  We performed a rigorous theoretical convergence analysis of the discrete
dipole approximation (DDA). We prove that errors in any measured quantity are
bounded by a sum of a linear and quadratic term in the size of a dipole d, when
the latter is in the range of DDA applicability. Moreover, the linear term is
significantly smaller for cubically than for non-cubically shaped scatterers.
Therefore, for small d errors for cubically shaped particles are much smaller
than for non-cubically shaped. The relative importance of the linear term
decreases with increasing size, hence convergence of DDA for large enough
scatterers is quadratic in the common range of d. Extensive numerical
simulations were carried out for a wide range of d. Finally we discuss a number
of new developments in DDA and their consequences for convergence.
