Enclosing Solutions of Singular Interval Systems Iteratively

Enclosing Solutions of Singular Interval Systems Iteratively Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yields to an iterative method x k+1 = Ax k + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x * = x * (x 0) 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]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

Enclosing Solutions of Singular Interval Systems Iteratively

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Publisher
Springer Journals
Copyright
Copyright © 2005 by Springer Science + Business Media, Inc.
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1007/s11155-005-3614-3
Publisher site
See Article on Publisher Site

Abstract

Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yields to an iterative method x k+1 = Ax k + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x * = x * (x 0) 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].

Journal

Reliable ComputingSpringer Journals

Published: Jan 1, 2005

References

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