zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Perturbation analysis of the maximal solution of the matrix equation $X+A^*X^{-1}A=P$. (English) Zbl 0992.15013
The nonlinear matrix equation $X+A^*X^{-1}A=P$ is considered where $A$, $P$ are $n\times n$ complex matrices with $P$ Hermitian positive definite and $A^*$ denoting the conjugate transpose of a matrix $A$. Such equations arise in control theory, ladder networks, dynamic programming, stochastic filtering, statistics etc. Hermitian positive definite solutions $X$ are of particular interest. The inequality $X\geq Y$ means that $X-Y$ is positive semidefinite. One defines the maximal and minimal solutions $X_L$ and $X_S$ such that $X_S\leq X\leq X_L$ for any Hermitian positive definite solution $X$. The paper contains a perturbation bound for the maximal solution of the above matrix equation and a computable error bound for approximate solutions; they are derived using a property of the maximal solution expressed in terms of matrix $2$-norms $\|.\|$. Perturbation properties are related to the condition number $\kappa (A,P)=(\frac{1}{2}-\|A\|\|P^{-1}\|)^{-1}$. The results are illustrated by numerical examples carried out using MATLAB on a PC Pentium III/500 computer, with machine epsilon $\varepsilon =2.2\times 10^{-16}$.

MSC:
15A24Matrix equations and identities
Software:
Matlab
WorldCat.org
Full Text: DOI
References:
[1] Anderson, W. N.; Morley, T. D.; Trapp, G. E.: Positive solutions to X=A-BX-1B\ast. Linear algebra appl. 134, 53-62 (1990) · Zbl 0702.15009
[2] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X+ATX-1A=I. Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013
[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\astX-1A=I. Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008
[4] Ran, A. C. M.; Rodman, L.: Stable Hermitian solutions of discrete algebraic Riccati equations. Math. control signals systems 5, 165-193 (1992) · Zbl 0771.93059
[5] Xu, S. F.: On the maximal solution of the matrix equation X+ATX-1A=I. Acta sci. Natur. univ. Pekinensis 36, 29-38 (2000)
[6] Zhan, X.: On the matrix equation X+ATX-1A=I. Linear algebra appl. 247, 337-345 (1996)
[7] Zhan, X.: Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044