  We consider the generating function (prepotential) for Gromov-Witten
invariants of rational elliptic surface. We apply the local mirror principle to
calculate the prepotential and prove a certain recursion relation, holomorphic
anomaly equation, for genus 0 and 1. We propose the holomorphic anomaly
equation for all genera and apply it to determine higher genus Gromov-Witten
invariants and also the BPS states on the surface. Generalizing G\"ottsche's
formula for the Hilbert scheme of $g$ points on a surface, we find precise
agreement of our results with the proposal recently made by Gopakumar and
Vafa(hep-th/9812127).
