%
% Written by:
% -- 
% John L. Weatherwax                2009-04-21
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

phi_1 = 1.2728;
phi_2 = -0.81;
rts = roots( [ -phi_2, -phi_1, 1 ] );
G_1inv = rts(1);
G_2inv = rts(2);
G_1 = 1/G_1inv;
G_2 = 1/G_2inv;

kArray = 0:3; % we only need the first three elements:
numer = G_1 * ( 1 - G_2^2 ) * G_1.^kArray - G_2 * ( 1 - G_1^2 ) * G_2.^kArray ;
denom = ( G_1 - G_2 ) * ( 1 + G_1 * G_2 ) ;
rho_k  = numer / denom 

% fill in the autocorrelation matrix: 
R_x = zeros(2,2);
R_x(1,1) = rho_k(1);
R_x(1,2) = rho_k(2)';
R_x(2,1) = rho_k(2);
R_x(2,2) = rho_k(1);
b_rhs = [ rho_k(2); rho_k(3) ];

% solve for the optimal coefficients: 
w = R_x \ b_rhs

% compute the eigenvalues of R_x:
er = eig(R_x)

1/max(er)