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Iterative solution of two matrix equations. (English) Zbl 0940.65036

Iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations \(X+ A^*X^{-1}A= Q\) and \(X- A^*A^{-1}A= Q\) are studied. Here \(Q\) is Hermitian positive definite and the ordering \(X\leq Y\) if \(X- Y\) is positive semidefinite. The convergence rates of various algorithms depend on the eigenvalues of \((X_+)^{-1}A\), where \(X_+\) is a solution. These eigenvalues are related to the eigenvalues of a matrix pencil which is independent of \(X_+\).

MSC:

65F10 Iterative numerical methods for linear systems
65F30 Other matrix algorithms (MSC2010)
15A22 Matrix pencils
15A24 Matrix equations and identities
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[1] W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, Positive solutions to \?=\?-\?\?\(^{-}\)\textonesuperior \?*, Linear Algebra Appl. 134 (1990), 53 – 62. · Zbl 0702.15009
[2] Jacob C. Engwerda, On the existence of a positive definite solution of the matrix equation \?+\?^{\?}\?\(^{-}\)\textonesuperior \?=\?, Linear Algebra Appl. 194 (1993), 91 – 108. · Zbl 0798.15013
[3] Jacob C. Engwerda, André C. M. Ran, and Arie L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \?+\?*\?\(^{-}\)\textonesuperior \?=\?, Linear Algebra Appl. 186 (1993), 255 – 275. · Zbl 0778.15008
[4] Augusto Ferrante and Bernard C. Levy, Hermitian solutions of the equation \?=\?+\?\?\(^{-}\)\textonesuperior \?*, Linear Algebra Appl. 247 (1996), 359 – 373. · Zbl 0876.15011
[5] J. D. Gardiner, A. J. Laub, J. J. Amato, and C. B. Moler, Solution of the Sylvester matrix equation \(AXB^T+CXD^T=E\), ACM Trans. Math. Software 18 (1992), 223-231. CMP 92:13 · Zbl 0893.65026
[6] Julio F. Fernández and Juan Rivero, A fast algorithm for the generation of random numbers with exponential and normal distributions, Computational physics (Granada, 1994) Lecture Notes in Phys., vol. 448, Springer, Berlin, 1995, pp. 201 – 209. · Zbl 0827.65001
[7] C.-H. Guo, Newton’s method for discrete algebraic Riccati equations when the closed-loop matrix has eigenvalues on the unit circle, SIAM J. Matrix Anal. Appl. 20 (1999), 279-294. CMP 99:02
[8] G. A. Hewer, An iterative technique for the computation of the steady-state gains for the discrete optimal regulator, IEEE Trans. Autom. Control 16 (1971), 382-384.
[9] Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991. · Zbl 0729.15001
[10] M. A. Krasnosel\(^{\prime}\)skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish.
[11] Peter Lancaster and Leiba Rodman, Algebraic Riccati equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. · Zbl 0836.15005
[12] Peter Lancaster and Miron Tismenetsky, The theory of matrices, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0558.15001
[13] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. · Zbl 0241.65046
[14] A. C. M. Ran and R. Vreugdenhil, Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems, Linear Algebra Appl. 99 (1988), 63 – 83. · Zbl 0637.15008
[15] P. Van Dooren, Erratum: ”A generalized eigenvalue approach for solving Riccati equations” [SIAM J. Sci. Statist. Comput. 2 (1981), no. 2, 121 – 135; MR0622709 (83h:65052)], SIAM J. Sci. Statist. Comput. 4 (1983), no. 4, 787. · Zbl 0524.65027
[16] Xingzhi Zhan, Computing the extremal positive definite solutions of a matrix equation, SIAM J. Sci. Comput. 17 (1996), no. 5, 1167 – 1174. · Zbl 0856.65044
[17] Xingzhi Zhan and Jianjun Xie, On the matrix equation \?+\?^{\?}\?\(^{-}\)\textonesuperior \?=\?, Linear Algebra Appl. 247 (1996), 337 – 345. · Zbl 0863.15005
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