% Octave commands
% NOTE:  after cutting-and-pasting into Octave, use RET to execute.
% Percent sign "%" is the Octave comment-line prefix.

% Inspired by Chapter 2 of Hugh L. Montgomery's "Early Fourier Analysis"

% M.V.Wickerhauser
%   Math 410 lecture notes
%   2017-09-08


%%%% Roots of unity
n=10;                           % Semicolon implies no printout.
k=0:(n-1);                      % Integers 0,1,...,n-1
exp(2* pi * i * k/n)            % kth root of z^n-1=0, ordered by argument.

rou=@(n,k) exp(2* pi * i * k/n); % "Anonymous" function
rou(n,k)                         % Another specification of the 10 10th roots of unity
sum(rou(50,1:50))               % Sum of the 50 50th roots of unity
% NOTE: the answers should be zero up to double precision truncation error 2e-16.

% I.M.Vinogradov's e-function
eIMV = @(x) exp(2*pi*i*x)       % "anonymous" function

% Theorem 2.1, p.34 example:
q=3; k=9;                       % ...so q| ("q divides k")
n=1:q;                          % ...so we sum from n=1 to n=q
sum(eIMV(k*n/q))                % This should be q.

q=5; k=9;                       % ...so q does not divide k
n=1:q;                          % ...so we sum from n=1 to n=q
sum(eIMV(k*n/q))                % This should be zero.

%%%% Fourier matrix
% Order 5 example, explicitly from roots of unity
n=5; k=0:(n-1);
fourier5 = [ rou(5,-0*k);       % 0th row = -0th powers of 5th roots of unity
             rou(5,-1*k);       % 1st row = -1st powers of 5th roots of unity
             rou(5,-2*k);       %  ...
             rou(5,-3*k); 
             rou(5,-4*k) ]      % last row = -(n-1)th powers of 
fourier5' * fourier5            % Should be 5*Identity

%%%% Function to produce a Fourier matrix of arbitrary order.
% Write this into a file "fouriermatrix.m"
function F=fouriermatrix(n)
  F=zeros(n,n);                 % Initialize an n*n matrix
  for(m=1:n)                    % row index
    for(k=1:n)                  % column index
      F(m,k)=exp(-2*pi*i*m*k/n); % can also use Vinogradov's function here
    endfor
  endfor  
end
%%%%%%%%%% end of "fouriermatrix.m"