A note on the fixed-point iteration for the matrix equations \(X \pm A^* X^{-1}A=I\). (English) Zbl 1166.65018

Fital, Sandra; Guo, Chun-Hua

Linear Algebra Appl. 429, No. 8-9, 2098-2112 (2008).

Consider the complex matrix equations \(X\pm A^*X^{-1} A = I\), where \(I\) is the identity matrix. Conditions for solvability and representation of the solutions to these equations are well known. Also, it has been observed that the rate of convergence of the simple fixed-point iteration \(X_{k+1} = I \mp A^*X_k^{-1}A\) strongly depends on the initial state \(X_0 = \gamma I\), where \(\gamma\) is a parameter. The authors discuss this phenomenon and explain the fast convergence when \(A\) is normal or nearly normal matrix.

Reviewer: Mihail M. Konstantinov (Sofia)

MSC:

65F30	Other matrix algorithms (MSC2010)
15A24	Matrix equations and identities

Keywords:

matrix equations; maximal solutions; fixed-point iteration; rate of convergence; nearly normal matrix

References:

[1]	Anderson, W.N.; Morley, T.D.; Trapp, G.E., Positive solutions to \(X = A - \mathit{BX}^{- 1} B^\ast\), Linear algebra appl., 134, 53-62, (1990) · Zbl 0702.15009
[2]	P. Benner, H. Fassbender, On the solution of the rational matrix equation \(X = Q + \mathit{LX}^{- 1} L^{\operatorname{T}}\), EURASIP J. Adv. Signal Process. 2007, 10 (Art. ID 21850). · Zbl 1168.15310
[3]	C.-Y. Chiang, E.K.-W. Chu, C.-H. Guo, T.-M. Huang, W.-W. Lin, S.-F. Xu, Convergence analysis of the doubling algorithm for several nonlinear matrix equations in the critical case, Preprint, 2008.
[4]	Engwerda, J.C., On the existence of a positive definite solution of the matrix equation \(X + A^{\mathsf{T}} X^{- 1} A = I\), Linear algebra appl., 194, 91-108, (1993) · Zbl 0798.15013
[5]	Engwerda, J.C.; Ran, A.C.M.; Rijkeboer, A.L., Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X + A^\ast X^{- 1} A = Q\), Linear algebra appl., 186, 255-275, (1993) · Zbl 0778.15008
[6]	Ferrante, A.; Levy, B.C., Hermitian solutions of the equation \(X = Q + \mathit{NX}^{- 1} N^\ast\), Linear algebra appl., 247, 359-373, (1996) · Zbl 0876.15011
[7]	Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009
[8]	Guo, C.-H., Convergence rate of an iterative method for a nonlinear matrix equation, SIAM J. matrix anal. appl., 23, 295-302, (2001) · Zbl 0997.65069
[9]	Guo, C.-H., Numerical solution of a quadratic eigenvalue problem, Linear algebra appl., 385, 391-406, (2004) · Zbl 1060.65039
[10]	Guo, C.-H.; Lancaster, P., Iterative solution of two matrix equations, Math. comput., 68, 1589-1603, (1999) · Zbl 0940.65036
[11]	Helton, J.W.; Sakhnovich, L.A., Extremal problems of interpolation theory, Rocky mountain J. math., 35, 819-841, (2005) · Zbl 1094.47026
[12]	Horn, R.A.; Johnson, C.R., Matrix analysis, (1990), Cambridge University Press Cambridge · Zbl 0704.15002
[13]	Ivanov, I.G.; Hasanov, V.I.; Uhlig, F., Improved methods and starting values to solve the matrix equations \(X \pm A^\ast X^{- 1} A = I\) iteratively, Math. comput., 74, 263-278, (2005) · Zbl 1058.65051
[14]	Lin, W.-W.; Xu, S.-F., Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. matrix anal. appl., 28, 26-39, (2006) · Zbl 1116.65051
[15]	Meini, B., New convergence results on functional iteration techniques for the numerical solution of \(M / G / 1\) type Markov chains, Numer. math., 78, 39-58, (1997) · Zbl 0889.65145
[16]	Meini, B., Efficient computation of the extreme solutions of \(X + A^\ast X^{- 1} A = Q\) and \(X - A^\ast X^{- 1} A = Q\), Math. comput., 71, 1189-1204, (2002) · Zbl 0994.65046
[17]	Ran, A.C.M.; Reurings, M.C.B., A nonlinear matrix equation connected to interpolation theory, Linear algebra appl., 379, 289-302, (2004) · Zbl 1039.15007
[18]	Zhan, X.; Xie, J., On the matrix equation \(X + A^{\mathsf{T}} X^{- 1} A = I\), Linear algebra appl., 247, 337-345, (1996)