% EXER24_3  Plot and compare  ||exp(t A)||_2  to  exp(t alpha(A)).

expers = 5;            % Tref & Bau call for 10 expers but this displays less cluttered
t = (0:.1:20)';
ynorm  = zeros(length(t),expers);
yother = zeros(length(t),expers);
for K=1:expers
  m = 10;  A = randn(m) - 2*eye(m);  % shift average real part of eigs to the left by 2
  lam = eig(A);
  alpha = max(real(lam));
  disp(['K = ' num2str(K) ':'])
  disp(lam(real(lam) > alpha - 1.0e-6)')  % show eigs with biggest real parts
  for j=1:length(t)
    ynorm(j,K)  = norm(expm(t(j) * A));
    yother(j,K) = exp(t(j) * alpha);
  end
end
%semilogy(t,ynorm,'-',t,yother,'k:')  % one way to view
semilogy(t,ynorm./yother)             % clear view!
legend('1','2','3','4','5')