# Chebychev differentiation

We have discussed that the differentiation matrix is at the core of our tool. In ost of the code, we use a spectral discretisation based on Chebychev polynomials. This is customary in studies of fluid stability problems since the shape of the variables are typically smooth, and spectral discretization is very good for smooth problems. It was developped in the Matlab differentiation matrix suite by Weigdeman & Reddy. You will find together with it many nice ways to build differentiation matrices for other spectral discretization. We use it extensivelly in easystab.

function [x, DM] = chebdif(N, M)

%  The function DM =  chebdif(N,M) computes the differentiation
%  matrices D1, D2, ..., DM on Chebyshev nodes.
%
%  Input:
%  N:        Size of differentiation matrix.
%  M:        Number of derivatives required (integer).
%  Note:     0 < M <= N-1.
%
%  Output:
%  DM:       DM(1:N,1:N,ell) contains ell-th derivative matrix, ell=1..M.
%
%  The code implements two strategies for enhanced
%  accuracy suggested by W. Don and S. Solomonoff in
%  SIAM J. Sci. Comp. Vol. 6, pp. 1253--1268 (1994).
%  The two strategies are (a) the use of trigonometric
%  identities to avoid the computation of differences
%  x(k)-x(j) and (b) the use of the "flipping trick"
%  which is necessary since sin t can be computed to high
%  relative precision when t is small whereas sin (pi-t) cannot.

%  J.A.C. Weideman, S.C. Reddy 1998.

I = eye(N);                          % Identity matrix.
L = logical(I);                      % Logical identity matrix.

n1 = floor(N/2); n2  = ceil(N/2);     % Indices used for flipping trick.

k = [0:N-1]';                        % Compute theta vector.
th = k*pi/(N-1);

x = sin(pi*[N-1:-2:1-N]'/(2*(N-1))); % Compute Chebyshev points.

T = repmat(th/2,1,N);
DX = 2*sin(T'+T).*sin(T'-T);          % Trigonometric identity.
DX = [DX(1:n1,:); -flipud(fliplr(DX(1:n2,:)))];   % Flipping trick.
DX(L) = ones(N,1);                       % Put 1's on the main diagonal of DX.

C = toeplitz((-1).^k);               % C is the matrix with
C(1,:) = C(1,:)*2; C(N,:) = C(N,:)*2;     % entries c(k)/c(j)
C(:,1) = C(:,1)/2; C(:,N) = C(:,N)/2;

Z = 1./DX;                           % Z contains entries 1/(x(k)-x(j))
Z(L) = zeros(N,1);                      % with zeros on the diagonal.

D = eye(N);                          % D contains diff. matrices.

for ell = 1:M
D = ell*Z.*(C.*repmat(diag(D),1,N) - D); % Off-diagonals
D(L) = -sum(D');                            % Correct main diagonal of D
DM(:,:,ell) = D;                                   % Store current D in DM
end