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)
Iterative methods for the extremal positive definite solution of the matrix equation $X+A^{*}X^{-\alpha}A=Q$. (English) Zbl 1118.65029
The authors propose two algorithms that avoid matrix inversion for every iteration, called inversion free variant of the basic point iteration. The first algorithm computes the maximal positive definite solution $X_{+}$ of the nonlinear equation $X+A^{\ast }X^{-\alpha }A=Q,$ where $A$ is a nonsingular matrix, $Q$ is a Hermitian positive definite matrix and $\alpha \in (0,1].$ The second algorithm computes the minimal positive definite solution $X_{-}$ of the same nonlinear equation with the case $\alpha \in \lbrack 1,\infty )$. Convergence theorems are provided. Some numerical examples are added to illustrate the convergence features.

65F30Other matrix algorithms
65F10Iterative methods for linear systems
15A24Matrix equations and identities
Full Text: DOI
[1] Jr., W. N. Anderson; Morley, T. D.; Trapp, G. E.: Positive solutions to X=A-BX-1B*. Linear algebra appl. 134, 53-62 (1990) · Zbl 0702.15009
[2] Bhatia, R.: Matrix analysis, graduate texts in mathematics. (1997)
[3] S.M. El-Sayed, Investigation of the special matrices and numerical methods for the special matrix equation, Ph.D. Thesis, Sofia, 1996 (in Bulgarian).
[4] El-Sayed, S. M.: Two sided iteration methods for computing positive definite solutions of a nonlinear matrix equation. J. austral. Math, soc. Ser. B. 44, 1-8 (2003) · Zbl 1054.65041
[5] El-Sayed, S. M.; Al-Dbiban, A. M.: On positive definite solutions of nonlinear matrix equation. Appl. math. Comput. 151, 533-541 (2004) · Zbl 1055.15022
[6] El-Sayed, S. M.; Al-Dbiban, A. M.: A new inversion free iteration for solving the equation X+A*X-1A=Q. J. comput. Appl. math. 181, 148-156 (2005) · Zbl 1072.65060
[7] El-Sayed, S. M.; El-Alem, M.: Some properties for the existence of a positive definite solution of matrix equation X+A*X-2mA=I. Appl. math. Comput. 128, 99-108 (2002) · Zbl 1031.15015
[8] El-Sayed, S. M.; Petkov, M. G.: Iterative methods for nonlinear matrix equations X+A*X-$\alpha $A=I. Linear algebra appl. 403, 45-52 (2005) · Zbl 1074.65057
[9] El-Sayed, S. M.; Ran, A. C. M.: On an iteration methods for solving a class of nonlinear matrix equations. SIAM J. Matrix anal. Appl. 23, 632-645 (2001) · Zbl 1002.65061
[10] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X+ATX-1A=I. Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013
[11] Ferrante, A.; Levy, B. C.: Hermitian solution of the matrix X=Q-NX-1N*. Linear algebra appl. 247, 359-373 (1996) · Zbl 0876.15011
[12] Furuta, T.: Operator inequalities associated with holder -- McCarthy and Kantorovich inequalities. J. inequal. Appl. 6, 137-148 (1998) · Zbl 0910.47014
[13] Guo, C. H.; Lancaster, P.: Iterative solution of two matrix equations. Math. comput. 228, 1589-1603 (1999) · Zbl 0940.65036
[14] V.I. Hasanov, Positive definite solutions of a nonlinear matrix equation, in: mathematics and education in mathematics, Proceedings of Twenty Eighth Spring Conference of the Union of Bulgarian Mathematics, 1999, pp. 107 -- 112.
[15] V.I. Hasanov, Solutions and perturbation theory of the nonlinear matrix equations, Ph.D. Thesis, Sofia, 2003 (in Bulgarian). · Zbl 1042.65057
[16] Hasanov, V. I.: Positive definite solutions of the matrix equations X$\pm $A*X-qa=Q. Linear algebra appl. 404, 166-182 (2005) · Zbl 1078.15012
[17] Hasanov, V.; Ivanov, I. G.: Solutions and perturbation estimates for the matrix equations X$\pm $A*X-na=Q. Appl. math. Comput. 156, 513-525 (2004) · Zbl 1063.15012
[18] Ivanov, I. G.; El-Sayed, S. M.: Properties of positive definite solutions of the equation X+A*X-2A=I. Linear algebra appl. 279, 303-316 (1998) · Zbl 0935.65041
[19] Ivanov, I. G.; Hasanov, V.; Minchev, B. V.: On matrix equations X$\pm $A*X-2A=I. Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007
[20] Lancaster, P.; Rodman, L.: Algebraic Riccati equations. (1995) · Zbl 0836.15005
[21] M. Parodi, La localisation des valeurs caracterisiques des matrices etses applications, Gauthiervillars, Paris, 1959. · Zbl 0087.01602
[22] Ramadan, M. A.; El-Shazly, N. M.: On the matrix equation X+ATX-12mA=I. Appl. math. Comput. 173, 992-1013 (2006) · Zbl 1089.65037
[23] Zhan, X.: Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044