Index-proper nonnegative splittings of matrices

Previous Article
Derivatives of eigenvalues and Jordan frames
NACO Home
This Issue
Next Article
On bounds of the Pythagoras number of the sum of square magnitudes of Laurent polynomials

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

Abstract
Related Papers

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 and 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, New York, 2003. doi: 10.1007/978-1-4612-0873-0.
[2]	A. K. Baliarsingh and L. Jena, A note on index-proper multisplittings of matrices, Banach J. Math. Anal., 9 (2015), 384-394. doi: 10.15352/bjma/09-4-19.
[3]	A. K. Baliarsingh and D. Mishra, Comparison results for proper nonnegative splittings of matrices, Results. Math., online, 2015. doi: 10.1007/s00025-015-0504-9.
[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.
[5]	L. Jena, Extensions of regular and weak regular splittings to real square singular matrices, submitted.
[6]	L. Jena and D. Mishra, B_D-splittings of matrices, Linear Algebra Appl., 437 (2012), 1162-1173. doi: 10.1016/j.laa.2012.04.009.
[7]	L. Jena and S. Pani, Index-range monotonicity and index proper splittings of matrices, Numer. Algebra Control Optim, 3 (2013), 379-388. doi: 10.3934/naco.2013.3.379.
[8]	L. Jena, D. Mishra and S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices, Calcolo, 51 (2014), 141-149. doi: 10.1007/s10092-013-0079-3.
[9]	I. Marek and D. B. Szyld, Comparison theorems for weak splittings of bounded operators, Numer. Math., 58 (1990), 387-397. doi: 10.1007/BF01385632.
[10]	D. Mishra, Nonnegative splittings for rectangular matrices, Comput. Math. Appl., 67 (2014), 136-144. doi: 10.1016/j.camwa.2013.10.012.
[11]	S. Q. Shen and T. Z. Huang, Convergence and comparison theorems for double splittings of matrices, Comput. Math. Appl., 51 (2006), 1751-1760. doi: 10.1016/j.camwa.2006.02.006.
[12]	Y. Song, Comparison theorems for splittings of matrices, Numer. Math., 92 (2002), 563-591. doi: 10.1007/s002110100333.
[13]	R. S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-1-4612-0873-0.
[14]	Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput., 95 (1998), 115-124. doi: 10.1016/S0096-3003(97)10098-4.

show all references

References:

[1]	A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4612-0873-0.
[2]	A. K. Baliarsingh and L. Jena, A note on index-proper multisplittings of matrices, Banach J. Math. Anal., 9 (2015), 384-394. doi: 10.15352/bjma/09-4-19.
[3]	A. K. Baliarsingh and D. Mishra, Comparison results for proper nonnegative splittings of matrices, Results. Math., online, 2015. doi: 10.1007/s00025-015-0504-9.
[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.
[5]	L. Jena, Extensions of regular and weak regular splittings to real square singular matrices, submitted.
[6]	L. Jena and D. Mishra, B_D-splittings of matrices, Linear Algebra Appl., 437 (2012), 1162-1173. doi: 10.1016/j.laa.2012.04.009.
[7]	L. Jena and S. Pani, Index-range monotonicity and index proper splittings of matrices, Numer. Algebra Control Optim, 3 (2013), 379-388. doi: 10.3934/naco.2013.3.379.
[8]	L. Jena, D. Mishra and S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices, Calcolo, 51 (2014), 141-149. doi: 10.1007/s10092-013-0079-3.
[9]	I. Marek and D. B. Szyld, Comparison theorems for weak splittings of bounded operators, Numer. Math., 58 (1990), 387-397. doi: 10.1007/BF01385632.
[10]	D. Mishra, Nonnegative splittings for rectangular matrices, Comput. Math. Appl., 67 (2014), 136-144. doi: 10.1016/j.camwa.2013.10.012.
[11]	S. Q. Shen and T. Z. Huang, Convergence and comparison theorems for double splittings of matrices, Comput. Math. Appl., 51 (2006), 1751-1760. doi: 10.1016/j.camwa.2006.02.006.
[12]	Y. Song, Comparison theorems for splittings of matrices, Numer. Math., 92 (2002), 563-591. doi: 10.1007/s002110100333.
[13]	R. S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-1-4612-0873-0.
[14]	Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput., 95 (1998), 115-124. doi: 10.1016/S0096-3003(97)10098-4.

[1]	Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062
[2]	Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083
[3]	Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
[4]	Litismita Jena, Sabyasachi Pani. Index-range monotonicity and index-proper splittings of matrices. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 379-388. doi: 10.3934/naco.2013.3.379
[5]	Haifeng Ma, Xiaoshuang Gao. Further results on the perturbation estimations for the Drazin inverse. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 493-503. doi: 10.3934/naco.2018031
[6]	Md. Ibrahim Kholil, Ziqi Sun. A uniqueness theorem for inverse problems in quasilinear anisotropic media. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022008
[7]	Ben Green, Terence Tao, Tamar Ziegler. An inverse theorem for the Gowers $U^{s+1}[N]$-norm. Electronic Research Announcements, 2011, 18: 69-90. doi: 10.3934/era.2011.18.69
[8]	Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
[9]	Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[10]	M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599
[11]	Jochen Brüning, Franz W. Kamber, Ken Richardson. The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations. Electronic Research Announcements, 2010, 17: 138-154. doi: 10.3934/era.2010.17.138
[12]	Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial and Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[13]	Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems and Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267
[14]	Meixin Xiong, Liuhong Chen, Ju Ming, Jaemin Shin. Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition. Electronic Research Archive, 2021, 29 (5) : 3383-3403. doi: 10.3934/era.2021044
[15]	Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113
[16]	Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
[17]	G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485
[18]	Muhammad Hamid, Wei Wang. A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1933-1948. doi: 10.3934/dcds.2021178
[19]	Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic and Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[20]	Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Chinmay Kumar Giri
on Google Scholar
Chinmay Kumar Giri

[Back to Top]