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Original Articles

Convergence of Nonstationary Iterative Methods for Solving Singular Linear Equations with Index One

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Pages 1507-1525 | Received 12 Apr 2016, Accepted 23 Jun 2017, Published online: 07 Sep 2017

References

  • A. Ben-Israel and T. Greville (2003). Generalized Inverses: Theory and Applications. 2nd ed., Springer-Verlag, New York.
  • A. Berman and B. Plemmons (1994). Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia.
  • P. Brown and H. Walker (1997). GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl. 18:37–51.
  • D. Calvetti, B. Lewis, and L. Reichel (2000). GMRES-type methods for inconsistent systems. Linear Algebra Appl. 316:157–169.
  • Z. Cao (2001). On the convergence of nonstationary iterative methods for symmetric positive (semi)definite systems. Appl. Numer. Math. 37:319–330.
  • Z. Cao (2002). On the convergence of iterative methods for solving singular linear systems. J. Comput. Appl. Math. 145:1–9.
  • Z. Cao (2008). On the convergence of general stationary linear iterative methods for singular linear systems. SIAM J. Matrix Anal. Appl. 29:1382–1388.
  • X. Cui, Y. Wei, and N. Zhang (2007). Quotient convergence and multi-splitting methods for solving singular linear equations. Calcolo 44:21–31.
  • X. Du and D. Szyld (2008). Inexact GMRES for singular linear systems. BIT Numer. Math. 48:511–531.
  • A. Frommer, R. Nabben, and D. Szyld (2008). Convergence of stationary iterative methods for Hermitian semidefinite linear systems and applications to Schwarz methods. SIAM J. Matrix Anal. Appl. 30:925–938.
  • A. Frommer and D. Szyld (2014). On necessary conditions for convergence of stationary iterative methods for Hermitian semidefinite linear systems. Linear Algebra Appl. 453:192–201.
  • N. Higham and P. Knight (1993). Finite precision behavior of stationary iteration for solving singular systems. Linear Algebra Appl. 192:165–186.
  • Y. -J. Lee, J. Wu, J. Xu, and L. Zikatanov (2006). On the convergence of iterative methods for semidefinite linear systems. SIAM J. Matrix Anal. Appl. 28:634–641.
  • L. Lin, Y. Wei, C. Woo, and J. Zhou (2008). On the convergence of splittings for semidefinite linear systems. Linear Algebra Appl. 429:2555–2566.
  • L. Lin, Y. Wei, and N. Zhang (2009). Convergence and quotient convergence of iterative methods for solving singular linear equations with index one. Linear Algebra Appl. 430:1665–1674.
  • G. Maess (1988). Projection methods solving rectangular systems of linear equations. J. Comput. Appl. Math. 24:107–119.
  • G. Marchuk and Yu. Kuznetsov (1972). Iterative Methods and Quadratic Functionals. Nauka, Novosibirsk (in Russian).
  • C. Meyer (1975). The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17:443–464.
  • V. Migallón, J. Penadés, and D. Szyld (1995). Block two-stage methods for singular systems and Markov chains. Numer. Linear Algebra Appl. 3:413–426.
  • V. Migallón, J. Penadés, and D. Szyld (2001). Nonstationary multisplittings with general weighting matrices. SIAM J. Matrix Anal. Appl. 22:1089–1094.
  • M. Neumann (1976). Subproper splitting for rectangular matrices. Linear Algebra Appl. 14:41–51.
  • R. Plemmons (1976). Regular splittings and the discrete Neumann problem. Numer. Math. 25:153–161.
  • E. Seneta (1981). Non-Negative Matrices and Markov Chains. Springer-Verlag, New York.
  • X. Shi, Y. Wei, and W. Zhang (2011). Convergence of general nonstationary iterative methods for solving singular linear equations. SIAM. J. Matrix Anal. Appl. 32:72–89.
  • A. Sidi (2001). DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular nonsymmetric linear systems. Linear Algebra Appl. 335:189–204.
  • G. Stewart (2011). On the numerical analysis of oblique projectors. SIAM. J. Matrix Anal. Appl. 32:309–348.
  • D. Szyld (1994). Equivalence of convergence conditions for iterative methods for singular equations. Numer. Linear Algebra Appl. 1:151–154.
  • H. vander Vorst (2003). Iterative Krylov Methods for Large Linear systems. Cambridge University Press, Cambridge.
  • G. Wang, Y. Wei, and S. Qiao (2004). Generalized Inverses: Theory and Computations. Science Press, Beijing.
  • Y. Wei (2000). Perturbation analysis of singular linear systems with index one. Int. J. Comput. Math. 74:483–491.
  • Y. Wei and H. Wu (2000). Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index. J. Comput. Appl. Math. 114:305–318.
  • N. Zhang (2010). A Note on preconditioned GMRES for solving singular linear systems. BIT Numer. Math. 50:207–220.
  • S. Zhang, Y. Oyanagi, and M. Su-gihara (2000). Necessary and suffcient conditions for the convergence of Orthomin(k) on singular and inconsistent linear systems. Numer. Math. 87:391–405.
  • N. Zhang and Y. Wei (2004). Solving EP singular linear systems. Int. J. Comput. Math. 81:1395–1405.
  • N. Zhang and Y. Wei (2010). On the convergence of general stationary iterative methods for Range-Hermitian singular linear systems. Numer. Linear Algebra Appl. 17:139–154.

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