DERCGL Spectral (Chebyshev Gauss Lobatto) derivative matrix
[D]=DERCGL(X,NP) returns the spectral derivative
matrix D at the NP CGL nodes X (in [-1,1]). NP-1 is the polynomial
degree used. Chebyshev Gauss Lobatto (CGL) grid ORDERED FROM LEFT TO RIGHT
Input: x = array of CGL nodes on [-1,1] (computed by xwcgl)
np = number of CGL nodes (=n+1, n=polynomial interpolation degree)
Output: d = spectral derivative matrix: formula (2.4.31), pag. 89 CHQZ2
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 [d] = dercgl(x,np) 0002 %DERCGL Spectral (Chebyshev Gauss Lobatto) derivative matrix 0003 % 0004 % [D]=DERCGL(X,NP) returns the spectral derivative 0005 % matrix D at the NP CGL nodes X (in [-1,1]). NP-1 is the polynomial 0006 % degree used. Chebyshev Gauss Lobatto (CGL) grid ORDERED FROM LEFT TO RIGHT 0007 % 0008 % 0009 % Input: x = array of CGL nodes on [-1,1] (computed by xwcgl) 0010 % np = number of CGL nodes (=n+1, n=polynomial interpolation degree) 0011 % 0012 % Output: d = spectral derivative matrix: formula (2.4.31), pag. 89 CHQZ2 0013 % 0014 % Reference: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, 0015 % "Spectral Methods. Fundamentals in Single Domains" 0016 % Springer Verlag, Berlin Heidelberg New York, 2006. 0017 0018 % Written by Paola Gervasio 0019 % $Date: 2007/04/01$ 0020 0021 0022 n=np-1; 0023 x=x(:); 0024 0025 if n==0 d=0; return; end; 0026 0027 c=[2;ones(n-1,1);2]; 0028 for j=2:n 0029 d(j,j)=-x(j)/(2*(1-x(j)^2)); 0030 end 0031 for j=1:np 0032 xj=x(j); 0033 for l=1:j-1 0034 d(j,l)=c(j)/c(l)*(-1)^(l+j)/(xj-x(l)); 0035 end 0036 for l=j+1:np 0037 d(j,l)=c(j)/c(l)*(-1)^(l+j)/(xj-x(l)); 0038 end 0039 end 0040 d(np,np)=(2*n^2+1)/6; 0041 d(1,1)=-d(np,np); 0042 d(1,np)=(-1)^(np+1)/(x(1)-x(np)); 0043 d(np,1)=-d(1,np); 0044 return