Optimal unit triangular factorization of symplectic matrices

We prove that any symplectic matrix can be factored into no more than 5 unit triangular matrices, moreover, is the optimal number. This result improves existing factorization of matrices which gives proof 9 factors. also show corresponding improved conclusions for structured subsets matrices. further provides an unconstrained optimization method on 2d-by-2d real group (a 2d2+d-dimensional Lie group) with 2d2+3d parameters.