Advances in Operator Theory

Comparison results for proper multisplittings of rectangular matrices

Chinmay Kumar Giri and Debasisha Mishra

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The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is practically very slow in many cases. Thus, works on improving the speed of the iteration scheme have attracted great interest. In this direction, comparison of the rate of convergence of the iteration schemes produced by two matrix splittings is very useful. But, in the case of matrices having many matrix splittings, this process is time-consuming. The main goal of the current article is to provide a solution to the above issue by using proper multisplittings. To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first. We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.

Article information

Adv. Oper. Theory, Volume 2, Number 3 (2017), 334-352.

Received: 5 January 2017
Accepted: 19 May 2017
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A09: Matrix inversion, generalized inverses
Secondary: 65F15: Eigenvalues, eigenvectors 65F20: Overdetermined systems, pseudoinverses

Moore–Penrose inverse proper splitting multisplittings convergence theorem comparison theorem


Giri, Chinmay Kumar; Mishra, Debasisha. Comparison results for proper multisplittings of rectangular matrices. Adv. Oper. Theory 2 (2017), no. 3, 334--352. doi:10.22034/aot.1701-1088.

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  • A. K. Baliarsingh and L. Jena, A note on index-proper multisplittings of matrices, Banach J. Math. Anal. 9 (2015), no. 4, 384–394.
  • A. K. Baliarsingh and D. Mishra, Comparison results for proper nonnegative splittings of matrices, Results. Math. 71 (2017), no.1-2, 93–109.
  • A. Ben-Israel and T. N. E. Greville, Generalized inverses. Theory and applications, Second edition. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 15. Springer-Verlag, New York, 2003.
  • A. Berman and R. J. Plemmons, Cones and iterative methods for best least squares solutions of linear systems, SIAM J. Numer. Anal. 11 (1974), no. 1, 145–154.
  • A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Revised reprint of the 1979 original. Classics in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994.
  • J.-J. Climent, A. Devesa, and C. Perea, Convergence results for proper splittings, Recent advances in applied and theoretical mathematics, 39–44, Math. Comput. Sci. Eng., World Sci. Eng. Soc. Press, Athens, 2000.
  • J.-J. Climent and C. Perea, Iterative methods for least square problems based on proper splittings, J. Comput. Appl. Math. 158 (2003), no. 1, 43–48.
  • L. Collatz, Functional analysis and numerical mathematics, Translated from the German by Hansjörg Oser Academic Press, New York-London, 1966.
  • L. Elsner, Comparisons of weak regular splittings and multisplitting methods, Numer. Math. 56 (1989), no. 2-3, 283–289.
  • L. Elsner, A. Frommer, R. Nabben, H. Schneider, and D. B. Szyld, Conditions for strict inequality in comparisons of spectral radii of splittings of different matrices, Linear Algebra Appl. 363 (2003), 65–80.
  • C. K. Giri and D. Mishra, Additional results on convergence of alternating iterations involving rectangular matrices, Numer. Funct. Anal. Optimiz. 38 (2017), no. 2, 160–180.
  • G. Golub, Numerical methods for solving linear least squares problem, Numer. Math. 7 (1965), no. 3, 206–216.
  • L. Jena, D. Mishra, and S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices, Calcolo 51 (2014), no. 1, 141–149.
  • D. Mishra, Further study of alternating iterations for rectangular matrices, Linear Multilinear Algebra 65 (2017), no. 8, 1566–1580.
  • D. Mishra, Nonnegative splittings for rectangular matrices, Comput. Math. Appl. 67 (2014), no. 1, 136–144.
  • D. Mishra and K. C. Sivakumar, On splitting of matrices and nonnegative generalized inverses, Oper. Matrices 6 (2012), no. 1, 85–95.
  • D. P. O'leary and R. E. White, Multisplitting of matrices and parallel solution of linear system, SIAM J. Alg. Disc. Meth. 6 (1985), no. 4, 630–640.
  • Y. Song, Comparisons of nonnegative splittings of matrices, Linear Algebra Appl. 154–156 (1991), 433–455.
  • Y. Song, Comparison theorems for splittings of matrices, Numer. Math. 92 (2002), no. 3, 563–591.
  • R. S. Varga, Matrix iterative analysis, Second revised and expanded edition. Springer Series in Computational Mathematics, 27, Springer-Verlag, Berlin, 2000.
  • Z. I. Woźnicki, Basic comparison theorems for weak and weaker matrix splittings, Electron. J. Linear Algebra 8 (2001), 53–59.