  We define Chern and Segre forms, or rather currents, associated with a
Griffiths positive singular hermitian metric $h$ with analytic singularities on
a holomorphic vector bundle $E$. The currents are constructed as pushforwards
of generalized Monge-Amp\`ere products on the projectivization of $E$. The
Chern and Segre currents represent the Chern and Segre classes of $E$,
respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where
$h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms
constructed by the first three authors and Ruppenthal in the cases when these
are defined.
