# 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)
Basic comparison theorems for weak and weaker matrix splittings. (English) Zbl 0981.65041
In iterative methods for linear systems, a matrix is split as $A=M-N$ and the iteration is $x^{(t+1)}=M^{-1}Nx^{(t)}+M^{-1}b$, $t\ge 0$. The splitting is called convergent if the iterative method converges, i.e., if $\rho(M^{-1}N)<1$. It is called a weak (weaker) splitting if $M$ is nonsingular and $M^{-1}N\ge 0$ and (or) $NM^{-1}\ge 0$. In the weaker case it is called of type 1 or 2 depending on whether the first or the second inequality holds. For two convergent splittings $A=M_1-N_1=M_2-N_2$, comparison theorems compare the spectral radii $\rho(M_1^{-1}N_1)$ and $\rho(M_2^{-1}N_2)$ under various conditions on the $M_i$ and $N_i$. This paper gives comparison theorems for weak and weaker splittings which may be of the same or of different types. See also {\it H. A. Jedrzejec} and {\it Z. I. Woźnicki} [Electron. J. Linear Algebra 8, 53-59 (2001; reviewed above)].

##### MSC:
 65F10 Iterative methods for linear systems
Full Text: