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.

On the Semigroup of Standard Symplectic Matrices and its Applications / Chu, M.; Del Buono, N.; Diele, F.; Politi, T.; Ragni, S.. - In: LINEAR ALGEBRA AND ITS APPLICATIONS. - ISSN 0024-3795. - STAMPA. - 389:1-3(2004), pp. 215-225. [10.1016/j.laa.2004.03.017]

On the Semigroup of Standard Symplectic Matrices and its Applications

Politi, T.;
2004-01-01

Abstract

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.
2004
On the Semigroup of Standard Symplectic Matrices and its Applications / Chu, M.; Del Buono, N.; Diele, F.; Politi, T.; Ragni, S.. - In: LINEAR ALGEBRA AND ITS APPLICATIONS. - ISSN 0024-3795. - STAMPA. - 389:1-3(2004), pp. 215-225. [10.1016/j.laa.2004.03.017]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/5453
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