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\title{Improving Error Estimation in Linear System Solving}
\author{Christian P. Ullrich\\
        Institut f\"ur Informatik, Universit\"at Basel\\
        Switzerland}

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Error estimations of linear system solutions are often unreliable,
while  verification algorithms always deliver guaranteed interval
inclusions of the solution. Since latter methods are frequently
considered inefficient and non applicable to real problems, the
development of numerical libraries has been proposed in which the
estimation is supplemented by efficient inclusion routines.

In former papers the feasibility of this approach was shown using
the libraries LINPACK, ITPACK and LAPACK. Verification routines
capable of computing an inclusion merely using information delivered
by the approximation methods were provided. Hence, the original
library routines were not changed allowing the user to decide
whether he wants to verify an approximation or not.

A strong drawback of the implemented verification routines is the
fact that in the iterative case interval-H-matrices are required and
in the case of direct methods the routines work the more
successfull, the better this property is fulfilled  by the problem
matrix. Therefore, new algorithms are presented which bound the
error of the solution in a highly reliable way without similar
restrictions. In a long series of tests runtime and accuracy results
confirmed efficiency and reliability of the implementated
algorithms.

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