For a diffeomorphism F on $/IR/sp2$, it is possible to find periodic orbits of period k by applying Newton 's method to the function F ^k - I , where I is the identity function. Here we use variations of Newton 's method which are more robust than the traditional Newton 's method. For an initial point x, we iterate Newton 's method many times. If the process converges to a point p which is a periodic point of F, we say x is in the Newton basin of p for period k. We investigate the size of the Newton basin and how it depends on p and k. We show that if p is an attracting periodic point, then there is an open neighborhood of p that is in the Newton basin of p for all k. If p is a repelling periodic point, it is possible that p is the only point which remains in the Newton basin for all k. It is when p is a periodic saddle point that the Newton basin has its most interesting behavior. We present numerical data which indicates that the area of the basin of a periodic saddle point p is proportional to $[1/over /lambda/sp[c]]$ where λ is the unstable eigenvalue of DF^k(p) and c is approximately 1 (c ≈ 0.84 in Figure 2.4). For long periods (k more than about 20), many orbits have λ so large that the basins are numerically undetectable. Our main result states that if p is a saddle point of F, the Newton basin of p includes a narrow neighborhood of the local stable manifold of p.