  We present a thorough analysis of computing the Generalized Alignment Index
(GALI), a rapid and effective chaos indicator, through a simple multi-particle
approach, which avoids the use of variational equations. We develop a
theoretical leading-order estimation of the error in the computed GALI for both
the variational method (VM) and the multi-particle method (MPM), and confirm
its predictions through extensive numerical simulations of two well-known
Hamiltonian models: the H\'enon-Heiles and the $\beta$-Fermi-Pasta-Ulam-Tsingou
systems. For these models the GALIs of several orders are computed and the MPM
results are compared to the VM outcomes. The dependence of the accuracy of the
MPM on the renormalization time, integration time step, as well as the
deviation vector size, is studied in detail. We find that the implementation if
the MPM in double machine precision ($\varepsilon \approx 10^{-16}$) is
reliable for deviation vector magnitudes centred around $d_0\approx
\varepsilon^{1/2}$, renormalization times $\tau \lesssim 1$, and relative
energy errors $E_r \lesssim \varepsilon^{1/2}$. These results are valid for
systems with many degrees of freedom and for several orders of the GALIs, with
the MPM particularly capturing very accurately the $\textrm{GALI}_2$ behavior.
Our results show that the computation of the GALIs by the MPM is a robust and
efficient method for investigating the global chaotic dynamics of autonomous
Hamiltonian systems, something which is of distinct importance in cases where
it is difficult to explicitly write the system's variational equation or when
these equations are too cumbersome.
