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-rw-r--r--buch/papers/multiplikation/code/MM.py23
-rw-r--r--buch/papers/multiplikation/code/meas_1024.pdfbin17660 -> 17653 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_1024.txt10
-rw-r--r--buch/papers/multiplikation/code/meas_128.pdfbin17961 -> 18120 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_128.txt10
-rw-r--r--buch/papers/multiplikation/code/meas_256.pdfbin18067 -> 19428 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_256.txt10
-rw-r--r--buch/papers/multiplikation/code/meas_32.pdfbin17078 -> 17964 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_32.txt10
-rw-r--r--buch/papers/multiplikation/code/meas_64.pdfbin17678 -> 17747 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_64.txt10
-rwxr-xr-xbuch/papers/multiplikation/loesungsmethoden.tex53
-rwxr-xr-xbuch/papers/multiplikation/main.tex22
-rwxr-xr-xbuch/papers/multiplikation/references.bib17
14 files changed, 127 insertions, 38 deletions
diff --git a/buch/papers/multiplikation/code/MM.py b/buch/papers/multiplikation/code/MM.py
index 626b82d..352771f 100644
--- a/buch/papers/multiplikation/code/MM.py
+++ b/buch/papers/multiplikation/code/MM.py
@@ -174,10 +174,11 @@ def test_perfomance(n):
plt.plot(n, t_mm_strassen, label='Strassen', lw=5)
plt.plot(n, t_wino, label='Winograd', lw=5)
plt.plot(n, t_np, label='NumPy A@B', lw=5)
+ plt.xscale('log', base=2)
plt.legend()
plt.xlabel("n")
plt.ylabel("time (s)")
- plt.grid(True)
+ plt.grid(True, which="both", ls="-")
plt.tight_layout()
# plt.yscale('log')
plt.legend(fontsize=19)
@@ -198,7 +199,7 @@ def plot(num):
plt.plot(n, t_mm, label='3 For Loops', lw=5)
plt.plot(n, t_mm_dc, label='Divide and Conquer', lw=5)
plt.plot(n, t_mm_strassen, label='Strassen', lw=5)
- # plt.plot(n, t_wino, label='Winograd', lw=5)
+ plt.plot(n, t_wino, label='Winograd', lw=5)
plt.plot(n, t_np, label='NumPy A@B', lw=5)
plt.legend()
plt.xlabel("n")
@@ -275,22 +276,22 @@ def plot_c_res(ave, num):
# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if __name__ == '__main__':
- plot_c_res(1, 4096)
+ # plot_c_res(1, 4096)
# plot(8)
- # n = np.logspace(1,10,10,base=2,dtype=(np.int))
+ n = np.logspace(1,8,8,base=2,dtype=(np.int))
# n = np.arange(1,50,2)
- A = np.random.randint(-10, 10, (5,3))
- B = np.random.randint(-10, 10, (3,5))
+ # A = np.random.randint(-10, 6, (5,3))
+ # B = np.random.randint(-10, 6, (3,5))
- C = winograd2(A, B)
- C_test = A@B
- print(C)
- print(C_test)
+ # C = winograd2(A, B)
+ # C_test = A@B
+ # print(C)
+ # print(C_test)
# print(np.equal(C, C_test))
- # t_np = test_perfomance(n)
+ t_np = test_perfomance(n)
# C = strassen(A, B)
# C_test = A@B
diff --git a/buch/papers/multiplikation/code/meas_1024.pdf b/buch/papers/multiplikation/code/meas_1024.pdf
index fd0a108..7b7a133 100644
--- a/buch/papers/multiplikation/code/meas_1024.pdf
+++ b/buch/papers/multiplikation/code/meas_1024.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/code/meas_1024.txt b/buch/papers/multiplikation/code/meas_1024.txt
index c5ce619..ab507a2 100644
--- a/buch/papers/multiplikation/code/meas_1024.txt
+++ b/buch/papers/multiplikation/code/meas_1024.txt
@@ -1,6 +1,6 @@
2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 1.280000000000000000e+02 2.560000000000000000e+02 5.120000000000000000e+02 1.024000000000000000e+03
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diff --git a/buch/papers/multiplikation/code/meas_128.pdf b/buch/papers/multiplikation/code/meas_128.pdf
index ed1ec63..c54648f 100644
--- a/buch/papers/multiplikation/code/meas_128.pdf
+++ b/buch/papers/multiplikation/code/meas_128.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/code/meas_128.txt b/buch/papers/multiplikation/code/meas_128.txt
index 976bbdf..f3a5beb 100644
--- a/buch/papers/multiplikation/code/meas_128.txt
+++ b/buch/papers/multiplikation/code/meas_128.txt
@@ -1,6 +1,6 @@
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diff --git a/buch/papers/multiplikation/code/meas_256.pdf b/buch/papers/multiplikation/code/meas_256.pdf
index 5f049dc..4ca7102 100644
--- a/buch/papers/multiplikation/code/meas_256.pdf
+++ b/buch/papers/multiplikation/code/meas_256.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/code/meas_256.txt b/buch/papers/multiplikation/code/meas_256.txt
index 15035c6..2ca4447 100644
--- a/buch/papers/multiplikation/code/meas_256.txt
+++ b/buch/papers/multiplikation/code/meas_256.txt
@@ -1,6 +1,6 @@
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diff --git a/buch/papers/multiplikation/code/meas_32.pdf b/buch/papers/multiplikation/code/meas_32.pdf
index 94c3731..b926095 100644
--- a/buch/papers/multiplikation/code/meas_32.pdf
+++ b/buch/papers/multiplikation/code/meas_32.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/code/meas_32.txt b/buch/papers/multiplikation/code/meas_32.txt
index afdb6d5..0fdc18d 100644
--- a/buch/papers/multiplikation/code/meas_32.txt
+++ b/buch/papers/multiplikation/code/meas_32.txt
@@ -1,6 +1,6 @@
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diff --git a/buch/papers/multiplikation/code/meas_64.pdf b/buch/papers/multiplikation/code/meas_64.pdf
index 3a90949..92af29b 100644
--- a/buch/papers/multiplikation/code/meas_64.pdf
+++ b/buch/papers/multiplikation/code/meas_64.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/code/meas_64.txt b/buch/papers/multiplikation/code/meas_64.txt
index ae6ff9b..b4fc7a1 100644
--- a/buch/papers/multiplikation/code/meas_64.txt
+++ b/buch/papers/multiplikation/code/meas_64.txt
@@ -1,6 +1,6 @@
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diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex
index 7ee0b6e..0f6aa6b 100755
--- a/buch/papers/multiplikation/loesungsmethoden.tex
+++ b/buch/papers/multiplikation/loesungsmethoden.tex
@@ -295,9 +295,58 @@ Die Implementation kann Algorithmus \ref{multiplikation:alg:winograd} entnommen
\end{algorithmic}
\end{algorithm}
-\subsection{Weitere Algorithmen}
+\subsection{Basic Linear Algebra Subprograms (BLAS)}
+
+die gebr\"uchlichen Methode f\"ur die Anwendung einer optimierten Matrizenmultiplikation ist die Verwendung einer Subrutine aus den \textit{Basic Linear Algebra Subprograms (BLAS)} \cite{multiplikation:BLAS}.
+Die meisten Numerischen Bibliotheken von High-Level Skriptsprachen wie \texttt{Matlab}, \texttt{NumPy (Python)}, \texttt{GNU Octave} oder \texttt{Mathematica} ben\"utzen eine Form von \textit{BLAS}.
+
+\textit{BLAS} sind dabei in drei unterschiedliche Levels aufgeteilt.
+
+\begin{itemize}
+ \item Level 1
+ \begin{itemize}
+ \item Operationen der Art: $\mathbf{y} \leftarrow \alpha \mathbf{x}+\mathbf{y}$
+ \item Dieses Level hat $\mathcal{O}(n)$ karakteristik
+ \end{itemize}
+ \item Level 2
+ \begin{itemize}
+ \item Operationen der Art: $\mathbf{y} \leftarrow \alpha \mathbf{A}\mathbf{x}+\beta \mathbf{y}$
+ \item Dieses Level hat $\mathcal{O}\left(n^2\right)$ karakteristik
+ \end{itemize}
+ \item Level 3
+ \begin{itemize}
+ \item Operationen der Art: $\mathbf{C} \leftarrow \alpha \mathbf{A}\mathbf{B}+\beta\mathbf{C}$
+ \item Dieses Level hat $\mathcal{O}\left(n^3\right)$ karakteristik
+ \end{itemize}
+\end{itemize}
+
+Die \textit{BLAS} sind auf die modernen Computer Prozessoren optimiert und k\"onnen dank einer ausgek\"ugelter Verwedung der Speicher Architektur zur erheblichen Leistungoprimierung f\"uhren.
+
+
+\subsubsection{General Matrix Multiplication (GEMM)}
+
+Die \textit{Double-GEMM} ist in \cite{multiplikation:DGEMM} definiert als:
+
+\textit{DGEMM performs one of the matrix-matrix operations}
+$$
+ C := \alpha \cdot op( A )\cdot op( B ) + \beta \cdot C,
+ $$
+ \textit{where op( X ) is one of}
+$$
+op( X ) = X \quad \text{ or } \quad op( X ) = X^T,
+$$
+ \textit{alpha and beta are scalars, and A, B and C are matrices, with op( A )
+ an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
+ }
+
+Die Implementaion von $\alpha\mathbf{A}\mathbf{B} + \beta \mathbf{C} = \mathbf{C}$, wobei $\alpha = 1.0$ und $\beta = 0.0$ in der \texttt{C}-Version von \textit{BLAS}, ist als
+\begin{lstlisting}[style=multiplikationC]
+cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans,
+ m, n, k, 1, A, m , B, k, 0, C, m);
+\end{lstlisting}
+definiert.
+
-\textcolor{red}{TODO: BLAS}
\section{Implementation}
\rhead{Implementation}
diff --git a/buch/papers/multiplikation/main.tex b/buch/papers/multiplikation/main.tex
index 8d0a8df..fb1908e 100755
--- a/buch/papers/multiplikation/main.tex
+++ b/buch/papers/multiplikation/main.tex
@@ -4,6 +4,28 @@
%
% (c) 2021 Hochschule Rapperswil
%
+\definecolor{mygreen}{RGB}{28,172,0} % color values Red, Green, Blue
+\definecolor{mylilas}{RGB}{170,55,241}
+\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
+\lstdefinestyle{multiplikationC}{
+ numbers=left,
+ belowcaptionskip=1\baselineskip,
+ breaklines=true,
+ frame=l,
+ framerule=0pt,
+ framesep=-1pt,
+ xleftmargin=1em,
+ language=C,
+ showstringspaces=false,
+ basicstyle=\ttfamily,
+ keywordstyle=\bfseries\color{green!40!black},
+ commentstyle=\itshape\color{purple!40!black},
+ identifierstyle=\color{blue},
+ stringstyle=\color{red},
+ numberstyle=\ttfamily\tiny,
+ backgroundcolor=\color{backcolour}
+}
+
\chapter{Schnelle Matrizen Multiplikation\label{chapter:multiplikation}}
\lhead{FMM}
\begin{refsection}
diff --git a/buch/papers/multiplikation/references.bib b/buch/papers/multiplikation/references.bib
index 63cb976..8815386 100755
--- a/buch/papers/multiplikation/references.bib
+++ b/buch/papers/multiplikation/references.bib
@@ -83,3 +83,20 @@
day = {28}
}
+@online{multiplikation:BLAS,
+ title = {BLAS (Basic Linear Algebra Subprograms)},
+ url = {http://www.netlib.org/blas/},
+ date = {2021-08-01},
+ year = {2021},
+ month = {8},
+ day = {01}
+}
+
+@online{multiplikation:DGEMM,
+ title = {DGEMM},
+ url = {http://www.netlib.org/lapack/explore-html/d1/d54/group__double__blas__level3_gaeda3cbd99c8fb834a60a6412878226e1.html#gaeda3cbd99c8fb834a60a6412878226e1},
+ date = {2021-08-01},
+ year = {2021},
+ month = {8},
+ day = {01}
+}