Title:
|
Self-correcting iterative methods for computing ${2}$-inverses (English) |
Author:
|
Stanimirović, Predrag S. |
Language:
|
English |
Journal:
|
Archivum Mathematicum |
ISSN:
|
0044-8753 (print) |
ISSN:
|
1212-5059 (online) |
Volume:
|
39 |
Issue:
|
1 |
Year:
|
2003 |
Pages:
|
27-36 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
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. (English) |
Keyword:
|
generalized inverses |
Keyword:
|
Moore–Penrose inverse |
Keyword:
|
error matrix |
MSC:
|
15A09 |
MSC:
|
15A24 |
MSC:
|
65F20 |
idZBL:
|
Zbl 1122.15301 |
idMR:
|
MR1982209 |
. |
Date available:
|
2008-06-06T22:41:08Z |
Last updated:
|
2012-05-10 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107851 |
. |
Reference:
|
[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. Zbl 0095.09401, MR 0126165 |
Reference:
|
[2] Ben-Israel, A.: An iterative method for computing the generalized inverse of an arbitrary matrix.Math. Comp. 19 (1965), 452–455. Zbl 0136.12703, MR 0179915 |
Reference:
|
[3] Ben-Israel, A.: A note on an iterative method for generalized inversion of matrices.Math. Comp. 20 (1966), 439–440. Zbl 0142.11603 |
Reference:
|
[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 0203917 |
Reference:
|
[5] Chen, Y.: Finite algorithms for $(2)$-generalized inverse $A_{T,S}^{(2)}$.Linear and Multilinear Algebra 40 (1995), 61–68. MR 1374491 |
Reference:
|
[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 0281330 |
Reference:
|
[7] Herzberger, J.: Using error-bounds hyperpower methods to calculate inclusions for the inverse of a matrix.BIT 30 (1990), 508–515. MR 1059311 |
Reference:
|
[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 1114976 |
Reference:
|
[9] Petryshyn, W. V.: On the inversion of matrices and linear operators.Proc. Amer. Math. Soc. 16 (1965), 893–901. Zbl 0151.19301, MR 0182121 |
Reference:
|
[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 0208381 |
Reference:
|
[11] Pierce, W. H.: A self-correcting matrix iteration for the Moore-Penrose inverse.Linear Algebra Appl. 244 (1996), 357–363. MR 1403289 |
Reference:
|
[12] Schulz, G.: Iterative Berechnung der reziproken Matrix.Z. Angew. Math. Mech. 13 (1933), 57–59. |
Reference:
|
[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 0341843 |
Reference:
|
[14] Stanimirović, P. S. and Djordjević, D. S.: Universal iterative methods for computing generalized inverses.Acta Math. Hungar. 79(3) (1998), 253–268. MR 1616062 |
Reference:
|
[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 1672776 |
Reference:
|
[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. Zbl 0327.15012, MR 0416001 |
Reference:
|
[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. |
Reference:
|
[18] Zielke, G.: Iterative refinement of generalized matrix inverses now practicable.SIGNUM Newsletter 13.4 (1978), 9–10. |
Reference:
|
[19] Zielke, G.: A survey of generalized matrix inverses.Computational Mathematics, Banach center Publications 13 (1984), 499–526. Zbl 0572.65026, MR 0798117 |
Reference:
|
[20] Zlobec, S.: On computing the generalized inverse of a linear operator.Glasnik Matematički 2(22) No 2 (1967), 265–271. Zbl 0149.35101, MR 0234967 |
. |