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)
Further results on iterative methods for computing generalized inverses. (English) Zbl 1190.65088
Let $A$ be a complex Banach algebra with unit $1$; $\mathcal{X,Y}$ two complex Banach spaces and $\mathcal{B(X,Y)}$ the set of all bounded linear operators from $\mathcal{X}$ to $\mathcal{Y}$. The main theorem is stated as follows: Define the sequence $$X_{k}=X_{k-1}+\beta Y(I_{\mathcal{Y}}-AX_{k-1}),\quad k=1,2,\dots,$$ where $\beta\in \mathbb{C} \setminus \{0\}$ and $X_{0}\in \mathcal{B(Y,X)}$ with $Y\neq YAX_{0}.$ Then the above iteration converges if and only if $\rho(I_{\mathcal{X}}-\beta YA)<1,$ equivalently, $\rho(I_{\mathcal{Y}}-\beta AY)<1.$ In this case suppose now $(\rho(I_{\mathcal{X}}-\beta YA)<1 $ and that $T, S$ are closed subspaces of $\mathcal{X,Y}.$ If moreover $\mathcal{R}(Y)=T$,\quad $\mathcal{N}(Y)=S \quad$and $\mathcal{R}(X_{0})\subset T$, then $A_{T,S}^{(2)}$ exists and $\{X_{k}\}$ converges to $A_{T,S}^{(2)}$ and if $q=\min(\|I_{\mathcal{X}}-\beta YA \|,\|I_{\mathcal{Y}}-\beta AY \|)<1$: $$\|A_{T,S}^{(2)}-X_{k}\|\leq\frac{|\beta|q^{k}}{1-q} \|Y\| \|I_{\mathcal{Y}}-AX_{0} \|.$$ Here $A_{T,S}^{(2)}$ denotes the generalized inverse. Several following theorems yield variants of the above theorem which state conditions under which the iterative procedures approximate the generalized inverse. The next section entitled “The generalized Drazin inverse of Banach algebra elements” defines an iteration approximating this inverse and different conditions under which this sequence converges to the Drazin inverse. Finally, the last section is devoted to a numerical example in which $A\in \mathbb{C}^{5\times 4}.$

MSC:
65J10Equations with linear operators (numerical methods)
47A05General theory of linear operators
15A09Matrix inversion, generalized inverses
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
WorldCat.org
Full Text: DOI
References:
[1] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses: theory and applications, (2003) · Zbl 1026.15004
[2] González, N. Castro: On the convergence of semiiterative methods to the Drazin inverse solution of linear equations in Banach spaces, Collect. math. 46, No. 3, 303-314 (1995) · Zbl 0848.65043 · http://www.mat.ub.es/Collectanea/1995.html
[3] Chen, Y.: Iterative methods for computing the generalized inverses AT,$S(2)$ of a matrix A, Appl. math. Comput. 75, No. 2--3, 207-222 (1996) · Zbl 0853.65044 · doi:10.1016/0096-3003(96)90063-6
[4] Chen, X.; Hartwig, R. E.: The hyperpower iteration revisited, Linear algebra appl. 233, 207-229 (1996) · Zbl 0848.65021 · doi:10.1016/0024-3795(94)00076-X
[5] Djordjević, D. S.: Iterative methods for computing generalized inverses, Appl. math. Comput. 189, No. 1, 101-104 (2007) · Zbl 1125.65046 · doi:10.1016/j.amc.2006.11.063
[6] Djordjević, D. S.; Stanimirović, P. S.: Iterative methods for computing generalized inverses related with optimization methods, J. aust. Math. soc. 78, No. 2, 257-272 (2005) · Zbl 1102.46035 · doi:10.1017/S1446788700008077
[7] Li, X.; Wei, Y.: Iterative methods for the Drazin inverse of a matrix with a complex spectrum, Appl. math. Comput. 147, 855-862 (2004) · Zbl 1038.65037 · doi:10.1016/S0096-3003(02)00817-2
[8] Mitra, S. K.; Hartwig, R. E.: Partial orders based on outer inverse, Linear algebra appl. 176, 3-20 (1992) · Zbl 0778.15003 · doi:10.1016/0024-3795(92)90206-P
[9] Načevska, B.: Iterative methods for computing generalized inverses and splittings of operators, Appl. math. Comput. 228, No. 1, 186-188 (2009) · Zbl 1160.65312 · doi:10.1016/j.amc.2008.11.030
[10] Nashed, M. Z.; Chen, X.: Convergence of Newton-like methods for singular operator equations using outer inverses, Numer. math. 66, 235-257 (1993) · Zbl 0797.65047 · doi:10.1007/BF01385696
[11] Wang, G.; Wei, Y.; Qiao, S.: Generalized inverses: theory and computations, (2004)
[12] Chen, Y.; Chen, X.: Representation and approximation of the outer inverse AT,$S(2)$ of a matrix A, Linear algebra appl. 308, 85-107 (2000) · Zbl 0957.15002 · doi:10.1016/S0024-3795(99)00269-4
[13] Djordjević, D. S.; Stanimirović, P. S.: On the generalized Drazin inverse and generalized resolvent, Czechoslovak math. J. 126, 617-634 (2001) · Zbl 1079.47501 · doi:10.1023/A:1013792207970
[14] Djordjević, D. S.; Stanimirović, P. S.: Splittings of operators and generalized inverses, Publ. math. Debrecen 59, 147-159 (2001) · Zbl 0981.47001
[15] Djordjević, D. S.; Stanimirović, P. S.; Wei, Y.: The representation and approximations of outer generalized inverses, Acta math. Hungar. 104, No. 1--2, 1-26 (2004) · Zbl 1071.65075 · doi:10.1023/B:AMHU.0000034359.98588.7b
[16] Sheng, X.; Chen, G.; Gong, Y.: The representation and computation of generalized inverse AT,$S(2)$, J. comput. Appl. math. 213, No. 1, 248-257 (2008) · Zbl 1135.65021 · doi:10.1016/j.cam.2007.01.009
[17] Wei, Y.: A characterization and representation of the generalized inverse AT,$S(2)$ and its applications, Linear algebra appl. 280, 87-96 (1998) · Zbl 0934.15003 · doi:10.1016/S0024-3795(98)00008-1
[18] Wei, Y.; Wu, H.: ${T,S}$ splitting methods for computing the generalized inverse AT,$S(2)$ and regular systems, Int. J. Comput. math. 77, No. 3, 401-424 (2001)$ · Zbl 0986.65038 · doi:10.1080/00207160108805075
[19] Wei, Y.; Wu, H.: The representation and approximation for the generalized inverse AT,$S(2)$, Appl. math. Comput. 135, 263-276 (2003) · Zbl 1027.65048 · doi:10.1016/S0096-3003(01)00327-7
[20] Wei, Y.; Zhang, N.: A note on the representation and approximation of the outer inverse AT,$S(2)$ of a matrix A, Appl. math. Comput. 147, 837-841 (2004) · Zbl 1040.15007 · doi:10.1016/S0096-3003(02)00815-9
[21] Yu, Y.; Wei, Y.: The representation and computational procedures for the generalized inverse AT,$S(2)$ of an operator A in Hilbert spaces, Numer. funct. Anal. optim. 30, No. 1--2, 168-182 (2009) · Zbl 1165.47004 · doi:10.1080/01630560902735314
[22] Yu, Y.; Wei, Y.: Determinantal representation of the generalized inverse AT,$S(2)$ over integral domains and its applications, Linear multilinear algebra 57, No. 6, 547-559 (2009) · Zbl 1182.15007 · doi:10.1080/03081080701871665
[23] Zheng, B.; Wang, G.: Representation and approximation for generalized inverse AT,$S(2)$, J. appl. Math. comput. 22, No. 3, 225-240 (2006) · Zbl 1112.15013 · doi:10.1007/BF02832049
[24] Liu, X.; Yu, Y.; Hu, C.: The iterative methods for computing the generalized inverse AT,$S(2)$ of the bounded linear operator between Banach spaces, Appl. math. Comput. 214, 391-410 (2009) · Zbl 1175.65063 · doi:10.1016/j.amc.2009.04.007
[25] Müller, V.: Spectral theory of linear operators and spectral systems in Banach algebras, (2007)
[26] Koliha, J. J.: A generalized Drazin inverse, Glasg. math. J. 38, 367-381 (1996) · Zbl 0897.47002 · doi:10.1017/S0017089500031803
[27] Barnes, B. A.: Common operator properties of the linear operators RS and SR, Proc. amer. Math. soc. 126, 1055-1061 (1998) · Zbl 0890.47004 · doi:10.1090/S0002-9939-98-04218-X
[28] Saad, Y.: Iterative methods for sparse linear systems, (2003) · Zbl 1031.65046