Self-correcting iterative methods for computing ${2}$-inverses
Archivum mathematicum, Tome 39 (2003) no. 1, pp. 27-36
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper we construct a few iterative processes for computing $\lbrace 2\rbrace $-inverses of a linear bounded operator. These algorithms are extensions of the corresponding algorithms introduced in [11] and a method from [8]. A few error estimates are derived.
In this paper we construct a few iterative processes for computing $\lbrace 2\rbrace $-inverses of a linear bounded operator. These algorithms are extensions of the corresponding algorithms introduced in [11] and a method from [8]. A few error estimates are derived.
Classification : 15A09, 15A24, 65F20
Keywords: generalized inverses; Moore–Penrose inverse; error matrix
@article{ARM_2003_39_1_a2,
     author = {Stanimirovi\'c, Predrag S.},
     title = {Self-correcting iterative methods for computing ${2}$-inverses},
     journal = {Archivum mathematicum},
     pages = {27--36},
     year = {2003},
     volume = {39},
     number = {1},
     mrnumber = {1982209},
     zbl = {1122.15301},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2003_39_1_a2/}
}
TY  - JOUR
AU  - Stanimirović, Predrag S.
TI  - Self-correcting iterative methods for computing ${2}$-inverses
JO  - Archivum mathematicum
PY  - 2003
SP  - 27
EP  - 36
VL  - 39
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ARM_2003_39_1_a2/
LA  - en
ID  - ARM_2003_39_1_a2
ER  - 
%0 Journal Article
%A Stanimirović, Predrag S.
%T Self-correcting iterative methods for computing ${2}$-inverses
%J Archivum mathematicum
%D 2003
%P 27-36
%V 39
%N 1
%U http://geodesic.mathdoc.fr/item/ARM_2003_39_1_a2/
%G en
%F ARM_2003_39_1_a2
Stanimirović, Predrag S. Self-correcting iterative methods for computing ${2}$-inverses. Archivum mathematicum, Tome 39 (2003) no. 1, pp. 27-36. http://geodesic.mathdoc.fr/item/ARM_2003_39_1_a2/

[1] Altman, M.: An optimum cubically convergent iterative method of inverting a linear bounded operator in Hilbert space. Pacific J. Math. 10 (1960), 1107–113. | MR | Zbl

[2] Ben-Israel, A.: An iterative method for computing the generalized inverse of an arbitrary matrix. Math. Comp. 19 (1965), 452–455. | MR | Zbl

[3] Ben-Israel, A.: A note on an iterative method for generalized inversion of matrices. Math. Comp. 20 (1966), 439–440. | Zbl

[4] Ben-Israel, A. and Cohen, D.: On iterative computation of generalized inverses and associated projectors. SIAM J. Numer. Anal. 3 (1966), 410–419. | MR

[5] Chen, Y.: Finite algorithms for $(2)$-generalized inverse $A_{T,S}^{(2)}$. Linear and Multilinear Algebra 40 (1995), 61–68. | MR

[6] Garnett, J., Ben-Israel, A. and Yau, S. S.: A hyperpower iterative method for computing matrix products involving the generalized inverse. SIAM J. Numer. Anal. 8 (1971), 104–109. | MR

[7] Herzberger, J.: Using error-bounds hyperpower methods to calculate inclusions for the inverse of a matrix. BIT 30 (1990), 508–515. | MR

[8] Pan, V. and Schreiber, R.: An improved Newton iteration for the generalized inverse of a matrix, with applications. SIAM. J. Sci. Stat. Comput. 12 (1991), 1109–1130. | MR

[9] Petryshyn, W. V.: On the inversion of matrices and linear operators. Proc. Amer. Math. Soc. 16 (1965), 893–901. | MR | Zbl

[10] Petryshyn, W. V.: On generalized inverses and on the uniform convergence of $(I-\beta K)^n$ with application to iterative methods. J. Math. Anal. Appl. 18 (1967), 417–439. | MR

[11] Pierce, W. H.: A self-correcting matrix iteration for the Moore-Penrose inverse. Linear Algebra Appl. 244 (1996), 357–363. | MR

[12] Schulz, G.: Iterative Berechnung der reziproken Matrix. Z. Angew. Math. Mech. 13 (1933), 57–59.

[13] Söderström, T. and Stewart, G. W.: On the numerical properties of an iterative method for computing the Moore-Penrose generalized inverse. SIAM J. Numer. Anal. 11 (1974), 61–74. | MR

[14] Stanimirović, P. S. and Djordjević, D. S.: Universal iterative methods for computing generalized inverses. Acta Math. Hungar. 79(3) (1998), 253–268. | MR

[15] Stanimirović, P. S.: Block representation of $\lbrace 2\rbrace $, $\lbrace 1,2\rbrace $ inverses and the Drazin inverse. Indian J. Pure Appl. Math. 29 (1998), 1159–1176. | MR

[16] Tanabe, K.: Neumann-type expansion of reflexive generalized inverses of a matrix and the hyperpower iterative method. Linear Algebra Appl. 10 (1975), 163–175. | MR | Zbl

[17] Wang, G.: The representations of the generalized inverses $(A\otimes B)_{T,S}^{(1,2)}$ and $(A\otimes B)_{T,S}^{(2)}$ and some applications. J. Shanghai Univ. (Natural Sciences) 24 (1995), 1–6.

[18] Zielke, G.: Iterative refinement of generalized matrix inverses now practicable. SIGNUM Newsletter 13.4 (1978), 9–10.

[19] Zielke, G.: A survey of generalized matrix inverses. Computational Mathematics, Banach center Publications 13 (1984), 499–526. | MR | Zbl

[20] Zlobec, S.: On computing the generalized inverse of a linear operator. Glasnik Matematički 2(22) No 2 (1967), 265–271. | MR | Zbl