function Q = invtoepl(r,M)
%
% Programmed by: Vinay K. Ingle, 1999
%
%-----------------------------------------------------------
% Copyright 2000, by Dimitris G. Manolakis, Vinay K. Ingle,
% and Stephen M. Kogon.  For use with the book
%
% "Statistical and Adaptive Signal Processing"
%
% McGraw-Hill Higher Education.
%
%-----------------------------------------------------------
%  Input: First col or row of symmetric Toeplitz matrix R
%         r(m), 0 <= m <= M-1 
% Output: Q = Inverse of R using Section 7.7.3

% Requires the function sympersym

M = length(r);

% Determine vector b using the durbin function
[a,k,Po] = durbin(r,M-1);
b = flipud(a(2:M));

% Determine P and q
P = r(1) + (r(M:-1:2))'*b;
q = b/P;

% Computation of Q matrix using eq. (7.7.24)
QM = zeros(M);                  % Initialize QM
QM(1:M-1,M) = q; QM(M,M) = 1/P; % Last column of QM
QM = sympersym(QM,QM(:,M));     % symmetry and persymmetry operations
Q = QM(1:M-1,1:M-1);            % Extract Q

Mend = ceil((M+1)/2);
for j = M-1:-1:Mend
   for i = (M-j+1):1:j
      % Use eq. (7.7.24)
      Q(i,j) = Q(M-j,M-i) - P*(q(M-j)*q(M-i)-q(i)*q(j));
   end
   % Perform symmetry and persymmetry operations
   QM = sympersym(QM,Q((M-j+1):1:j,j));
   Q = QM(1:M-1,1:M-1);
end
Q = QM;