From a051deba942d276f5dd0703a652f6c54322ffbf4 Mon Sep 17 00:00:00 2001 From: bugrahan sahin Date: Wed, 22 Feb 2017 10:16:27 +0300 Subject: [PATCH] final --- Fibonacci.ipynb | 133 +++++++++++++++++++++++++++++++++++++++++------- 1 file changed, 114 insertions(+), 19 deletions(-) diff --git a/Fibonacci.ipynb b/Fibonacci.ipynb index 7cabb0e..6ed8e7c 100644 --- a/Fibonacci.ipynb +++ b/Fibonacci.ipynb @@ -108,12 +108,24 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, "metadata": { "collapsed": true }, "outputs": [], - "source": [] + "source": [ + "import numpy as np\n", + "import math\n", + "import matplotlib.pyplot as plt\n", + "A = [[1.,1.],[1.,0.]]\n", + "A=np.matrix(A)\n", + "B=[[1.],[0.]]\n", + "B=np.matrix(B)\n", + "\n", + "\n", + "def fib(N):\n", + " return ((A**(N-1))*B)[0,0]" + ] }, { "cell_type": "markdown", @@ -131,12 +143,35 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, "metadata": { - "collapsed": true + "collapsed": false }, - "outputs": [], - "source": [] + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "yt=[]\n", + "\n", + "\n", + "for i in range(1,100):\n", + " yt.append(fib(i+1)/fib(i));\n", + "\n", + "plt.plot(range(len(yt)),yt)\n", + "plt.show()\n", + "\n" + ] }, { "cell_type": "markdown", @@ -152,6 +187,66 @@ "Initialize $F_0 = 2$ and $F_1 = F_1 = 1 - \\sqrt{5}$, then plot the $\\dfrac{F_{n+1}}{F_{n}}$ values for $ 1 \\leq n \\leq 100$. If we would represent $\\sqrt{5}$ exactly in our floating point arithmetic, then as $n \\to \\infty$, we expect $\\dfrac{F_{n+1}}{F_{n}} \\to \\dfrac{1 - \\sqrt{5}}{2}$, but for the very large values of $n$, this ratio unexpectedly converges to $\\dfrac{\\sqrt{5} + 1}{2}$." ] }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/bugrahansahin/anaconda/lib/python2.7/site-packages/ipykernel/__main__.py:10: RuntimeWarning: divide by zero encountered in double_scalars\n", + "/Users/bugrahansahin/anaconda/lib/python2.7/site-packages/ipykernel/__main__.py:10: RuntimeWarning: invalid value encountered in double_scalars\n" + ] + }, + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "B=np.matrix([[1.-math.sqrt(5)],[2.]])\n", + "\n", + "yt=[]\n", + "xt=[]\n", + "\n", + "a=B[0,0]\n", + "b=B[1,0]\n", + "\n", + "for i in range(1,100):\n", + " yt.append(fib(i+1)/fib(i));\n", + " c=a+b;\n", + " xt.append(c/a)\n", + " b=a;\n", + " a=c;\n", + "\n", + "#this one gives wrong plot due to numerical overflow\n", + "plt.plot(range(len(yt)),yt)\n", + "plt.show()\n", + "\n", + "plt.plot(range(len(xt)),xt)\n", + "plt.show()" + ] + }, { "cell_type": "code", "execution_count": null, @@ -164,21 +259,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Octave", - "language": "octave", - "name": "octave" + "display_name": "Python 2", + "language": "python", + "name": "python2" }, "language_info": { - "file_extension": ".m", - "help_links": [ - { - "text": "MetaKernel Magics", - "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" - } - ], - "mimetype": "text/x-octave", - "name": "octave", - "version": "0.16.1" + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.13" } }, "nbformat": 4,