CONVERGENCE OF NEWTON'S METHOD FOR SOLVING A CLASS OF QUADRATIC MATRIX EQUATIONS

- Journal title : Honam Mathematical Journal
- Volume 30, Issue 2, 2008, pp.399-409
- Publisher : The Honam Mathematical Society
- DOI : 10.5831/HMJ.2008.30.2.399

Title & Authors

CONVERGENCE OF NEWTON'S METHOD FOR SOLVING A CLASS OF QUADRATIC MATRIX EQUATIONS

Kim, Hyun-Min;

Kim, Hyun-Min;

Abstract

We consider the most generalized quadratic matrix equation, Q(X) = , where X is m n, , and are p m, is n m, , and are n q and is p q matrices with complex elements. The convergence of Newton's method for solving some different types of quadratic matrix equations are considered and we show that the elementwise minimal positive solvents can be found by Newton's method with the zero starting matrices. We finally give numerical results.

Keywords

quadratic matrix equation;solvent;Newton's method;M-matrix;

Language

English

Cited by

1.

SOLVING A MATRIX POLYNOMIAL BY NEWTON'S METHOD,;;

Journal of the Korea Society for Industrial and Applied Mathematics, 2010. vol.14. 2, pp.113-124

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References

1.

Peter Benner and Ralph Byers, An exact line search method for solving generalized continuous-time algebraic Riccati equations, IEEE Trans. Automat. Control. 43 (1998), 101-107.

2.

Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1994.

3.

George J. Davis, Numerical solution of a quadratic matrix equation, SIAM J. Sci. Stat. Comput. 2 (1981), 164-175.

4.

George J. Davis, Algorithm 598: An algorithm to compute solvents of the matrix equation $AX^2$ + BX + C = 0, ACM Trans. Math. Software 9 (1983), 246-254.

5.

J. E. Dennis, Jr., J. F. Traub, and R. P. Weber, The algebraic theory of matrix polynomials, SIAM J. Numer. Anal. 13 (1976), 831-845.

6.

M. Fiedler and V. Ptak, On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J. 12 (1962), 382-400.

7.

Chun-Hua Guo, Newton's method for discrete algebraic Riccati equation when the closed-loop matrix has eigenvalues on the unit circle, SIAM J. Matrix Anal. Appl. 20 (1998), 279-294.

8.

Chun-Hua Guo and Peter Lancaster, Analysis and modification of Newton's method for algebraic Riccati equations, Math. Comp. 67 (1998), 1089-1105.

9.

Chun-Hua Guo and Alan J. Laub, On the iterative solution of a class of non-symmetric algebraic Riccati equations, SIAM J. Matrix Anal. Appl. 22 (2000), 376-391.

10.

Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1996.

11.

Nicholas J. Higham and Hyun-Min Kim, Numerical analysis of a quadratic matrix equation, IMA J. Numer. Anal. 20 (2000), 499-519.

12.

Nicholas J. Higham and Hyun-Min Kim, Solving a quadratic matrix equation by Newton's method with exact line searches, SIAM J. Matrix Anal. Appl. 23 (2001), 303-316.

13.

Roger A. Horn and Charles R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.

14.

Peter Lancaster, Lambda-Matrices and Vibrating Systems, Pergamon Press, Oxford, 1966.

15.

Peter Lancaster and Leiba Rodman, Algebraic Riccati Equations, Oxford University Press, (1995).

16.

Guy Latouche, Newton's iteration for non-linear equations in Markov chains, IMA J. Numer. Anal. 14 (1994), 583-598.

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), 148-162.

18.

H. A. Smith, R. K. Singh, and D. C. Sorensen, Formulation and solution of the non-linear, damped eigenvalue problem for skeletal systems, Int. J. Numer. Methods Eng. 38 (1995), 3071-3085.