mirror of
https://github.com/janishutz/eth-summaries.git
synced 2025-11-25 10:34:23 +00:00
68 lines
3.2 KiB
TeX
68 lines
3.2 KiB
TeX
\newpage
|
||
\subsection{Matrix Multiplication}
|
||
\subsubsection{Strassen's Algorithm}
|
||
\begin{definition}[]{Strassen’s Algorithm}
|
||
The \textbf{Strassen’s Algorithm} is an efficient algorithm for matrix multiplication, reducing the asymptotic complexity compared to the standard method.
|
||
\end{definition}
|
||
|
||
\begin{properties}[]{Characteristics of Strassen’s Algorithm}
|
||
\begin{itemize}
|
||
\item \textbf{Standard Multiplication:} Requires \tco{n^3} time for two $n \times n$ matrices.
|
||
\item \textbf{Strassen’s Approach:} Reduces the complexity to \tco{n^{\log_2 7}} (approximately \tco{n^{2.81}}).
|
||
\item \textbf{Idea:} Uses divide-and-conquer to reduce the number of scalar multiplications from $8$ to $7$ in each recursive step.
|
||
\item Useful for applications involving large matrix multiplications.
|
||
\end{itemize}
|
||
\end{properties}
|
||
|
||
\begin{usage}[]{Strassen's Algorithm}
|
||
Strassen's algorithm is used for matrix multiplication, reducing the computational complexity from \(O(n^3)\) to approximately \(O(n^{2.81})\).
|
||
|
||
\begin{enumerate}
|
||
\item \textbf{Divide Matrices:}
|
||
\begin{itemize}
|
||
\item Split the input matrices \(A\) and \(B\) into four submatrices each:
|
||
\[
|
||
A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, \quad
|
||
B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix}
|
||
\]
|
||
\end{itemize}
|
||
|
||
\item \textbf{Compute 7 Products:}
|
||
\begin{itemize}
|
||
\item Calculate seven intermediate products using combinations of the submatrices:
|
||
\begin{align*}
|
||
M_1 & = (A_{11} + A_{22})(B_{11} + B_{22}) \\
|
||
M_2 & = (A_{21} + A_{22})B_{11} \\
|
||
M_3 & = A_{11}(B_{12} - B_{22}) \\
|
||
M_4 & = A_{22}(B_{21} - B_{11}) \\
|
||
M_5 & = (A_{11} + A_{12})B_{22} \\
|
||
M_6 & = (A_{21} - A_{11})(B_{11} + B_{12}) \\
|
||
M_7 & = (A_{12} - A_{22})(B_{21} + B_{22})
|
||
\end{align*}
|
||
\end{itemize}
|
||
|
||
\item \textbf{Combine Results:}
|
||
\begin{itemize}
|
||
\item Use the intermediate products to compute the submatrices of the result \(C\):
|
||
\[
|
||
C_{11} = M_1 + M_4 - M_5 + M_7, \quad
|
||
C_{12} = M_3 + M_5
|
||
\]
|
||
\[
|
||
C_{21} = M_2 + M_4, \quad
|
||
C_{22} = M_1 - M_2 + M_3 + M_6
|
||
\]
|
||
\end{itemize}
|
||
|
||
\item \textbf{Repeat Recursively:}
|
||
\begin{itemize}
|
||
\item If the submatrices are larger than a certain threshold, repeat the process recursively.
|
||
\end{itemize}
|
||
|
||
\item \textbf{End:}
|
||
\begin{itemize}
|
||
\item The resulting matrix \(C\) is the product of \(A\) and \(B\).
|
||
\end{itemize}
|
||
\end{enumerate}
|
||
\end{usage}
|