In iterative methods for linear systems, a matrix is split as and the iteration is , . The splitting is called convergent if the iterative method converges, i.e., if . It is called a weak (weaker) splitting if is nonsingular and and (or) . 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 , comparison theorems compare the spectral radii and under various conditions on the and .
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)].