  The convergence of the iterative solutions of the transport equations of
cosmic muon and tau neutrinos propagating through Earth is studied and
analyzed. For achieving a fast convergence of the iterative solutions of the
coupled transport equations of nu_tau, nubar_tau and the associated tau^{\pm}
fluxes, a new semi-analytic input algorithm is presented where the peculiar
tau-decay contributions are implemented already in the initial zeroth order
input. Furthermore, the common single transport equation for muon neutrinos is
generalized by taking into account the contributions of secondary nu_mu and
nubar_mu fluxes due to the prompt tau-decay tau -> nu_mu initiated by the
associated tau flux. Differential and total nadir angle integrated upward-going
mu^- + mu^+ event rates are presented for underground neutrino telescopes and
compared with the muon rates initiated by the primary nu_mu, nu_tau and tau
fluxes.
