We compute the temperature-dependent spin-wave spectrum and the magnetization
for a spin system using the unified decoupling procedure for the high-order
Green's functions for the exchange coupling and anisotropy, both in the
classical and quantum case. Our approach allows us to establish a clear
crossover between quantum-mechanical and classical methods by developing the
classical analog of the quantum Green's function technique. The results are
compared with the classical spectral density method and numerical modeling
based on the stochastic Landau-Lifshitz equation and the Monte Carlo technique.
As far as the critical temperature is concerned, there is a full agreement
between the classical Green's functions technique and the classical spectral
density method. However, the former method turns out to be more straightforward
and more convenient than the latter because it avoids any \emph{a priori}
assumptions about the system's spectral density. The temperature-dependent
exchange stiffness as a function of magnetization is investigated within
different approaches