%{
# Fourier differentiation matrix

%}

function [x, DM] = fourdif(N,m)
%
% The function [x, DM] = fourdif(N,m) computes the m'th derivative Fourier 
% spectral differentiation matrix on grid with N equispaced points in [0,2pi)
% 
%  Input:
%  N:        Size of differentiation matrix.
%  M:        Derivative required (non-negative integer)
%
%  Output:
%  x:        Equispaced points 0, 2pi/N, 4pi/N, ... , (N-1)2pi/N
%  DM:       m'th order differentiation matrix
%
% 
%  Explicit formulas are used to compute the matrices for m=1 and 2. 
%  A discrete Fouier approach is employed for m>2. The program 
%  computes the first column and first row and then uses the 
%  toeplitz command to create the matrix.

%  For m=1 and 2 the code implements a "flipping trick" to
%  improve accuracy suggested by W. Don and A. Solomonoff in 
%  SIAM J. Sci. Comp. Vol. 6, pp. 1253--1268 (1994).
%  The flipping trick is necesary since sin t can be computed to high
%  relative precision when t is small whereas sin (pi-t) cannot.
%
%  S.C. Reddy, J.A.C. Weideman 1998.  Corrected for MATLAB R13 
%  by JACW, April 2003.
 

    x=2*pi*(0:N-1)'/N;                       % gridpoints
    h=2*pi/N;                                % grid spacing
    zi=sqrt(-1);
    kk=(1:N-1)';
    n1=floor((N-1)/2); n2=ceil((N-1)/2);
    if m==0,                                 % compute first column
      col1=[1; zeros(N-1,1)];                % of zeroth derivative
      row1=col1;                             % matrix, which is identity

    elseif m==1,                             % compute first column
      if rem(N,2)==0                         % of 1st derivative matrix
	topc=cot((1:n2)'*h/2);
        col1=[0; 0.5*((-1).^kk).*[topc; -flipud(topc(1:n1))]]; 
      else
	topc=csc((1:n2)'*h/2);
        col1=[0; 0.5*((-1).^kk).*[topc; flipud(topc(1:n1))]];
      end;
      row1=-col1;                            % first row

    elseif m==2,                             % compute first column  
      if rem(N,2)==0                         % of 2nd derivative matrix
	topc=csc((1:n2)'*h/2).^2;
        col1=[-pi^2/3/h^2-1/6; -0.5*((-1).^kk).*[topc; flipud(topc(1:n1))]];
      else
	topc=csc((1:n2)'*h/2).*cot((1:n2)'*h/2);
        col1=[-pi^2/3/h^2+1/12; -0.5*((-1).^kk).*[topc; -flipud(topc(1:n1))]];
      end;
      row1=col1;                             % first row 

    else                                     % employ FFT to compute
      N1=floor((N-1)/2);                     % 1st column of matrix for m>2
      N2 = (-N/2)*rem(m+1,2)*ones(rem(N+1,2));  
      mwave=zi*[(0:N1) N2 (-N1:-1)];
      col1=real(ifft((mwave.^m).*fft([1 zeros(1,N-1)])));
      if rem(m,2)==0,
	row1=col1;                           % first row even derivative
      else
	col1=[0 col1(2:N)]'; 
	row1=-col1;                          % first row odd derivative
      end;
    end;
    DM=toeplitz(col1,row1);