On the convergence of splittings for semidefinite linear systems

Abstract Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634–641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281–290]. In view of their results, we consider here stipulations on a splitting A = M - N , which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore–Penrose solution to the system Ax = b . Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme.

[1]  H. Keller.  On the Solution of Singular and Semidefinite Linear Systems by Iteration , 1965 .

[2]  A. Berman,et al.  Cones and Iterative Methods for Best Least Squares Solutions of Linear Systems , 1974 .

[3]  Daniel B. Szyld,et al.  Schwarz Iterations for Symmetric Positive Semidefinite Problems , 2006, SIAM J. Matrix Anal. Appl..

[4]  Jinchao Xu,et al.  On the Convergence of Iterative Methods for Semidefinite Linear Systems , 2006, SIAM J. Matrix Anal. Appl..

[5]  Hebing Wu,et al.  Subproper and regular splittings for restricted rectangular linear system , 2003, Appl. Math. Comput..

[6]  M. Neumann,et al.  Subproper splitting for rectangular matrices , 1976 .

[7]  Hebing Wu,et al.  (T,S) splitting methods for computing the generalized inverse and rectangular systems , 2001, Int. J. Comput. Math..

[8]  ZHI-HAO CAO.  On the Convergence of General Stationary Linear Iterative Methods for Singular Linear Systems , 2007, SIAM J. Matrix Anal. Appl..

[9]  L. M. Lawson.  Computational methods for generalized inverse matrices arising from proper splittings , 1975 .

[10]  Yimin Wei,et al.  Condition number related with generalized inverse A T,S (2) and constrained linear systems , 2003 .

[11]  Hebing Wu,et al.  On the perturbation and subproper splittings for the generalized inverse AT,S(2) of rectangular matrix A , 2001 .

[12]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[13]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[14]  Zhi-Hao Cao.  A note on properties of splittings of singular symmetric positive semidefinite matrices , 2001, Numerische Mathematik.

[15]  Guorong Wang,et al.  A Cramer rule for minimum-norm (T) least-squares (S) solution of inconsistent linear equations , 1986 .

[16]  Ying-Tung Chen.  Iterative methods for linear least squares problems. , 1974 .

[17]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[18]  I. Marek,et al.  On the solution of singular linear systems of algebraic equations by semiiterative methods , 1988 .

[19]  Martin Hanke,et al.  Preconditionings and splittings for rectangular systems , 1990 .

[20]  Abraham Berman,et al.  Consistency and Splittings , 1976 .

[21]  R. Plemmons.  Monotonicity and iterative approximations involving rectangular matrices , 1972 .