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\title{Accelerating Matrix Multiplication}

\author{Namrata Priyadarshani \\ \href{mailto:cs1170353@cse.iitd.ac.in}{cs1170353@cse.iitd.ac.in}\and Musunuru Saurav \\ \href{mailto:cs1170352@cse.iitd.ac.in}{cs1170352@cse.iitd.ac.in}}


\begin{document}
\maketitle

\section{Linear Algebra Libraries}
We tried to accelerate matrix multiplication using different linear algebra libraries. We used Intel MKL\cite{mklLink} and linked it using
\cite{mklDocumnent} which provides highly optimized, threaded and vectorized maths function that enhances performance on Intel processor architecture. It consists of Intel MKL DGEMM function which is highly optimized for matrix-multiplication accelerating the matrix-multiplication over 30 times the normal matrix-multiplication. We also implemented matrix-multiplication using OpenBLAS\cite{openBlas} function cblas\_sgemm  which also had a comparable performance with Intel MKL. These libraries highly optimizes the matrix-multiplication.

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\section{Performance Comparison}
\vskip 0.2in
\begin{center}
\begin{tabular}{|p{4cm}|p{4cm}|p{4cm}|}
\hline\hline                        %inserts double horizontal lines
\multicolumn{3}{|c|}{Mean of latencies over different matrix sizes} \\
\hline\hline
Size of Matrix & Open BLAS & MKL \\ [2 ex]
\hline
500 & 2123 & 2155  \\% inserting body of the table
750 & 4921 & 5001 \\[1ex]
900 & 6026  & 6578  \\[1ex]
1000 & 78078 & 79034 \\[1ex]
\hline
\end{tabular}

\vskip 0.2in

\begin{tabular}{|p{4cm}|p{4cm}|p{4cm}|}
\hline\hline                        %inserts double horizontal lines
\multicolumn{3}{|c|}{Standard Deviation of latencies over different matrix sizes} \\
\hline\hline
Size of Matrix & Open BLAS & MKL \\ [2 ex]
\hline
500 & 11 & 27  \\% inserting body of the table
750 & 134 & 53  \\[1ex]
900 & 19  & 51  \\[1ex]
1000  & 367 & 146 \\[1ex]
\hline
\end{tabular}
\vskip 0.2in

In the above tables with matrix size 1000 we used a kernel of 10 and a kernel size of 3 with others

\includegraphics[width=5in]{time1000.ps} 
% \includegraphics[width=.5\textwidth]{time500.ps}
\end{center}

\section{Acceleration with pthreads}
\vskip 0.2 in
\begin{itemize}
  \item In normal matrix we were carrying out single operation in one loop cycle.
  \item Using pthread we divided matrix into quarters and all 4 threads were computing the 4 parts in parallel, optimizing the performance compared to naive matrix-multiplication.
  \item Matrix-multiplication using pthread is nearly 3 times faster than normal matrix-multiplication.
\end{itemize}

\includegraphics[width=5in]{time500.ps} 

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