LinearAlgebra.jl Tutorial

LinearAlgebra.ipynb

{
 "cells": [
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  LinearAlgebra.jl\n",
    "\n",
    "Juliaでは簡単に行列の固有値を求めることができる. このノートでは 斎藤正彦『線型代数入門』(1966, 東京大学出版会) の問題を題材として, 固有値ソルバの簡単なベンチマークテストを行った.\n",
    "\n",
    "© 2021 Shuhei Ohno\n",
    "<br>Source: https://gist.github.com/ohno\n",
    "<br>License: https://opensource.org/licenses/MIT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.6.2\n",
      "Commit 1b93d53fc4 (2021-07-14 15:36 UTC)\n",
      "Platform Info:\n",
      "  OS: Windows (x86_64-w64-mingw32)\n",
      "  CPU: Intel(R) Core(TM) i7-4650U CPU @ 1.70GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-11.0.1 (ORCJIT, haswell)\n"
     ]
    }
   ],
   "source": [
    "versioninfo()\n",
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ベンチマーク\n",
    "\n",
    "[斎藤正彦『線型代数入門』(1966, 東京大学出版会)]( http://www.utp.or.jp/book/b302039.html) より, 第5章の問題1をベンチマークとする. 対角形の各対角成分は元の行列の固有値である.\n",
    "\n",
    "<table>\n",
    "    <tr>\n",
    "      <th>問題1</th>\n",
    "      <th>行列</th>\n",
    "      <th>対角形</th>\n",
    "    </tr>\n",
    "    <tr>\n",
    "      <td>イ)</td>\n",
    "      <td>\n",
    "        \\begin{pmatrix}\n",
    "            -4 & 0 & 6 \\\\\n",
    "            -3 & 2 & 3 \\\\\n",
    "            -3 & 0 & 5 \\\\\n",
    "        \\end{pmatrix}\n",
    "      </td>\n",
    "      <td>\n",
    "        \\begin{pmatrix}\n",
    "            2 & 0 &  0 \\\\\n",
    "            0 & 2 &  0 \\\\\n",
    "            0 & 0 & -1 \\\\\n",
    "        \\end{pmatrix}\n",
    "      </td>\n",
    "    </tr>\n",
    "    <tr>\n",
    "      <td>ロ)</td>\n",
    "      <td>\n",
    "        \\begin{pmatrix}\n",
    "             1 &  0 & -2 \\\\\n",
    "             2 & -1 & -2 \\\\\n",
    "            -2 &  2 &  0 \\\\\n",
    "        \\end{pmatrix}\n",
    "      </td>\n",
    "      <td>\n",
    "        \\begin{pmatrix}\n",
    "            1 &  0 & 0 \\\\\n",
    "            0 & -1 & 0 \\\\\n",
    "            0 &  0 & 0 \\\\\n",
    "        \\end{pmatrix}\n",
    "      </td>\n",
    "    </tr>\n",
    "    <tr>\n",
    "      <td>ハ)</td>\n",
    "      <td>\n",
    "        \\begin{pmatrix}\n",
    "            -3 & -2 & -2 &  1 \\\\\n",
    "             2 &  3 &  2 &  0 \\\\\n",
    "             3 &  1 &  2 & -1 \\\\\n",
    "            -4 & -2 & -2 &  2 \\\\\n",
    "        \\end{pmatrix}\n",
    "      </td>\n",
    "      <td>\n",
    "        \\begin{pmatrix}\n",
    "            0 & 0 & 0 & 0 \\\\\n",
    "            0 & 2 & 0 & 0 \\\\\n",
    "            0 & 0 & 1 & 0 \\\\\n",
    "            0 & 0 & 0 & 1 \\\\\n",
    "        \\end{pmatrix}\n",
    "      </td>\n",
    "    </tr>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 行列の宣言\n",
    "\n",
    "行列は次のように宣言する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       " -4  0  6\n",
       " -3  2  3\n",
       " -3  0  5"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [-4 0 6;\n",
    "     -3 2 3;\n",
    "     -3 0 5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       "  1   0  -2\n",
       "  2  -1  -2\n",
       " -2   2   0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = [ 1  0 -2;\n",
    "      2 -1 -2;\n",
    "     -2  2  0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Int64}:\n",
       " -3  -2  -2   1\n",
       "  2   3   2   0\n",
       "  3   1   2  -1\n",
       " -4  -2  -2   2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = [-3 -2 -2  1;\n",
    "      2  3  2  0;\n",
    "      3  1  2 -1;\n",
    "     -4 -2 -2  2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 固有値\n",
    "\n",
    "事前に`using LinearAlgebra`を宣言しておく必要がある. 行列を`eigvals()`に渡すことで固有を格納した配列が得られる. なお, `eigvals()`は固有値を小さいから順に求めるため, 対角形の順序とは異なる. また, Juliaに限らず, コンピュータで実数を扱うことは不可能なので, 代わりに[浮動小数点数](https://ja.wikipedia.org/wiki/%E6%B5%AE%E5%8B%95%E5%B0%8F%E6%95%B0%E7%82%B9%E6%95%B0)が採用される. そのため`-1`が`-0.999999999999998`となったり, `0`が`-2.539596429704816e-15`となったりする. 前述のベンチマークに矛盾しない結果が得られている."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " -1.0\n",
       "  2.0\n",
       "  2.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigvals(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " -0.999999999999998\n",
       " -2.539596429704816e-15\n",
       "  1.0000000000000024"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigvals(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 4.440892098500626e-15\n",
       " 0.9999999999999977\n",
       " 0.9999999999999997\n",
       " 1.9999999999999996"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigvals(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 固有ベクトル\n",
    "\n",
    "固有値は`eigvec()`で求められる. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " -0.816497  0.0  -0.707107\n",
       " -0.408248  1.0   0.0\n",
       " -0.408248  0.0  -0.707107"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigvecs(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " -0.666667  0.666667  -0.707107\n",
       " -0.333333  0.666667  -0.707107\n",
       " -0.666667  0.333333   3.66027e-16"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigvecs(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " -0.57735       0.235702   0.0116258  -0.408248\n",
       " -8.46545e-16   0.471405  -0.712657    0.816497\n",
       "  0.57735      -0.707107   0.701031   -1.46869e-15\n",
       " -0.57735       0.471405   0.0232516  -0.408248"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigvecs(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 固有値と固有ベクトルを同時に求める\n",
    "\n",
    "`eigen()`に行列を渡すと, 固有値と固有ベクトルを同時に求める. 固有値と固有ベクトルの配列はそれぞれ`values`と`vectors`の名前が付けられており, 個別に取り出すこともできる."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
       "values:\n",
       "3-element Vector{Float64}:\n",
       " -1.0\n",
       "  2.0\n",
       "  2.0\n",
       "vectors:\n",
       "3×3 Matrix{Float64}:\n",
       " -0.816497  0.0  -0.707107\n",
       " -0.408248  1.0   0.0\n",
       " -0.408248  0.0  -0.707107"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " -1.0\n",
       "  2.0\n",
       "  2.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen(A).values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " -0.816497  0.0  -0.707107\n",
       " -0.408248  1.0   0.0\n",
       " -0.408248  0.0  -0.707107"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen(A).vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 参考文献\n",
    "- [斎藤 正彦『線型代数入門』(1966, 東京大学出版会)]( http://www.utp.or.jp/book/b302039.html) \n",
    "- https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/"
   ]
  }
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