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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)].

65F10Iterative methods for linear systems
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