Abstract
Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yields to an iterative method xk+1 = Axk + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x* = x* (x0) if and only if A is semi-convergent. In order to enclose such vectors we consider the corresponding interval iteration \([x]^{k+1} = [A][x]^k+[b]\) with ρ(|[A]|) = 1 where |[A]| denotes the absolute value of the interval matrix [A]. If |[A]| is irreducible we derive a necessary and sufficient criterion for the existence of a limit \([x]^* = [x]^*([x]^0)\) of each sequence of interval iterates. We describe the shape of \([x]^*\) and give a connection between the convergence of (\([x]^k\)) and the convergence of the powers \([A]^k\) of [A].
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Dedicated to Professor G. Maeβ on the occasion of his 65th birthday
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Alefeld, G., Mayer, G. Enclosing Solutions of Singular Interval Systems Iteratively. Reliable Comput 11, 165–190 (2005). https://doi.org/10.1007/s11155-005-3614-3
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DOI: https://doi.org/10.1007/s11155-005-3614-3