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%\title{On the Stability of a Partitioning Algorithm
%for Banded Linear Systems}
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%           new title
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\title{Backward Stability of a Parallel Partitioning Algorithm
for Banded Linear Systems}

\author{ Plamen Yalamov and Velisar Pavlov
\\ Center of Applied
Mathematics and Informatics, \\
University of Rousse,
7017 Rousse, Bulgaria}



\begin{abstract}

Banded systems of linear equations appear in many problems and are the
computing time consuming kernels of various applications.
It is known that the partition methods give efficient parallel algorithms
for solving such systems.
A typical member of the partition methods for solving tridiagonal
systems is the method of Wang \cite{Wa}.
Full roundoff error analysis for the whole algorithm in the case of nonsingular
tridiagonal matrices is presented in \cite{siam}.

In the present paper we consider the
generalized partition method of Wang for nonsingular band systems,
and in this case we present a full  componentwise error analysis.
Bounds on the
equivalent perturbations are obtained depending on three constants.
Then bounds on the forward error are presented as well depending on two
types of condition numbers. Estimates on the first constant come directly
from the roundoff error analysis of the Gaussian elimination for
banded linear systems \cite{ha4}.
In the present paper the second and the third constant are bounded
for some special classes of matrices,
i.e. row diagonally dominant, symmetric positive definite,
and $M$-matrices.


\end{abstract}


\begin{thebibliography}{99}

\bibitem{ha4}
N. J. Higham,
{\em Accuracy and Stability of Numerical Algorithms},
SIAM, Philadelphia, 1996.

\bibitem{Wa}
H.\,H.\,Wang,
A parallel method for tridiagonal linear systems,
{\em ACM Transactions on Mathematical Software}, 7 (1981), pp. 170--183.


\bibitem{siam}
P. Yalamov and V. Pavlov,
Stability of a partitioning algorithm for trdiagonal systems,
{\em SIAM Journal Matrix Anal. Appl.}, 1998. (to appear)

\end{thebibliography}

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