  In this note, we propose a symplectic algorithm for the stable manifolds of
the Hamilton-Jacobi equations combined with an iterative procedure in
[Sakamoto-van~der Schaft, IEEE Transactions on Automatic Control, 2008]. Our
algorithm includes two key aspects. The first one is to prove a precise
estimate for radius of convergence and the errors of local approximate stable
manifolds. The second one is to extend the local approximate stable manifolds
to larger ones by symplectic algorithms which have better long-time behaviors
than general-purpose schemes. Our approach avoids the case of divergence of the
iterative sequence of approximate stable manifolds, and reduces the computation
cost. We illustrate the effectiveness of the algorithm by an optimal control
problem with exponential nonlinearity.
