On the Semigroup of Standard Symplectic Matrices and its Applications

A matrix Z is an element of R-2n x 2n is said to be in the standard symplectic form if Z enjoys a block LU-decomposition in the sense of [(A)(-H) (0)(1)] z = [(1)(0) T-A(G)], where A is nonsingular and both G and H are symmetric and positive definite in R-n x n. Such a structure arises naturally in the discrete algebraic Riccati equations. This note contains two results. First, by means of a parameter representation it is shown that the set of all 2n x 2n standard symplectic matrices is closed under multiplication and, thus, forms a semigroup. Secondly, block LU-decompositions of powers of Z can be derived in closed form which, in turn, can be employed recursively to induce an effective structure-preserving algorithm for solving the Riccati equations. The computational cost of doubling and tripling of the powers is investigated. It is concluded that doubling is the better strategy.