Previous |  Up |  Next

Article

Keywords:
generalized inverses; Moore–Penrose inverse; error matrix
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.
References:
[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 0126165 | Zbl 0095.09401
[2] Ben-Israel, A.: An iterative method for computing the generalized inverse of an arbitrary matrix. Math. Comp. 19 (1965), 452–455. MR 0179915 | Zbl 0136.12703
[3] Ben-Israel, A.: A note on an iterative method for generalized inversion of matrices. Math. Comp. 20 (1966), 439–440. Zbl 0142.11603
[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
[5] Chen, Y.: Finite algorithms for $(2)$-generalized inverse $A_{T,S}^{(2)}$. Linear and Multilinear Algebra 40 (1995), 61–68. MR 1374491
[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
[7] Herzberger, J.: Using error-bounds hyperpower methods to calculate inclusions for the inverse of a matrix. BIT 30 (1990), 508–515. MR 1059311
[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
[9] Petryshyn, W. V.: On the inversion of matrices and linear operators. Proc. Amer. Math. Soc. 16 (1965), 893–901. MR 0182121 | Zbl 0151.19301
[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
[11] Pierce, W. H.: A self-correcting matrix iteration for the Moore-Penrose inverse. Linear Algebra Appl. 244 (1996), 357–363. MR 1403289
[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 0341843
[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
[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
[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 0416001 | Zbl 0327.15012
[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 0798117 | Zbl 0572.65026
[20] Zlobec, S.: On computing the generalized inverse of a linear operator. Glasnik Matematički 2(22) No 2 (1967), 265–271. MR 0234967 | Zbl 0149.35101
Partner of
EuDML logo