Example 6.12

Calculation of Derivatives Using Chebyshev Derivative Matrix Operator

Contents

Initialization

close all;
clear N u j X U D j k

N  = 5;


u  = @(x) 4*x.^2.*(1-x.^2).*exp(-x/2);

j  = [0:N]';
X  = cos(j*pi/N);
U  = u(X);

D  = zeros(N+1,N+1);

Creation of the matrix operator

for j = [1:N+1]
    for k = [1:N+1]
        if j==k
            switch j
                case 1
                    D(j,k) = (2*N^2+1)/6;
                case N+1
                    D(j,k) = -(2*N^2+1)/6;
                otherwise
                    D(j,k) = -X(j)/(2*(1-X(j)^2));
            end;
        else
            if or(j==1,j==N+1) cj = 2; else cj = 1; end;
            if or(k==1,k==N+1) ck = 2; else ck = 1; end;
            D(j,k) = cj*(-1)^(j+k)/(ck*(X(j)-X(k)));
        end;
    end;
end;

Display the results

X
U
D
D*U

X =

   1.000000000000000
   0.809016994374947
   0.309016994374947
  -0.309016994374947
  -0.809016994374947
  -1.000000000000000


U =

                   0
   0.603582782590411
   0.296029630345352
   0.403217672421526
   1.355465472313722
                   0


D =

  Columns 1 through 5

   8.500000000000000 -10.472135954999581   2.894427190999916  -1.527864045000421   1.105572809000084
   2.618033988749895  -1.170820393249937  -2.000000000000000   0.894427190999916  -0.618033988749895
  -0.723606797749979   2.000000000000000  -0.170820393249937  -1.618033988749895   0.894427190999916
   0.381966011250105  -0.894427190999916   1.618033988749895   0.170820393249937  -2.000000000000000
  -0.276393202250021   0.618033988749895  -0.894427190999916   2.000000000000000   1.170820393249936
   0.500000000000000  -1.105572809000084   1.527864045000421  -2.894427190999916  10.472135954999574

  Column 6

  -0.500000000000000
   0.276393202250021
  -0.381966011250105
   0.723606797749979
  -2.618033988749894
  -8.500000000000000


ans =

  -4.581460762243603
  -1.775821173923400
   1.716542943367882
  -2.702927992451481
   2.501699686116691
  12.812522829475302

Published with MATLAB® 7.5