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NumPy Tutorial メモ6(Linear Algebra)
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| { | |
| "cells": [ | |
| { | |
| "metadata": { | |
| "toc": "true" | |
| }, | |
| "cell_type": "markdown", | |
| "source": "# Table of Contents\n <p><div class=\"lev1\"><a href=\"#NumPy-Tutorial-6-1\"><span class=\"toc-item-num\">1 </span>NumPy Tutorial 6</a></div><div class=\"lev2\"><a href=\"#Linear-Algebra（線形代数）-1.1\"><span class=\"toc-item-num\">1.1 </span>Linear Algebra（線形代数）</a></div><div class=\"lev3\"><a href=\"#Simple-Array-Operations（シンプルな配列演算）-1.1.1\"><span class=\"toc-item-num\">1.1.1 </span>Simple Array Operations（シンプルな配列演算）</a></div><div class=\"lev3\"><a href=\"#行列クラス-1.1.2\"><span class=\"toc-item-num\">1.1.2 </span>行列クラス</a></div><div class=\"lev3\"><a href=\"#添字アクセス：行列と2次元配列の比較-1.1.3\"><span class=\"toc-item-num\">1.1.3 </span>添字アクセス：行列と2次元配列の比較</a></div><div class=\"lev3\"><a href=\"#Vector-Stacking（ベクトルのスタック）-1.1.4\"><span class=\"toc-item-num\">1.1.4 </span>Vector Stacking（ベクトルのスタック）</a></div><div class=\"lev3\"><a href=\"#Histograms-1.1.5\"><span class=\"toc-item-num\">1.1.5 </span>Histograms</a></div><div class=\"lev2\"><a href=\"#参考リンク-1.2\"><span class=\"toc-item-num\">1.2 </span>参考リンク</a></div>" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "# NumPy Tutorial 6" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "- [NumPy Tutorial メモ1(Basic)](http://nbviewer.jupyter.org/gist/Cartman0/9fa48b89664dc08ef82a55877767b5a0)\n- [NumPy Tutorial メモ2(ShapeManipulation)](http://nbviewer.jupyter.org/gist/Cartman0/bc062d1a162589e40a02c21e0ada7776)\n- [NumPy Tutorial メモ3(Copies and Views)](http://nbviewer.jupyter.org/gist/Cartman0/cd39c1bfc88f5edccd8f4523fbd33b91)\n- [NumPy Tutorial メモ4(Less Basic)](http://nbviewer.jupyter.org/gist/Cartman0/e9f50b0f0dbc28f40938b11088ac211d)\n- [NumPy Tutorial メモ5(Fancy indexing and index tricks)](http://nbviewer.jupyter.org/gist/Cartman0/5d24bdc3234a97bfc7c9fed4355a1580)" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "## Linear Algebra（線形代数）" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "Work in progress. \nBasic linear algebra to be included here.\n（基礎の線形代数がここに含まれる。）" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### Simple Array Operations（シンプルな配列演算）" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "See `linalg.py` in numpy folder for more." | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "import numpy as np\na = np.array([\n [1.0, 2.0], \n [3.0, 4.0]\n ])\nprint(a)", | |
| "execution_count": 1, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[ 1. 2.]\n [ 3. 4.]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "a.transpose() # 転置", | |
| "execution_count": 2, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 1., 3.],\n [ 2., 4.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 2 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "a.transpose() == a.T # Tも同じ", | |
| "execution_count": 3, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ True, True],\n [ True, True]], dtype=bool)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 3 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.linalg.inv(a) # 逆行列", | |
| "execution_count": 4, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[-2. , 1. ],\n [ 1.5, -0.5]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 4 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "$$\nA * A^{-1} = I\n$$\n\n$\nA = \\left(\n \\begin{array}{cc}\n a & b \\\\\n c & d \n \\end{array}\n \\right)\n$ , $\nA^{-1} = \\left(\n \\begin{array}{cc}\n a^{-1} & b^{-1} \\\\\n c^{-1} & d^{-1} \n \\end{array}\n \\right)\n$ として、\n\n$$\n\\left(\n \\begin{array}{cc}\n a & b \\\\\n c & d \n \\end{array}\n \\right) \n \\left(\n \\begin{array}{cc}\n a^{-1} & b^{-1} \\\\\n c^{-1} & d^{-1} \n \\end{array}\n \\right)\n = \n \\left(\n \\begin{array}{cc}\n 1 & 0 \\\\\n 0 & 1 \n \\end{array}\n \\right)\n$$\n\n\\begin{eqnarray}\na * a^{-1} + b * c{^-1} = 1\\\\\na * b^{-1} + b * d^{-1} = 0\\\\\nc * a^{-1} + d * c^{-1} = 0\\\\\nc * b^{-1} + d * d^{-1} = 1\\\\\n\\end{eqnarray}\n\n$$\nc * a^{-1} + d * c^{-1} = 0 \\\\\nc^{-1} = - \\frac{c*a^{-1}}{d}\n$$\n\n\\begin{eqnarray}\na * a^{-1} + b * c{^-1} &=& 1 \\\\\na * a^{-1} - b * \\frac{c*a^{-1}}{d} &=& 1 \\\\\n(a - \\frac{bc}{d})a^{-1} &=&1\\\\\na^{-1} &=& \\frac{1}{(a - \\frac{bc}{d})}\\\\\n&=& \\frac{d}{(ad - bc)}\\\\\n\\end{eqnarray}\n\n\\begin{eqnarray}\nc^{-1} &=& - \\frac{c}{d}*\\frac{d}{(ad - bc)}\\\\\n&=& - \\frac{c}{(ad - bc)}\\\\\n\\end{eqnarray}\n\n\\begin{eqnarray}\nc^{-1} &=& - \\frac{c}{d}*\\frac{d}{(ad - bc)}\\\\\n&=& - \\frac{c}{(ad - bc)}\\\\\n\\end{eqnarray}\n\n\\begin{eqnarray}\na * b^{-1} + b * d^{-1} &=& 0 \\\\\nd^{-1} &=& -\\frac{a * b^{-1}}{b}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nc * b^{-1} + d * d^{-1} &=& 1 \\\\\nc * b^{-1} + d * -\\frac{a * b^{-1}}{b} &=& 1 \\\\\n(c - d\\frac{a}{b})b^{-1} &=& 1\\\\\nb^{-1} &=& - \\frac{1}{d\\frac{a}{b} - c}\\\\\n &=& - \\frac{b}{ad - bc}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nd^{-1} &=& -\\frac{a * b^{-1}}{b}\\\\\n&=& -\\frac{a}{b} * - \\frac{b}{ad - bc}\\\\\n&=& \\frac{a}{ad - bc}\\\\\n\\end{eqnarray}\n\n$$\nA^{-1} = \\frac{1}{ad-bc} \\left(\n \\begin{array}{cc}\n d & -b \\\\\n -c & a \n \\end{array}\n \\right)\n$$" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "u = np.eye(2) # unit 2x2 matrix; \"eye\" represents \"I\" 単位行列\nu", | |
| "execution_count": 5, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 1., 0.],\n [ 0., 1.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 5 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "j = np.array([\n [0.0, -1.0], \n [1.0, 0.0]\n ])\nj", | |
| "execution_count": 6, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 0., -1.],\n [ 1., 0.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 6 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.dot(j, j) # matrix product", | |
| "execution_count": 7, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[-1., 0.],\n [ 0., -1.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 7 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "j.dot(j)", | |
| "execution_count": 8, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[-1., 0.],\n [ 0., -1.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 8 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.trace(u) # trace", | |
| "execution_count": 9, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "2.0" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 9 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "y = np.array([\n [5.], \n [7.]\n ])\ny", | |
| "execution_count": 10, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 5.],\n [ 7.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 10 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.linalg.solve(a, y) # 2元連立方程式 A^-1 * y", | |
| "execution_count": 11, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[-3.],\n [ 4.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 11 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "固有値の計算\n\nhttp://docs.scipy.org/doc/numpy-1.10.0/reference/generated/numpy.linalg.eig.html" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "scrolled": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "eig_val, eig_vec = np.linalg.eig(j)\neig_val, eig_vec", | |
| "execution_count": 12, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "(array([ 0.+1.j, 0.-1.j]),\n array([[ 0.70710678+0.j , 0.70710678-0.j ],\n [ 0.00000000-0.70710678j, 0.00000000+0.70710678j]]))" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 12 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "cell_type": "markdown", | |
| "source": "### 行列クラス" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "[numpy.matrix](http://docs.scipy.org/doc/numpy-1.10.1/reference/generated/numpy.matrix.html)" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A = np.matrix('1.0 2.0; 3.0 4.0')\nA", | |
| "execution_count": 13, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 1., 2.],\n [ 3., 4.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 13 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "type(A) # クラスを定義しているファイル", | |
| "execution_count": 14, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "numpy.matrixlib.defmatrix.matrix" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 14 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A.T # 転置", | |
| "execution_count": 15, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 1., 3.],\n [ 2., 4.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 15 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "X = np.matrix('5.0 7.0')\nX", | |
| "execution_count": 16, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 5., 7.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 16 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "Y = X.T\nY", | |
| "execution_count": 17, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 5.],\n [ 7.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 17 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A * Y) # 行列の乗算", | |
| "execution_count": 18, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[ 19.]\n [ 43.]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A.I) # 逆行列", | |
| "execution_count": 19, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[-2. 1. ]\n [ 1.5 -0.5]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.linalg.solve(A, Y) # 線形方程式を解く", | |
| "execution_count": 20, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[-3.],\n [ 4.]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 20 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### 添字アクセス：行列と2次元配列の比較" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "NumPy の配列と行列の間にはいくつかの重要な差異があることに注意。\n\nNumPy は、\n- N次元配列オブジェクトと\n- ユニバーサル関数オブジェクト\n\nという2種類の基本的なオブジェクトを用意している。\nその他のオブジェクトはこれら2つの上に構築されている。\n\n特に、行列はNumPyのarrayオブジェクトを継承した2次元配列オブジェクトである。\n配列も行列も、添字は以下の1つもしくは複数の適切な組み合わせで構成されていなければならない：整数スカラー値、省略記号、整数または真偽値のリスト、整数または真偽値のタプル、整数または真偽値の1次元配列。\n行列を行列のindex として用いることもできるが、普通は配列、リスト、あるいは他の形で事が足りる。\nPythonでもそうであるように、index は0ベースである。\n伝統的に我々は2次元配列や行列を行と列の長方形の配列として表す。\n軸0に沿った動きは行を横切り、軸1に沿った動きは列を横切る。" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A = np.arange(12)\nA", | |
| "execution_count": 21, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 21 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A.shape = (3,4)\nA", | |
| "execution_count": 22, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 22 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "M = np.mat(A.copy())\nM", | |
| "execution_count": 23, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 23 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(type(A), ' ', type(M))", | |
| "execution_count": 24, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "<class 'numpy.ndarray'> <class 'numpy.matrixlib.defmatrix.matrix'>\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A)", | |
| "execution_count": 25, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[ 0 1 2 3]\n [ 4 5 6 7]\n [ 8 9 10 11]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(M)", | |
| "execution_count": 26, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[ 0 1 2 3]\n [ 4 5 6 7]\n [ 8 9 10 11]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "ここで、いくつか単純なスライスを取ってみる。\n基本的なスライシングにはスライスオブジェクトまたは整数の組を用いる。\n\n例えば、`A[:]` と `M[:]` の評価はPython の index付けに似たものに見えるのだが、重要なのはNumPy配列のスライシングはデータをコピーしないという点である。\nスライシングは同じデータに対し新しいview を提供する。" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A[:])", | |
| "execution_count": 27, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[ 0 1 2 3]\n [ 4 5 6 7]\n [ 8 9 10 11]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A[:].shape)", | |
| "execution_count": 28, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "(3, 4)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(M[:]) ", | |
| "execution_count": 29, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[ 0 1 2 3]\n [ 4 5 6 7]\n [ 8 9 10 11]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(M[:].shape)", | |
| "execution_count": 30, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "(3, 4)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "Python のindex付けと異なるが、\n複数の軸に沿って同時にindex付けする場合にカンマ区切りのindex が使える。" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A[:, 1])", | |
| "execution_count": 31, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[1 5 9]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(A[:, 1].shape)", | |
| "execution_count": 32, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "(3,)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(M[:, 1])", | |
| "execution_count": 33, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "[[1]\n [5]\n [9]]\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "print(M[:, 1].shape)", | |
| "execution_count": 34, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "(3, 1)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "最後の2つの結果は異なる。\n- 2次元配列に対しコロンを1つ使うと1次元配列ができる\n- 一方、行列に対してだと2次元行列ができる。\n\n行列のスライスは常に行列を生み出す。\n例えばスライス `M[2, :]` は `shape (1, 4)` の行列になるが、\n対照的に、配列のスライスは常に可能な限り一番低い次元の配列になる。\n\n例えば `C` が3次元配列だとすると、\n- `C[..., 1]` は2次元配列に、\n- `C[1, :, 1]` は1次元配列になる。" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "配列の1番目と3番目の列が欲しいとしよう。\nリストを使ってスライスするのが1つの方法である：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A", | |
| "execution_count": 35, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 35 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[:, [1, 3]]", | |
| "execution_count": 36, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 1, 3],\n [ 5, 7],\n [ 9, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 36 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "`take()` メソッドを使うというのもある：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A.take([1, 3], axis=1)", | |
| "execution_count": 37, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 1, 3],\n [ 5, 7],\n [ 9, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 37 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "最初の行(第0軸の0番目)を飛ばしたいなら以下のようになる：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[1:,].take([1, 3], axis=1)", | |
| "execution_count": 38, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 5, 7],\n [ 9, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 38 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "あるいは単に `A[1:, [1, 3]]` としてもよい。\n上のをスライスするもう1つの方法は、`numpy.ix_` の利用がある：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.ix_((1, 2), (1, 3))", | |
| "execution_count": 39, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "(array([[1],\n [2]]), array([[1, 3]]))" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 39 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A", | |
| "execution_count": 40, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 40 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[np.ix_((1, 2), (1, 3))]", | |
| "execution_count": 41, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 5, 7],\n [ 9, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 41 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "1行目が1より大きい全ての列を保持したいととする。\nまずは真偽値のindex を作る方法：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[0, :] > 1", | |
| "execution_count": 42, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([False, False, True, True], dtype=bool)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 42 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "しかし行列をindex付けするのはあまり便利ではない。" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "M[0, :] > 1", | |
| "execution_count": 43, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[False, False, True, True]], dtype=bool)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 43 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "scrolled": true, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "M[:, M[0, :] > 1]", | |
| "execution_count": 44, | |
| "outputs": [ | |
| { | |
| "output_type": "error", | |
| "ename": "IndexError", | |
| "evalue": "too many indices for array", | |
| "traceback": [ | |
| "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", | |
| "\u001b[1;31mIndexError\u001b[0m Traceback (most recent call last)", | |
| "\u001b[1;32m<ipython-input-44-d1c10536344f>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mM\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mM\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", | |
| "\u001b[1;32mC:\\Miniconda3\\lib\\site-packages\\numpy\\matrixlib\\defmatrix.py\u001b[0m in \u001b[0;36m__getitem__\u001b[1;34m(self, index)\u001b[0m\n\u001b[0;32m 316\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 317\u001b[0m \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 318\u001b[1;33m \u001b[0mout\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mN\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mndarray\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m__getitem__\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mindex\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 319\u001b[0m \u001b[1;32mfinally\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 320\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_getitem\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;32mFalse\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", | |
| "\u001b[1;31mIndexError\u001b[0m: too many indices for array" | |
| ] | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "問題としているのは、行列スライスをスライスすることで行列が生成されてしまう点である。\nしかし行列には便利な'A'属性があって、この値はその行列の配列表現になっているので、代わりにこうするだけで良い：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "M[:, M.A[0, :] > 1]", | |
| "execution_count": 45, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 2, 3],\n [ 6, 7],\n [10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 45 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "行列を条件によって2つの方向にスライスしたいのであれば、\n戦略をわずかに補正しなければならない。" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A", | |
| "execution_count": 46, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 46 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[:, 0] > 2", | |
| "execution_count": 47, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([False, True, True], dtype=bool)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 47 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[0, :] > 1", | |
| "execution_count": 48, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([False, False, True, True], dtype=bool)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 48 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "scrolled": true, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[A[:, 0] > 2, A[0, :] > 1]", | |
| "execution_count": 49, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([ 6, 11])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 49 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "の代わりに、外積 `'ix_'` を用いる：" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "np.ix_(A[:, 0] > 2, A[0, :] > 1)", | |
| "execution_count": 50, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "(array([[1],\n [2]], dtype=int64), array([[2, 3]], dtype=int64))" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 50 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "scrolled": true, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "A[np.ix_(A[:, 0] > 2, A[0, :] > 1)]", | |
| "execution_count": 51, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[ 6, 7],\n [10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 51 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "M[np.ix_(M.A[:, 0] > 2, M.A[0, :] > 1)]", | |
| "execution_count": 52, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "matrix([[ 6, 7],\n [10, 11]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 52 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### Vector Stacking（ベクトルのスタック）" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "How do we construct a 2D array from a list of equally-sized row vectors? \n（サイズの等しい行ベクトルのリストから2次元配列を構築するにはどうするか？）\n\nIn MATLAB this is quite easy（MATLABではこれはとても簡単）:\nif `x` and `y` are two vectors of the same length you only need do `m = [x; y]`. \n（`x` と `y` の2つが同じ長さのベクトルなら、`m=[x;y]` とするだけで良い。）\n\nIn NumPy this works via the functions `column_stack`, \n`dstack`, `hstack` and `vstack`, \ndepending on the dimension in which the stacking is to be done. \n（NumPyでは、関数 `column_stack`, `dstack`, `hstack`, `vstack` で同じことができる、\n関数の選択はスタックが行われる次元に依存する。）\n\nFor example:" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": true, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = np.arange(0, 10, 2) # x=([0,2,4,6,8])\ny = np.arange(5)# y=([0,1,2,3,4])", | |
| "execution_count": 53, | |
| "outputs": [] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "m = np.vstack([x, y]) # m=([[0,2,4,6,8],\nm # [0,1,2,3,4]]), 2 * 5", | |
| "execution_count": 54, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([[0, 2, 4, 6, 8],\n [0, 1, 2, 3, 4]])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 54 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "m.shape", | |
| "execution_count": 55, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "(2, 5)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 55 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "xy = np.hstack([x, y]) # xy =([0,2,4,6,8,0,1,2,3,4]) # 1 * 10\nxy", | |
| "execution_count": 56, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "array([0, 2, 4, 6, 8, 0, 1, 2, 3, 4])" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 56 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "xy.shape", | |
| "execution_count": 57, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "(10,)" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 57 | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "See alse:\n\n- [Numpy for Matlab users](https://docs.scipy.org/doc/numpy-dev/user/numpy-for-matlab-users.html)" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "### Histograms" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "The NumPy `histogram` function applied to an array returns a pair of vectors（NumPy の`histogram`関数は、2つのベクトルを返す配列に適用される）： \nthe histogram of the array and the vector of bins. \n（配列のヒストグラムと、瓶(bins)のベクトル、が返ってくる。）\n\nBeware: `matplotlib` also has a function to build histograms (called `hist`, as in Matlab) that differs from the one in NumPy. \n（注意：`matplotlib` にもヒストグラムを構築する関数がある\n（Matlabと同様 hist と呼ばれる）が、NumPyのものとは異なる。）\n\nThe main difference is that `pylab.hist` plots the histogram automatically, \nwhile `numpy.histogram` only generates the data.\n（主な違いは、`pylab.hist` はヒストグラムを自動的に描画するが、\n`numpy.histogram` はデータを生成するところまでしか行わない点である。" | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "import numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\n# Build a vector of 10000 normal deviates with variance 0.5^2 and mean 2\n\n(mu, sigma) = 2, 0.5\nv = np.random.normal(mu, sigma, 10000)\n# Plot a normalized histogram with 50 bins\nplt.hist(v, bins=50, normed=1) # matplotlib version (plot)\nplt.show()", | |
| "execution_count": 58, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<matplotlib.figure.Figure at 0x25ab9ec54e0>", | |
| "image/png": 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| |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": false, | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "%matplotlib inline\n\n# Compute the histogram with numpy and then plot it\n(n, bins) = np.histogram(v, bins=50, normed=True) # NumPy version (no plot)\nprint('n ', n)\nprint('bins', bins)\nprint('bins[:-1]', bins[:-1])\nplt.plot(.5 * (bins[:-1] + bins[1:]), n)", | |
| "execution_count": 59, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "n [ 0.00547374 0.00136843 0.00547374 0.01231591 0.01505278 0.02326339\n 0.02189496 0.04242148 0.0547374 0.07526392 0.09305357 0.16284375\n 0.17379123 0.21073898 0.28052916 0.31337159 0.41874108 0.56926892\n 0.57611109 0.66505936 0.70748085 0.72663893 0.77453416 0.69242806\n 0.81969251 0.79779755 0.73484954 0.75400763 0.69242806 0.56653205\n 0.54053179 0.43242543 0.36810899 0.31200316 0.25452889 0.20115993\n 0.17515967 0.12042227 0.07526392 0.07115862 0.05063209 0.03421087\n 0.01778965 0.01505278 0.01505278 0. 0.00136843 0.00821061\n 0.00273687 0.00136843]\nbins [ 0.22548126 0.29855745 0.37163363 0.44470981 0.517786 0.59086218\n 0.66393836 0.73701455 0.81009073 0.88316691 0.9562431 1.02931928\n 1.10239546 1.17547165 1.24854783 1.32162401 1.3947002 1.46777638\n 1.54085256 1.61392875 1.68700493 1.76008111 1.8331573 1.90623348\n 1.97930966 2.05238585 2.12546203 2.19853821 2.2716144 2.34469058\n 2.41776676 2.49084295 2.56391913 2.63699531 2.7100715 2.78314768\n 2.85622386 2.92930005 3.00237623 3.07545241 3.1485286 3.22160478\n 3.29468096 3.36775715 3.44083333 3.51390951 3.5869857 3.66006188\n 3.73313806 3.80621425 3.87929043]\nbins[:-1] [ 0.22548126 0.29855745 0.37163363 0.44470981 0.517786 0.59086218\n 0.66393836 0.73701455 0.81009073 0.88316691 0.9562431 1.02931928\n 1.10239546 1.17547165 1.24854783 1.32162401 1.3947002 1.46777638\n 1.54085256 1.61392875 1.68700493 1.76008111 1.8331573 1.90623348\n 1.97930966 2.05238585 2.12546203 2.19853821 2.2716144 2.34469058\n 2.41776676 2.49084295 2.56391913 2.63699531 2.7100715 2.78314768\n 2.85622386 2.92930005 3.00237623 3.07545241 3.1485286 3.22160478\n 3.29468096 3.36775715 3.44083333 3.51390951 3.5869857 3.66006188\n 3.73313806 3.80621425]\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "data": { | |
| "text/plain": "[<matplotlib.lines.Line2D at 0x25abbc7f780>]" | |
| }, | |
| "metadata": {}, | |
| "execution_count": 59 | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "<matplotlib.figure.Figure at 0x25abba57f98>", | |
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| |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "cell_type": "markdown", | |
| "source": "## 参考リンク" | |
| }, | |
| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "- [QuickStartTutorial](https://docs.scipy.org/doc/numpy-dev/user/quickstart.html)\n- [私訳「暫定的 NumPy チュートリアル」](http://naoyat.hatenablog.jp/entry/2011/12/29/021414)" | |
| } | |
| ], | |
| "metadata": { | |
| "hide_input": false, | |
| "kernelspec": { | |
| "name": "python3", | |
| "display_name": "Python 3", | |
| "language": "python" | |
| }, | |
| "language_info": { | |
| "name": "python", | |
| "mimetype": "text/x-python", | |
| "nbconvert_exporter": "python", | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 3 | |
| }, | |
| "pygments_lexer": "ipython3", | |
| "file_extension": ".py", | |
| "version": "3.5.1" | |
| }, | |
| "toc": { | |
| "toc_window_display": false, | |
| "toc_cell": true, | |
| "toc_number_sections": true, | |
| "toc_threshold": "6" | |
| }, | |
| "gist": { | |
| "id": "", | |
| "data": { | |
| "description": "NumPy Tutorial メモ6(Linear Algebra)", | |
| "public": true | |
| } | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 0 | |
| } |
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