diff options
Diffstat (limited to 'buch')
| -rw-r--r-- | buch/papers/multiplikation/code/MM.py | 23 | ||||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_1024.pdf | bin | 17660 -> 17653 bytes | |||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_1024.txt | 10 | ||||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_128.pdf | bin | 17961 -> 18120 bytes | |||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_128.txt | 10 | ||||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_256.pdf | bin | 18067 -> 19428 bytes | |||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_256.txt | 10 | ||||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_32.pdf | bin | 17078 -> 17964 bytes | |||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_32.txt | 10 | ||||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_64.pdf | bin | 17678 -> 17747 bytes | |||
| -rw-r--r-- | buch/papers/multiplikation/code/meas_64.txt | 10 | ||||
| -rwxr-xr-x | buch/papers/multiplikation/loesungsmethoden.tex | 53 | ||||
| -rwxr-xr-x | buch/papers/multiplikation/main.tex | 22 | ||||
| -rwxr-xr-x | buch/papers/multiplikation/references.bib | 17 |
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 Binary files differindex fd0a108..7b7a133 100644 --- a/buch/papers/multiplikation/code/meas_1024.pdf +++ b/buch/papers/multiplikation/code/meas_1024.pdf 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 -1.502037048339843750e-05 6.628036499023437500e-05 4.780292510986328125e-04 2.713203430175781250e-03 2.115225791931152344e-02 1.758832931518554688e-01 1.338865518569946289e+00 1.009106445312500000e+01 8.192077994346618652e+01 7.835870332717895508e+02 -6.675720214843750000e-06 7.200241088867187500e-05 5.540847778320312500e-04 3.144979476928710938e-03 2.545046806335449219e-02 2.083067893981933594e-01 1.659256219863891602e+00 1.319160294532775879e+01 1.046767003536224365e+02 9.679818902015686035e+02 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2.217054367065429688e-03 1.880884170532226562e-02 diff --git a/buch/papers/multiplikation/code/meas_128.pdf b/buch/papers/multiplikation/code/meas_128.pdf Binary files differindex ed1ec63..c54648f 100644 --- a/buch/papers/multiplikation/code/meas_128.pdf +++ b/buch/papers/multiplikation/code/meas_128.pdf 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 @@ 2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 1.280000000000000000e+02 -1.978874206542968750e-05 1.134872436523437500e-04 4.298686981201171875e-04 2.815246582031250000e-03 2.616596221923828125e-02 1.767718791961669922e-01 1.293319463729858398e+00 -6.675720214843750000e-06 1.251697540283203125e-04 4.818439483642578125e-04 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1.725149154663085938e-02 1.314754486083984375e-01 1.121860027313232422e+00 8.884316682815551758e+00 +3.147125244140625000e-05 6.675720214843750000e-06 4.768371582031250000e-06 7.867813110351562500e-06 2.574920654296875000e-05 5.888938903808593750e-05 2.071857452392578125e-04 6.518363952636718750e-04 diff --git a/buch/papers/multiplikation/code/meas_32.pdf b/buch/papers/multiplikation/code/meas_32.pdf Binary files differindex 94c3731..b926095 100644 --- a/buch/papers/multiplikation/code/meas_32.pdf +++ b/buch/papers/multiplikation/code/meas_32.pdf 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 @@ 2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 -1.215934753417968750e-05 5.459785461425781250e-05 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8.058547973632812500e-05 3.998279571533203125e-04 2.514839172363281250e-03 1.842117309570312500e-02 +1.215934753417968750e-05 8.583068847656250000e-06 6.675720214843750000e-06 2.694129943847656250e-05 2.789497375488281250e-05 diff --git a/buch/papers/multiplikation/code/meas_64.pdf b/buch/papers/multiplikation/code/meas_64.pdf Binary files differindex 3a90949..92af29b 100644 --- a/buch/papers/multiplikation/code/meas_64.pdf +++ b/buch/papers/multiplikation/code/meas_64.pdf 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 @@ 2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 -1.645088195800781250e-05 7.295608520507812500e-05 3.807544708251953125e-04 2.672195434570312500e-03 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1.807212829589843750e-04 6.821155548095703125e-04 4.796504974365234375e-03 2.968001365661621094e-02 2.291278839111328125e-01 +3.504753112792968750e-05 1.106262207031250000e-04 4.322528839111328125e-04 2.696514129638671875e-03 2.188420295715332031e-02 1.477701663970947266e-01 +3.218650817871093750e-05 1.144409179687500000e-05 7.390975952148437500e-06 4.625320434570312500e-05 3.814697265625000000e-05 5.435943603515625000e-05 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} +} |