  It is demonstrated is this letter that linear multistep methods for
integrating ordinary differential equations can be used to develop a family of
fast forward scattering algorithms with higher orders of convergence. Excluding
the cost of computing the discrete eigenvalues, the nonlinear Fourier transform
(NFT) algorithm thus obtained has a complexity of $O{KN+C_pN\log^2N}$ such that
the error vanishes as $O{N^{-p}}$ where $p\in\{1,2,3,4\}$ and $K$ is the number
of eigenvalues. Such an algorithm can be potentially useful for the recently
proposed NFT based modulation methodology for optical fiber communication. The
exposition considers the particular case of the backward differentiation
formula ($C_p=p^3$) and the implicit Adams method ($C_p=(p-1)^3$) of which the
latter proves to be the most accurate family of methods for fast NFT.
