×

zbMATH — the first resource for mathematics

Extensions of theory of regular and weak regular splittings to singular matrices. (English) Zbl 1390.15012
Summary: Matrix splittings are useful in finding a solution of linear systems of equations, iteratively. In this note, we present some more convergence and comparison results for recently introduced matrix splittings called index-proper regular and index-proper weak regular splittings. We then apply to theory of double index-proper splittings.

MSC:
15A09 Theory of matrix inversion and generalized inverses
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] A. K. Baliarsingh and L. Jena, A note on index-proper multisplittings of matrices, Banach J. Math. Anal 9 (2015), 384-394. · Zbl 1312.65039
[2] A. K. Baliarsingh and D. Mishra, Comparison results for proper nonnegative splittings of matrices, Results in Math 71 (2017), 93-109. · Zbl 1360.65097
[3] A. Ben-Israel and T. N. E. Greville, Generalized inverses. Theory and applications, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 15. New York, NY: Springer, 2003. · Zbl 0227.15004
[4] A. Berman and R. J. Plemmons, Cones and iterative methods for best square least squares solutions of linear systems, SIAM J. Numer. Anal. 11 (1974), 145-154. · Zbl 0244.65024
[5] L. Collatz, Functional analysis and numerical mathematics, Translated from the German by Hansjörg Oser Academic Press, New York-London, 1966.
[6] L. Jena and D. Mishra, \(B_{D}\)-splittings of matrices, Linear Algebra Appl. 437 (2012), 1162-1173.
[7] L. Jena and S. Pani, Index-range monotonicity and index proper splittings of matrices, Numer. Algebra Control Optim. 3 (2013), 379-388. · Zbl 1264.15005
[8] M. A. Krasnosel’skij, Je. A. Lifshits, and A. V. Sobolev, Positive linear systems, Translated from the Russian by Jürgen Appell. Sigma Series in Applied Mathematics, 5, Heldermann Verlag, Berlin, 1989.
[9] L. Jena, D. Mishra, and S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices, Calcolo 51 (2014), 141-149. · Zbl 1318.65017
[10] M. R. Kannan and K. C. Sivakumar, Moore-Penrose inverse positivity of interval matrices, Linear Algebra Appl. 436 (2012), 571-578. · Zbl 1236.15012
[11] I. Marek and D. B. Szyld, Comparison theorems for weak splittings of bounded operators, Numer. Math. 56 (1989), 283-289. · Zbl 0694.65023
[12] D. Mishra, Nonnegative splittings for rectangular matrices, Comput. Math. Appl. 67 (2014), 136-144. · Zbl 1350.65025
[13] W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl. 30 (1980), 149-153. · Zbl 0436.15002
[14] S.-Q. Shen and T.-Z. Huang, Convergence and comparison theorems for double splittings of matrices, Comput. Math. Appl. 51 (2006), 1751-1760. · Zbl 1134.65341
[15] R. S. Varga, Matrix iterative analysis, Second revised and expanded edition, Springer Series in Computational Mathematics, 27, Springer-Verlag, Berlin, 2000.
[16] Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput. 95 (1998), 115-124. · Zbl 0942.15003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.