%  [a,ia,js] = SPBuild_a(h,qcols,neq)
%
%  Build the companion matrix, deleting inessential lags.
%  Solve for x_{t+nlead} in terms of x_{t+nlag},...,x_{t+nlead-1}.

function [a,ia,js] = SPBuild_a(h,qcols,neq)

left  = 1:qcols;
right = qcols+1:qcols+neq;
hs=SPSparse(h);

hs(:,left) = -hs(:,right)\hs(:,left);

%  Build the big transition matrix.

a = zeros(qcols,qcols);

if(qcols > neq)
  eyerows = 1:qcols-neq;
  eyecols = neq+1:qcols;
  a(eyerows,eyecols) = eye(qcols-neq);
end
hrows      = qcols-neq+1:qcols;
a(hrows,:) = hs(:,left);

%  Delete inessential lags and build index array js.  js indexes the
%  columns in the big transition matrix that correspond to the
%  essential lags in the model.  They are the columns of q that will
%  get the unstable left eigenvectors. 

js       = 1:qcols;
zerocols = sum(abs(a)) == 0;
while( any(zerocols) )
  a(:,zerocols) = [];
  a(zerocols,:) = [];
  js(zerocols)  = [];
  zerocols = sum(abs(a)) == 0;
end
ia = length(js);