Elyseeva, J. A transformation for symplectic systems and the definition of a focal point. (English) Zbl 1049.93056 Comput. Math. Appl. 47, No. 1, 123-134 (2004). Linear time-varying discrete-time systems with system matrices having the symplectic property are considered. State variables are collected into a \(2n\times n\) block matrix instead of the usual \(n\)-dimensional vector. Although no mention of potential applications is given, such system structures are useful in fast calculation of the solutions to Riccati difference equations in the form of so-called “doubling algorithms”. A related Riccati operator is defined for the matrix difference equation and used to state Sturm’s separation theorems. Reviewer: Edwin Engin Yaz (Milwaukee) Cited in 6 Documents MSC: 93C55 Discrete-time control/observation systems 39A12 Discrete version of topics in analysis 37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010) 93B40 Computational methods in systems theory (MSC2010) Keywords:discrete version of topics in analysis; symplectic mappings; fixed points; discrete-time systems; focal point; Riccati difference operator; conjoined basis; linear time-varying systems; fast calculation; doubling algorithms; Sturm’s separation theorems PDFBibTeX XMLCite \textit{J. Elyseeva}, Comput. Math. Appl. 47, No. 1, 123--134 (2004; Zbl 1049.93056) Full Text: DOI References: [1] Nelson, P.; Ray, A.; Wing, G., On the effectiveness of the inverse Riccati transformation in the matrix case, J. Math. Anal. Appl., 65, 201-210 (1978) · Zbl 0392.65035 [2] Bohner, M.; Došlý, O., Disconjugacy and transformations for symplectic systems, Rocky Mountain J. Math., 27, 3, 707-743 (1997) · Zbl 0894.39005 [3] Bohner, M., Riccati matrix difference equations and linear Hamiltonian difference system, Dynam. Conlin. Discrete Impuls, Systems, 2, 2, 147-159 (1996) [4] Kratz, W., Discrete oscillation, J. Difference Equations and Appl., 9, 1, 135-147 (2003) · Zbl 1038.39010 [5] Golub, G.; Van Loan, C., Matrix Computations (1996), The Johns Hopkins University Press · Zbl 0865.65009 [6] Zelikin, M., Uniform Spaces and the Riccati Equation in Calculus of Variations (1998), Factorial: Factorial Moscow · Zbl 0951.49001 [7] Eliseeva, Y., An algorithm for solving the matrix difference Riccati equation, Comp. Math. and Math. Phys., 39, 2, 187-194 (1999) [8] Taufer, J., On factorization method, Aplikace Matematiky, 11, 427-451 (1966) · Zbl 0149.04803 [9] Eliseeva, Y., On an algorithm for solving the symplectic matrix Riccati equation, Moscow University Comp. Math. Cyber., 2, 14-19 (1990) · Zbl 0704.34032 [10] Gantmacher, F., (The Theory of Matrices, Volume 1 (1959), Chelsea Publishing: Chelsea Publishing New York) · Zbl 0085.01001 [11] Bohner, M., Discrete Sturmian theory, Math. Inequal. Appl., 1, 3, 375-383 (1998) · Zbl 0907.39019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.