zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 matrix equation X-A * X -p A=Q. (English) Zbl 1173.15006
The authors consider the matrix equation X-A * X -p A=Q with 0<p1, with X, A and Q being n×n-complex matrices, Q being positive definite. They prove (using a new method) existence and uniqueness of a positive definite solution X. Except a generalization of known results for arbitrary p(0,1], a sharper perturbation bound and backward error of an approximation of this solution are evaluated. Explicit expressions of the condition number for the unique positive definite solution are obtained. The results are illustrated by numerical examples.
15A24Matrix equations and identities
65F30Other matrix algorithms
15A12Conditioning of matrices
[1]Zhan, X.; Xie, J.: On the matrix equation X+A*X-1A=I, Linear algebra appl. 247, 337-345 (1996) · Zbl 0863.15005 · doi:10.1016/0024-3795(95)00120-4
[2]Zhan, X.: Computing the extremal positive definite solutions of a matrix equations, SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044 · doi:10.1137/S1064827594277041
[3]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 · doi:10.1016/0024-3795(93)90115-5
[4]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+ATX-1A=Q, Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008 · doi:10.1016/0024-3795(93)90295-Y
[5]Guo, C. H.; Lancaster, P.: Iterative solution of two matrix equations, Math. comp. 68, 1589-1603 (1999) · Zbl 0940.65036 · doi:10.1090/S0025-5718-99-01122-9
[6]Ferrante, A.: Hermitian solutions of the equation X=Q+NX-1N*, Linear algebra appl. 247, 359-373 (1996) · Zbl 0876.15011 · doi:10.1016/0024-3795(95)00121-2
[7]Ivanov, I. G.; El-Sayed, S. M.: Properties of positive definite solution of the equation X+A*X-2A=I, Linear algebra appl. 279, 303-316 (1998) · Zbl 0935.65041 · doi:10.1016/S0024-3795(98)00023-8
[8]Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.: On matrix equations X±a*X-2A=I, Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007 · doi:10.1016/S0024-3795(00)00302-5
[9]Zhang, Y. H.: 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
[10]Zhang, Y. H.: On Hermitian positive definite solutions of matrix equation X-A*X-2A=I, J. comput. Math. 23, 408-418 (2005) · Zbl 1087.15020
[11]Liu, X. G.; Gao, H.: 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
[12]Xu, S. F.: Perturbation analysis of the maximal solution of the matrix equation X+A*X-1A=P, Linear algebra appl. 336, 61-70 (2001) · Zbl 0992.15013 · doi:10.1016/S0024-3795(01)00300-7
[13]Sun, J. G.; Xu, S. F.: Perturbation analysis of the maximal solution of the matrix equation X+A*X-1A=P. II, Linear algebra appl. 362, 211-228 (2003) · Zbl 1020.15012 · doi:10.1016/S0024-3795(02)00490-1
[14]Hasanov, V. I.; Ivanov, I. G.; Uhlig, F.: Improved perturbation estimates for the matrix equation X±a*X-1A=Q, Linear algebra appl. 379, 113-135 (2004) · Zbl 1039.15005 · doi:10.1016/S0024-3795(03)00424-5
[15]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
[16]Hasanov, V. I.: Positive definite solutions of the matrix equations X±a*X-qa=Q, Linear algebra appl. 404, 166-182 (2005) · Zbl 1078.15012 · doi:10.1016/j.laa.2005.02.024
[17]Li, J.: The Hermitian positive definite solutions and perturbation analysis of the matrix equation X-A*X-1A=Q, Math. numer. Sinica 30, 129-142 (2008) · Zbl 1174.65385
[18]M.C.B. Reurings, Symmetric matrix equation, Ph.D. Thesis, Vrije Universiteit, Amsterdan, 2003.
[19]Rice, J. R.: A theory of condition, SIAM J. Numer. anal. 3, 287-310 (1966) · Zbl 0143.37101 · doi:10.1137/0703023