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\section*{Introduction}

Since the introduction of BLAS 1 in the early 70s'~\cite{BLAS1}, the
numerical linear algebra community has made tremendous progress in the
generation of high-performance libraries. Nowadays BLAS consists of
three levels, corresponding to routines for vector-vector,
matrix-vector and matrix-matrix operations~\cite{BLAS2,BLAS3}.  From a
mathematical perspective, it might appear that this structure
introduces unnecessary duplication: For instance, a matrix-matrix
multiplication (a level 3 routine) can be expressed in terms of
matrix-vector products (level 2), which in turn can be expressed in
terms of inner products (level 1).  Beyond convenience, the layered
structure is motivated by the efficiency of the routines: Due to the
increasing ratio between operation count and number of memory
accesses---about 1/2, 2 and $n$/2 for BLAS 1, 2, and 3,
respectively---, higher-level operations offer better opportunity to
amortize the costly memory accesses with calculations.  In practice,
this means that level 3 routines attain the best performance and
should be preferred whenever possible. 


\section*{\vdots}
\section*{Conclusions}


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\begin{thebibliography}{99}

\bibitem{BLAS1}
C.~L. Lawson, R.~J. Hanson, R.~J. Kincaid, and F.~T. Krogh.
{\sl Basic linear algebra subprograms for {F}ortran usage}.
ACM Transactions on Mathematical Software, 5 (1979), pp.~308--323.

\bibitem{BLAS2}
J.~J. Dongarra, J.~D. Croz, S.~Hammarling, and R.~J. Hanson.
{\sl An extended set of {F}ortran basic linear algebra subprograms}.
ACM Transactions on Mathematical Software, 14 (1988), pp.~1--17.

\bibitem{BLAS3}
J.~Dongarra, J.~D. Croz, I.~Duff, and S.~Hammarling.
{\sl A set of {L}evel 3 {B}asic {L}inear {A}lgebra {S}ubprograms}.
ACM Transactions on Mathematical Software, 16 (1990), pp.~1--17.

\end{thebibliography}

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  \section*{Appendix 1}
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