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)
On the matrix equation X-A * X -n A=I. (English) Zbl 1081.65036

The authors consider the nonlinear matrix equation

X-A X -n A=I,

where X is an unknown matrix, I is the m×m identity matrix and n is a positive integer. The equations X+A X -1 A=Q and X-A X -1 A=Q have many applications, and iterative procedures for solving the equation X-A X -1 A=Q have been proposed [see C.-H. Guo and P. Lancaster, Math. Comput. 68, 1589–1603 (1999; Zbl 0940.65036) and A. Ferrante and B. C. Levy, Linear Algebra Appl. 247, 359–373 (1996; Zbl 0876.15011 )]. The iterative positive definite solutions and the properties of the equations X-A X -2 A=I, and X+A X -2 A=I have been discussed by I. G. Ivanov and S. M. El-Sayed [Linear Algebra Appl. 279, 303–316 (1998; Zbl 0935.65041)], and by I. G. Ivanov, V. I. Hasanov and B. V. Minchov [Linear Algebra Appl. 326, No. 1–3, 27–44 (2001; Zbl 0979.15007)].

In this paper, the authors review the existing methods for solving the equation X-A X -n A=I, and derive a sufficient condition for this equation to have a unique positive definite solution. Moreover, the convergence of the iterative methods proposed by S. M. El-Sayed [Comput. Math. Appl. 41, 579–588 (2001; Zbl 0984.65043)] is proved under weaker restrictions for the matrix A.

65F30Other matrix algorithms
15A24Matrix equations and identities
65F10Iterative methods for linear systems