  We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of
particles which dynamically interacts via the scattering matrix of affine Toda
field theory and whose statistical interaction is of a general Haldane type. Up
to the first leading order, we provide general approximated analytical
expressions for the solutions of these equations from which we derive general
formulae for the ultraviolet scaling functions for theories in which the
underlying Lie algebra is simply laced. For several explicit models we compare
the quality of the approximated analytical solutions against the numerical
solutions. We address the question of existence and uniqueness of the solutions
of the TBA-equations, derive precise error estimates and determine the rate of
convergence for the applied numerical procedure. A general expression for the
Fourier transformed kernels of the TBA-equations allows to derive the related
Y-systems and a reformulation of the equations into a universal form.
