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

Index-proper nonnegative splittings of matrices

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.
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

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[2]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[3]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[4]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[5]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[6]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[7]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[8]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[9]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[10]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[11]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[12]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[13]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[14]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[15]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[16]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[17]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[18]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[19]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[20]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

 Impact Factor: 

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]