*(English)*Zbl 0981.65041

In iterative methods for linear systems, a matrix is split as $A=M-N$ and the iteration is ${x}^{(t+1)}={M}^{-1}N{x}^{\left(t\right)}+{M}^{-1}b$, $t\ge 0$. The splitting is called convergent if the iterative method converges, i.e., if $\rho \left({M}^{-1}N\right)<1$. It is called a weak (weaker) splitting if $M$ is nonsingular and ${M}^{-1}N\ge 0$ and (or) $N{M}^{-1}\ge 0$. 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 $\rho \left({M}_{1}^{-1}{N}_{1}\right)$ and $\rho \left({M}_{2}^{-1}{N}_{2}\right)$ 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 *H. A. Jedrzejec* and *Z. I. Woźnicki* [Electron. J. Linear Algebra 8, 53-59 (2001; reviewed above)].

##### MSC:

65F10 | Iterative methods for linear systems |