Abstract
The systematic analysis of convergence conditions, used in comparison theorems proven for different matrix splittings, is presented. The central idea of this analysis is the scheme of condition implications derived from the properties of regular splittings of a monotone matrix A = M1 − N1 = M2 − N2. An equivalence of some conditions as well as an autonomous character of the conditions and A−1N2 ≥ A−1N1 ≥ 0 are pointed out. The secondary goal is to discuss some essential topics related with existing comparison theorems.
Highlights
The main objective of this expository paper is to present the systematic analysis of convergence conditions derived from their implications for the regular splitting case and discussed in the subsequent sections
As was stated in the previous section, weak nonnegative splittings, determined by conditions (2.31) and either (2.33) or (2.35), are convergent if and only if A−1 ≥ 0, which means that both conditions A−1 ≥ 0 and (M−1N) = (NM−1) < 1 are equivalent
In the case of weak and weaker splittings (based on conditions (2.33) and/or (2.35)), the assumption A−1 ≥ 0 is not a sufficient condition in order to ensure the convergence of a given splitting of A; it is possible to construct a convergent weak or weaker splitting when A−1 ≥ 0
Summary
The main objective of this expository paper is to present the systematic analysis of convergence conditions derived from their implications for the regular splitting case and discussed in the subsequent sections. The secondary goal is to survey, compare and further develop properties of matrix splittings in order to present more clearly some aspects related with the results known in the literature
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