.

    [x,w]=xwlg(np) returns the np weigths and nodes 
    of the corresponding Legendre Gauss quadrature 
    formula in the reference interval (-1,1).

    [x,w]=xwlg(np,a,b) returns the np weigths and the nodes 
    of the corresponding Legendre Gauss quadrature 
    formula in the  interval (a,b).

 Input: np = number of nodes
        a, b = extrema of the interval

 Output: x(np,1) = LG nodes  (CHQZ2, (2.3.10), pag. 76)
         w(np,1) = LG weigths (CHQZ2, (2.3.10), pag. 76)


 Reference: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
                    "Spectral Methods. Fundamentals in Single Domains"
                    Springer Verlag, Berlin Heidelberg New York, 2006.

0001 function [x,w] = xwlg(np,a,b)
0002 %XWLG  Computes nodes and weights of the Legendre-Gauss  quadrature formula.
0003 %
0004 %    [x,w]=xwlg(np) returns the np weigths and nodes
0005 %    of the corresponding Legendre Gauss quadrature
0006 %    formula in the reference interval (-1,1).
0007 %
0008 %    [x,w]=xwlg(np,a,b) returns the np weigths and the nodes
0009 %    of the corresponding Legendre Gauss quadrature
0010 %    formula in the  interval (a,b).
0011 %
0012 % Input: np = number of nodes
0013 %        a, b = extrema of the interval
0014 %
0015 % Output: x(np,1) = LG nodes  (CHQZ2, (2.3.10), pag. 76)
0016 %         w(np,1) = LG weigths (CHQZ2, (2.3.10), pag. 76)
0017 %
0018 %
0019 % Reference: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
0020 %                    "Spectral Methods. Fundamentals in Single Domains"
0021 %                    Springer Verlag, Berlin Heidelberg New York, 2006.
0022 
0023 %   Written by Paola Gervasio
0024 %   $Date: 2007/04/01$
0025 
0026 
0027 n=np-1;
0028 if np<=1
0029   x=0;w=2;
0030   return
0031 end
0032 x=jacobi_roots(np,0,0);
0033 w=2./(pnleg1(x,np).^2.*(1-x.^2));
0034 
0035 
0036 
0037 %
0038 % map on (a,b)
0039 %
0040 if nargin == 3
0041   bma=(b-a)*.5;
0042   bpa=(b+a)*.5;
0043   x=bma*x+bpa;
0044   w=w*bma;
0045 end
0046 return