# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 65F10 Iterative methods for linear systems
Full Text:
##### 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