  Quantum chaos manifests itself also in algorithmical complexity of methods,
including the numerical ones, in solving the Schr\"odinger equation. In this
contribution we address the problem of calculating the eigenenergies and the
eigenstates by various numerical methods applied to 2-dim generic billiard
systems. In particular we analyze the dependence of the accuracy (errors) on
the density of discretization of the given numerical method. We do this for
several different billiard shapes, especially for the Robnik billiard. We study
the numerical error of the boundary integral method and the plane wave
decomposition method as a function of the discretization parameter $b$ which by
definition is the number of discretization nodes on the boundary per one de
Broglie wavelength (arclength) interval. For boundary integral method, we
discover that at each $\lambda$ the error scales as a power law $<|\Delta E|> =
A b^{-\alpha}$, where $\alpha$ is a strong function of $\lambda$: In the
KAM-like regime $0\le \lambda\le 1/4$ it is large and close to 3.5, but close
to $\lambda = 1/4$ it changes almost discontinuously becoming hardly any larger
than zero. This is because the billiard becomes nonconvex beginning at
$\lambda=1/4$. For the plane wave decomposition method, we found at a fixed $b$
that the average absolute value of the error $<|\Delta E|>$ correlates strongly
with the classical chaos, where the error $<|\Delta E|>$ does increase sharply
with increasing classical chaos. This property is valid also for Bunimovich
stadium and the Sinai billiard.
