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Convergence and comparison results for double splittings of Hermitian positive definite matrices. (English) Zbl 1150.65008
The authors give sufficient conditions for convergence of a 2-step stationary iteration based on a double splitting of the matrix, i.e. $A=P-R-S$, when the aim is solution of $Ax=b$. This condition is weaker than an earlier result by two of the same authors. Moreover, a comparison result for two double splittings of the same matrix is proved.

MSC:
65F10Iterative methods for linear systems
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References:
[1] 1. Cvetković, L.J.: Two-sweep iterative methods. Nonlinear Anal. 30, 25--30 (1997) · Zbl 0889.65025
[2] 2. Elsner, L.: Comparisons of weak regular splittings and multisplitting methods. Numer. Math. 56, 283--289 (1989) · Zbl 0673.65018
[3] 3. Golub, G.H., Varga, R.S.: Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. I., II. Numer. Math. 3, 147--156, 157--168 (1961) · Zbl 0099.10903
[4] 4. Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge: Cambridge University Press 1985 · Zbl 0576.15001
[5] 5. Miller, J.J.H.: On the location of zeros of certain classes of polynomials with applications to numerical analysis. J. Inst. Math. Appl. 8, 397--406 (1971) · Zbl 0232.65070
[6] 6. Nabben, R.: A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices. Linear Algebra Appl. 233, 67--80 (1996) · Zbl 0841.65019
[7] 7. Ortega, J.M.: Introduction to parallel and vector solution of linear systems. New York: Plenum Press 1988 · Zbl 0669.65017
[8] 8. Shen, S.-Q., Huang, T.-Z.: Convergence and comparison theorems for double splittings of matrices. Comput. Math. Appl. 51, 1751--1760 (2006) · Zbl 1134.65341
[9] 9. Song, Y.Z.: Comparison theorems for splittings of matrices. Numer. Math. 92, 563--591 (2002) · Zbl 1012.65028
[10] 10. Varga, R.S.: Matrix iterative analysis. Englewood Cliffs, NJ: Prentice-Hall 1962
[11] 11. Woźnicki, Z.I.: Estimation of the optimum relaxation factors in partial factorization iterative methods. SIAM J. Matrix Anal. Appl. 14, 59--73 (1993) · Zbl 0767.65025
[12] 12. Woźnicki, Z.I.: Nonnegative splitting theory. Japan J. Indust. Appl. Math. 11, 289--342 (1994) · Zbl 0805.65033
[13] 13. Woźnicki, Z.I.: Basic comparison theorems for weak and weaker matrix splittings. Electron. J. Linear Algebra 8, 53--59 (2001) · Zbl 0981.65041