×
Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.

On matrix equations \(X\pm A^*X^{-2}A=I\). (English) Zbl 0979.15007

Linear Algebra Appl. 326, No. 1-3, 27-44 (2001).
Solutions of the matrix equations \(X+A^*X^{-2} A=I\) and \(X-A^*X^{-2} A=I\) are studied \((I\) is the \(n\times n\) unit matrix and \(A\) is an \(n\times n\) invertible matrix). Iterative methods are used to find positive definite solutions of these equations. Sufficient conditions are found for the existence of two different solutions of the former equation and sufficient conditions are derived for the existence of positive definite solutions of the latter equations. The equations arise in dynamic programming, stochastic filtering, control theory, and statistics. Numerical examples are discussed.
Reviewer: Václav Burjan (Praha)

Cited in 2 Reviews
Cited in 37 Documents

MSC:

15A24 Matrix equations and identities
65F10 Iterative numerical methods for linear systems
65F30 Other matrix algorithms (MSC2010)

Keywords:

iterative methods; numerical examples; matrix equations; positive definite solutions
PDF BibTeX XML Cite
\textit{I. G. Ivanov} et al., Linear Algebra Appl. 326, No. 1--3, 27--44 (2001; Zbl 0979.15007)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Buzbee, B.L.; Golub, G.H.; Nielson, C.W., On direct methods for solving Poisson’s equations, SIAM J. numer. anal., 7, 627-656, (1970) · Zbl 0217.52902
[2] Guo, C.; Lancaster, P., Iterative solution of two matrix equations, Math. comput., 68, 1589-1603, (1999) · Zbl 0940.65036
[3] 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 \(X+A\^{}\{*\}X\^{}\{−1\}A=Q\), Linear algebra appl., 186, 255-275, (1993) · Zbl 0778.15008
[4] Engwerda, J.C., On the existence of a positive definite solution of the matrix equation \(X+A\^{}\{T\}X\^{}\{−1\}A=I\), Linear algebra appl., 194, 91-108, (1993) · Zbl 0798.15013
[5] Ferrante, A.; Levy, B.C., Hermitian solutions of the the equation \(X=Q+NX\^{}\{−1\}N\^{}\{*\}\), Linear algebra appl., 247, 359-373, (1996) · Zbl 0876.15011
[6] Ivanov, I.G.; El-Sayed, S.M., Properties of positive definite solution of the equation \(X+A\^{}\{*\}X\^{}\{−2\}A=I\), Linear algebra appl., 279, 303-316, (1998) · Zbl 0935.65041
[7] Householder, A.S., The theory of matrices in numerical analysis, (1964), Blaisdell New York · Zbl 0161.12101
[8] Lancaster, P., Theory of matrices, (1969), Academic Press New York · Zbl 0186.05301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.