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)
Solutions and perturbation estimates for the matrix equation X s +A * X -t A=Q. (English) Zbl 1179.15015
This paper is concerned with the matrix equation X s +A * X -t A=Q, for s and t positive integers, where Q is Hermitian and positive definite. Necessary and sufficient conditions for the existence of a Hermitian and positive definite solution are established. An iterative method for computing the solution and perturbation estimates are also considered. Some numerical examples illustrate the presented theory.
MSC:
15A24Matrix equations and identities
15A45Miscellaneous inequalities involving matrices
65F30Other matrix algorithms
15B48Positive matrices and their generalizations; cones of matrices
References:
[1]Engwerda, J. C.: On the existence of a positive definite solution of the matrix eqation X+ATX-1A=I, Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013 · doi:10.1016/0024-3795(93)90115-5
[2]Lancaster, P.; Rodman, L.: Algebraic Riccati equations, (1995) · Zbl 0836.15005
[3]Zhan, X.: Computing the extremal positive definite solutions of a matrix equation, SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044 · doi:10.1137/S1064827594277041
[4]Ivanov, I. G.; Hasanov, V. I.; Uhlig, F.: Improved methods and starting values to solve the matrix equations X±a*X-1A=I iteratively, Math. comp. 74, No. 249, 263-278 (2004) · Zbl 1058.65051 · doi:10.1090/S0025-5718-04-01636-9
[5]Chen, X.; Li, W.: The solutions and perturbation analysis of the matrix equation X+A*X-1A=P, Math. numer. Sin. 27, 303-310 (2005)
[6]Sun, J. G.; Xu, S. F.: Perturbation analysis of the special solution of the matrix equation X+A*X-1A=PΠ, Linear algebra appl. 362, 211-228 (2003) · Zbl 1020.15012 · doi:10.1016/S0024-3795(02)00490-1
[7]Hasanov, V. I.; Ivanov, I. G.: On two perturbation estimates of the extreme solutions to the equations X±a*X-1A=Q, Linear algebra appl. 413, 81-92 (2006) · Zbl 1087.15016 · doi:10.1016/j.laa.2005.08.013
[8]Ivanov, I. G.: Perturbation analysis for solutions of X±a*X-na=Q, Linear algebra appl. 395, 313-331 (2005) · Zbl 1076.15015 · doi:10.1016/j.laa.2004.08.017
[9]Hasanov, V. I.; Ivanov, I. G.: Solutions and perturbation estimates for the matrix equations X±a*X-na=Q, Appl. math. Comput. 156, 513-525 (2004) · Zbl 1063.15012 · doi:10.1016/j.amc.2003.08.007
[10]El-Sayed, Salah M.; Al-Dbiban, A. M.: On positive definite solutions of the nonlinear matrix equation X+A*X-na=I, Appl. math. Comput. 151, 533-541 (2004) · Zbl 1055.15022 · doi:10.1016/S0096-3003(03)00360-6
[11]Ivanov, I. G.: On positive definite solutions of the family of matrix equations X+A*X-na=Q, J. comput. Appl. math. 193, 277-301 (2006) · Zbl 1096.15003 · doi:10.1016/j.cam.2005.06.007
[12]Guo, Xiao-Xia: On Hermitian positive definite solution of nonlinear matrix equation X+A*X-2A=Q, J. comput. Math. 23, No. 5, 513-526 (2005) · Zbl 1081.15008
[13]Zhang, Yuhai: On Hermitian positive definite solutions of matrix equation X+A*X-2A=I, Linear algebra appl. 372, 295-304 (2003) · Zbl 1035.15017 · doi:10.1016/S0024-3795(03)00530-5
[14]Peng, Zhen-Yun; El-Sayed, Salah M.: On positive definite solution of a nonlinear matrix equation, Numer. linear algebra appl. 14, 99-113 (2007) · Zbl 1199.65145 · doi:10.1002/nla.510
[15]Zhen-yun Peng, Salah M. El-Sayed, Xiang-lin Zhang, Iterative methods for the extremal positive definite solution of the matrix equation X+Anbsp;X-nbsp;A=Q, J. Comput. Appl. Math. 2000 (2007) 520-527. · Zbl 1118.65029 · doi:10.1016/j.cam.2006.01.033
[16]Liu, Xin-Guo; Gao, Hua: On the positive definite solutions of the matrix equations xs±ATX-ta=In, Linear algebra appl. 368, 83-97 (2003) · Zbl 1025.15018 · doi:10.1016/S0024-3795(02)00661-4
[17]Du, Sh.; Hou, J.: Positive definite solutions of operator equations xm+A*X-na=I, Linear and multilinear algebra 51, 163-173 (2003) · Zbl 1046.47019 · doi:10.1080/0308108031000068958
[18]Yueting, Yang: The iterative method for solving nonlinear matrix equation xs+A*X-ta=Q, Appl. math. Comput. 188, 46-53 (2007) · Zbl 1131.65039 · doi:10.1016/j.amc.2006.09.085
[19]Zhang, Shisheng: Fixed point theory and its applications, (1984)
[20]Ran, A. C. M.; Reurings, M. C. B.: On the matrix equation X+A*F(X)A=Q: solution and perturbation theory, Linear algebra appl. 346, 15-26 (2002) · Zbl 1086.15013 · doi:10.1016/S0024-3795(01)00508-0