Convergence and comparison theorems for a generalized alternating iterative method

Using as a principal tool the convergence results of standard iterative process for the solution of linear systems, alternating iterative methods are studied. We extend the convergence theorem for the stationary alternating iterative method of Benzi and Szyld [Numererische Mathematik 76 (1997) 309], for weak nonnegative splittings of the first type of a monotone matrix to weak nonnegative splitting of the second type. On the other hand, we introduce a more general method, the nonstationary alternating iterative method, establishing convergence results for weak nonnegative splittings of a monotone matrix, and for P-regular splittings of a symmetric positive definite matrix.

[1]  Joan-Josep Climent,et al.  SOME COMPARISON THEOREMS FOR WEAK NONNEGATIVE SPLITTINGS OF BOUNDED OPERATORS , 1998 .

[2]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[3]  Richard S. Varga,et al.  Comparisons of regular splittings of matrices , 1984 .

[4]  Numerische,et al.  Comparisons of Weak Regular Splittings and Multisplitting Methods , 2005 .

[5]  Valerie A. Miller,et al.  A note on comparison theorems for nonnegative matrices , 1985 .

[6]  L. Elsner,et al.  Models of parallel chaotic iteration methods , 1988 .

[7]  Yongzhong Song,et al.  Some comparison theorems for nonnegative splittings of matrices , 1993 .

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

[9]  Y. Wallach,et al.  Alternating methods for sets of linear equations , 1979 .

[10]  Reinhard Nabben,et al.  A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices , 1996 .

[11]  Joan-Josep Climent,et al.  Comparison theorems for weak nonnegative splittings of K-monotone matrices , 1999 .

[12]  D. Rose,et al.  Convergence of nested classical iterative methods for linear systems , 1990 .

[13]  Daniel B. Szyld,et al.  Comparison theorems for weak splittings of bounded operators , 1990 .

[14]  Michele Benzi,et al.  Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods , 1997 .

[15]  Daniel B. Szyld,et al.  Weighted max norms, splittings, and overlapping additive Schwarz iterations , 1999, Numerische Mathematik.

[16]  G. Marchuk.  Splitting and alternating direction methods , 1990 .

[17]  S. Venit,et al.  Numerical Analysis: A Second Course. , 1974 .

[18]  Yongzhong Song,et al.  COMPARISONS OF NONNEGATIVE SPLITTINGS OF MATRICES , 1991 .

[19]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[20]  Z. I. Wozniki,et al.  NONNEGATIVE SPLITTING THEORY , 1994 .

[21]  J. Miller.  Numerical Analysis , 1966, Nature.