Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Analysis and modificaton of Newton's method
for algebraic Riccati equations


Authors: Chun-Hua Guo and Peter Lancaster
Journal: Math. Comp. 67 (1998), 1089-1105
MSC (1991): Primary 65H10; Secondary 15A24, 93B40
DOI: https://doi.org/10.1090/S0025-5718-98-00947-8
MathSciNet review: 1459388
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: When Newton's method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Fréchet derivative is invertible at the solution. In this paper we examine the behaviour of the Newton iteration when the derivative is not invertible at the solution. We find that a simple modification can improve the performance of the Newton iteration dramatically.


References [Enhancements On Off] (What's this?)

  • 1. R. H. Bartels and G. W. Stewart, Solution of the matrix equation $AX+XB=C$, Comm. ACM 15 (1972), 820-826.
  • 2. P. Benner and R. Byers, An exact line search method for solving generalized continuous-time algebraic Riccati equations, IEEE Trans. Autom. Control (to appear).
  • 3. P. Benner, A. J. Laub and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: continuous-time case, Technical Report SPC 95-22, Fakultät für Mathematik, Technische Universität Chemnitz-Zwickau, FRG, 1995.
  • 4. W. A. Coppel, Matrix quadratic equations, Bull. Austral. Math. Soc. 10 (1974), 377-401. MR 51:3623
  • 5. D. W. Decker, H. B. Keller and C. T. Kelley, Convergence rates for Newton's method at singular points, SIAM J. Numer. Anal. 20 (1983), 296-314. MR 84d:65041
  • 6. D. W. Decker and C. T. Kelley, Newton's Method at singular points I, SIAM J. Numer. Anal. 17 (1980), 66-70. MR 81k:65065a
  • 7. -, Convergence acceleration for Newton's method at singular points, SIAM J. Numer. Anal. 19 (1982), 219-229. MR 83e:65090
  • 8. I. Gohberg, P. Lancaster and L. Rodman, On Hermitian solutions of the symmetric algebraic Riccati equation, SIAM J. Control Optimization 24 (1986), 1323-1334. MR 88f:93041
  • 9. G. H. Golub, S. Nash and C. Van Loan, A Hessenberg-Schur method for the problem $AX+XB=C$, IEEE Trans. Autom. Control 24 (1979), 909-913. MR 81a:65046
  • 10. L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces, Pergamon, New York, 1964. MR 35:4699
  • 11. C. T. Kelley, A Shamanskii-like acceleration scheme for nonlinear equations at singular roots, Math. Comp. 47 (1986), 609-623. MR 87m:65100
  • 12. C. T. Kelley and R. Suresh, A new acceleration method for Newton's method at singular points, SIAM J. Numer. Anal. 20 (1983), 1001-1009. MR 85c:65063
  • 13. D. L. Kleinman, On an iterative technique for Riccati equation computations, IEEE Trans. Autom. Control 13 (1968), 114-115.
  • 14. P. Lancaster and L. Rodman, Algebraic Riccati equations, Oxford University Press, 1995. MR 97b:93003
  • 15. A. Linnemann, Numerische methoden für lineare regelungssysteme, BI Wissenschafts Verlag, Mannheim, 1993. MR 94g:93001
  • 16. V. L. Mehrmann, The autonomous linear quadratic control problem, Lecture Notes in Control and Information Sciences, Vol. 163, Springer Verlag, Berlin, 1991. MR 93d:93004
  • 17. J. M. Ortega and W. C. Rheinboldt, Iterative solutions of nonlinear equations in several variables, Academic Press, New York, 1970. MR 42:8686
  • 18. G. W. Reddien, On Newton's method for singular problems, SIAM J. Numer. Anal. 15 (1978), 993-996. MR 80b:65064
  • 19. V. Sima, An efficient Schur method to solve the stabilizing problem, IEEE Trans. Autom. Control 26 (1981), 724-725. MR 82j:93032
  • 20. H. K. Wimmer, Monotonicity of maximal solutions of algebraic Riccati equations, Syst. Control Lett. 5 (1985), 317-319. MR 86f:93083

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65H10, 15A24, 93B40

Retrieve articles in all journals with MSC (1991): 65H10, 15A24, 93B40


Additional Information

Chun-Hua Guo
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: guo@math.ucalgary.ca

Peter Lancaster
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: lancaste@math.ucalgary.ca

DOI: https://doi.org/10.1090/S0025-5718-98-00947-8
Received by editor(s): February 18, 1997
Additional Notes: Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society