CONVERGENCE OF NEWTONS METHOD FOR SOLVING A NONLINEAR MATRIX EQUATION

Title & Authors
CONVERGENCE OF NEWTONS METHOD FOR SOLVING A NONLINEAR MATRIX EQUATION
Meng, Jie; Lee, Hyun-Jung; Kim, Hyun-Min;

Abstract
We consider the nonlinear matrix equation $X^p+AX^qB+CXD+E Keywords Matrix equation;Elementwise nonnegative solution;Newton`s method;M-matrix; Language English Cited by References 1. W.N. Anderson, G.D. Kleindorfer, P.R. Kleindorfer, M.B. Woodroofe, Consistent estimates of the parameters of a linear system, Ann. Math. Statist. 40 (1969) 2064-2075. 2. D.A. Bini, G. Latouche, B. Meini, Numerical methods for structured Markov chains, Oxford University Press, 2005. 3. Ph. Bougerol, Kalman filtering with random coeffcients and contractions, SIAM J. Control Optim. 31 (1993) 942-959. 4. G.J. Davis, Numerical solution of a quadratic matrix equation, SIAM J. Sci. Statist. Comput. 2 (2) (1981) 164-175. 5. G.J. Davis, Algorithm 598: an algorithm to compute solvent of the matrix equation$AX^2$+ BX + C = 0, ACM Trans. Math. Software 9 (2) (1983) 341-345. 6. X. Duan, A. Liao, On the existence of Hermitian positive definite solutions of the matrix equation$X^s+A^*X^{-t}A$= Q, Linear Algebra Appl. 429 (2008) 673-687. 7. C.-H. Guo, Convergence rate of an iterative method for nonlinear matrix equations, SIAM J. Control Optim. 31 (1993) 942-959. 8. C.-H. Guo, Newton's method for the discrete algebraic Riccati equations when the closed loop matrix has eigenvalues on the unit circle, SIAM J. Matrix Anal. Appl. 20 (1999) 279-294. 9. C.-H. Guo, P. Lancaster, Iterative solution of two matrix equations, Math. Comp. 68 (1999) 1589-1603. 10. C.-H. Guo, N.J. Higham, Iterative solution of a nonsymmetric algebraic Riccati equation, SIAM J. Matrix Anal. Appl. 29 (2007) 396-412. 11. C.-H. Guo, A.J. Laub, On the iterative solution of a class of nonsymmetric algebraic Riccati equations, SIAM J. Matrix Anal. Appl. 67 (1998) 1089-1105. 12. G.A. Hewer, An iterative technique for the computation of the steady-state gains for the discrete optimal regular, IEEE Trans. Autom. Control 16 (1971) 382-384. 13. N.J. Higham, H.-Y. Kim, Numercial analysis of a quadratic matrix equation, IAM J. Numer. Anal. 20 (4) (2000) 499-519. 14. N.J. Higham, H.-Y. Kim, Solving a quadratic matrix eqaution by Newton's method with exact line searches, SIAM J. Matrix Anal. Appl. 23 (2001) 303-416. 15. Z. Jia, M. Zhao, M. Wang, S. Ling, Solvability theory and iteration method for one self-adjoint polynomial matrix equation, J. Appl. Math. 2014, Art. ID 681605, 7 pp. 16. Z. Jia, M. Wei, Solvability and sensitivity theory of polynomial matrix equation$X^s+A^TX^tA$= Q, Appl. Math. Comput. 209 (2009) 230-237. 17. C. Jung, H.-M. Kim, Y. Lim, On the solution of the nonlinear matrix equation$X^n$= f(X), Linear Algebra Appl. 430 (2009) 2042-2052. 18. H.-M. Kim, Convergence of Newtion's method for solving a class of quadratic matrix equaitons, Honam Math. J. 30 (2) (2008) 399-409. 19. W. Kratz, E. Stickel, Numerical solution of matrix polynomial equations by Newton's method, IMA J. Numer. Anal. 7 (1987) 355-369. 20. G. Latouche, Newton's iteration for nonlinear equations in Markov chains, IMA J. Numer. Anal. 7 (1987) 355-369. 21. A.J. Lauh, Invariant Subspace Methods for the Numerical Solution of Riccati equations, Berlin: Springer-Verlag, 1991, 163-196. 22. X. Liu, H. Gao, On the positive definite solutions of the matrix equations$X^s{\pm}A^TX^{-t}A=I_n$, Linear Algebra Appl. 368 (2003) 83-97. 23. J. Meng, H.-M. Kim, The positive definite solution to a nonlinear matrix equation, Linear and Multilinear Algebra, 2015. . 24. Z.-Y. Peng, S.M. EL-Sayed, X.-L. Zhang, Iterative mthods for the extremal positive definite solution of the matrix equation$X+A_*X^{-{\alpha}}A=Q\$, J. Comput. Appl. Math. 200 (2007) 520-527.

25.
G. Poole, T. Boullion, A survey on M-matrices, SIAM Rev. 16 (4) (1974) 419-427.

26.
J.-H. Seo, H.-M. Kim, Convergence of pure and relaxed Newton methods for solving a matrix polynomial equation arising in stochastic models, Linear Algebra Appl. 440 (2014) 34-49.

27.
X. Zhan, Computing the extremal positive definite solutions of a matrix equation, SIAM J. Sci, Comput. 17 (1996) 1167-1174.