% GET_EIGEN_BLOCK_A     Obtain the eigen decomposition of the Hermitian
% matrix, R, for the generating function such that
%
% R=[-T  I]
%   [ I  0]
%
% where T is the positive-definite Toeplitz-like Block matrix
%
% [G,J,rowcol] = GET_EIGEN_BLOCK_A(M,hyp)
%
% Inputs:   M       : Time-series Inputs of Gaussian process, size nx1
%           hyp     : hyperparameters of the covariance function [g v]
%
% Outouts:  G       = Generates the Generating function (matrix)
%           J       = J-unitary matrix, J=[I 0;0 -I]
%           rowcol  = block-position 
%
% Note that the covariance function used, is the usual convention, of
%
%       C(Z|a,g,v) = exp(-0.5*g*[Zi-Zj]^2) + v
%
%
% (C) Gaussian Process Schur Toolbox 1.0
% (C) Copyright 2005-2007, Keith Neo <keith.neo@nuim.ie>
%                          http://www.hamilton.ie

function [G,J,rowcol]=get_eigen_block_a(M,hyp)

[g_row,e_dec,g_toe]=eigen_tool();

rowcol=g_row(M);

n=size(M,1);
rk=size(rowcol,2);
A=zeros(rk);
B=zeros(2*n,rk);
B(1:n,1)=-g_toe(M,hyp,1);
B(n+1,1)=1;
for k=2:rk
    p=rowcol(k);
    tmp1=g_toe(M,hyp,p);
    tmp3=g_toe(M,hyp,p-1);
    tmp2=[0;tmp3(1:end-1,:)];
    B(1:n,k)=tmp2-tmp1;
end
tmp0=zeros(2*n,rk);
for h=1:rk
    p=rowcol(h);
    A(h,:)=B(p,:);
    tmp0(p,h)=1;
end

[G,J]=e_dec(A,B,tmp0);