% GET_EIGEN_BLOCK_TRACEQQ_A1     Obtain the eigen decomposition of
% the Hermitian matrix, R, for the generating function such that
%
%      [ T+rI  T   0]
%  R = [  T    0   I]
%      [  0    I   0]
%
% where T is the positive-definite Toeplitz-like Block matrix, specifically
% to obtain the Schur complement of inv(T*T)
%
% [G,J,rowcol] = GET_EIGEN_BLOCK_TRACEQQ_A1(M,hyp,r)
%
% Inputs:   M       : Time-series Inputs of Gaussian process, size nx1
%           hyp     : hyperparameters of the covariance function [g v]
%           r       : coefficient, fudge factor
%
% 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|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_traceqq_a1(M,hyp,coef)

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

rowcol=g_row(M);

n=size(M,1);
nn=3*n;
rk2=size(rowcol,2);
rk=rk2*2;
A=zeros(rk);
B=zeros(nn,rk);
tmp1=g_toe(M,hyp,1);
B(n+1:2*n,1)=tmp1;
B(1:n,rk2+1)=tmp1;
B(1:n,1)=tmp1;
B(1,1)=B(1,1)+coef;
B(2*n+1,rk2+1)=1;
for k=2:rk2
    p=rowcol(k);
    tmp1=g_toe(M,hyp,p);
    tmp3=g_toe(M,hyp,p-1);
    tmp2=[0;tmp3(1:end-1,:)];
    tmp3=tmp1-tmp2;
    B(n+1:2*n,k)=tmp3;
    B(1:n,rk2+k)=tmp3;
    B(1:n,k)=tmp3;
end
tmp0=zeros(nn,rk);
for k=1:rk2
    p=rowcol(k);
    A(k,:)=B(p,:);
    tmp0(p,k)=1;
    p=n+p;
    A(rk2+k,:)=B(p,:);
    tmp0(p,rk2+k)=1;
end

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