{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# フィボナッチ数列\n", "\n", "授業担当：劉雪峰" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. 数列の定義と作成\n", "\n", "フィボナッチ数列（Fibonacci sequence）は、次のように再帰的に定義されます。\n", "\n", "- $F_1 = 1$\n", "- $F_2=1$\n", "- $F_n = F_{n-1} + F_{n-2}\\quad \\quad (n\\geq 3)$\n", "\n", "フィボナッチ数列の項を具体的に書くと、\n", "$$\n", "1,1,2,3,5,8,13, \\cdots\n", "$$\n", "\n", "
\n", "\n", "

フィボナッチ数列と兎の問題

\n", "\n", "\n", "フィボナッチ数列は、以下の兎の問題に例えられることが多いです。\n", "\n", "\n", "毎月のつがい数（F(n)）について、これまでにいる兎（前月のつがい数=F(n-1)）＋2ヶ月前の兎が新しく産む兎（2ヶ月前のつがい数=F(n-2)）となっています。即ち、$F(n)=F(n-1)+F(n-2)$です。よって、フィボナッチ数列が現れていることがわかります。\n", "\n", "
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## フィボナッチ数列を作成するプログラム\n", "\n", "フィボナッチ数列の第10項まで作成してみます。\n", "\n", "- ``\n", "F=[1,1]\n", "``\n", "は2つの成分をもつ配列を作成します。配列の各成分は１です。\n", "\n", "- 配列の最初の値と2番目の値はそれぞれ $F[0], F[1]$ で表されます。" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "第1月のつがい数は1です。\n", "第2月のつがい数は1です。\n", "第3月のつがい数は2です。\n", "第4月のつがい数は3です。\n", "第5月のつがい数は5です。\n", "第6月のつがい数は8です。\n", "第7月のつがい数は13です。\n", "第8月のつがい数は21です。\n", "第9月のつがい数は34です。\n", "第10月のつがい数は55です。\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#演習1\n", "import matplotlib.pyplot as plt\n", "\n", "N=10\n", "n_list=range(1,N+1)\n", "F=[1,1]\n", "print(\"第%d月のつがい数は%dです。\"%(1,F[0]))\n", "print(\"第%d月のつがい数は%dです。\"%(2,F[1]))\n", "\n", "#第3月からのつがい数(F[2],F[3],...,F[N-1])はforループで作成します。\n", "for n in range(2,N):\n", " F.append(F[n-1]+F[n-2])\n", " print(\"第%d月のつがい数は%dです。\"%(n+1,F[n]))\n", " \n", "plt.plot(n_list,F,'r.-')\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習 1\n", "\n", "$N=10,15,20,25$に対して数列を考察しなさい。\n", "\n", "- 第$N$項までのフィボナッチ数列$F$を作りなさい。\n", "- 数列$F$のグラフを描きなさい。\n", "\n", "以下のことを考えてください。\n", "\n", "- 数列は発散しますか？\n", "- フィボナッチ数列のグラフは次のどの関数のグラフに似ていますか？　\n", "\n", "$$f(x)=x^a, f(x)=a^x, f(x)=\\log(x)$$ \n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "#ここにコードを書いてください。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. フィボナッチ数列の式を推定する\n", "\n", "以下のコードでは、数列Fのlog値を計算してグラフを描いています。$\\log(F)$のグラフはどのような関数のグラフに近いでしょうか？" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#演習2\n", "import matplotlib.pyplot as plt\n", "import math\n", "\n", "N=10\n", "F=[1,1]\n", "n_list=[1,2]\n", "for n in range(2,N):\n", " F.append(F[n-1]+F[n-2])\n", " n_list.append(n+1)\n", "\n", "log_F=[]\n", "for n in range(N):\n", " log_F.append(math.log(F[n]))\n", "plt.plot(n_list, log_F, marker='o')\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習 2\n", "\n", "- (2.1) $\\log(F)$に当てはまる直線(フィッティング直線)を求めてください。即ち、$\\log(F)$のグラフに近い直線$y=ax+b$の係数$a$と$b$を求めてください。解く方法は自由です。\n", "\n", "$$\n", "\\log(F_n)\\approx a\\times n+b\n", "$$\n", "\n", "- (2.2) $n$ ~ $\\log(F_n)$ のグラフは直線に近いので、$F_n$を以下のように近似できます。\n", "\n", "$$\n", "\\log(F_n)\\approx a \\times n+b \\Rightarrow \n", "F_n \\approx \\exp(a)^n \\times \\exp(b)\n", "$$\n", "- (2.3) $c:=\\exp(b), r:=\\exp(a)$とおく。$n$の値が大きくなる時、$c$と$r$の値を計算してください。また、$r$の値と黄金比との関係を検討してください。\n", "\n", "$$\n", "\\mbox{黄金比}=\\frac{\\sqrt{5}+1}{2}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 補足 1：多くの点に当てはまる直線 y=ax+b の係数 a と b を求める方法\n", "\n", "良さそうな$a,b$に対応する直線$y=ax+b$のグラフを描いて、それを$n$ ~ $\\log(F_n)$のグラフと比較し、より良い近似になる係数$a,b$の値を見つけます。\n", "\n", "例えば、「$a=0.45,b=-0.40$」に対応する直線のグラフを描いてみます。\n", "係数$a,b$の値を調整しながら、もっと良い$a,b$の値を推定してください。\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#補足1\n", "x_list=range(1,N+1)\n", "a=0.45; b=-0.40\n", "y_list=[]\n", "for x in x_list:\n", " y_list.append(a*x+b)\n", "plt.scatter(n_list, log_F, marker='o')\n", "plt.plot(x_list, y_list, \"r-\")\n", "plt.grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 補足 2：2点を通る直線 y=ax+b の係数 a と b を求める方法\n", "\n", "2点$(x_1,y_1), (x_2,y_2)$を通る直線を求めます。\n", " \n", " $$\n", " y = y_1 + (x-x_1)\\frac{y_2-y_1}{x_2-x_1} = \\left(\\frac{y_2-y_1}{x_2-x_1} \\right) x + \\left( y_1 -x_1 \\frac{y_2-y_1}{x_2-x_1} \\right)\n", " $$\n", " \n", " よって、2点の座標を使って、$y=ax+b$の係数$a,b$を求めることができます。\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#補足2\n", "x1=2; x2=7\n", "y1=math.log(F[1]); y2=math.log(F[6])\n", "a=(y2-y1)/(x2-x1)\n", "b=y1-x1*a\n", "\n", "y_list=[]\n", "x_list=[x1,x2]\n", "for x in x_list:\n", " y_list.append(a*x+b)\n", "\n", "plt.plot(n_list, log_F, 'o')\n", "plt.plot(x_list, y_list, \"r.-\")\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. [オプション] 最小二乗法による点列に当てはまる直線の計算方法\n", "\n", "\n", "### 最小二乗法とは \n", "\n", "*与えられた点列$(x_i,y_i)_{i=1}^N$に対して、一次関数$y=ax+b$を用いた良い近似となるように、残差の二乗和を最小とするような係数を決定する方法です。*\n", "\n", "### 定式化 ###\n", "\n", "数式で考えるとき、以下の式を最小化する$a,b$を求める問題となります。\n", "\n", "$$\n", "V(a,b)=\\sum_{i=1}^N (ax_i + b - y_i)^2\n", "$$\n", "\n", "### 解法 ###\n", "\n", "行列を利用して、上記の最小化問題を考えます。\n", "\n", "$$\n", "\\mathbf{x}=\\left(\\begin{array}{c} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_N \\end{array}\\right),\\quad \n", "\\mathbf{y}=\\left(\\begin{array}{c} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_N \\end{array}\\right)\n", "$$\n", "\n", "$$\n", "V(a,b)= (a \\mathbf{x} + b - \\mathbf{y})^T ( a \\mathbf{x} + b - \\mathbf{y})\n", "$$\n", "さらに、以下の$A$と$\\alpha$を定義します。\n", "$$\n", "\\mathbf{A}=\\left(\\begin{array}{cc} x_1 & 1 \\\\ x_2 & 1 \\\\ \\vdots & \\vdots \\\\ x_N & 1 \\end{array}\\right),\\quad \n", "\\alpha = \\left(\\begin{array}{c} a \\\\ b \\end{array}\\right)\n", "$$\n", "このとき、$a \\mathbf{x} + b = A\\alpha $なので、\n", "\n", "$$\n", "V(\\alpha)=(A \\alpha - \\mathbf{y})^T ( A \\alpha- \\mathbf{y})\n", "$$\n", "\n", "2変数関数$V(\\alpha)$が最小値を取るとき、$V$の$\\alpha=(a,b)$に関する微分は$0$になります。即ち、以下の式が成り立ちます。\n", "\n", "$$\n", "(A^TA) \\alpha - A^T \\mathbf{y} = 0\n", "$$\n", "\n", "$A^TA$が正則であれば、$\\alpha$を次のように算出できます。\n", "\n", "$$\n", "\\alpha = (A^TA)^{-1} (A^T\\mathbf{y})\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習 3（オプション）\n", "\n", "1) 以下の点列に対して、点列に当てはまる直線$y=ax+b$を求めなさい。\n", "\n", "- x=1, y=4.44357\n", "- x=2, y=5.07900\n", "- x=3, y=6.53377\n", "- x=4, y=8.57194\n", "\n", "2) $n$~$\\log(F_n)$のグラフに当てはまる直線を求めなさい。\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "以下は演習3の解答例です。" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a= 1.3839000000000024 b= 2.697499999999991\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#演習3\n", "import matplotlib.pyplot as plt\n", "\n", "import numpy as np\n", "\n", "A=np.ones((4,2)) #全ての成分が1であるような4*2の行列を作成する。\n", "y=np.zeros((4,1)) #全ての成分が0であるような4*1の行列を作成する。\n", "\n", "A[:,0]=np.array([1,2,3,4])\n", "y[:,0]=np.array([4.444, 5.079, 6.534, 8.572])\n", "\n", "x_list=A[:,0] #行列の第1列を取って、xにする。\n", "y_list=y[:,0]\n", "plt.plot(x_list,y_list,'o')\n", "\n", "alpha=np.linalg.inv(A.T@A)@(A.T@y) #A.T@y:行列Aの転置とベクトルyの積、@:行列の積計算の演算子\n", "\n", "a=alpha[0,0]; b=alpha[1,0];\n", "print(\"a=\",a,\" b=\",b)\n", "\n", "new_y_list=a*x_list+b #Numpyの配列は一体として四則演算などの数学関数の計算ができます。\n", "\n", "plt.plot(x_list,new_y_list,'r-.')\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. [オプション] フィボナッチ数列の式を求める\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $F_n$の式を推定（１） ###\n", "\n", "$F_n = cr^n$を仮定して、この式を$F_n=F_{n-1}+F_{n-2}$に代入すると、$r$に関する方程式が得られます。\n", "\n", "$$\n", "r^2=r+1\n", "$$\n", "\n", "よって、\n", "$$\n", "r=\\frac{1+\\sqrt{5} }{2}, \\quad r=\\frac{1-\\sqrt{5} }{2}\n", "$$\n", "\n", "数列$F_n$は無限大に発散するので、$r$は正の値を取ります。よって、\n", "$$\n", "F_n = c \\left[\\frac{1+\\sqrt{5} }{2}\\right]^n\n", "$$\n", "上記の式はフィボナッチ数列になりませんが、フィボナッチ数列の主要項となります。\n", "\n", "### $F_n$の式を推定（２） ###\n", "\n", "$Z_n:=F_n - c \\left[ \\frac{1+\\sqrt{5} }{2}\\right]^n$を考察します。\n", "\n", "$Z_n$は次の式を満たします。\n", "\n", "$$\n", "Z_n =Z_{n-1} + Z_{n-2}\n", "$$\n", "よって、\n", "$Z_n$も以下の形式を持っていると推定できます。\n", "$$\n", "Z_n = \\tilde{c} \\tilde{r}^n\n", "$$\n", "この式を満たす$\\tilde{r}$は$\\tilde{r}^2=\\tilde{r}+1$の解となりますので、\n", "$$\n", "\\tilde{r} = \\frac{1-\\sqrt{5} }{2}\n", "$$\n", "よって、$F_n$に関して新しい式が考えられます。\n", "$$\n", "F_n = c \\left[ \\frac{1+\\sqrt{5} }{2} \\right] ^n + \\tilde{c} \\left[\\frac{1-\\sqrt{5} }{2}\\right]^n\n", "$$\n", "条件$F_1=1,F_2=1$を利用すると、以下のことが分かります。\n", "\n", "$$\n", "c = \\frac{1}{\\sqrt{5}}, \\quad \n", "\\tilde{c} = - \\frac{1}{\\sqrt{5}}\n", "$$\n", "\n", "即ち、\n", "\n", "$$\n", "F_n = \\frac{1}{\\sqrt{5}} \\left( \\left[ \\frac{1+\\sqrt{5} }{2} \\right] ^n - \\left[\\frac{1-\\sqrt{5} }{2}\\right]^n \\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 5. コーヒーブレイク\n", "\n", "フィボナッチ数列によるゴールデン･スパイラルの図を描く。Nを他の値にしてグラフを描いてみよう。\n", "" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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nuGnQCKdLlA4kNDSEESOG8/nnOpoPpPafVK2gurqKxYsX07NnL26//XbXDJHsFteVG9KzAOgc2Yl+vftQUnqMjN59Se/Vx+HqpCMaPnwEu3cXcPr0GadLcQ2F/FWcPXuWRYsWk5aWxuTJf4VbH8138NhhduzdxbD+NzhdinRgkZERDBkymM2bNzldimso5K+gsrKSRYsW0T8jg4kTb8WtAV9ZfZb7f/YYv3jsH4iO0qyA4qysrCwKCgqdLsM1FPKXce5cHb/73etkZmaSM348bg34c/XnuP9nj3LPhBlMGzvF6XJE6NGjB9XVVZw6pZujAkEhf0mWt956i65dExk37hani2kx1loe//f/Sb/efXl85jynyxEBwBhDv379KCwscLoUV1DIX8Lqz1ZTUVHB1KlTcesRPMD6nZtZ+qc3+OzLdYx99DbGPnobH278hJVr3mfgfaPYtPsLZv3kIe566rtOlyodTEZGBoWFOmUTCJoL92sKCgrYsnUL8+bNIzjY3d0zOms4vg8PXHLbt3TqRhyUmprGG2+8QVVVtdOltHs6kr/AiePHWblyJffOupdOnXQBUsQpISHBJCcnc+DAfqdLafcU8k2qqs6yJDeX2267jaTuesCwiNO6JnSltLTU6TLaPYU8jVMGL1v2ewZlZZGVleV0OSICxMXHKeQDQCGP5b333iM8PIzxEyY4XYyINImLi6esTPPYNFeHD/mtW7Zy+PAh7rzzLtdMVyDiBvHxcZSWlmGtdbqUdq1Dh/zp0xV89PFH3HPPPYSFaU54kbYkIiKC4GAPx44dc7qUdq0Dh7zlnXfeZeTIUcTHJzhdjIhcQkxMDPv3a4RNc7h7IPgV7Ny5C5+vnFmzZjldishlnSwt5e9//SvKfRVgDNMnjmfW7R3nHoaQkBBqamqcLqNd65AhX11dzQcfvM+sWbPweDxOlyMdjLXg99fj9/ux1hIcHILH4+FSl4Q8QR6e+O59ZKSlcra6moef/HuGDx5Eas8erV+4AzyeYGpra50uo13rkCG/atUqBgwYSM+evZwuRdqRL9MrvvF7z9We4z8e/g9qa2qpra4lYUACicMT8ZX78Pl81NfX4/F4uO9sOFFVRwBLp06dSE5OPv8VHx9PXXgaOWke4BAAN6bG0LW+gJvi/WysvLdlPmgb0hCxXQMimqnDhfyB/fspKtrHggULnC5FXMhaS3FxMfk784kYHoGtsaT3TmfX0l0MunkQ46aPw+v1EhERAcCC8sYfHH6/H1+Fj0MHD3Hg4AHWrFmDMYa+9/4TNcE1hIeHc+BYOV/sPcrIAR3n4CTURjldQrvXoUK+vv4cK99+m9tvv4PQUI2mkcDx+/3s3LmTDRs2UF1dzZAbhnDXzLtISkqivraeAysOkJqWSlJS0iW/3+PxEBcbR1xsHDfeeCMAR44cYdO5c/z6P3/N0OGjeDJ3B79acAfRUeGt+dEcVVNTQ3R0tNNltGsdKuQ//fQzkpKS6Nevn9OliEtYa8nPz2fVqlXEx8dzyy23kJ6eTlBQEA3+Bp6e8zQnD59k3L3jSB2Uek377tGjB19VxjB37kNM/LuFDOgWxl8NS2mZD9IGWaCsrIz09HSnS2nXOkzIl548yRdfbOXRRx9zuhRxiTNnzvDWW29RWVnJPffcQ69eF59GCfIE8aOlP6LqTBUL/24hR/cepXvfa5sXyVrLD/5fHhNGDWFGVhSvvPIKD9z/AJ07u38CvaqzZzHGEB8f73Qp7VqHCfk1a9cwatRoOnXq5HQp4gJHjhxhyZIlZA/P5uaxN19xlFZk50j6Zfdj1+e7rjnk83cX8uqqLxiU1o1PtxmKjpTyw7eeJq1nAi/9p7svvJaWlREXH68Lr83UIUK+osLHnj1fMWXKbU6XIi5w+PBhcnNzmTZtGhkZGZd8z5lTZ/CEeIjsHEldTR0FGwqYNHfSNbc1aGB/7Me/OP/6021FfPLRh7y60f2Pxjt44AA9uneMoaItqUOE/Lp16xk6dCjh4R3ngpW0jKNHj5Kbm8udd95J3759L/u+06WnWfwPi2loaMA2WIZOGsqgWwY1u/1xQ9KICBrP85++hsXNzy2DgsJCJk269h+McjHXh3xVVRXbt2/TkElpNr/fz5tvvsltt912xYAH6NGvB0/lPtUidSQkNE7DceL4cRITE1ukDaedPn0aX3k5yb17O11Ku+f6uWs2btzIwIGZetKTNNuaNWuIiYkhMzPT0ToMhpCQECpOn3a0jpa0a9fu86OUpHlcfSRfV1fHpk2bmPfww06XIu1cTU0N69at47HHHgvshcDxVz7aP7HyA/6jfuhF607VH+ZE/csEe9z539ff0MD6DeuZeffdTpfiCi3+Y9IYM8UYU2iM2WuMebKl27vQ1q1bSU1NJSY2tjWbFRfau3cvvXv3pkuXLk6Xwt69+2iwDfROduepjJ35O4nxeunZs6fTpbhCi4a8McYD/Ba4DRgIzDHGDGzJNv/M7/ezbt06xo4Z0xrNicsVFhaS0f/SI2laj+W3T85j2dPfx19VwYx5t7Diw2WtXsWSt15i9vemMOfxKfzkl39LbV3gJhCz1rJ27RrGjB0bsH12dC39+94IYK+1tgjAGJMLTAd2tXC75O/YQUJCPN0ucxu5yLUoKSmhdy9njpxPnjxJUVERuwt2Y3vewN9+76f06tWLMTWtf7rmRNkxlq5cRO5vPyA8LJwf/csTrFq9kqm3zgzI/rdu/YLQ0FD69OkTkP1Jy4d8D+DwBa+LgZEt3CYA23dsZ/jwEa3RlHQA8fHxlJr6Vm3z+eefp5pQfrdyE2lpfRgxfAT9+2cQFOTs9Nj+hnpq62oIDg6mpraa+NjAjPA5c+YMH3/8Efc/8ICrh4a2NtOSz080xswEplhr5zW9/i4w0lr7+AXveQR4BCAxMXFYbm7udbXlP1N3ftk2WI4fP05it0RH75araqghMkhj8y+nPfXPieMniInytPLFTkt1SAgR5y79w6UuzJnrA2+tXM7i118hNDSMoTcM5Qd/F5hLbadOnSI4JJjozhdPSBbZJfSy31NZWam72IHx48dvsdZmX2pbS/+LPQJcOKFHz6Z151lrFwILAbKzs21OTs51NVSx6uD55fwdOziTf5Spc8Zd174CZXNNIdnhTp/HbbvaU/8s/CCPgX0t3Vv5DsxtiYkMOX78ktv2pwxr1VoATldWsGXjl7z5wmd0jormqX95nE2f5nPb+BnN2u+2bdtZvWYTj/71XxMcdHEsjchJu+z35eXlcb2Z0VG09OiaTUC6MSbVGBMKzAZWtHCbFBQW0v8yt5uLXI9u3bpx+PDhq7/R5TZ9uZbuib2I6RJHcHAI40dPZkfB1mbts7i4mA9XfcisWbMIDnbnsFAntWjIW2vrgceBD4DdwDJr7c6WbNPv97Nv3z76ZWg6YQmcjIz+FBQUOl2G4xITupNf+CU1tdVYa9m07XNSel3/RdLTZ86wbNkypk2bRtemO3klsFr8x6a19l3g3ZZu588OHjhAQkI8UVE6TyeBk5aWRm7JUaqrq88/1am5hv2vn9IpLIygoCCCg4JY9YMfBmS/LSkr4wYmjJnC/d+fhsfjoV9aJjMmz76ufdXU1JC7ZAkjRowkQ894aDGu+92ooLCQjIz+TpchLhMSEkJKcgp7937FoEGDA7bfN574G+La2YXDR+77Po/c9/1m7aO6uppXX3uN3r2TGTNW97K0JJdNDGEbb1rRqRppAZmZmWzatImWHJHWEZytqmLRokWkpaYyecpkDZdsYa4K+WMlxwgJCdaTZKRFZA3KwlrL1q1bArI/A8x69rdM/Nd/ZfHatQHZZ1tX7vOx6JVXyMjoz60TJyrgW4GrTtfs3buX9PR+uHuWbXFKkAli2rRpvPLKItL79fuL8dzXauX3/xtJXi8nz5zhnt8+Q3piIqOvMoVxe7ZvXxHLl7/BzTffzMiRrXJPpOCyI/mTpSfp5tL5taVtSEjoSnZ2Nu+9+16z95Xk9Tbus3Nnbh88hK0HD175G9opC6xZs5Y331zOzJkzFfCtzFUhX1paRlx8nNNliMvdfMvNlJaWsnnz5uvex9naWipras4v5xUUMMCF8yydPXuWpUuXUlCwm/nz55OSkuJ0SR2Oi07XWMrKSomL0/l4aVnBnmDmzJnD4sWLsbbhuuZIOnnmDHNffAFonD/9rmHZTBjYKhO0tgoL7Ny5k/fff58bhtzAzJkzCb7Cw86l5bgm5CsrK/F4ggM2hlnkSmJjY3lg7gMsXrwYv9/PqFGjr+n7U+LjyXuyZR4P6DSfz8e7776Lr6KC2ffeq3nhHeaakC8rLSNep2qkFcV4Y5j7wFwWL15Mfb2fsR18DvSq6mrWrF7NF19+yU2jb+Lee0fj0dG741wT8qVlZTpVI62uS5cuzJ3bGPSnTpUxecoUwkLDnC6rVdXV1bF+/QbWb1hP5sBMFixYQOd2doOXm7nmwmtZWanGx4sjOnfuzLz58wDDc889x6FD7hwl83Vnq6pYvXo1v/nNbzhZepJ5Dz/MHXfcroBvY9xzJF9aSkpyitNlSAcVFhrGtGnTKCws5A9//CN90vowcdJEoiKjnC4t4I4dP86GDRso2L2bAQMG8J3vfpfErl2dLksuw0Uhr+GT4ryMjAxSUlLIy8vj2WefZdTIkQwdNqzdh31VdTW7d+9m+/btlJeXMzx7OI8/8QRRkZFOlyZX4ZqQP3PmNNHRzjwpR+RCYWFhTJ48maFDb+Tzz9fxzG+eYcCA/owcOYrEdnSzXm1dHYUFheTvzOfQwYP06duXUaNG0a9fPzxBrjnT63quCHlrLX6/n+BgXcmXtiMhoSvTp09n4qSJbN2yhddff524uFhGjhxFenp6mxt54vf7KT5yhAP7D7D/wH5Kjh4lOSWFrKws7r77bsJCL/8YPmm7XBHy9fX1GGMwRkcX0vZERUZx8823cNNNY9i9exeff/45y99cTnLv3qSlpZGW1oeErgmYVp5zqaGhgZKSEvbvP8CBA/s5XFxMfFwcKSmpjB0zlt69exGqYG/3XBHytbW1eFr1Acsi187j8ZCVNYisrEFUVVdxYP9+9u0rYsOGjdT760lLTSM5OZnY2Fi8MV6io5s3ARo0Ppij3OfDV+7D5yunvLy86XU5vooKYmNiSE1NJXv4cO6eOZOI8PbxYHX55lyRjHV1dW3uV1+RK4mMiGTgwEwGDszEYikvL6doXxGHDh1k27YvKff5qDpbxci//wmvvroYrzeG6OjOeDwePB4P9Z5gjhyPw+/3N3014Pf7qa+vp+J0Bb6mMG/w+/HGxBDj9eL1xhAXF0+fvn2bXnt1pN4BuCLkvV4v/33p/6ZLXNsaXePJ20+XnGSny2iz2l3/THq8xXbdFfj6o+dramr49NNPif3Zz9i/fz/bS0qoq6vj3LlzhIWFUVW5gZCQEEJDQwmJDCEkJISIsDD69UohNXU8qampxMfHY4ym3u7IXBHyQUFBxLWxgBdprvDwcMLCwsjJyXG6FGnHdKVSRMTFFPIiIi6mkBcRcTGFvIiIiynkRURcTCEvIuJiCnkRERdTyIuIuJhCXkTExRTyIiIuppAXEXExhbyIiIsp5EVEXEwhLyLiYgp5EREXU8iLiLiYQl5ExMUU8iIiLtaskDfG3GOM2WmMaTDGZH9t21PGmL3GmEJjzOTmlSkiItejuc94zQfuAp6/cKUxZiAwG8gEugN/Msb0s9b6m9meiIhcg2YdyVtrd1trCy+xaTqQa62ttdbuB/YCI5rTloiIXLuWOiffAzh8wevipnUiItKKrnq6xhjzJ6DbJTb92Fr7VnMLMMY8AjwCkJiYSF5eXnN32WZUVla66vMEmvrn6tRHV6b+ubqrhry1duJ17PcI0OuC1z2b1l1q/wuBhQDZ2dk2JyfnOpprm/Ly8nDT5wk09c/VqY+uTP1zdS11umYFMNsYE2aMSQXSgY0t1JaIiFxGc4dQ3mmMKQZGA+8YYz4AsNbuBJYBu4D3ge9pZI2ISOtr1hBKa+1yYPlltv0c+Hlz9i8iIs2jO15FRFxMIS8i4mIKeRERF1PIi4i4mEJeRMTFFPIiIi6mkBcRcTGFvIiIiynkRURcTCEvIuJiCnkRERdTyIuIuJhCXkTExRTyIiIuppAXEXExhbyIiIsp5EVEXEwhLyLiYgp5EREXU8iLiLiYQl5ExMUU8iIiLqaQFxFxMYW8iIiLKeRFRFxMIS8i4mIKeRERF1PIi4i4mEJeRMTFFPIiIi6mkBcRcTGFvIiIiynkRURcTCEvIuJiCnkRERdTyIuIuJhCXkTExRTyIiIuppAXEXGxZoW8MeaXxpgCY8x2Y8xyY4z3gm1PGWP2GmMKjTGTm12piIhcs+Yeya8Csqy1g4E9wFMAxpiBwGwgE5gCPGuM8TSzLRERuUbNCnlr7YfW2vqml+uBnk3L04Fca22ttXY/sBcY0Zy2RETk2gUHcF8PAUublnvQGPp/Vty07i8YYx4BHgFITEwkLy8vgCU5q7Ky0lWfJ9DUP1enProy9c/VXTXkjTF/ArpdYtOPrbVvNb3nx0A98Pq1FmCtXQgsBMjOzrY5OTnXuos2Ky8vDzd9nkBT/1yd+ujK1D9Xd9WQt9ZOvNJ2Y8xcYCpwq7XWNq0+AvS64G09m9aJiEgrau7ominAD4Fp1tqqCzatAGYbY8KMMalAOrCxOW2JiMi1a+45+WeAMGCVMQZgvbX2UWvtTmPMMmAXjadxvmet9TezLRERuUbNCnlrbd8rbPs58PPm7F9ERJpHd7yKiLiYQl5ExMUU8iIiLqaQFxFxMYW8iIiLKeRFRFxMIS8i4mIKeRERF1PIi4i4mEJeRMTFFPIiIi6mkBcRcTGFvIiIiynkRURcTCEvIuJiCnkRERdTyIuIuJhCXkTExRTyIiIuppAXEXExhbyIiIsp5EVEXEwhLyLiYgp5EREXU8iLiLiYQl5ExMUU8iIiLqaQFxFxMYW8iIiLKeRFRFxMIS8i4mIKeRERFzPWWqdrOM8YcxI46HQdARQPlDpdRBum/rk69dGVqX8aJVtrEy61oU2FvNsYYzZba7OdrqOtUv9cnfroytQ/V6fTNSIiLqaQFxFxMYV8y1rodAFtnPrn6tRHV6b+uQqdkxcRcTEdyYuIuJhCXkTExRTyAWaM+aUxpsAYs90Ys9wY471g21PGmL3GmEJjzGQHy3SUMeYeY8xOY0yDMSb7a9vUR4AxZkpTH+w1xjzpdD1tgTHmJWPMCWNM/gXrYo0xq4wxXzX9GeNkjW2RQj7wVgFZ1trBwB7gKQBjzEBgNpAJTAGeNcZ4HKvSWfnAXcBnF65UHzVq+sy/BW4DBgJzmvqmo3uFxn8XF3oS+Mhamw581PRaLqCQDzBr7YfW2vqml+uBnk3L04Fca22ttXY/sBcY4USNTrPW7rbWFl5ik/qo0Qhgr7W2yFpbB+TS2DcdmrX2M+DU11ZPBxY1LS8CZrRmTe2BQr5lPQS817TcAzh8wbbipnXyX9RHjdQP31yitbakafkYkOhkMW1RsNMFtEfGmD8B3S6x6cfW2rea3vNjoB54vTVrayu+SR+JBJK11hpjNCb8axTy18FaO/FK240xc4GpwK32v25EOAL0uuBtPZvWudLV+ugyOlQfXYH64Zs7boxJstaWGGOSgBNOF9TW6HRNgBljpgA/BKZZa6su2LQCmG2MCTPGpALpwEYnamzD1EeNNgHpxphUY0wojRejVzhcU1u1AnigafkBQL8lfo2O5APvGSAMWGWMAVhvrX3UWrvTGLMM2EXjaZzvWWv9DtbpGGPMncBvgATgHWPMl9bayeqjRtbaemPM48AHgAd4yVq70+GyHGeMWQLkAPHGmGLgp8DTwDJjzMM0TlM+y7kK2yZNayAi4mI6XSMi4mIKeRERF1PIi4i4mEJeRMTFFPIiIi6mkBcRcTGFvIiIi/1/RRPdQY/lpLsAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import math\n", "N=7\n", "N_list=range(1,N+1)\n", "F=[]\n", "x_indx=[]\n", "y_indx=[]\n", "for k in range(0,N+1):\n", " if(k<2):\n", " F.append(1) \n", " else:\n", " F.append(F[k-1] +F[k-2]) \n", " y_indx.append(pow(-1,int(k/2)))\n", " x_indx.append(pow(-1,int((k+1)/2)))\n", " \n", "import matplotlib.patches as pat\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111) \n", "\n", "x=[0];y=[0];\n", "#text( -1,0.3, '1');\n", "for k in range(0,N):\n", " x.append(x[k]+ x_indx[k]*F[k+1])\n", " y.append(y[k]+ y_indx[k]*F[k+1])\n", " s1=90*(k-1)\n", " x0=x[k]-F[k+1]*math.cos(s1/180*math.pi)\n", " y0=y[k]-F[k+1]*math.sin(s1/180*math.pi)\n", " c=pat.Arc( xy= (x0, y0), width=2*F[k+1], height=2*F[k+1],theta1=s1, theta2=s1+90)\n", " ax.add_patch(c)\n", " x_list = [x[k],x[k+1],x[k+1],x[k]]; #x座標のリスト\n", " y_list = [y[k],y[k],y[k+1],y[k+1]]; #y座標のリスト\n", " plt.fill(x_list,y_list,alpha=0.5)\n", " plt.text((x[k]+x[k+1])/2, (y[k]+y[k+1])/2,'%d'%F[k+1])\n", "#plt.plot(x,y)\n", "\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "** Created by Xuefeng LIU, 2017/04/23, modified on 2018/04/22. **\n", "\n", "** The original matlab code is revised to python language by Wada Kaoru. 2020/04 **\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "計算機演習A・Bのノート　授業担当：劉雪峰\n", "
" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "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" } }, "nbformat": 4, "nbformat_minor": 4 }