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On the semigroup of standard symplectic matrices and its applications (Articolo in rivista)

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On the semigroup of standard symplectic matrices and its applications (Articolo in rivista) (literal)

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Chu M.T.; Del Buono N.; Diele F.; Politi T.; Ragni S. (2004)
On the semigroup of standard symplectic matrices and its applications
in Linear algebra and its applications
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CHU Moody T., North Carolina State University, USA DEL BUONO Nicoletta, Università degli Studi di Bari POLITI Tiziano, Politecnico di Bari RAGNI Stefania, Università degli Studi di Bari (literal)

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Abstract

A matrix Z ? R2n×2n is said to be in the standard symplectic form if Z enjoys a block LU-decomposition in the sense of ? A 0 -H I ? Z = ? I G 0 AT ? , where A is nonsingular and both G and H are symmetric and positive definite in Rn×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 × 2n standard symplectic matrices is closed undermultiplication 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. (literal)

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