legendre_tr_eval

 LEGENDRE_TR_EVAL     Evaluates Discrete Legendre Transform

            formula (4.4.17)  pag. 118, Quarteroni-Valli, Springer 1997
            with coefficients computed with formula (4.4.18)  
                   pag. 118, Quarteroni-Valli, Springer 1997

  [u_int]=legendre_tr_eval(x,u,x_int) returns the evaluation at nodes x_int
       of the expansion of (I_N u)(x) with respect to Legendre polynomials

       (I_n u)(x)=\Sum_{k=0}^n u_k L_k(x)             (*)

       where L_k(x) are the Legendre polynomials

 Input: x = array of np Legendre Gauss Lobatto (LGL) nodes on [-1,1]
        u = array of np values of u at LGL nodes x : u_j=u(x_j)
            size(u)=[np,nc], where nc is the number of transforms which
            should be computed.
        If u is a 2-indices array, then every column of u is transformed
        following formula (*)
        x_int = array of a second set of nodes in [-1,1]

 Output: u_int = array of the values of  the
              expansion of I_Nu(x) with respect to Legendre polynomials.

 Reference: A. Quarteroni, A. Valli:
            "Numerical Approximation of Partial Differential Equations"
            Springer Verlag / 1997 (2nd Ed)

0001 function [u_int]=legendre_tr_eval(x,u,x_int);
0002 % LEGENDRE_TR_EVAL     Evaluates Discrete Legendre Transform
0003 %
0004 %            formula (4.4.17)  pag. 118, Quarteroni-Valli, Springer 1997
0005 %            with coefficients computed with formula (4.4.18)
0006 %                   pag. 118, Quarteroni-Valli, Springer 1997
0007 %
0008 %  [u_int]=legendre_tr_eval(x,u,x_int) returns the evaluation at nodes x_int
0009 %       of the expansion of (I_N u)(x) with respect to Legendre polynomials
0010 %
0011 %       (I_n u)(x)=\Sum_{k=0}^n u_k L_k(x)             (*)
0012 %
0013 %       where L_k(x) are the Legendre polynomials
0014 %
0015 % Input: x = array of np Legendre Gauss Lobatto (LGL) nodes on [-1,1]
0016 %        u = array of np values of u at LGL nodes x : u_j=u(x_j)
0017 %            size(u)=[np,nc], where nc is the number of transforms which
0018 %            should be computed.
0019 %        If u is a 2-indices array, then every column of u is transformed
0020 %        following formula (*)
0021 %        x_int = array of a second set of nodes in [-1,1]
0022 %
0023 % Output: u_int = array of the values of  the
0024 %              expansion of I_Nu(x) with respect to Legendre polynomials.
0025 %
0026 % Reference: A. Quarteroni, A. Valli:
0027 %            "Numerical Approximation of Partial Differential Equations"
0028 %            Springer Verlag / 1997 (2nd Ed)
0029 %
0030 
0031 %   Written by Paola Gervasio
0032 %   $Date: 2007/04/01$
0033 
0034 
0035 n1=length(x_int);
0036 np=length(x); n=np-1;
0037 
0038 uk=legendre_tr_coef(x,u);
0039 pn=pnleg_all(x_int,n);
0040 u_int=pn*uk;
0041 
0042 return