Consider the ODEs 
and 
. In each case we compute 
, so the condition number bound is 
in both problems. However, they behave quite differently. In the case of exponential growth, , the bound is the actual condition number.
and . In each case we compute , so the condition number bound is in both problems. However, they behave quite differently. In the case of exponential growth, , the bound is the actual condition number.for u0 = [0.7 1 1.3]
fplot(@(t) exp(t)*u0,[0 3]), hold on
end
xlabel('t'), ylabel('u(t)') % ignore this line
title('Exponential divergence of solutions') % ignore this line
But with , solutions actually get closer together with time.
, solutions actually get closer together with time.clf
for u0 = [0.7 1 1.3]
fplot(@(t) exp(-t)*u0,[0 3]), hold on
end
xlabel('t'), ylabel('u(t)') % ignore this line
title('Exponential convergence of solutions') % ignore this line
In this case the actual condition number is one, due to the difference of solutions at the initial time.