  We extend the key idea behind the generalized Petviashvili method of Ref.
\cite{gP} by proposing a novel technique based on a similar idea. This
technique systematically eliminates from the iteratively obtained solution a
mode that is "responsible" either for the divergence or the slow convergence of
the iterations. We demonstrate, theoretically and with examples, that this mode
elimination technique can be used both to obtain some nonfundamental solitary
waves and to considerably accelerate convergence of various iteration methods.
As a collateral result, we compare the linearized iteration operators for the
generalized Petviashvili method and the well-known imaginary-time evolution
method and explain how their different structures account for the differences
in the convergence rates of these two methods.
