Miyajima, Shinya Fast verified computation for stabilizing solutions of discrete-time algebraic Riccati equations. (English) Zbl 1360.65127 J. Comput. Appl. Math. 319, 352-364 (2017). Summary: Fast iterative algorithms for computing interval matrices containing solutions of discrete-time algebraic Riccati equations are proposed. These algorithms involve only cubic complexity per iteration. The stabilizability and uniqueness of the contained solution can moreover be verified by these algorithms. Numerical results show the properties of the algorithms. Cited in 5 Documents MSC: 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 93C55 Discrete-time control/observation systems 93C05 Linear systems in control theory 65G30 Interval and finite arithmetic 65F10 Iterative numerical methods for linear systems Keywords:stabilizing solution; verified computation; iterative algorithms; interval matrices; discrete-time algebraic Riccati equations; numerical results Software:DAREX; mftoolbox; INTLAB PDFBibTeX XMLCite \textit{S. Miyajima}, J. Comput. Appl. Math. 319, 352--364 (2017; Zbl 1360.65127) Full Text: DOI References: [1] Bini, D. A.; Iannazzo, B.; Meini, B., Numerical Solution of Algebraic Riccati Equations (2012), SIAM Publications: SIAM Publications Philadelphia · Zbl 1244.65058 [2] Emami-Naeini, A.; Franklin, G., Comments on the numerical solution of the discrete-time algebraic Riccati equation, IEEE Trans. Automat. Control, 25, 1015-1016 (1980) · Zbl 0475.93039 [3] Lancaster, P.; Rodman, L., Algebraic Riccati Equations (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0836.15005 [5] Luther, W.; Otten, W., Verified calculation of the solution of algebraic Riccati equation, (Csendes, T., Developments in Reliable Computing (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 105-118 · Zbl 0951.65040 [7] Miyajima, S., Fast verified computation for solutions of continuous-time algebraic Riccati equations, Japan J. Indust. Appl. Math., 32, 529-544 (2015) · Zbl 1329.65093 [8] Haqiri, T.; Poloni, F., Methods for verified solutions to continuous-time algebraic Riccati equations, J. Comput. Appl. Math., 313, 515-535 (2017) · Zbl 1353.65034 [9] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0801.15001 [10] Higham, N. J., Functions of Matrices: Theory and Computation (2008), SIAM Publications: SIAM Publications Philadelphia · Zbl 1167.15001 [11] Meyer, C. D., Matrix Analysis and Applied Linear Algebra (2000), SIAM Publications: SIAM Publications Philadelphia [13] Miyajima, S., Fast enclosure for all eigenvalues and invariant subspaces in generalized eigenvalue problems, SIAM J. Matrix Anal. Appl., 35, 1205-1225 (2014) · Zbl 1307.65044 [14] Ionescu, V.; Weiss, M., On computing the stabilizing solution of the discrete-time Riccati equation, Linear Algebra Appl., 174, 229-238 (1992) · Zbl 0756.15020 [15] Caprani, O.; Madsen, K., Iterative methods for interval inclusion of fixed points, BIT, 18, 42-51 (1978) · Zbl 0401.65035 [16] Rump, S. M., Verification methods for dense and sparse systems of equations, (Herzberger, J., Topics in Validated Computations - Studies in Computational Mathematics (1994), Elsevier: Elsevier Amsterdam), 63-136 · Zbl 0813.65072 [17] Arndt, H., On the interval systems \([x] = [A] [x] + [b]\) and the powers of interval matrices in complex interval arithmetics, Reliab. Comput., 13, 245-259 (2007) · Zbl 1115.65033 [19] Rump, S. M., INTLAB - INTerval LABoratory, (Csendes, T., Developments in Reliable Computing (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 77-104 · Zbl 0949.65046 [20] Bouhamidi, A.; Heyouni, M.; Jbilou, K., Block Arnoldi-based methods for large scale discrete-time algebraic Riccati equations, J. Comput. Appl. Math., 236, 1531-1542 (2011) · Zbl 1236.93054 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.