The algorithm being developed here is based on the generating function approach for finite-time H control and application of canonical transformation of linear Hamiltonian system. First, an equivalent finite-time H control law in terms of the third-type generating function is derived. Then, by using symplectic structure of the Hamiltonian system's state transition matrix, a group of matrix recursive formulae are deduced for the evaluation of the finite-time H control law. Combining with a matrix singularity testing procedure, this recursive algorithm verifies the existence condition of sub-optimal H controllers and gives the minimum H norm of finite-time control systems. Inherited from the canonical transformation, the matrix recursive formulae have a standard symplectic form; this structure-preserving property helps facilitate reliable and effective computation. Numerical results show the effectiveness of the proposed algorithm.