ITERATIONS FOR APPROXIMATING LIMIT REPRESENTATIONS OF GENERALIZED INVERSES
Abstract
$\lim\limits _{\alpha \to 0} \left( \alpha I+A^*A\right) ^{-1} A^*$ from
[\v Zukovski, Lipcer, On recurent computation of normal solutions of linear algebraic equations, \v Z. Vicisl. Mat. i Mat. Fiz. 12 (1972), 843--857] and
[\v Zukovski, Lipcer, On computation pseudoinverse matrices, \v Z. Vicisl. Mat. i Mat. Fiz. 15 (1975), 489--492]. The iterative process for the implementation of the general limit formula
$\lim\limits _{\alpha \to 0}(\alpha I+R^*S) ^{-1}R^*$
was defined in [P.S. Stanimirovi\'c, Limit representations of generalized inverses and related methods, Appl. Math. Comput. 103 (1999), 51--68].
In this paper we develop an improvement of this iterative process.
The iterative method defined in such a way is able to produce the result in a predefined number of iterative steps. Convergence properties of defined iterations are further investigated.
Full Text:
PDFReferences
O.M. Baksalary, G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 (2010), 681–697.
A. Ben-Israel, On matrices of index zero or one, SIAM J. Appl. Math. 17 (1969), 1118–1121.
C.G. den Broeder Jr., A. Charnes, Contributions to the theory of generalized inverses for matrices, Technical Report, Purdue University, Department of Mathematics,
Lafayette, IN, 1957.
D.A. Harville, Generalized inverses and ranks of modified matrices, Jour. Ind. Soc. Ag. Statistics 49 (1996-97), 67–78.
J. Ji, An alternative limit expression of Drazin inverse and its applications, Appl. Math. Comput. 61 (1994), 151–156.
X. Liu, Y. Yu, J. Zhong, Y. Wei, Integral and limit representations of the outer inverse in Banach space, Linear Multilinear Algebra, 60 (2012), 333–347.
C.D. Meyer, Limits and the index of a square matrix, SIAM J. Appl. Math. 26 (1974), 469–478.
C.D. Meyer, Generalized inversion of modified matrices, SIAM J. Appl. Math. 24 (1973), 315–323.
K.M. Prasad, An introduction to generalized inverse. In: Bapat RB, Kirkland S, Manjunatha Prasad K, Puntanen S, editors. Lectures on matrix and graph methods.
Manipal: Manipal University Press; 2012. p. 43–60.
K.M. Prasad, K.S. Mohana, Core-EP inverse, Linear Multilinear Algebra 62 (2014) 792-802.
P.S. Stanimirovi´ c, Limit representations of generalized inverses and related methods,
Appl. Math. Comput. 103 (1999), 51–68.
G.R. Wang, Y. Wei, Limiting expression for generalized inverse A(2)T,S and corresponding projectors, Numerical Mathematics, J. Chinese Mathematics 4 (1995), 25–30.
Y. Wei, A characterization and representation of the generalized inverse A(2)T,S and its applications, Linear Algebra Appl. 280 (1998), 79–86.
G.R. Wang, Y.M. Wei, S.Z. Qiao, Generalized Inverses: Theory and Computations, Second edition. Developments in Mathematics, 53. Springer, Singapore; Science Press
Beijing, Beijing, 2018.
S. Wolfram, The Mathematica Book, 5th ed., Wolfram Media/Cambridge University Press, Champaign, IL 61820, USA, 2003.
M.M. Zhou, J.L. Chen, T.T. Li, D.G. Wang, Three limit representations of the core-EP inverse, arXiv: 1084.05206v1, 2018.
E.L. Zukovski, R.S. Lipcer, On recurent computation of normal solutions of linear algebraic equations,
Z. Vicisl. Mat. i Mat. Fiz. 12 (1972), 843–857, In Russian.
E.L. Zukovski, R.S. Lipcer, On computation pseudoinverse matrices, Z. Vicisl. Mat.
i Mat. Fiz. 15 (1975), 489–492, In Russian.
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)