American Institute of Mathematical Sciences

Advanced
2016, 6(2): 103-113. doi: 10.3934/naco.2016002

Index-proper nonnegative splittings of matrices

Chinmay Kumar Giri 1,

1. 

Department of Mathematics, National Institute of Technology Raipur, Raipur - 492 010, India

Received  November 2014 Revised  April 2016 Published  June 2016

The theory of splitting is a useful tool for finding solution of a system of linear equations. Many woks are going on for singular system of linear equations. In this article, we have introduced a new splitting called index-proper nonnegative splitting for singular square matrices. Several convergence and comparison results are also established. We then apply the same theory to double splitting.
Keywords: Drazin inverse; group inverse; non-negativity; Index-proper splitting; convergence theorem; comparison theorem..
Mathematics Subject Classification: 15A09, 65F15, 65F2.
Citation: Chinmay Kumar Giri. Index-proper nonnegative splittings of matrices. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 103-113. doi: 10.3934/naco.2016002
References:
[1]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications,, Springer-Verlag, (2003).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[2]

A. K. Baliarsingh and L. Jena, A note on index-proper multisplittings of matrices,, Banach J. Math. Anal., 9 (2015), 384.  doi: 10.15352/bjma/09-4-19.  Google Scholar

[3]

A. K. Baliarsingh and D. Mishra, Comparison results for proper nonnegative splittings of matrices,, Results. Math., (2015).  doi: 10.1007/s00025-015-0504-9.  Google Scholar

[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.   Google Scholar

[5]

L. Jena, Extensions of regular and weak regular splittings to real square singular matrices,, submitted., ().   Google Scholar

[6]

L. Jena and D. Mishra, BD-splittings of matrices,, Linear Algebra Appl., 437 (2012), 1162.  doi: 10.1016/j.laa.2012.04.009.  Google Scholar

[7]

L. Jena and S. Pani, Index-range monotonicity and index proper splittings of matrices,, Numer. Algebra Control Optim, 3 (2013), 379.  doi: 10.3934/naco.2013.3.379.  Google Scholar

[8]

L. Jena, D. Mishra and S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices,, Calcolo, 51 (2014), 141.  doi: 10.1007/s10092-013-0079-3.  Google Scholar

[9]

I. Marek and D. B. Szyld, Comparison theorems for weak splittings of bounded operators,, Numer. Math., 58 (1990), 387.  doi: 10.1007/BF01385632.  Google Scholar

[10]

D. Mishra, Nonnegative splittings for rectangular matrices,, Comput. Math. Appl., 67 (2014), 136.  doi: 10.1016/j.camwa.2013.10.012.  Google Scholar

[11]

S. Q. Shen and T. Z. Huang, Convergence and comparison theorems for double splittings of matrices,, Comput. Math. Appl., 51 (2006), 1751.  doi: 10.1016/j.camwa.2006.02.006.  Google Scholar

[12]

Y. Song, Comparison theorems for splittings of matrices,, Numer. Math., 92 (2002), 563.  doi: 10.1007/s002110100333.  Google Scholar

[13]

R. S. Varga, Matrix Iterative Analysis,, Springer-Verlag, (2000).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[14]

Y. Wei, Index splitting for the Drazin inverse and the singular linear system,, Appl. Math. Comput., 95 (1998), 115.  doi: 10.1016/S0096-3003(97)10098-4.  Google Scholar

show all references

References:
[1]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications,, Springer-Verlag, (2003).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[2]

A. K. Baliarsingh and L. Jena, A note on index-proper multisplittings of matrices,, Banach J. Math. Anal., 9 (2015), 384.  doi: 10.15352/bjma/09-4-19.  Google Scholar

[3]

A. K. Baliarsingh and D. Mishra, Comparison results for proper nonnegative splittings of matrices,, Results. Math., (2015).  doi: 10.1007/s00025-015-0504-9.  Google Scholar

[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.   Google Scholar

[5]

L. Jena, Extensions of regular and weak regular splittings to real square singular matrices,, submitted., ().   Google Scholar

[6]

L. Jena and D. Mishra, BD-splittings of matrices,, Linear Algebra Appl., 437 (2012), 1162.  doi: 10.1016/j.laa.2012.04.009.  Google Scholar

[7]

L. Jena and S. Pani, Index-range monotonicity and index proper splittings of matrices,, Numer. Algebra Control Optim, 3 (2013), 379.  doi: 10.3934/naco.2013.3.379.  Google Scholar

[8]

L. Jena, D. Mishra and S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices,, Calcolo, 51 (2014), 141.  doi: 10.1007/s10092-013-0079-3.  Google Scholar

[9]

I. Marek and D. B. Szyld, Comparison theorems for weak splittings of bounded operators,, Numer. Math., 58 (1990), 387.  doi: 10.1007/BF01385632.  Google Scholar

[10]

D. Mishra, Nonnegative splittings for rectangular matrices,, Comput. Math. Appl., 67 (2014), 136.  doi: 10.1016/j.camwa.2013.10.012.  Google Scholar

[11]

S. Q. Shen and T. Z. Huang, Convergence and comparison theorems for double splittings of matrices,, Comput. Math. Appl., 51 (2006), 1751.  doi: 10.1016/j.camwa.2006.02.006.  Google Scholar

[12]

Y. Song, Comparison theorems for splittings of matrices,, Numer. Math., 92 (2002), 563.  doi: 10.1007/s002110100333.  Google Scholar

[13]

R. S. Varga, Matrix Iterative Analysis,, Springer-Verlag, (2000).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[14]

Y. Wei, Index splitting for the Drazin inverse and the singular linear system,, Appl. Math. Comput., 95 (1998), 115.  doi: 10.1016/S0096-3003(97)10098-4.  Google Scholar

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