On the iterative solution of a class of nonsymmetric algebraic Riccati equations

SIAM Journal on Matrix Analysis and Applications

Volumn 22, Issue 2, 2001, Pages 376-391

  Guo, Chun Hua ^a,b   Laub, Alan J ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF REGINA   (Canada)

^c University of California   (United States)

Author keywords

Convergence rate; Fixed point iterations; M matrices; Minimal positive solution; Newton's method; Nonsymmetric algebraic Riccati equations

Indexed keywords

EID: 0041410017     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S089547989834980X     Document Type: Article

References (21)

1. R. H. BARTELS AND G. W. STEWART, Solution of the matrix equation AX + XB = C, Comm. ACM, 15 (1972), pp. 820-826.
2. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
3. D. W. DECKER AND C. T. KELLEY, Newton's method at singular points. I, SIAM J. Numer. Anal., 17 (1980), pp. 66-70.
4. J. W. DEMMEL, Three methods for refining estimates of invariant subspaces, Computing, 38 (1987), pp. 43-57.
5. M. FIEDLER AND V. PTAK, On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J., 12 (1962), pp. 382-400.
6. G. H. GOLUB, S. NASH, AND C. VAN LOAN, A Hessenberg-Schur method for the problem AX + XB = C, IEEE Trans. Automat. Control, 24 (1979), pp. 909-913.
7. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
8. C.-H. GUO AND P. LANCASTER, Analysis and modification of Newton's method for algebraic Riccati equations, Math. Comp., 67 (1998), pp. 1089-1105.
9. J. JUANG, Existence of algebraic matrix Riccati equations arising in transport theory, Linear Algebra Appl., 230 (1995), pp. 89-100.
10. J. JUANG AND W.-W. LIN, Nonsymmetric algebraic Riccati equations and Hamiltonian-like. matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 228-243.
11. L. V. KANTOROVICH AND G. P. AKILOV, Functional Analysis in Normed Spaces, Pergamon, New York, 1964.
12. C. T. KELLEY, A Shamanskii-like acceleration scheme for nonlinear equations at singular roots, Math. Comp., 47 (1986), pp. 609-623.
13. M. A. KRASNOSELSKII, G. M. VAINIKKO, P. P. ZABREIKO, YA. B. RUTITSKII, AND V. YA. STETSENKO, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen, The Netherlands, 1972.
14. P. LANCASTER AND L. RODMAN, Algebraic Riccati Equations, Oxford University Press, London, 1995.
15. P. LANCASTER AND M. TISMENETSKY, The Theory of Matrices, 2nd ed., Academic Press, Orlando, FL, 1985.
16. A. J. LAUB, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, 24 (1979), pp. 913-921.
17. J. A. MEIJERINK AND H. A. VAN DER VORST, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.
18. H.-B. MEYER, The matrix equation AZ + B - ZCZ - ZD = 0, SIAM J. Appl. Math., 30 (1976), pp. 136-142.
19. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
20. J. S. VANDERGRAFT, Newton's method for convex operators in partially ordered spaces, SIAM J. Numer. Anal., 4 (1967), pp. 406-432.
21. R. S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.