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NEWTON`S METHOD FOR SOLVING A QUADRATIC MATRIX EQUATION WITH SPECIAL COEFFICIENT MATRICES
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  • Journal title : Honam Mathematical Journal
  • Volume 35, Issue 3,  2013, pp.417-433
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2013.35.3.417
 Title & Authors
NEWTON`S METHOD FOR SOLVING A QUADRATIC MATRIX EQUATION WITH SPECIAL COEFFICIENT MATRICES
Seo, Sang-Hyup; Seo, Jong-Hyun; Kim, Hyun-Min;
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 Abstract
We consider the iterative solution of a quadratic matrix equation with special coefficient matrices which arises in the quasibirth and death problem. In this paper, we show that the elementwise minimal positive solvent of the quadratic matrix equations can be obtained using Newton`s method if there exists a positive solvent and the convergence rate of the Newton iteration is quadratic if the Frchet derivative at the elementwise minimal positive solvent is nonsingular. Although the Frchet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.
 Keywords
quadratic matrix equation;elementwise positive solvent;elementwise nonnegative solvent;M-matrix;Newton`s method;convergence rate;
 Language
English
 Cited by
 References
1.
Dario A. Bini, Guy Latouche, and Beatrice Meini, Numerical methods for structured Markov chains, Oxford University Press Oxford, 2005.

2.
B.J. Broxson, The Kronecker product, Master's thesis, University of North Florida, 2006.

3.
Chun-Hua Guo and Nicholas J. Higham, Iterative solution of a nonsymmetric algebraic RICCATI equation, SIAM Journal on Matrix Analysis and Applications, 29 (2007), 396-412. crossref(new window)

4.
Chun-Hua Guo and Peter Lancaster, Analysis and modification of newton's method for algebraic RICCATI equations, Mathematics of Computation, 67 (223) (1998), 1089-1105. crossref(new window)

5.
S. Hautphenne, G. Latouche, and Marie-Ange Remiche, Newton's iteration for the extinction probability of a Markovian binary tree, Linear Algebra and its Applications, 428 (2008), 2791-2804. crossref(new window)

6.
C. He, B. Meini, N. H. Rhee, and K. Sohraby, A quadratically convergent Bernoulli-like algorithm for solving matrix polynomial equations in markov chains, Electronic transactions on numerical analysis, 17 (2004), 151-167.

7.
Qi-Ming He and Marcel F. Neuts, On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains, Journal of Applied Probability, 38(2) (2001), 519-541. crossref(new window)

8.
Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

9.
C. T. Kelley, A shamanskii-like acceleration scheme for nonlinear equations at singular roots, Mathematics of Computation, 47(176) (1986), 609-623.

10.
Hyun-Min Kim, Numerical methods for solving a quadratic matrix equation, PhD thesis, Department of Mathematics, University of Manchester, 2000.

11.
W. Kratz and E. Stickel, Numerical solution of matrix polynomial equations by Newton's method, IMA Journal of Numerical Analysis, 7 (1987), 355-369. crossref(new window)

12.
G. Latouche, Newton's iteration for nonlinear equations in Markov chains, IMA Journal of Numerical Analysis, 14 (1994), 583-598. crossref(new window)

13.
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM, 1999.

14.
George Poole and Thomas Boullion, A survey on M-matrices, SIAM review, 16(4) (1974), 419-427. crossref(new window)

15.
Jong Hyeon Seo and Hyun-Min Kim, Solving matrix polynomials by Newton's method with exact line searches, Journal of KSIAM, 12(2) (2008), 55-68.

16.
G. Strang, Linear Algebra and Its Applications, Thomson, Brooks/Cole, 4th edition, 2006.

17.
David M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971.