stammtisch ß6.05.2020
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Steffen Fritz
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5 changed files with 904 additions and 1 deletions
318
post/motivation_matrix_mult.ipynb
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318
post/motivation_matrix_mult.ipynb
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Motivation\n",
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"## How to solve matrix-matrix multiplication"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import pandas as pd"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Simple approach in C\n",
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"\n",
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"~~~\n",
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" for (c = 0; c < m; c++) {\n",
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" for (d = 0; d < q; d++) {\n",
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" for (k = 0; k < p; k++) {\n",
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" sum = sum + first[c][k]*second[k][d];\n",
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" }\n",
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" \n",
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" multiply[c][d] = sum;\n",
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" sum = 0;\n",
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" }\n",
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" }\n",
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"~~~"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"def mymatrixmult(A,B):\n",
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" y = np.zeros((A.shape[0], B.shape[1]))\n",
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" for i in range(A.shape[0]):\n",
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" for j in range(B.shape[1]):\n",
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" for k in range(A.shape[0]):\n",
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" y[i][j] += A[i][k]*B[k][j]\n",
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" return y\n",
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" \n",
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"m = np.arange(40000).reshape(200,200)\n",
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"m1 = m/np.average(m)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Takes a while (200**3) = 8 MFLOPS"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[6.61708085e-03, 1.65675784e-02, 2.65180759e-02, ...,\n",
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" 1.96686509e+00, 1.97681559e+00, 1.98676609e+00],\n",
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" [1.65675784e-02, 4.65190759e-02, 7.64705735e-02, ...,\n",
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" 5.91701260e+00, 5.94696409e+00, 5.97691559e+00],\n",
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" [2.65180759e-02, 7.64705735e-02, 1.26423071e-01, ...,\n",
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" 9.86716010e+00, 9.91711260e+00, 9.96706510e+00],\n",
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" ...,\n",
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" [1.96686509e+00, 5.91701260e+00, 9.86716010e+00, ...,\n",
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" 7.80145924e+02, 7.84096071e+02, 7.88046219e+02],\n",
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" [1.97681559e+00, 5.94696409e+00, 9.91711260e+00, ...,\n",
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" 7.84096071e+02, 7.88066220e+02, 7.92036368e+02],\n",
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" [1.98676609e+00, 5.97691559e+00, 9.96706510e+00, ...,\n",
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" 7.88046219e+02, 7.92036368e+02, 7.96026518e+02]])"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"mymatrixmult(m1, m1.T)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[6.61708085e-03, 1.65675784e-02, 2.65180759e-02, ...,\n",
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" 1.96686509e+00, 1.97681559e+00, 1.98676609e+00],\n",
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" [1.65675784e-02, 4.65190759e-02, 7.64705735e-02, ...,\n",
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" 5.91701260e+00, 5.94696409e+00, 5.97691559e+00],\n",
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" [2.65180759e-02, 7.64705735e-02, 1.26423071e-01, ...,\n",
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" 9.86716010e+00, 9.91711260e+00, 9.96706510e+00],\n",
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" ...,\n",
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" [1.96686509e+00, 5.91701260e+00, 9.86716010e+00, ...,\n",
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" 7.80145924e+02, 7.84096071e+02, 7.88046219e+02],\n",
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" [1.97681559e+00, 5.94696409e+00, 9.91711260e+00, ...,\n",
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" 7.84096071e+02, 7.88066220e+02, 7.92036368e+02],\n",
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" [1.98676609e+00, 5.97691559e+00, 9.96706510e+00, ...,\n",
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" 7.88046219e+02, 7.92036368e+02, 7.96026518e+02]])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"m1.dot(m1.T)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Now numpy: (2000**3) = 8 GFLOPS"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"M = np.arange(4000000).reshape(2000,2000) \n",
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"M1 = M/np.average(M) "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[6.66167083e-04, 1.66566758e-03, 2.66516808e-03, ...,\n",
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" 1.99666867e+00, 1.99766817e+00, 1.99866767e+00],\n",
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" [1.66566758e-03, 4.66516908e-03, 7.66467058e-03, ...,\n",
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" 5.99167016e+00, 5.99466966e+00, 5.99766917e+00],\n",
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" [2.66516808e-03, 7.66467058e-03, 1.26641731e-02, ...,\n",
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" 9.98667166e+00, 9.99167116e+00, 9.99667067e+00],\n",
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" ...,\n",
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" [1.99666867e+00, 5.99167016e+00, 9.98667166e+00, ...,\n",
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" 7.98001466e+03, 7.98400966e+03, 7.98800466e+03],\n",
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" [1.99766817e+00, 5.99466966e+00, 9.99167116e+00, ...,\n",
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" 7.98400966e+03, 7.98800666e+03, 7.99200366e+03],\n",
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" [1.99866767e+00, 5.99766917e+00, 9.99667067e+00, ...,\n",
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" 7.98800466e+03, 7.99200366e+03, 7.99600267e+03]])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"M1.dot(M1.T)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11],\n",
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" [ 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23],\n",
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" [ 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35],\n",
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" [ 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47],\n",
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" [ 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59],\n",
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" [ 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71],\n",
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" [ 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83],\n",
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" [ 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95],\n",
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" [ 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107],\n",
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" [108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119],\n",
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" [120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131],\n",
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" [132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143]])"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"n = 12\n",
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"np.arange(n*n).reshape(n,n)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Splitting the matrix\n",
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"\n",
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"Two reasons:\n",
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"1. optimize cache usage\n",
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"2. using SIMD power\n",
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"\n",
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"<img src=\"pics/matrix.png\">"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## ?\n",
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"\n",
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"No idea, about the following: $y = tanh(M)$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"y = np.tanh(M1.dot(M1.T))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(1000, 1000)"
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]
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},
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"execution_count": 24,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"y.shape"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 51,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"16.0"
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]
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},
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"execution_count": 51,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"((4*128)**3)*16/((128)**3*64)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# das wichtig"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.9"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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