  This paper presents a novel {\em Interpolated Factored Green Function} method
(IFGF) for the accelerated evaluation of the integral operators in scattering
theory and other areas. Like existing acceleration methods in these fields, the
IFGF algorithm evaluates the action of Green function-based integral operators
at a cost of $\mathcal{O}(N\log N)$ operations for an $N$-point surface mesh.
The IFGF strategy, which leads to an extremely simple algorithm, capitalizes on
slow variations inherent in a certain Green function {\em analytic factor},
which is analytic up to and including infinity, and which therefore allows for
accelerated evaluation of fields produced by groups of sources on the basis of
a recursive application of classical interpolation methods. Unlike other
approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT),
and is thus better suited than other methods for efficient parallelization in
distributed-memory computer systems. Only a serial implementation of the
algorithm is considered in this paper, however, whose efficiency in terms of
memory and speed is illustrated by means of a variety of numerical experiments
-- including a 43 min., single-core operator evaluation (on 10 GB of peak
memory), with a relative error of $1.5\times 10^{-2}$, for a problem of
acoustic size of 512$\lambda$.
