% Proof of Lemma 7.1

startintlab; format long e; intvalinit('DisplayInfsup')

n = 1142; p_n = 9209;
A = spdiags(primes(p_n)',0,n,n); e = ones(n,2);
for k=2.^(0:floor(log2(n))), A = A + spdiags(e,[-k k],n,n); end
[V,D] = eig(full(A)); V = intval(V);
R = V*D-A*V;
for i=1:n, lambda(i) = midrad(D(i,i),norm(R(:,i))/norm(V(:,i))); end
[lambda0,j] = min(inf(lambda));
lambda_min = infsup(lambda(j))

lambda0 = intval(1.120651470854673);
alpha2 = 11*100; lambda1 = 9221-100;
lambdaF = infsup(2*(lambda0*lambda1-alpha2)/...
                     (lambda0+lambda1+sqrt(4*alpha2+(lambda1-lambda0)^2)))