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{\Large On a Verification Algorithm for Tridiagonal
        Circulant Symmetric Linear Systems }
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{\large N. S. Dimitrova}
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{\bf Abstract}
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Many practical problems such as finite difference solution of
elliptic equations with periodic boundary conditions, approximation
of
periodic functions using splines etc. \cite{chen}, \cite{rojo},  lead
to linear
systems with tridiagonal circulant symmetric matrices. In the present
paper
an algorithm with result verification for solving such
linear systems is proposed. It consists of two phases -- an
approximation step and a verification phase \cite{DimUll}. The first
step is
based on a known method \cite{rojo} and produces an approximate
solution of the
problem. It is further extended by a verification phase to bound the
effect
of rounding errors during the computations. As a result a vector with
interval
components is delivered which encloses the exact solution of the
linear system.
The verification phase of the algorithm is not mandatory. The user
decides if
he needs to verify the approximate solution or not. The
algorithm is implemented in the computer algebra system Maple V
Release 3
and numerical results are reported.

\begin{thebibliography}{9}

\bibitem{chen} Chen, M., {\em On the Solution of Circulant Linear
Systems},
SIAM J. Numer. Analysis, Vol. 24, No. 3, 1987, 668--683.

\bibitem{DimUll}
Dimitrova, N. S.,  Ullrich, C. P.: {\it Verified Solving of Linear
Systems with Uncertainties in Maple}, Numerical Analysis and Its
Applications,
L. Vulkov, J. Wa\'{s}niewski, P. Yalamov (eds.),
Lecture Notes in Computer Science, vol. 1196, Springer, 1997,
142--149.

\bibitem{rojo}
Rojo, O.: {\it A New Method for Solving Symmetric Circulant
Tridiagonal Systems of Linear Equations}, Computers Math. Applic.
Vol. 20,
No. 12, 1990, 61--67.

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Address of the Author: \\
Section Biomathematics \\
Institute for Mathematics and Informatics \\
Bulgarian Academy of Sciences \\
Acad. G. Bonchev str., bl. 8 \\
BG-1113 Sofia,
Bulgaria  \\
e-mail: {\tt nelid@iph.bio.acad.bg}
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}

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