function spectralmarkov(m0,n0)
global m n
randomize
m = m0;
n = n0;

global pstr
pstr = 'spectralmarkov'
global lstr rstr

global C
C=ones(m,1)/m
%C=rand(m,1); C=C/sum(C);

pdim = m*m;
%fdim = m;
%xdim = pdim + fdim;
xdim = pdim;
LB = zeros(xdim,1);
LB(pdim+1:end) = 0*LB(pdim+1:end) - 1e6;
UB = LB*0;
UB(1:pdim) = 0*UB(1:pdim) + 1;
UB(pdim+1:end) = 0*UB(pdim+1:end) + 1e6;

[Aeq0, Beq] = normab(m,m);
%Aeq = assembleblock({Aeq0,ones(1,fdim)});
%Beq(end+1) = 0;
Aeq = Aeq0;

global maxr
maxr = 0;
while 1
  A = pnormdim(rand(m),1);
  %f = randn(m,1);
  %x = [A(:);f(:)];
  x = [A(:)];
  y = fmincon(@radrat,x,[],[],Aeq,Beq,LB,UB);
  fprintf('%s: n=%d; m=%d; rat = %s/%s;  maxr = %1.9f \n',pstr,n,m,lstr,rstr,maxr);
end

return


function y = radrat(x)
global m n
global lstr rstr
global pstr
global C
pdim = m*m;
%fdim = m;
%xdim = pdim + fdim;
xdim = pdim;

A = makepos(reshape(x(1:pdim),[m m]));
%f = x(pdim+1:end);

P = markfillprob(A,C,n);

ae = sort(abs(eig(A)));
lam2 = ae(end-1);
slam = sum(ae(1:end-1));
tha = stricthh(A);
mtv = mintv(A);


%rstr = 'tha';
%rstr = 'mtv';
%lstr = 'lam2';


Ln = sum(lam2.^(0:n-1));
%Ln = sum(sqrt(lam2).^(0:n-1));
[hh,Ht,Hn] = gethhn(P,m,n);
hh = hh+eye(n);
H2 = (max(eig(hh'*hh)));


%lstr = 'TV(A*f)';
%rstr = 'lam2*TV(f)';

lstr = 'Hn';
%lstr = 'H2';
%rstr = 'Ln';
%rstr = '1/(1-lam2)';
rstr = '1/(1-slam)';



lhs = eval(lstr);
rhs = eval(rstr)+1e-12;
r = lhs/rhs;
y = -r;
global maxr
if r > maxr
  global optA optC optf
  optA = A;
  optC = C;
  %optf = f;
  maxr = r;
  fprintf('%s: n=%d; m=%d; rat = %s/%s;  maxr = %1.9f \n',pstr,n,m,lstr,rstr,maxr);
end


return