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Hua Dai^{1}, ZhongZhi Bai^{2}
Hua Dai, ZhongZhi Bai. ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS[J]. Journal of Computational Mathematics, 2011, 29(3): 341366.
[1] B.D.O. Anderson, Secondorder convergent algorithms for the steadystate Riccati equation,Intern. J. Control, 28 (1978), 295306. [2] B.D.O. Anderson and J.B. Moore, Optimal Filtering, PrenticeHall, Englewood Cliffs, New Jersey,1979. [3] W.F. Arnold III and A.J. Laub, Generalized eigenproblem algorithms and software for algebraicRiccati equations, Proc. IEEE, 72 (1984), 17461754. [4] A.Y. Barraud, A numerical algorithm to solve A^{T}XAX = Q, IEEE Trans. Automat. Control,AC22 (1977), 883885. [5] P. Benner, A.J. Laub and V. Mehrmann, A Collection of Benchmark Examples for the NumericalSolution of Algebraic Riccati Equations II: DiscreteTime Case, Technical Report SPC 9523,Faculty of Mathematics, TU ChemnitzZwickau, D09107 Chemnitz, December 1995. [6] P. Benner, V. Mehrmann and H.G. Xu, A numerically stable, structure preserving method forcomputing the eigenvalues of real Hamiltonian or symplectic pencils, Numer. Math., 78 (1998),329358. [7] H. Dai, The Theory of Matrices, Science Press, Beijing, 2001. [8] R. Davies, P. Shi and R. Wiltshire, New upper solution bounds of the discrete algebraic Riccatimatrix equation, J. Comput. Appl. Math., 213 (2008), 307315. [9] J. Garloff, Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunovequations and the continuous Lyapunov equation, Intern. J. Control, 43 (1986), 423431. [10] G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd Edition, The Johns Hopkins UniversityPress, Baltimore and London, MD, 1996. [11] C.H. Guo, Newton's method for discrete algebraic Riccati equations when the closedloop matrixhas eigenvalues on the unit circle, SIAM J. Matrix Anal. Appl., 20 (1998), 279294. [12] T.M. Hwang, E.K.W. Chu and W.W. Lin, A generalized structurepreserving doubling algorithm for generalized discretetime algebraic Riccati equations, Intern. J. Control, 78 (2005),10631075. [13] S.W. Kim, P. Park and W.H. Kwon, Lower bounds for the trace of the solution of the discretealgebraic Riccati equation, IEEE Trans. Automat. Control, 38 (1993), 312314. [14] M. Kimura, Convergence of the doubling algorithm for the discretetime algebraic Riccati equation,Intern. J. Systems Sci., 19 (1988), 701711. [15] G. Kitagawa, An algorithm for solving the matrix equation X = FXF^{T} + S, Intern. J. Control,25 (1977), 745753. [16] N. Komaroff, Upper bounds for the eigenvalues of the solution of the Lyapunov matrix equation,IEEE Trans. Automat. Control, 35 (1990), 737739. [17] N. Komaroff, Upper bounds for the solution of the discrete Riccati equation, IEEE Trans.Automat. Control, 37 (1992), 13701372. [18] N. Komaroff, Iterative matrix bounds and computational solutions to the discrete algebraic Riccatiequation, IEEE Trans. Automat. Control, 39 (1994), 16761678. [19] N. Komaroff and B. Shahian, Lower summation bounds for the discrete Riccati and Lyapunovequations, IEEE Trans. Automat. Control, 37 (1992), 10781080. [20] V.S. Kouikoglou and Y.A. Phillis, Trace bounds on the covariances of continuoustime systemswith multiplicative noise, IEEE Trans. Automat. Control, 38 (1993), 138142. [21] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, John Wiley & Sons, New York,1972. [22] B.H. Kwon, M.J. Youn and Z. Bien, On bounds of the Riccati and Lyapunov matrix equations,IEEE Trans. Automat. Control, AC30 (1985), 11341135. [23] W.H. Kwon, Y.S. Moon and S.C. Ahn, Bounds in algebraic Riccati and Lyapunov equations: asurvey and some new results, Intern. J. Control, 64 (1996), 377389. [24] P. Lancaster and L. Rodman, Algebraic Riccati Equations, The Clarendon Press, Oxford andNew York, 1995. [25] A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat.Control, AC24 (1979), 913921. [26] C.H. Lee, On the matrix bounds for the solution matrix of the discrete algebraic Riccati equation,IEEE Trans. Circuits SystemsI: Fundamental Theory Appl., 43 (1996), 402407. [27] C.H. Lee, Upper matrix bound of the solution for the discrete Riccati equation, IEEE Trans.Automat. Control, 42 (1997), 840842. [28] C.H. Lee, Upper and lower bounds of the solutions of the discrete algebraic Riccati and Lyapunovmatrix equations, Intern. J. Control, 68 (1997), 579598. [29] C.H. Lee, Simple stabilizability criteria and memoryless state feedback control design for timedelay systems with timevarying perturbations, IEEE Trans. Circuits SystemsI: FundamentalTheory Appl., 45 (1998), 12111215. [30] C.H. Lee, Matrix bounds of the solutions of the continuous and discrete Riccati equationsaunified approach, Intern. J. Control, 76 (2003), 635642. [31] W.W. Lin and C.S. Wang, On computing stable Lagrangian subspaces of Hamiltonian matricesand symplectic pencils, SIAM J. Matrix Anal. Appl., 18 (1997), 590614. [32] W.W. Lin and S.F. Xu, Convergence analysis of structurepreserving doubling algorithms forRiccatitype matrix equations, SIAM J. Matrix Anal. Appl., 28 (2006), 2639. [33] L.Z. Lu and W.W. Lin, An iterative algorithm for the solution of the discretetime algebraicRiccati equation, Linear Algebra Appl., 188/189 (1993), 465488. [34] L.Z. Lu, W.W. Lin, and C.E.M. Pearce, An efficient algorithm for the discretetime algebraicRiccati equation, IEEE Trans. Automat. Control, 44 (1999), 12161220. [35] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, AcademicPress, New York and London, 1979. [36] V. Mehrmann, The Autonomous Linear Quadratic Control Problem: Theory and NumericalSolution, in Lecture Notes in Control and Information Sciences 163, SpringerVerlag, Berlin,1991. [37] T. Mori and I.A. Derese, A brief summary of the bounds on the solution of the algebraic matrixequations in control theory, Intern. J. Control, 39 (1984), 247256. [38] T. Mori, N. Fukuma and M. Kuwahara, On the discrete Riccati equation, IEEE Trans. Automat.Control, AC32 (1987), 828829. [39] J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables,SIAM, Philadelphia, PA, 2000. [40] T. Pappas, A.J. Laub and N.R. Sandell, On the numerical solution of the discretetime algebraicRiccati equation, IEEE Trans. Automat. Control, AC25 (1980), 631641. [41] G. Schulz, Iterative berechnung der reziproken matrix, Z. Angew. Math. Mech., 13 (1933), 5759. [42] M.T. Tran and M.E. Sawan, On the discrete Riccati matrix equation, SIAM J. Alg. Disc. Meth.,6 (1985), 107108. [43] S.S. Wang, B.S. Chen and T.P. Lin, Robust stability of uncertain timedelay systems, Intern.J. Control, 46 (1987), 963976. [44] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, 2nd Edition, SpringerVerlag, New York and Berlin, 1979. [45] K. Yasuda and K. Hirai, Upper and lower bounds on the solution of the algebraic Riccati equation,IEEE Trans. Automat. Control, AC24 (1979), 483487. [46] X.Z. Zhan, Computing the extremal positive definite solutions of a matrix equation, SIAM J.Sci. Comput., 17 (1996), 11671174. 
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