  Harmonic Balance is one of the most popular methods for computing periodic
solutions of nonlinear dynamical systems. In this work, we address two of its
major shortcomings: First, we investigate to what extent the computational
burden of stability analysis can be reduced by consistent use of Chebyshev
polynomials. Second, we address the problem of a rigorous error bound, which,
to the authors' knowledge, has been ignored in all engineering applications so
far. Here, we rely on Urabe's error bound and, again, use Chebyshev polynomials
for the computationally involved operations. We use the error estimate to
automatically adjust the harmonic truncation order during numerical
continuation, and confront the algorithm with a state-of-the-art adaptive
Harmonic Balance implementation. Further, we rigorously prove, for the first
time, the existence of some isolated periodic solutions of the forced-damped
Duffing oscillator with softening characteristic. We find that the effort for
obtaining a rigorous error bound, in its present form, may be too high to be
useful for many engineering problems. Based on the results obtained for a
sequence of numerical examples, we conclude that Chebyshev-based stability
analysis indeed permits a substantial speedup. Like Harmonic Balance itself,
however, this method becomes inefficient when an extremely high truncation
order is needed as, e.g., in the presence of (sharply regularized)
discontinuities.
