Basic comparison theorems for weak and weaker matrix splittings (Q2716356)

In iterative methods for linear systems, a matrix is split as \(A=M-N\) and the iteration is \(x^{(t+1)}=M^{-1}Nx^{(t)}+M^{-1}b\), \(t\geq 0\). 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 \(M^{-1}N\geq 0\) and (or) \(NM^{-1}\geq 0\). 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 \(N_i\). NEWLINENEWLINENEWLINEThis paper gives comparison theorems for weak and weaker splittings which may be of the same or of different types. See also \textit{H. A. Jedrzejec} and \textit{Z. I. Woźnicki} [Electron. J. Linear Algebra 8, 53-59 (2001; reviewed above)].