{"cells":[{"cell_type":"markdown","metadata":{},"source":["<br />\n","\n","<div style=\"text-align: center;\">\n","<font size=\"7\">数値計算試験問題</font>\n","</div>\n","<br />\n","<div style=\"text-align: right;\">\n","<font size=\"4\">2024/07/10 実施</font>\n","<br />\n","<font size=\"4\">cc by Shigeto R. Nishitani 2024</font>\n","</div>"]},{"cell_type":"markdown","metadata":{},"source":[" # 1 簡単な行列計算:25点 \n"," \n"," 次の行列\n","  $\n","  A = \\left(\\begin{array}{ccc}\n","    0 & -2 & 0 \\\\\n","    1 & 3 & 0 \\\\\n","    -1 & 0 & 3\n","  \\end{array}\n","  \\right)\n","  $\n","  の固有値と固有ベクトルを求めよ．\n","  \n","  また，固有ベクトルで構成される対角化行列$P$ を用いて，ドット演算 により$P^{-1}.A.P$が対角化されることを確かめよ．\n"]},{"cell_type":"markdown","metadata":{},"source":["# 2 Gauss-Seidelの収束性:25点\n","\n","初期値を$[0,0,0]^{t}$として，\n","$A(tt)x=b$ \n","にガウス・ザイデルによる連立一次方程式の反復解法プログラムを適用する．\n","ただし，\n","\\begin{equation}\n","A(tt)=\n","\\left(\n","\\begin{array}{ccc}\n","1&tt&tt \\\\\n","tt&1&tt \\\\\n","tt&tt&1\n","\\end{array}\n","\\right)\n",", \\, \n","b=\n","\\left(\n","\\begin{array}{c}\n","4 \\\\\n","4 \\\\\n","4 \\\\\n","\\end{array}\n","\\right)\n","\\end{equation}\n","\n","である．\n","$tt=0.25,0.5,0.75$ に対して有効数字6桁の解を得るための反復回数を求めよ．\n","\n","(E.クライツィグ著「数値解析」(培風館,2003), p.89, 問題2.3-9)"]},{"cell_type":"markdown","metadata":{},"source":["# 3 家の価格\n","\n","回帰モデルとして，家の売却価格の予測を行う．家の特徴量として，\n","x: 家の面積\n","y: 寝室の数\n","をとる．\n","z: 家の売却価格\n","とすると，５軒の家の表は次のようになる．\n","\n","| 家番号|x面積|y:寝室数|z:売却価格|予想価格\n","|---|---|---|---|---|\n","|1 |0.846|1|115 |120.52\n","|2 |1.324|2|234.5\n","|3 |1.150|3|198\n","|4 | 3.037|4|528\n","|5| 3.084|5|572.5\n","\n","2次元曲面のフィッティングを参考にして，\n","$$\n","z = a_0 + a_1 x + a_2 y\n","$$\n","の平面にフィッティングして，予想価格を求めよ．\n","(ステファン・ボイド，リーヴェン・ヴァンデンベルグ 「スタンフォード ベクトル・行列 からはじめる 最適化数学」 講談社 ２０２１年, c2 pp60-2)"]},{"cell_type":"markdown","metadata":{},"source":["# 4 ページランク改(感染症の推移):25点\n","\n","ページランクの元になった線形動的システムの代表例である\n","単純な感染症モデルを考える．\n","実際の感染症（例えばcovid19）では成り立たなかったのですが，\n","この問題では単純モデルを仮定します．\n","未感染，感染中，回復(免疫あり)と死亡の４種の状態がある．\n","これらの状態にある人には，毎日次のことが起こると仮定する．\n","\n","- 未感染の人の5%がこの病気にかかる\n","    - 残りの95%は未感染のままである．\n","- 感染している人のうち\n","    - 1%が死亡する\n","    - 10%が回復して免疫を持ち\n","    - 4%は回復するが免疫を持たない（未感染の状態に戻る）\n","    - 残りの85%は感染状態のままである\n","\n","なお回復して免疫を持った人と，\n","死亡した人はそのままの状態にとどまり続ける．\n","未感染，感染中，回復(免疫あり)と死亡のそれぞれ\n","$i$ 日目の状態の人の割合を$x_i$ベクトルで表す．\n","$i+1$ 日目の状態ベクトル$x_{i+1}$ は，\n","\\begin{equation}\n","x_{i+1}=\n","\\left(\n","\\begin{array}{cccc}\n","0.95&0.04&0 &0 \\\\\n","A&B&0&0 \\\\\n","0&0.10&1&0 \\\\\n","0&0.01&0&1 \\\\\n","\\end{array}\n","\\right)\n","x_i\n","\\end{equation}\n","という関係で表される\n","\n","## (1) 遷移行列の完成\n","A,Bに入る数値は何か？\n","## (2) 手計算\n","初日の状態を\n","\\begin{equation}\n","x_{0}=\n","\\left(\n","\\begin{array}{c}\n","1 \\\\\n","0 \\\\\n","0 \\\\\n","0 \\\\\n","\\end{array}\n","\\right)\n","\\end{equation}\n","\n","とすると2日目および3日目はどうなるか？\n","## (3) loopあるいは．．．\n","200日目はどういう状態に落ち着くか？\n","\n","初日から２００日目までの，状態の変化をプロットしてください．ボーナスで10点プラスします．\n","\n","(ステファン・ボイド，リーヴェン・ヴァンデンベルグ 「スタンフォード ベクトル・行列 からはじめる 最適化数学」 講談社 ２０２１年)"]}],"metadata":{"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.8.5"},"toc":{"base_numbering":1,"nav_menu":{},"number_sections":true,"sideBar":true,"skip_h1_title":false,"title_cell":"Table of Contents","title_sidebar":"Contents","toc_cell":false,"toc_position":{},"toc_section_display":true,"toc_window_display":false}},"nbformat":4,"nbformat_minor":4}