Predavanje 07 - Interpolacija.ipynb

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 "cells": [
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   "source": [
    "<font size=\"2\"> [github.com/jankoslavic/pypinm](http://github.com/jankoslavic/pypinm)</font>\n",
    "\n",
    "[Prof. dr. Janko Slavič](http://www.ladisk.si/~slavic)\n",
    "\n",
    "<center>\n",
    "<img width=300 src=\"./fig/ladisk.png\">\n",
    "<br>\n",
    "<font size=\"7\" color=\"f00e0e\" face=\"garamond\"><b>Interpolacija</b></font>\n",
    "<img src=\"\">\n",
    "<font size=\"2\">September 2021</font>\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "toc": "true"
   },
   "source": [
    "<h1>Kazalo<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Uvod\" data-toc-modified-id=\"Uvod-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Uvod</a></span></li><li><span><a href=\"#Interpolacija-s-polinomom\" data-toc-modified-id=\"Interpolacija-s-polinomom-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Interpolacija s polinomom</a></span></li><li><span><a href=\"#Lagrangeva-metoda\" data-toc-modified-id=\"Lagrangeva-metoda-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Lagrangeva metoda</a></span><ul class=\"toc-item\"><li><span><a href=\"#Ocena-napake\" data-toc-modified-id=\"Ocena-napake-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Ocena napake</a></span></li><li><span><a href=\"#Zgled\" data-toc-modified-id=\"Zgled-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Zgled</a></span></li><li><span><a href=\"#Zgled-ocene-napake\" data-toc-modified-id=\"Zgled-ocene-napake-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>Zgled ocene napake</a></span></li><li><span><a href=\"#Interpolacija-z-uporabo-scipy\" data-toc-modified-id=\"Interpolacija-z-uporabo-scipy-3.4\"><span class=\"toc-item-num\">3.4&nbsp;&nbsp;</span>Interpolacija z uporabo <code>scipy</code></a></span></li></ul></li><li><span><a href=\"#Kubični-zlepki\" data-toc-modified-id=\"Kubični-zlepki-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Kubični zlepki</a></span><ul class=\"toc-item\"><li><span><a href=\"#Naravni-kubični-zlepki\" data-toc-modified-id=\"Naravni-kubični-zlepki-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Naravni kubični zlepki</a></span></li><li><span><a href=\"#Numerična-implementacija\" data-toc-modified-id=\"Numerična-implementacija-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>Numerična implementacija</a></span></li></ul></li><li><span><a href=\"#Nekaj-vprašanj-za-razmislek!\" data-toc-modified-id=\"Nekaj-vprašanj-za-razmislek!-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Nekaj vprašanj za razmislek!</a></span><ul class=\"toc-item\"><li><span><a href=\"#Dodatno\" data-toc-modified-id=\"Dodatno-5.1\"><span class=\"toc-item-num\">5.1&nbsp;&nbsp;</span>Dodatno</a></span></li><li><span><a href=\"#Nekaj-komentarjev-modula-scipy.interpolate\" data-toc-modified-id=\"Nekaj-komentarjev-modula-scipy.interpolate-5.2\"><span class=\"toc-item-num\">5.2&nbsp;&nbsp;</span>Nekaj komentarjev modula <code>scipy.interpolate</code></a></span></li><li><span><a href=\"#Odvajanje,-integriranje-...-zlepkov\" data-toc-modified-id=\"Odvajanje,-integriranje-...-zlepkov-5.3\"><span class=\"toc-item-num\">5.3&nbsp;&nbsp;</span>Odvajanje, integriranje ... zlepkov</a></span></li></ul></li></ul></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Uvod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pri **interpolaciji** izhajamo iz tabele (različnih) vrednosti $x_i, y_i$:\n",
    "\n",
    "| $\\mathbf{x}$  | $\\mathbf{y}$  |\n",
    "|:-:|:-:|\n",
    "| $x_0$  | $y_0$  |\n",
    "| $x_1$  | $y_1$  |\n",
    "| $\\dots$   | $\\dots$ |\n",
    "| $x_{n-1}$| $y_{n-1}$|\n",
    "\n",
    "določiti pa želimo vmesne vrednosti. Če želimo določiti vrednosti zunaj območja $x$ v tabeli, govorimo o **ekstrapolaciji**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V okviru **interpolacije** (angl. *interpolation*) točke povežemo tako, da predpostavimo neko funkcijo in dodamo pogoj, da funkcija *mora* potekati skozi podane točke. \n",
    "\n",
    "Pri **aproksimaciji** (angl. *approximation* ali tudi *curve fitting*) pa predpostavimo funkcijo, ki se čimbolj (glede na izbrani kriterij) prilega podatkom. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer:\n",
    "\n",
    "| $\\mathbf{x}$  | $\\mathbf{y}$  |\n",
    "|:-:|:-:|\n",
    "| 1.0  | 0.54030231  |\n",
    "| 2.5  | -0.80114362  |\n",
    "| 4.0   | -0.65364362 |\n",
    "\n",
    "Pri interpolaciji izhajamo iz tabele vrednosti. Da bomo pozneje lahko enostavno prikazali napako, smo zgornjo tabelo generirali s pomočjo izraza $y = \\cos(x)$!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo numerični zgled; najprej uvozimo pakete:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato pripravimo tabelo ter prikaz:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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      "image/png": 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ciYiQFO1s50Tg58DMiPgkX01ELAYWA2QymTa3ZWZmxesw8CNiSlvrJL0r6cSI2JYL9PfaqDsWeAy4ISLWdrq3ZmbWaUlP6SwHZuamZwKPtC6QdATwL8B9EfFgwvbMzKyTkgb+AmCqpNeBKbl5JGUk3ZWruRyYCFwlaUPuMTJhu2ZmViRFlOep8kwmE3V1dT3dDTOzXkXSuojI5FvnV9qamaWEA9/MLCUc+GZmKeHANzNLCQe+mVlKOPDNzFLCgW9mlhIOfDOzlHDgm5mlhAPfzCwlHPhmZinhwDczSwkHvplZSjjwzcxSwoFvZpYSDnwzs5Rw4JuZpYQD38wsJRz4ZmYp4cA3M0sJB76ZWUokCnxJn5X0pKTXc1+Pb6f2WEkNkv4pSZtmZtY5SY/w5wGrImIosCo335a/B55J2J6ZmXVS0sCfBizJTS8BpucrkjQGGAI8kbA9MzPrpKSBPyQituWm3yEb6oeR9BngfwPXdLQxSbMl1Umqa2pqStg1MzNrqV9HBZKeAj6XZ9UNLWciIiRFnrrvACsiokFSu21FxGJgMUAmk8m3LTMz66QOAz8iprS1TtK7kk6MiG2STgTey1M2AThX0neAo4EjJO2NiPbO95uZWYl1GPgdWA7MBBbkvj7SuiAivn5oWtJVQMZhb2bW/ZKew18ATJX0OjAlN4+kjKS7knbOzMxKRxHleao8k8lEXV1dT3fDzKxXkbQuIjL51vmVtmZmKeHANzNLCQe+mVlKOPDNzFLCgW9mlhIOfDOzlHDgm5mlhAPfzCwlHPhmZinhwDczSwkHvplZSjjwzcxSwoFvZpYSDnwzs5Rw4JuZpYQD38wsJRz4ZmYp4cA3M0sJB76ZWUo48M3MUiJR4Ev6rKQnJb2e+3p8G3WnSHpC0iuS6iXVJGnXzMyKl/QIfx6wKiKGAqty8/ncByyMiDOBccB7Cds1M7MiJQ38acCS3PQSYHrrAknDgH4R8SRAROyNiA8TtmtmZkVKGvhDImJbbvodYEiemj8Hdkl6WNJ6SQslVeTbmKTZkuok1TU1NSXsmpmZtdSvowJJTwGfy7PqhpYzERGSoo02zgVGAf8BLAWuAu5uXRgRi4HFAJlMJt+2zMyskzoM/IiY0tY6Se9KOjEitkk6kfzn5huADRHxRu45y4Dx5Al8MzPrOklP6SwHZuamZwKP5Kl5ARgkqTI3fz5Qn7BdMzMrUtLAXwBMlfQ6MCU3j6SMpLsAIuIgcA2wStJGQMCdCds1M7MidXhKpz0RsR24IM/yOmBWi/kngdokbZmZ9XXL1jeycOUmtu7ax0mDBnLthaczfVRVybafKPDNzKw0lq1v5PqHN7Jv/0EAGnft4/qHNwKULPT91gpmZmVg4cpNzWF/yL79B1m4clPJ2nDgm5mVga279hW1vDMc+GZmZeCkQQOLWt4ZDnwzszJw7YWnM7D/4W9CMLB/BddeeHrJ2vBFWzOzMnDowqzv0jEzS4Hpo6pKGvCt+ZSOmVlKOPDNzFLCgW9mlhIOfDOzlHDgm5mlhAPfzCwlHPhmZinhwDczSwkHvplZSjjwzcxSwoFvZpYSDnwzs5Rw4JuZpYQD38wsJRK9PbKkzwJLgRrgLeDyiNiZp+4fgK+S/QPzJDA3IiJJ22Y9Ydn6xi59v3KzrpT0CH8esCoihgKrcvOHkfQF4BygFhgOjAXOS9iuWbdbtr6R6x/eSOOufQTQuGsf1z+8kWXrG3u6a2YFSRr404AlueklwPQ8NQEMAI4AjgT6A+8mbNes2y1cuYl9+w8etmzf/oMsXLmph3pkVpykgT8kIrblpt8BhrQuiIh/A1YD23KPlRHxSr6NSZotqU5SXVNTU8KumZXW1l37ilpuVm46DHxJT0n6Q57HtJZ1uXPynzovL+nPgDOBaqAKOF/SufnaiojFEZGJiExlZWWnBmTWVU4aNLCo5WblpsPAj4gpETE8z+MR4F1JJwLkvr6XZxN/AayNiL0RsRd4HJhQykGYdYdrLzydgf0rDls2sH8F1154eg/1yKw4SU/pLAdm5qZnAo/kqfkP4DxJ/ST1J3vBNu8pHbNyNn1UFT/62tlUDRqIgKpBA/nR1872XTrWayjJ3ZGSTgAeAE4BtpC9LXOHpAzw7YiYJakC+Akwkewpn3+NiKs72nYmk4m6urpO983MLI0krYuITL51ie7Dj4jtwAV5ltcBs3LTB4G/SdKOmZkl51fampmlhAPfzCwlHPhmZinhwDczS4lEd+l0JUlNZO/86azBwPsl6k5P6ivjAI+lXPWVsfSVcUCysZwaEXlfuVq2gZ+UpLq2bk3qTfrKOMBjKVd9ZSx9ZRzQdWPxKR0zs5Rw4JuZpURfDvzFPd2BEukr4wCPpVz1lbH0lXFAF42lz57DNzOzw/XlI3wzM2vBgW9mlhK9OvAl3SPpPUl/aGO9JN0uabOklySN7u4+FqqAsUyS9IGkDbnHTd3dx0JIOlnSakn1kl6WNDdPTa/YLwWOpez3i6QBkn4v6d9z4/hBnpojJS3N7ZPnJdX0QFc7VOBYrpLU1GKfzOqJvhZKUoWk9ZIezbOutPslInrtg+xbLo8G/tDG+q+Q/cAVAeOB53u6zwnGMgl4tKf7WcA4TgRG56aPAV4DhvXG/VLgWMp+v+S+z0fnpvsDzwPjW9V8B7gjN30lsLSn+51gLFcB/9TTfS1iTFcD/5zv56jU+6VXH+FHxDPAjnZKpgH3RdZaYNChT+gqNwWMpVeIiG0R8WJueg/ZD7tp/QkhvWK/FDiWspf7Pu/NzfbPPVrfrTENWJKbfhC4QJK6qYsFK3AsvYakauCrwF1tlJR0v/TqwC9AFfB2i/kGeuEvbAsTcv/KPi7prJ7uTEdy/36OInsU1lKv2y/tjAV6wX7JnTbYQPZjSJ+MiDb3SUQcAD4ATujWThaogLEAXJI7XfigpJO7t4dFuQ34PvBJG+tLul/6euD3JS+SfY+MEcCPgWU92532SToaeAj4u4jY3dP9SaKDsfSK/RIRByNiJFANjJM0vIe71GkFjOXXQE1E1AJP8qcj5LIi6b8C70XEuu5qs68HfiPQ8q97dW5ZrxMRuw/9KxsRK4D+kgb3cLfyyn128UPALyLi4TwlvWa/dDSW3rRfACJiF7AauKjVquZ9IqkfcBywvVs7V6S2xhIR2yPiP3OzdwFjurlrhToHuFjSW8D9wPmS/m+rmpLul74e+MuBv8rdFTIe+CAitvV0pzpD0ucOnbuTNI7sviu7X8hcH+8GXomIf2yjrFfsl0LG0hv2i6RKSYNy0wOBqcCrrcqWAzNz05cCT0fuSmE5KWQsra4HXUz22kvZiYjrI6I6ImrIXpB9OiL+slVZSfdLos+07WmSfkn2LonBkhqA/0X2Ig4RcQewguwdIZuBD4Fv9ExPO1bAWC4F/rukA8A+4Mpy/IUke9Ty34CNufOsAP+T7Afd97b9UshYesN+ORFYIqmC7B+kByLiUUk3A3URsZzsH7afS9pM9uaBK3uuu+0qZCx/K+li4ADZsVzVY73thK7cL35rBTOzlOjrp3TMzCzHgW9mlhIOfDOzlHDgm5mlhAPfzCwlHPhmZinhwDczS4n/DyfeK02lAGDYAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 3\n",
    "x = np.linspace(1, 4, n)\n",
    "f = np.cos # posplošimo interpolirano funkcijo (lahko spremenite v drugo funkcijo)\n",
    "f_ime = f.__str__().split('\\'')[1] # avtomatsko vzamemo ime funkcije\n",
    "y = f(x)# + np.random.rand(len(x))*0.3\n",
    "plt.plot(x, y, 'o', label='Interpolacijske točke');\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Interpolacija s polinomom"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interpolacija s polinomom se zdi najbolj primerna, saj je enostavna! \n",
    "\n",
    "Polinom stopnje $n-1$:\n",
    "$$y = a_0\\,x^{n-1} +a_1\\,x^{n-2} +\\cdots + a_{n-2}\\,x + a_{n-1}.$$\n",
    "\n",
    "je definiran z $n$ konstantami $a_i$. Da določimo $n$ konstant, potrebujemo $n$ (različnih) enačb. Za vsak par $x_i, y_i$ lahko torej zapišemo:\n",
    "$$y_i = a_{0}\\,x_i^{n-1} +a_{1}\\,x_i^{n-2} +\\cdots + a_{n-2}\\,x_i + a_{n-1}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ker imamo podanih $n$ parov, lahko določimo $n$ neznanih konstant $a_i$, ki definirajo polinom stopnje $n-1$. Sistem $n$ linearnih enačb lahko zapišemo:\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "x_{0}^{n-1}&x_{0}^{n-2}&\\dots&x_{0}^0\\\\\n",
    "x_{1}^{n-1}&x_{1}^{n-2}&\\dots&x_{1}^0\\\\\n",
    "&&\\vdots&&\\\\\n",
    "x_{n-1}^{n-1}&x_{n-1}^{n-2}&\\dots&x_{n-1}^0\\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{pmatrix}\n",
    "a_{0}\\\\\n",
    "a_{1}\\\\\n",
    "\\vdots\\\\\n",
    "a_{n-1}\n",
    "\\end{pmatrix}=\n",
    "\\begin{pmatrix}\n",
    "y_{0}\\\\\n",
    "y_{1}\\\\\n",
    "\\vdots\\\\\n",
    "y_{n-1}\n",
    "\\end{pmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sistem linearnim enačb zapišemo v obliki:\n",
    "$$\\mathbf{M}\\,\\mathbf{a}=\\mathbf{b}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo matriko koeficientov $\\mathbf{M}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  ,  1.  ,  1.  ],\n",
       "       [ 6.25,  2.5 ,  1.  ],\n",
       "       [16.  ,  4.  ,  1.  ]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.asarray([[_**p for p in reversed(range(len(x)))] for _ in x])\n",
    "M  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunamo koeficiente $a_{0}, a_{1},\\dots$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.33087687, -2.05236633,  2.26179176])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resitev = np.linalg.solve(M, y)\n",
    "resitev"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo interpolacijski polinom kot Pythonovo funkcijo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def y_function(x, resitev):\n",
    "    A = np.asarray([[_**p for p in reversed(range(len(resitev)))] for _ in x])\n",
    "    return A.dot(resitev)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izris interpolacijskega polinoma pri bolj gosti mreži točk:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xint = np.linspace(np.min(x), np.max(x), 50)\n",
    "yint = y_function(xint, resitev)\n",
    "plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "plt.plot(xint, yint, '.',label='Interpolacija')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Slabosti zgornjega postopka so:\n",
    "\n",
    " * število numeričnih operacij raste sorazmerno z $n^3$,\n",
    " * problem je lahko slabo pogojen (z večanjem stopnje polinoma slaba pogojenost naglo narašča):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "71.3022787031108"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.cond(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Navodilo*: vrnite se par vrstic nazaj in spremenite število interpolacijskih točk $n$ na višjo vrednost (npr. 10)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Lagrangeva metoda"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "*Lagrangeva* metoda ne zahteva reševanja sistema enačb in je s stališča števila računskih operacij (narašča sorazmerno z $n^2$ ([vir](http://www.ams.org/journals/mcom/1970-24-109/S0025-5718-1970-0258240-X/S0025-5718-1970-0258240-X.pdf))) boljša od predhodno predstavljene polinomske interpolacije (število operacij narašča sorazmerno z $n^3$), kjer smo reševali sistem linearnih enačb. Rešitev pa je seveda popolnoma enaka!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/jpeg": 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      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"800\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/c_TbU8XBHsw\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x1aceefac370>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('c_TbU8XBHsw', width=800, height=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Lagrangev interpolacijski polinom stopnje $n-1$ je definiran kot:\n",
    "$$P_{n-1}(x)=\\sum_{i=0}^{n-1}y_i\\,l_i(x),$$\n",
    "kjer je $l_i$ Lagrangev polinom:\n",
    "$$l_i(x)=\\prod_{j=0, j\\ne i}^{n-1} \\frac{x-x_j}{x_i-x_j}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si interpolacijo za zgoraj prikazane $x$ in $y$ podatke.\n",
    "\n",
    "Definirajmo najprej Lagrangeve polinome $l_i(x)=\\prod_{j=0, j\\ne i}^{n-1} \\frac{x-x_j}{x_i-x_j}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lagrange(x, x_int, i):\n",
    "    \"\"\" Vrne vrednosti i-tega Lagrangevega polinoma\n",
    "    \n",
    "    x: neodvisna spremenljivka (skalar ali numerično polje)\n",
    "    x_int: seznam interpolacijskih točk\n",
    "    i: indeks polinoma\n",
    "    \"\"\"\n",
    "    Lx = 1.0\n",
    "    for j in range(len(x_int)):\n",
    "        if j != i:\n",
    "            Lx *= (x-x_int[j]) / (x_int[i]-x_int[j])\n",
    "    return Lx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def slika(i=0):\n",
    "    xint = np.linspace(np.min(x), np.max(x), 30)\n",
    "    plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "    plt.axhline(0, color='k', linewidth=0.3);\n",
    "    plt.plot(xint, lagrange(xint, x_int=x, i=i), '.',label=f'Lagrangev polinom i={i}');\n",
    "    for _ in x:\n",
    "        plt.axvline(_, color='r', linewidth=0.5);\n",
    "    plt.legend()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "slika(i=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Opazimo, da ima $i$-ti Lagrangev polinom v $x_i$ vrednost 1, v ostalih podanih točkah pa nič!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Če torej Lagrangev polinom za $i=0$ pomnožimo z $y_0$, bomo pri $x=x_0$ dobili pravo vrednost, v ostalih interpolacijskih točkah pa nič; implementirajmo torej Lagrangev interpolacijski polinom:\n",
    "$$P_{n-1}(x)=\\sum_{i=0}^{n-1}y_i\\,l_i(x),$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def lagrange_interpolacija(x, x_int, y_int):\n",
    "    \"\"\" Vrne vrednosti Lagrangeve interpolacije\n",
    "    \n",
    "    x: neodvisna spremenljivka (skalar ali numerično polje)\n",
    "    x_int: abscisa interpolacijskih točk\n",
    "    y_int: ordinata interpolacijskih točk\n",
    "    \"\"\"\n",
    "    y = 0.\n",
    "    for i in range(len(x_int)):\n",
    "        Lx = 1.0\n",
    "        for j in range(len(x_int)):\n",
    "            if j != i:\n",
    "                Lx *= (x-x_int[j]) / (x_int[i]-x_int[j])\n",
    "        y += y_int[i] * Lx\n",
    "    return y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo sliko:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def slika(i=0):\n",
    "    xint = np.linspace(np.min(x), np.max(x), 30)\n",
    "    plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "    plt.plot(xint, lagrange(xint, x_int=x, i=i), '.',label=f'Lagrangev polinom i={i}');\n",
    "    plt.plot(xint, lagrange_interpolacija(xint, x_int=x, y_int=y), '.',label=f'Lagrangev int polinom');\n",
    "    plt.axhline(0, color='k', linewidth=0.3);\n",
    "    for _ in x:\n",
    "        plt.axvline(_, color='r', linewidth=0.5);\n",
    "    plt.legend()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Iz `ipywidgets` uvozimo ``interact``, ki je močno orodje za avtomatsko generiranje (preprostega) uporabniškega vmesnika znotraj `jupyter` okolja. Tukaj bomo uporabili relativno preprosto interakcijo s sliko; za pregled vseh zmožnosti pa radovednega bralca naslavljamo na [dokumentacijo](http://ipywidgets.readthedocs.io/en/latest/examples/Using%20Interact.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uvoz funkcije ``interact``"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "from ipywidgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5681b8e8528c422d86eb4d350c210dfd",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=0, description='i', max=1), Output()), _dom_classes=('widget-interact',)…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(slika);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Iz slike vidimo, da ima Lagrangev polinom $i$ samo pri $x_i$ vrednost 1 v ostalih točkah $\\ne i$ pa ima vrednosti nič; ko Lagrangev polinom $l_i(x)$ pomnožimo z $y_i$ zadostimo $i$-ti točki iz tabele. Posledično Lagrangeva interpolacija z vsoto Lagrangevih polinomov interpolira tabelo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Polinomska interpolacija pri velikem številu točk je lahko slabo pogojena naloga in zato jo odsvetujemo. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Ocena napake"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Če je $f(x)$ funkcija, ki jo interpoliramo in je $P_{n-1}(x)$ interpolacijski polinom stopnje $n-1$, potem se lahko pokaže (glejte npr.: Burden, Faires, Burden: Numerical Analysis), da je napaka interpolacije s polinomom:\n",
    "$$e=f(x)-P_{n-1}(x)=\\frac{f^{(n)}(\\xi)}{n!}\\,(x-x_0)\\,(x-x_1)\\,\\cdots\\,(x-x_{n-1}),$$\n",
    "kjer je $f^{(n)}$ odvod funkcije, $n-1$ stopnja interpolacijskega polinoma in $\\xi$ vrednost na interpoliranem intervalu $[x_0, x_{n-1}]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Zgled"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tukaj si bomo ogledali interpolacijo točk:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([0.        ,  0.52359878,  1.04719755,  1.57079633])\n",
    "y = np.array([ 0.       ,  0.5      ,  0.8660254,  1.       ])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Točke prikažimo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Linearna interpolacija za vrednost pri `x=1.57079633/2`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6830127"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_linearna = lagrange_interpolacija(x=x[-1]/2, x_int=x[1:3], y_int=y[1:3])\n",
    "y_linearna"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Kvadratna:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6997595236464176"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_kvadratna = lagrange_interpolacija(x=x[-1]/2, x_int=x[0:3], y_int=y[0:3])\n",
    "y_kvadratna"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Kubična"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.705889286844634"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_kubična = lagrange_interpolacija(x=x[-1]/2, x_int=x, y_int=y)\n",
    "y_kubična"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Zgled ocene napake"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pri interpolaciji ponavadi funkcije $f(x)$ ne poznamo in napako ocenimo s pomočjo formule:\n",
    "\n",
    "$$e=\\frac{f^{(n)}(\\xi)}{n!}\\,(x-x_0)\\,(x-x_1)\\,\\cdots\\,(x-x_n).$$\n",
    "\n",
    "Pri tem vrednost $\\xi$  ni znana; ker je v primeru linearne interpolacije ($n=2$) drugi odvod sinusne funkcije ($f^{(n)}$) med -1 in +1, velja:\n",
    "$$|e|\\le\\left|\\frac{-1}{2!}\\,(\\pi/4-\\pi/6)\\,(\\pi/4-\\pi/3)\\right|=\\frac{1}{2}\\,\\frac{\\pi}{12}\\,\\frac{\\pi}{12}=\\frac{\\pi^2}{288}=0,034$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poleg **Lagrangeve metode** bi si tukaj lahko pogledali še **Newtonovo metodo** interpolacije."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Interpolacija z uporabo `scipy`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Poglejmo si interpolacijo v okviru modula `scipy.interpolate` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/interpolate.html))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uporabili bomo funkcijo za interpoliranje tabele z zlepki, `scipy.interpolate.interp1d` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.interp1d.html#scipy.interpolate.interp1d)):\n",
    "\n",
    "```python\n",
    "interp1d(x, y, kind='linear', axis=-1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)\n",
    "```\n",
    "\n",
    "Podati moramo vsaj dva parametra: seznama interpolacijskih točk `x` in `y`. Privzeti parameter `kind='linear'` pomeni, da inteproliramo z odsekoma linearno funkcijo. `interp1d` vrne funkcijo `f`, ki jo kličemo (npr. `y = f(x)`) za izračun interpolirane vrednosti. \n",
    "\n",
    "Parameter `kind` je lahko npr. tudi: `'zero'`, `'slinear'`, `'quadratic'` in `'cubic'`; takrat se uporabi interpolacijski zlepek (ang. *spline*) reda 0, 1, 2 oz. 3. Zlepke si bomo pogledali v naslednjem poglavju."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.interpolate import interp1d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo tabelo podatkov:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([ 1.        ,  2.14285714,  3.28571429,  4.42857143,  5.57142857,\n",
    "        6.71428571,  7.85714286,  9.        ])\n",
    "\n",
    "y = np.array([ 0.84147098,  0.84078711, -0.14362322, -0.95999344, -0.65316501,\n",
    "        0.41787078,  0.999995  ,  0.41211849])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.interpolate import interp1d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = interp1d(x, y, kind='linear')\n",
    "x_g = np.linspace(x[0], x[-1], 20*len(x)-1)\n",
    "plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "plt.plot(x_g, f(x_g), '-', label='Interpolacija')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Kubični zlepki"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Preden gremo v teorijo zlepkov, si poglejmo rezultat, ki ga dobimo s klicanjem funkcije `interp1d` s parametrom `kind='cubic'` (rezultat je kubični zlepek)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = interp1d(x, y, kind='cubic')\n",
    "plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "plt.plot(x_g, f(x_g), '-', label='Kubični zlepek')\n",
    "plt.plot(x_g, np.sin(x_g), label='sin', alpha=0.5)\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Kubični zlepki so pogost način interpolacije. \n",
    "\n",
    "Zahtevamo, da je: $x_0<x_1< \\cdots <x_n$.\n",
    "\n",
    "Od točke $x_i$ do $x_{i+1}$ naj bo zlepek \n",
    "polinom:\n",
    "\n",
    "$$f_{i,i+1}(x)= a_{i,3}\\,x^3+a_{i,2}\\,x^2+a_{i,1}\\,x+a_{i,0},$$\n",
    "\n",
    "pri čemer so neznane vrednosti konstant $a_{i,j}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če imamo na primer $n+1$ točk, potem je treba določiti $n$ polinomov. \n",
    "\n",
    "Celotni zlepek čez $n+1$ točk je definiran z:\n",
    "$$f(x) = \\left\\{\n",
    "\\begin{array}{rcl}\n",
    "f_{0,1}(x); && x\\in[x_0, x_1)\\\\\n",
    "f_{1,2}(x); && x\\in[x_1, x_2)\\\\\n",
    "&\\vdots&\\\\\n",
    "f_{n-1,n}(x); && x\\in[x_{n-1}, x_n]\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "\n",
    "Vsak polinom $f_{i,i+1}$ je definiran s 4 konstantami $a_{i,j}$; skupaj torej moramo izračunati $4n$ konstant $a_{i,j}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Kako določimo konstante $a_{i,j}$?**\n",
    "\n",
    "Za določitev $4n$ neznak potrebujemo $4n$ enačb. Poglejmo si, kako jih dobimo:\n",
    "\n",
    "* $n$ enačb dobimo iz interpolacijskega pogoja: $$y_i=f_{i,i+1}(x_i),\\quad i=0,1,2,\\dots,n-1$$\n",
    "* 1 enačbo iz zadnje točke: $$y_n=f_{n-1,n}(x_n)$$\n",
    "* $3(n-1)$ enačb dobimo iz pogoja $C^2$zveznosti: \n",
    "$$\\lim_{x\\rightarrow x_i^-}f(x)=\\lim_{x\\rightarrow x_i^+}f(x),$$\n",
    "$$\\lim_{x\\rightarrow x_i^-}f'(x)=\\lim_{x\\rightarrow x_i^+}f'(x)$$\n",
    "in\n",
    "$$\\lim_{x\\rightarrow x_i^-}f''(x)=\\lim_{x\\rightarrow x_i^+}f''(x).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Skupaj imamo definiranih $4n-2$ enačbi, manjkata torej še dve! \n",
    "\n",
    "Različni tipi zlepkov se ločijo po tem, kako ti dve enačbi določimo. V nadaljevanju si bomo pogledali *naravne kubične zlepke*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Naravni kubični zlepki"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Naravni kubični zlepki* temeljijo na ideji Eulerjevega nosilca:\n",
    "$$E\\,I\\,\\frac{\\textrm{d}^4y}{\\textrm{d}x^4}=q(x),$$\n",
    "kjer je $E$ elastični modul, $I$ drugi moment preseka in $q(x)$ zunanja porazdeljena sila. Ker zunanje porazdeljene sile ni ($q(x)=0$), velja:\n",
    "$$E\\,I\\,\\frac{\\textrm{d}^4y}{\\textrm{d}x^4}=0.$$\n",
    "Sledi, da lahko v vsaki točki tanek nosilec popišemo s polinomom tretje stopnje. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$C^2$ zveznost je zagotovljena v kolikor so vmesne podpore nosilca členki (moment zato nima nezvezne spremembe).\n",
    "\n",
    "Manjkajoči 2 neznanki pri naravnih kubičnih zlepkih določimo iz pogoja, da je moment na koncih enak nič (členkasto vpetje):\n",
    "\n",
    "$$f''(x_{0})=0\\qquad\\textrm{in}\\qquad f''(x_{n})=0$$\n",
    "\n",
    "Izpeljava je natančneje prikazana v knjigi Kiusalaas J: Numerical Methods in Engineering with Python 3, 2013, stran 120 (glejte tudi J. Petrišič: Interpolacija, Fakulteta za strojništvo, 1999); podrobna izpeljava presega namen te knjige. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Tukaj si bomo pogledali samo končni rezultat, ki ga lahko izpeljemo ob zgornjih pogojih. V primeru ekvidistantne delitve $h=x_{i+1}-x_i$ tako izpeljemo sistem enačb ($i=1,\\dots,n-1$):\n",
    "\n",
    "$$k_{i-1} + 4 k_{i} + k_{i+1}  =  \\frac{6}{h^2} \\left(y_{i-1} -2 y_{i} +y_{i+1} \\right).$$\n",
    "\n",
    "kjer je neznanka $k_i$ drugi odvod odsekovne funkcije $k_i = f''_{i,i+1}(x_i)$.\n",
    "\n",
    "Rešljiv sistem enačb dobimo, če dodamo še robna pogoja za naravne kubične zlepke:\n",
    "\n",
    "$$k_0=k_n=0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ko določimo neznake $k_i$, jih uporabimo v odsekoma definirani funkciji:\n",
    "\n",
    "$$\n",
    "f_{i,i+1}(x)=\\frac{k_i}{6}\\left(\\frac{(x-x_{i+1})^3}{h}-(x-x_{i+1})\\,h\\right)\n",
    "-\\frac{k_{i+1}}{6}\\left(\\frac{(x-x_{i})^3}{h}-(x-x_{i})\\,h\\right)\n",
    "+\\frac{y_i\\,(x-x_{i+1})-y_{i+1}\\,(x-x_{i})}{h}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Numerična implementacija"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Najprej pripravimo funkcijo, katera za podane interpolacijske točke reši sistem linearnih enačb in vrne koeficiente $k_i$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def kubicni_zlepki_koeficient(x, y):\n",
    "    \"\"\" Vrne koeficiente kubičnih zlepkov `k`, matriko koeficientov `A` in konstant.\n",
    "    \n",
    "    x in y predstavljata seznam znanih vrednosti; x mora biti ekvidistanten.\n",
    "    \"\"\"\n",
    "    n = len(x)\n",
    "    A = np.zeros((n, n)) # pripravimo matriko koeficientov\n",
    "    h = x[1]-x[0] # korak h\n",
    "    for i in range(n):\n",
    "        if i==0 or i==n-1:\n",
    "            A[i,i] = 1. # k_0 in k_n sta nič zato tukaj damo 1\n",
    "                       # pri vektorju konstant pa bomo dali 0, k_0 in k_n bosta torej 0\n",
    "        else:\n",
    "            A[i, i-1:i+2] = np.asarray([1., 4., 1.])\n",
    "    b = np.zeros(n)\n",
    "    b[1:-1] = (6/h**2)*(y[:-2] - 2*y[1:-1] + y[2:]) # desna stran zgornje enačbe\n",
    "    k = np.linalg.solve(A,b)\n",
    "    return k, A, b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Opomba: pri zgornjem linearnem problemu, lahko izračun zelo pohitrimo, če upoštevamo tridiagonalnost matrike koeficientov! (Glejte [odgovor na stackoverflow.com](https://stackoverflow.com/questions/23120164/optimize-ax-b-solution-for-a-tridiagonal-coefficient-matrix))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer izračuna koeficientov:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrika koeficientov A lin. sistema:\n",
      " [[1. 0. 0. 0. 0.]\n",
      " [1. 4. 1. 0. 0.]\n",
      " [0. 1. 4. 1. 0.]\n",
      " [0. 0. 1. 4. 1.]\n",
      " [0. 0. 0. 0. 1.]]\n",
      "Vektor konstant b lin. sistema:      [  0. -12.  12. -12.   0.]\n",
      "Koeficienti k so: [ 0.         -4.28571429  5.14285714 -4.28571429  0.        ]\n"
     ]
    }
   ],
   "source": [
    "x = np.asarray([1, 2, 3, 4, 5])\n",
    "y = np.asarray([0, 1, 0, 1, 0])\n",
    "\n",
    "k, A, b = kubicni_zlepki_koeficient(x, y)\n",
    "print('Matrika koeficientov A lin. sistema:\\n', A)\n",
    "print('Vektor konstant b lin. sistema:     ', b)\n",
    "print('Koeficienti k so:', k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato potrebujemo še kubični polinom v določenem intervalu; implementirajmo izraz:\n",
    "\n",
    "$$\n",
    "f_{i,i+1}(x)=\\frac{k_i}{6}\\left(\\frac{(x-x_{i+1})^3}{h}-(x-x_{i+1})\\,h\\right)\n",
    "-\\frac{k_{i+1}}{6}\\left(\\frac{(x-x_{i})^3}{h}-(x-x_{i})\\,h\\right)\n",
    "+\\frac{y_i\\,(x-x_{i+1})-y_{i+1}\\,(x-x_{i})}{h}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def kubicni_zlepki(k, x, y, x_najdi):\n",
    "    \"\"\" Vrne kubicni zlepek pri delitvi `xint`.\n",
    "    \n",
    "    :param k: koeficienti kubičnih zlepkov\n",
    "    :param x in y: znane vrednosti, x mora biti ekvidistanten\n",
    "    :param x_najdi: vrednosti kjer želimo izračunati kubični zlepek\n",
    "    \"\"\"  \n",
    "    h = x[0] - x[1]\n",
    "    i = int((x_najdi-x[0])//(-h))\n",
    "    if i >= len(k)-1:\n",
    "        i = len(k)-2\n",
    "    out = ((x_najdi - x[i+1])**3/h - (x_najdi - x[i+1])*h)*k[i]/6.0 \\\n",
    "        - ((x_najdi - x[i])**3/h - (x_najdi - x[i])*h)*k[i+1]/6.0 \\\n",
    "        + (y[i]*(x_najdi - x[i+1]) \\\n",
    "        - y[i+1]*(x_najdi - x[i]))/h\n",
    "    return out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunamo interpolirane vrednosti:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "xint = np.linspace(np.min(x), np.max(x), 50)\n",
    "yint = np.asarray([kubicni_zlepki(k, x, y, _) for _ in xint])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "plt.plot(xint, yint, label='Naravni kubični zlepek')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Nekaj vprašanj za razmislek!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "1. Preštudirajte Lagrangevo polinomsko interpolacijo in pripravite funkcijo za Lagrangeve polinome. Pojasnite (z grafičnim prikazom) Lagrangeve polinome.\n",
    "2. Definirajte funkcijo za Lagrangevo polinomsko interpolacijo. Na primeru pojasnite, kako deluje.\n",
    "3. Pojasnite teoretično ozadje naravnih kubičnih zlepkov.\n",
    "4. Naravne kubične zlepke smo izpeljali pod pogojem, da momenta na koncu ni; včasih želimo drugačne pogoje na koncih (npr. znani naklon ali znani moment). Modificirajte na predavanjih predstavljeno kodo za primer, da je na koncih moment $\\ne 0$ (predpostavite neko numerično vrednost).\n",
    "5. Podatke:\n",
    "    ```python\n",
    "    x = np.linspace(0, 10, 10)\n",
    "    y = np.random.rand(10)-0.5\n",
    "    ```\n",
    "  interpolirajte z uporabo ``scipy.InterpolatedUnivariateSpline``. Podatke prikažite.\n",
    "6. Za zgoraj definirane podatke preučite pomoč in najdite vse ničle. Prikažite jih na predhodni sliki.\n",
    "7. Za zgoraj definirani zlepek izračunajte prvi odvod in ga prikažite.\n",
    "8. Za zgoraj definirani zlepek izračunajte določeni integral od začetka do konca.\n",
    "9. Za zgoraj definirane podatke z uporabo vgrajenih funkcij prikažite izračun linearnega in kvadratnega zlepka. Prikažite na sliki.\n",
    "10. Preučite pomoč za funkcijo ``scipy.interpolate.lagrange`` in k predhodni sliki dodajte Lagrangev interpolacijski polinom. Komentirajte rezultate.\n",
    "11. Preučite pomoč za funkcijo ``scipy.interpolate.interp1d`` in k predhodni sliki dodajte kvadratni zlepek.\n",
    "12. Preučite pomoč za funkcijo ``scipy.interpolate.BarycentricInterpolator`` in pojasnite ter prikažite njeno prednost.\n",
    "13. Preučite pomoč za funkcijo ``scipy.interpolate.KroghInterpolator`` in pojasnite njeno prednost."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Dodatno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 2D interpolacija: https://www.youtube.com/watch?v=_cJLVhdj0j4\n",
    "* Strojno prevajanje: https://pypi.org/project/goslate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Nekaj komentarjev modula ``scipy.interpolate``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SciPy ima implementiranih večje število različnih interpolacij ([glejte dokumentacijo](https://docs.scipy.org/doc/scipy/reference/interpolate.html)). S stališča uporabe se bomo tukaj dotaknili  objektne implementacije `scipy.interpolate.InterpolatedUnivariateSpline` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.InterpolatedUnivariateSpline.html)) (starejši pristop temelji na funkcijskem programiranju, glejte  [dokumentacijo](https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.splrep.html) `scipy.interpolate.splrep`):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```python\n",
    "InterpolatedUnivariateSpline(x, y, w=None, bbox=[None, None], k=3, ext=0, check_finite=False)\n",
    "```\n",
    "\n",
    "Pri inicializaciji objekta `InterpolatedUnivariateSpline` moramo posredovati interpolacijske točke `x` in `y`. Argument `k` s privzeto vrednostjo `k=3` definira red intepolacijskega zlepka (`1<=k<=5`). Pomemben opcijski parameter je tudi `w`, ki definira uteži posameznim interpolacijskim točkam (uporabimo ga, če želimo določenim področjem dati večji poudarek)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from scipy.interpolate import InterpolatedUnivariateSpline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "spl = InterpolatedUnivariateSpline(x, y, k=3) # poglejte opcije!\n",
    "plt.plot(x, y, 'o', label='Interpolacijske točke')\n",
    "plt.plot(xint, spl(xint), label='Kubični zlepek');\n",
    "plt.plot(spl.get_knots(), spl(spl.get_knots()), 'r.', label='Vozli B-zlepkov')\n",
    "plt.plot(xint, yint, label='Naravni kubični zlepki - lastna koda');\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ker gre za B-zlepke, je rezultat drugačen kot tisti, ki smo ga izpeljali z naravnimi kubičnimi zlepki. V nasprotju z naravnimi kubičnimi zlepki, ki imajo vozle (angl. *knots*) v interpolacijskih točkah, se vozli B-zlepkov prilagodijo podatkom. V konkretnem primeru so vozli v točkah:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 3., 5.])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spl.get_knots()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Odvajanje, integriranje ... zlepkov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zlepke lahko odvajamo in integriramo, saj so polinomi. Objekt ``InterpolatedUnivariateSpline`` je tako že pripravljen za odvajanje, integriranje, iskanje korenov (ničel), vozlov ... (glejte [dokumentacijo](https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.InterpolatedUnivariateSpline.html)).\n",
    "\n",
    "Za prvi odvod zlepka v objektu `spl` na primer uporabimo metodo ``spl.derivative(1)``, ki vrne nov objekt zlepka (njen red je sedaj za 1 nižji):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pxtSau9fV+/SWmioNDcU5JaXC9vJPPm4uTlwZ1YoxcSH07+CPs5Pt3qAahX2/wo/3QW4q9Punsaqbq4fZUV1QbkExi7en8vWmw2xKzjy7vaqWG926BSr1REOGWCs7Dmfz9vL9LN2ZVuG9nuHNef76GCJaeZkQ2cVJsnZckqyBo1mn+c93CfyWePyc7c09XbljQFsm9g2v2Mydmwb/i4W2g2DcFw0YbTXMmwdTp0L+3312hW4e/Ouqe/guqmKfYhsfD0b3COamuGDCWjRtyEgbv5IiWPoviP8IAiKNp+ngWLOjqpLWmnUHM/h6UwpLdqRxuri0wj5j9q7i2cVv4lZ+TISbM4xwhUl3wPBXwLVJA0ZdO/uO5TJr5QG+33qU0nKDQFydFdMHd+CeIe1xd7GtqV+SrB2XQyfrUovm07VJvPLzHk6V6/MN8HJn2qXtGdcrBE+3C8xn/uN1YzrXLQuhw+X1H3BNzJtXYbRu+sgb+WlHKgs2HWbHkcr78AZG+DOhdxhXdG7pEEUm6tWpE/DlRDi0xpirfNl/bLYyWFZ+EQs2Hebz9Yc4eKJi+VIXJ8VlkS25oUcwgzsF4PHV/HN/v559BtokwepXjWpqY+YaVdHsQEpGPu+tOsD8DSmUlEva7QOa8uLorvQMt53xBJKsHZe1RoN/AQwG/IFjwBNa6w+r2t8WkvWetFz+b+F2thzKOmf7hN6h/OvqyOpVDispNAqlOLvB9D/B2X76gHen5vB1/GG+23qEjFNFFd5v7e3B2F4hjO0ZSmsf226utUlpCfDFODh1HEa9DTE3mh1RBVprth3O5rN1yfy47SiFlUwTjGztxY2xwVzXPQj/ZtX4oLH3Z1g41SiTet070HlEPUReP6q6J4zvHcr/VfeeUM8kWTsuhyuKorXmoz+TeOGn3db5FL1nCXwx1lida8AM6wbbAIpKLPyeeJyv41NYvud4hSlhzk6KoZ1bMalfGH3btbDpwTc2Y/ePsPAu8PCBsfMgqIfZEZ2joLiUH7YeZe66JBKOVBzV7eXuwnXdg7gpLpiYIJ+a/5tnJhvleVO3GoMwL38CnOyjlabUovlsXTIvL008p7Wtlbc770yIJTbM3FHjkqwdl0Ml64LiUv797Q4Wbj5ydpurs+LuwR24uy79U1/eYgwgmr7G5gab1cSRrNN8sf4Q8zemcCKvsML7nVp5cWu/cK7vHkQTN9vqy7MJWsOqV2D5c0bFr7HzwKu12VGddSTrNJ+tS2b+hkNk5hdXeL9LoDcT+4Qx8pLAC3f/VEdxgdFXv+kT6DoWRr1lVy1PR7NO8/j3CSzb/fc4FjdnJ569LpoxPUNMi0uSteNymGSdll3AXZ/Gs63cfMtuwT68elO3uo/8zEk1msPbdIVbf7TOAgsmKiqx8PPOND5bl8z6vzIqvO/TxJWbe4YwsU8YIX6eJkRog0qLy2YILDCS07X/tYnR3lprNvyVwSdrkvh5Z1qFlhN3FydGdA3klj6hXBLia92WE62NpWWXPwsdr4abPraLgWdnaK1ZkpDGv7/dQVa5Dze39g3j/42IwtWEMR2SrB2XQyTrTckZTPtsM+nlpmSNiQvmmeuirTfaM/5jWHQ/jHwLeky0zjFtwL5jucxdm8w3mw+fU3gFjBrlV3RuxW39wx27ibykCBZMNup6X/Gk0fRr8rU409T98ZokdqdWbOoObt6ESX3DGBMXUv+lNze8Dz89bCy9Oe4Lu1sTPiUjnylz40lMyz27rU87P94e34MW1enHtyJJ1o6r0Sfr+RsO8Z/vEyguNX5OZyfF4yOimNQ3zLrJxWKBOSOMNYLv2Qherax3bBuQU1DM1/GHmbs2qdIyjpGtvZjcP5xRlwQ1upWOLqik0Oif3bsUhr8KvaaYGk5adgGfrkviiw0plQ4c7N+hBbf1a8tlkS0bdm79jgXw7V3Qqosxe6K+VxCzsvyiEh7+ejuLd6Se3Rbk24TZk2LpEujTYHFIsnZcjTZZa615/qfdvL/6r7Pb/Jq68fb4HvRt36J+TnpiH8zqD52uhjFz6uccJrNYNCv2HufjP5NYva9iAYzmnq6M6xXKxL5htPGxnybPWik+bYxX2L8MRrwBcZNNCUNrzeZDWXz8518sTUg7Z+AkGDW5b+gRxK39wuloZrGPvb8YH2x8gmHit+BrXt9vbWiteWfFAV79Zc/Z5W89XJ2YNSGWIZEtGyQGSdaOq1Ema601T/24i0/WJJ3d1rmNN7MnxtZ/H+uqV+H3Z2DsFxA5vH7PZbL9x3OZsyaZBZsOVyie4eykuKpLKyb1Dad3W7/G10RelA/zx8HBlTDyf6Z0fRQUl/LjtqPMXZtc6bz5IN8m3NovjJvjQm2ntGzyWvj8ZnD3gsmLoXm42RHV2G+7j3Hf/K3klS1k4ubsxKxbenB55/pvTZNk7bgaXbKuLFFf1aUVr998Sd1HuFZHaTG8dymczoR71ttd/1xtZJ8u5quNKcxZm8ThzIpLFUa29mJS33Cu626FUca2oDDPmK6X/KdRkeyScQ16+ouN6u7d1o/J/cO5onMr2yxsk7od5lxrrDB2xy/QrGGeSq1p//E8Jn+ygZQM4/fd1Vnx7i2x9Z6wJVk7rkaVrLXWPPnDTuas/XuRkGu6tuG/N1/SsDetw5vgg8uh5x1wzcyGO6/JSi2aX3cd45M1f7HuYMVR5N4eLoyJC2Fc71DaBzQzIUIrKMqHz0ZDyjq4fjZ0valBTmuxaP48cILP1iXz665jFUZ1u7k4MapbILf1D2/QPtRaS9kIc0caUx1vW2zMSbczR7JOM272Og5lGGM4XJ0VsybEckVU/SVsSdaOq9Eka601T/ywk7lmJ+ozlj4K696ByUsgrF/Dn99ke9Jymbs2iYWbj1RaX7pPOz/G9w7jqi6tbK7+cpUspUb50D0/wY0fQvToej9lem4hCzYd5osNh84mhfKCfJswsW8YN8eF0LxpPY/qtrb9y+DzsRDSC275xq6mdZ1xNOs0Y89L2O9MiGVoPSVsSdaOq1Eka601j3+/k0/X/Z2oR3RtwxtmJWowmkrf7Q/aAtP+sMsnB2vIPl3Mgk1VjyL3a+rGjbHBjOsVSlt/G15ERGtY8ghsmA1Xvwy976q3U1ksmrUHT/L5+kP8sivt7EyG8vp3aMGtfcO5vHMr+14xbccC+ObOskGZn4Kz/XWTHM06zbj31539/XZ1Vrw9vgdXdrF+QRxJ1o7L7pN1ZU/U13YL5PUx3czvr0vZAB8NM+pC3zDb3FhMZrFoVu5L5/P1h/g98fg5qxydERfWnNGxwQyPaVNxlTOzrfkf/PL/oO+9cNVz9XKKv06c4tvNh1m45Uilff/eHi7c0COYCb1DbXYJx1rZ8D789BBcMsGoo26HgxFTs40n7PIJuz76sCVZOy67T9YfrD7Is4t3n/16ZLdAXrOFRH3GipdgxfNwwwcN1r9p69KyC/gqPoX5Gw5xNLugwvtuLk4MjWrF6B5BDIwIMKVS1DkSFhpFT6Kugxs/tmqd6+z8YhbtOMrCzUfOWTO6vNiw5ozvFco1Xds03jnsK16EFS8YH4aufNZuE/a42etIKkvYTVyd+XpaX6KDrNeqJsnacdl1sv5t9zHunBt/ds6j6U3flSktgU+ugeO7jObw5mFmR2QzSi2alXuP8/n6Qyzfk17p07Z/MzeGRbfmqi6t6dOuRcMn7uQ1MHcUBMXCxO+sUkI0O7+Y3xKPsTQhjRV70ikqrbjalU8TV667JJDxvcPo1LoRPUVXpXw3w7CXoM80syOqlbTsAsa8t/ZsH3YbHw++v7c/Lb2sU3pWkrXjsttknZiWw+h31pxdGScurDnzpvS2zcFKmcnw7gBoGWWMfLXDfrn6diKvkB+2HmXhlsOVrgQFRgK7vHNLhnVpzaCOAfX/lJm+Fz4cCk0DjClGnrVf1/h4TgG/7DrGzzvTWHvgZIXCJWCsGT0ksiWjewQxJLKlbf4u1yeLxSgys3cpTPoO2g4yO6Ja2X88j+vf+ZPcAmMedrcQX76c2scqv6+SrB2XXSbrE3mFjHrrT45kGf16Qb5N+P7e/tVbb9cs27+ChVNgyP+DSx82Oxqbtictl4VbDvPdliMcy6m4+hcYTYz92regb/sW9GvvT2RrL5ysOdAqLx0+uMyoUnbnshoX7ygoLmVTciZrDpzgz/0n2XY4i6r+q3UN9mF0j2BGdG3T4LWmbU5BjjHtMf8kTF0BvqFmR1Qrq/amM/mTjWdbi67tFsibYy+pc3EgSdaOy+6SdWFJKePfX3+2f6+pmzPf3N2PyNZ2UHzkmzuN/s87foFg+f92MaUWTXxSBkt3pvFzQlql/dtn+DV1o287I3n3DPejXUDT2jeZlxYbTd9HNhtVtoJiL/oteYUl7E7NYf3Bk/y5/ySbDmVSVFKxefuMbiG+DOvSmqu6tKKdvc45ry8n9sH7lxkfkG7/Gdzsc2W3OWuSeOKHnWe/fmBoR/55eUSdjinJ2nFZJVkrpYYB/wWcgQ+01i9eaP9aJet589D//jekpHDEy5+XB03ix+ghfHhrHJdF2smiGaez4N2B4OQM01YbJRdFtWitSTiSw9KdqSxJSONg+qkL7u/m4kTHVs2IauNN5zbeRLXxpkPLZvg1dav86WbePHjsMTh0CAK8YEARPDEHuo45Z7fiUgvHcgrYk5bLrqM57E7LYdfRnLODiqripKB32xYMi27NlV1aNf666XW192ejLGnMTcZMCjsccKa15j/fJ/DZukMAjNy5nBc3zccz7SiEhsJzz8GECTU6piRrx1XnZK2Ucgb2AkOBw8BGYJzWeldV31PjZD1vHkydCvl/3xDzXdxZ9+gLXPb0jNqGbo7kNcaAs+jRcMP7dnkTsgVJJ07x54ETrDlwkrUHTla6wlRlXJ0VAc3caentQUsvd1p6u9N33c9c+cZ/cC38+8m91M2Fn+5/ntU9r+RYTiHHcwtJzy3g5KmiKpuzz9c+oCn92vvTv0MLerdtYX9FS8y28hVjLeyrnoe+95gdTa0Ul1qY/PFG/L7/mheXvoVnSbluHU9PmD27RglbkrXjskay7gs8qbW+quzrRwG01i9U9T01Ttbh4ZCcXGGzDg1FVbLd5q16BX5/Fq58Dvrda3Y0ds9i0ew9nsuf+0+y7uBJdh7JvmCT+fn+mDWZ4Jz0CtsPewcwYPrH1TqGs5OiQ0Azugb70K+D0Y/eyts6I4AdlsUCX0+CxMXGKl3tBpsdUa1k5xdzOiiE1lnHKr4ZFgZJSdU+liRrx2WNYclBQEq5rw8Dvc/fSSk1FZgKEBpaw0Ejhw5VulmlpFS63eYNfMhYzODX/xjr+7YfYnZEds3JSRHZ2pvI1t7cMaAtAFn5RexKzWF3allzdWoOKZn5Z0folheYU3Gpz6q2KwUtmrrRLsBoYo9q401UoNHE3mjnQJvFyQmumwUfXAFf32YMOLPDVbp8PF3xzj5e+ZtV3NuEOJ81nqxvBIZpre8s+3oi0FtrXeUjo7WerGv6qdSmFOYZN6G8NLu9Cdmj00WlHM8t4HhuIcdzCjmWU8CNo/vjfTy1wr45LQP58Yc1tPTyoJW3Oy29PGjRzM38Ii2O5uQBeH8I+LU3Bmc621h1u+qw0j1MnqwdlzXuOkeA8qvIB5dts57nnjP6d8rz9DS22yv3ZjB2nlE7fP4tUHThAVPCOpq4ORPWoik9w/24pmsbbh/QFu/JfeD8+7+nJ96vvcyE3mEMjWpF12BfWvt4SKI2Q4v2cO2bcHSzUenMHjXGe5hoUNa482wEIpRSbZVSbsBY4AcrHPdvEyYYAzHCwox2yLCwGg/MsEkt2sPoj+BYAnx/L9UeuSSsJ+Eb8PgN/nFV4/v9aky6XAfdb4HVMyHpT7OjqbnGeg8TDcZaU7eGA29gTN36SGt9wY+L9bWetd3643VY9iQMfRr632d2NI4jea0xnzqwO9z6A7g4eEESW1eYB+8NNObBT/sDmviaHVGDk2Zwx2WVNj2t9U9a645a6/YXS9SiEv3vhy7XGwl7/zKzo3EMJ/bBF2ONClnjvpBEbQ/cmxkL4uQchcUPSEuUcCjSAWcLlDKWBmwZBV9PNkaKi/qTdxw+G20MVJrwdZ1qfosGFhwLQx41ui+2f2V2NEI0GEnWtsKtKYz/Ety94bMbjBGwwvqKTsHnY+BUunG9/dqaHZGoqQEPQGhfWPwgZCaZHY0QDUKStS3xCTaKP1hK4dPrIKfidCJRB6UlsOB2SN1mrEtdjZrfwgY5OZeVIHWCb6YY/65CNHKSrG1NQEe4ZQHkZxhP2KczzY6ocdAaljxsLL84/FXoNMzsiERd+IbCiNfg8AZY/arZ0QhR7yRZ26KgWGMO9sn9xmIGRRdeJEJUw+pXIf4jYzBfzzvMjkZYQ8yNEDPGKN97bOfF9xfCjkmytlXtBhsLfaRsgK8mGdNVRO2sLKvF3vVmuPwJs6MR1nT1S+DhCz/8w+g+EqKRkmRty7pcByNeh/2/wnfT5WZUU1rD8heMlZu6jjXqTDvJr3yj4ulnJOwjm2DDbLOjEaLeyJ3L1sVNNp4Gd3xtLGZQfok9UTWt4fdnYOWLcMkEuO4dY2CSaHyiR0OHofDbM5AlC2OIxkmStT0Y+ABc9QLs/sGYdlSYZ3ZEtk1r+PVxozRlj1th5FuSqBszpYzBZgCLpFiKaJwkWduLvncbzbh/rTZKZOZnmB2RbdIafn4M1rwJcXfAiDek6dsR+IbC5Y8bXUY7FpgdjRBWJ3cxe3LJeLj5U0jbAR9fbZRdFH+zlMJPD8O6t6H3NLhmpiRqR9JrCgTFwdJ/wamTZkcjhFXJnczeRF5jzMPOPgwfXSWVzs44nWV0EWx8H/r9A4a9aDSPCsfh5Awj34SCbPjlMbOjEcKqJFnbo7aD4NYfjb7rj4YZ07sc2Yn98MEVcHCF0ex95bOSqB1Vqy4wYAZs+wL2/2Z2NEJYjSRrexXUA25fCm6eRpP4ulmOObBm/2/w/mVwOgMm/WCMnheObeBD0CICFt0PxafNjkYIq5Bkbc8COsHUlRBxJSz9P1gwGQpzzY6qYWgNa9+BeTeCbwhMWQ7h/c2OStgCVw+jPkHWIVjzP7OjEcIqJFnbuya+MPZzuOIp2PU9zB4Cx3ebHVX9KjoF398LPz8KnYbD7T9D8zCzoxK2pO1AiBoFq18zxncIYefqlKyVUjcppXYqpSxKqThrBSVqSCkYcL/Rj12QbTQLb/vS7KjqR9IfMKs/bP0MBj0CYz4F92ZmRyVs0dBnAA2/SolZYf/q+mSdANwArLJCLKKuwgfAtNUQ2B2+nWosB5mbZnZU1lGYZ0zL+uQaQMOti+Cyx2Rqlqha8zDo909IWADJa82ORog6qdOdTmu9W2u9x1rBCCvwam0MtLr0/2D3j/BWT1j3rn2v+XtwJczqBxveh97TYfoao5lTiIsZcD94B8GSR6S2vrBrDfZYopSaqpSKV0rFp6enN9RpHZOzCwx5FO5eB8FlRSLeHwwpG82OrGZOZxrlI+eOBCcXmLwErn4R3JqaHZmwF25NYejTkLYdtnxmdjRC1JrSF5nuo5RaBrSu5K3HtNbfl+2zAnhIax1fnZPGxcXp+Phq7SrqSmtj4NnSRyH3qFEr+/InoGkLsyOr2uksWPeOMR2tMBf63gNDHjOmqQlRU1ob0xtP7IN/bgYPH7MjqjWl1CattYwPckAuF9tBa31FQwQi6olSxlKbHS6HFS8aCXDH18ZKVH3vBr92Zkf4t4JsI76170BhNnS+1mjObx1tdmTCnillVLSbPRhWvgxXPWd2RELU2EWTtWgk3L2Mm1T3icYiF5s+gY0fQOcR0PcfENrbvNjyM4xY1r5lJOzIEXDpv6BNV/NiEo1L4CXQYyKsfxdibwP/CLMjEqJGLtoMfsFvVup64H9AAJAFbNVaX3Wx75NmcBuQkwobZkP8R1CQBcG9jIUQIoZCk+b1f/7CXEj8CRK+gQO/gaXEmDM9+P+gTbf6P79wPHnp8L8eENLbqK9vh6QZ3HHVKVnXliRrG1KYB1vnwdq3ISsZlLNxM+t4pVEZrWWU9epsF+UbSxgmfAN7f4aSAvAOhujroetYae4W9W/NW8YiH7d8Ax3sr4dPkrXjkmQtDJZSOLLJSKL7fjaW4QQjmbYfDP4doXm48fINMyqnVUVryDsGaQnGKNxjCcbxTu4HbYGmAdDleogebTzRy1xp0VBKiuCtWPDwNUr12tnvniRrxyXJWlQu5yjs+xX2/QLJa4yFMsrz8DVqcoNxAywte5UUGk/MRXl/7+sTCq1jjCfn0L4QPtCYXiaEGbZ9aRQNuvEj4wOjHZFk7bgkWYvqKciGzGTITPr7lX0YlBO4uIGze9mfZX9vHm4k51ZdGqYPXIjqsljgvYFQnA/3bABnV7MjqjZJ1o5LHm9E9Xj4GKOzZYS2sHdOTnD54/D5GNjyKcTdbnZEQlyUfXXYCCGENURcaXTJrHjJGPgohI2TZC2EcDxKGZX88tJgw3tmRyPERUmyFkI4prC+0HEY/PG6UYdeCBsmyVoI4bgu+w8U5MCf/zU7EiEuSJK1EMJxtY6GrmOMZWRzUs2ORogqSbIWQji2wY8a5W5XvWx2JEJUSZK1EMKx+bWFuMmwaQ6cPGB2NEJUSpK1EEIMehhc3I0lNIWwQZKshRCiWUujOMqOr+XpWtgkSdZCCAHQ7x9G6dE/XjM7EiEqkGQthBAAXq2hx62wbb5RB18IG1KnZK2UekUplaiU2q6U+lYp5WuluIQQouH1v89YnObPN8yORIhz1PXJ+lcgWmvdFdgLPFr3kIQQwiQ+QdD9FtjyGWQfMTsaIc6qU7LWWv+itS4p+3IdEFz3kIQQwkQDZoC2SFUzYVOs2Wd9O7CkqjeVUlOVUvFKqfj09HQrnlYIIazINxS6jYXNcyA3zexohACqkayVUsuUUgmVvEaV2+cxoASYV9VxtNaztdZxWuu4gIAA60QvhBD1YeCDUFoMa/5ndiRCAOBysR201ldc6H2l1G3ACOByrbW2UlxCCGEev3YQcxPEf2Q0izf1Nzsi4eDqOhp8GPAIMFJrLSu4CyEaj0EPQfFpWPuW2ZEIUec+67cAL+BXpdRWpdS7VohJCCHM5x8B0TfAhvchP8PsaISDq+to8A5a6xCt9SVlr2nWCkwIIUw38CEoyoN1s8yORDg4qWAmhBBVaRUFkSNg4/tQdMrsaIQDk2QthBAX0u+fcDoTtlQ52UWIeifJWgghLiS0NwT3MgaaWUrNjkY4KEnWQghxMf3/CVnJsPsHsyMRDkqStRBCXEyn4cbc6z/fBCknIUwgyVoIIS7GyRn63gNHN0PyGrOjEQ7oohXMhBBCAN3Gw/LnjRKk4f3NjoZNmza1dHFx+QCIRh68GgMLkFBSUnJnbGzs8fPflGQthBDV4eYJPafAyhchfQ8EdDI1HBcXlw9at27dOSAgINPJyUna5u2cxWJR6enpUWlpaR8AI89/Xz6NCSFEdfWaAi4etlKCNDogICBHEnXj4OTkpAMCArIxWkoqvt/A8QghhP1q6g+XjIdt8yH3mNnROEmiblzK/j0rzcuSrIUQoib63mssn7lhttmRCAciyVoIIWqiRXuIvAY2fuDwJUjnzp3rGxkZGVX+5eTkFPvSSy8FREREdLH2+YKCgmJSU1MdcqyVJGshhKipfv+EgizY8pnZkZhq0qRJWYmJibvOvO68887jsbGxeddee2222bE1Ng75CUUIIerkbAnSt6HnncY8bBOF/9/i2Po6dtKL12yqzn7bt293f+WVVwL/+OOPRIvFcnZ7SUkJ99xzT/Cff/7pVVRUpKZMmXL84YcfPrFo0SKvJ598MrBZs2alSUlJHv369cv59NNPDzk7O7Nw4ULvp59+OrCoqEiFhYUVzp8/P8nHx+fsQfPy8tTw4cM7jBo1KvPBBx88UQ8/ts2RJ2shhKiNPtONEqT7fjU7EtMVFhaq8ePHt3v22WdTIiIiisq/98Ybb/j7+PiUJiQk7N62bdvuOXPmBCQmJroB7Nixo+k777xzaP/+/QlJSUnuc+fObZ6amury/PPPt1m1atXeXbt27e7Ro0f+M8880+rM8XJycpyuvPLKiDFjxmQ4SqIGebIWQoja6XwteLWBDe9Bp2FmR2OqGTNmBHbs2PH0lClTMs9/b9myZd6JiYmeP/zwQ3OA3Nxc5127dnm4ubnpmJiYU1FRUUUAY8aMyVi9enUzDw8Py4EDBzx69eoVCVBcXKxiY2Pzzhxv5MiRHe6///606dOnZzTUz2cL6pSslVLPAKMwKq8cB27TWh+1RmBCCGHTnF0h7nZY/hyc2Af+EaaFUt2m6vqwaNEir8WLFzffunXrrsre11qrmTNnHho9enTO+d+nlDpnX6UUWmsGDBiQ8+OPP/5V2fF69uyZ9/PPP/vcddddGU5OjtM4XNef9BWtdVet9SXAIuDxuockhBB2oset4OQKG943OxJTpKenO991113hH3744V/Nmze3VLbP0KFDs2fNmhVQWFiowOjbzsnJcQKjGTwxMdGttLSUBQsW+A0cODB38ODBp+Lj45slJCS4g9HsvX37dvczx3vllVeO+vr6lkyaNCm0IX5GW1GnZK21Lv9JqSkgE/SFEI7DqxV0uR62fg6FuWZH0+Bee+21gIyMDJd77703rPz0rblz5/qd2WfGjBknIiMjC2JiYjpHRER0mTJlSlhxcbECiI6OPjVt2rTQ9u3bR4eGhhZOnDgxKzAwsOS9995LGjt2bLuOHTtGxcXFRe7YscOj/Hk/+uijlIKCAqdp06YFN/TPbBal67jcm1LqOWASkA0M0VqnV7HfVGAqQGhoaGxycnKdziuEEDYhZSN8eAUMf9UoR1qPlFKbtNZxANu2bUvq1q2b3Q6wWrRokdfMmTNbLV++fL/ZsdiSbdu2+Xfr1i38/O0XfbJWSi1TSiVU8hoFoLV+TGsdAswD7q3qOFrr2VrrOK11XEBAQB1+FCGEsCHBcRDY3ahoJmtdi3py0WSttb5Cax1dyev783adB4yunzCFEMJGKQW9psKJvXBwhdnR2I0RI0bkylN19dWpz1opVX744yggsW7hCCGEHepyA3i2cNiBZqL+1XU0+ItlTeLbgSuB+6wQkxBC2BdXD2Nk+N4lkCnjcYT11XU0+OiyJvGuWutrtdZHrBWYEELYlZ53GH/Gf2huHKJRcpwZ5UIIUZ98go3VuDbPheLTZkcjGhlJ1kIIYS297oLTmbBjgdmRNAhnZ+fYyMjIqIiIiC5XX311u9zc3GrnlKSkJNdhw4a1q+25R48eHf7xxx83r+33n/Hmm2+2sIcCK5KshRDCWsIHQMsoo164A0zjcnd3tyQmJu7at2/fTldXVz1z5sxz5uUWFxdX+b3h4eHFS5cuPVjvQTYSspCHEEJYi1LGkpmLH4DD8RDSs2HO+909IRzf5WnVY7aMyue6t1Oqu/uAAQPytm/f3mTRokVeTzzxRKCPj0/pwYMHPYYPH54ZEhJS9Oijj6YDPPDAA4HNmjUrnTBhQuaIESMi9u3bt7P8cSwWC9OnTw/+/ffffZRS+uGHH06dMmVKpsVi4bbbbgtdtWqVd2BgYJGrq6sFYMGCBd4ffvih/5IlSw7CucVW3nvvPb+ZM2e21lqrK664ImvWrFlHAP773/+2eP3119t4eXmVdunSJd/Nzc3mP1nJk7UQQlhTzE3g2hQ2f2J2JA2muLiYn3/+2TsmJuY0wK5duzzfeeedQ0lJSQkTJkzIWLhw4dnyo99//33zSZMmVbli1ty5c3137NjRZPfu3Tt/++23vY8//nhwcnKy66effuq7f/9+9/379yd8/vnnf23evLkZwKhRo3K2bt3a9Ey98S+++KL5TTfdlJGUlOT65JNPBq1YsWLvrl27dm7ZsqXpp59+6pucnOz64osvBq5ZsyZx48aNiXv37m1S39fHGuTJWgghrMnDG2JGG/3WVz0PHj71f84aPAFbU2FhoVNkZGQUQO/evXPvu+++E8uWLWvWtWvXU5GRkUUA/fv3P33y5EmXpKQk19TUVBcfH5/SDh06FO/Zs8etsmOuXr3aa8yYMRkuLi6EhISU9O7dO++PP/7wXLly5dnt4eHhxX379s0FcHV1ZfDgwTnz58/3mTx5cubvv//u89Zbbx1evHixd58+fXIDAwNLAG6++eaMlStXNgMov/2GG27I2Lt3r0dlsdgSSdZCCGFtPW4zRoXv+NpoFm+kzvRZn7/d09PznBW4Ro4cmfnZZ581T0tLc73hhhusvg71uHHjMt56662W/v7+pTExMflVrQBmz6QZXAghrC2oB7SKgU2fOMRAs4u55ZZbMr755hu/RYsWNZ84cWLmhfYdNGhQ7oIFC/xKSko4evSoy4YNG5oNHDjw1KWXXnp2e3Jysuu6deu8znzP8OHDc3fu3On5/vvv+48ZMyYDYODAgafWr1/vlZqa6lJSUsLXX3/tN3jw4LxBgwadWr9+vVdaWppzYWGh+vbbb+s8orwhyJO1EEJYm1IQeyv89BAc3WIkbwcWFxdXcOrUKadWrVoVhYWFVT1EHJg4cWLWmjVrmnXu3LmLUko/9dRTh0NDQ0smTpyY9dtvv3l36NAhOjAwsLB79+55Z77HxcWFyy+/PHvBggUtvvrqqySAsLCw4ieeeOLIpZde2vHMALNbbrklC+Bf//rX0T59+nT28vIqjY6Ozq/Pn91a6rxEZm3ExcXp+Pj4Bj+vEEI0mIJseLUTdLsZrv2vVQ7ZmJbIFJWr9RKZQgghasHDB6JvMAaaFeaaHY2wc5KshRCivsTeBkV5kPCN2ZEIOyfJWggh6ktwT6Oi2aY5Zkci7JwkayGEqC9KGUtnHt0MqdvNjkbYMUnWQghRn7qOARcP2CxP16L2JFkLIUR98vSDqOtg+1dQdMrsaISdskqyVko9qJTSSil/axxPCCEaldjboDAHdn5rXgzvvutHYGAMTk6xBAbG8O67fhf/pqrNnTvXNzIyMqr8y8nJKfarr77ytlbIZrHW8ptn7Nmzx+3dctd71apVnrfddltITY5R52StlAoBrgQO1fVYQgjRKIX2Af9O5g00e/ddP2bMCCM11Q2tITXVjRkzwuqSsCdNmpSVmJi468zrzjvvPB4bG5s3evToHGuGbi0XWq6zvu3bt8/9yy+/PHutBw0alP/JJ5/UqJ67NZ6sXwceAaSmnhBCVOZMRbPDG+BYhVLa9e/pp4MoKDj3fl9Q4MTTTwdZ4/Dbt293f+WVVwI///zzv5ydnVm0aJFXr169Og0bNqxd27Ztu4wcObKtxWKU637ooYfaREdHd46IiOgybty4sDPbe/Xq1Wn69OlBMTExncPDw6OXLl3aDKCkpIS77rorODo6unPHjh2jXnnllQotuHfffXfQCy+8cHYt7QceeCDw8ccfb7Vo0SKv2NjYTpdddlmHiIiI6KqOZbFYmDRpUmh4eHh0v379Op44ceJsdc+goKCYGTNmBEZFRXXu2LFj1JYtWzwAjh075nzFFVe079ixY1S3bt0i169f3wRg8eLFzc60NHTu3DkqMzPT6bHHHguKj49vFhkZGfXUU0+1XLRokdeQIUM61OQa1ylZK6VGAUe01tuqse9UpVS8Uio+PT29LqcVQgj7020ctL8cSgsb/txpaZWucFXl9hooLCxU48ePb/fss8+mREREFJ3Zvnv37iZvv/12yv79+3ceOnTI/ddff20G8PDDDx9PSEjYvW/fvp2nT592mj9//tllyUpKStSOHTt2v/TSSylPP/10IMAbb7zh7+PjU5qQkLB727Ztu+fMmROQmJh4TtwXWoaz/HKdVR2rquU3z/D39y/ZtWvX7ttvvz39xRdfbAXwyCOPBHbr1i1/7969u5555pkjt956a1uAmTNntn7zzTeTExMTd61bty6xWbNmlueee+5IXFxcXmJi4q4nnnjieG2u80WTtVJqmVIqoZLXKODfwOPVOZHWerbWOk5rHRcQEHDxbxBCiMbE0w8mLoTA7g1/7tati2q0vQZmzJgR2LFjx9NTpkw5Z4GOmJiYU+3bty92dnamS5cu+QcOHHADWLJkiVfXrl0jO3bsGLVmzRqvhISEs+tJ33TTTZkA/fr1O3X48GE3gGXLlnl/9dVXLSIjI6O6d+/eOTMz02XXrl3nLGlZfhnOtWvXNjmzDCdA+eU6qzpWVctvnjF+/PhMgF69euWnpKS4A2zYsMHrjjvuOAkwcuTI3KysLJeMjAynPn365D300EMhzz77bMsTJ044u7q61vUSA9VYyENrfUVl25VSMUBbYJtSCiAY2KyU6qW1TrNKdEIIIeru8cePMGNG2DlN4R4eFh5//EhdDrto0SKvxYsXN9+6dWuFtn13d/ezXaPOzs6UlJSo/Px89eCDD4atX79+V4cOHYofeOCBwIJyMXl4eGgwFuYoLS1VAFprNXPmzEMX6wuvahnO8st1VnWsRYsWXXDR8XJx6ZKSEnWhfZ9//vm06667Lvv777/3GThwYOTixYv3XWj/6qp1M7jWeofWuqXWOlxrHQ4cBnpIohZCCBszbVoGr7+eTJs2RSgFbdoU8frryUybVuu1pdPT053vuuuu8A8//PCv6q4fnZ+f7wTQunXrkuzsbKcff/zxoiOuhw4dmj1r1qyAwsJCBUb/eE5OToXcVZ1lOKs61oWW36xK7969cz/++OMWYHxoad68eYmfn59l586d7r169Tr93HPPpXXt2vVUQkKCh4+PT2leXp7zxY55IbJEphBCOIJp0zLqkpzP99prrwVkZGS43HvvvWHltz/44IOpbdq0Kanse/z9/UsnTJiQ3rlz5y4BAQEl3bp1u+jE8xkzZpxISkpyj4mJ6ay1Vn5+fsU//fTTgfP3q84ynFUd60LLb1blpZdeOjphwoTwjh07RjVp0sTyySef/AXw8ssvt1yzZo23Ukp36tTp9I033pjt5OSEs7Oz7tSpU9T48eNPxMbGnr7Y8c8nS2QKIYSdkCUyGz9ZIlMIIYSwU5KshRBCCBsnyVoIIeyTxWKxXHBksrAvZf+elQ7Wk2QthBD2KSE9Pd1HEnbjYLFYVHp6ug+QUNn7pgwwU0qlA8m1/HZ/wBYHVUhcNSNx1YzEVTO2GhfULbYwrXUAwKZNm1q6uLh8AEQjD16NgQVIKCkpuTM2NrZClTNTknVdKKXiz4yGtCUSV81IXDUjcdWMrcYFth2bsF3yaUwIIYSwcZKshRBCCBtnj8l6ttkBVEHiqhmJq2Ykrpqx1bjAtmMTNsru+qyFEEIIR2OPT9ZCCCGEQ5FkLYQQQtg4m0zWSqmPlFLHlVKVTw43vKmU2q+U2q6U6mEjcQ1WSmUrpbaWvR5voLhClFLLlVK7lFI7lVL3VbJPg1+zasbV4NdMKeWhlNqglNpWFtdTlezjrpT6sux6rVdKhdtIXLcppdLLXa876zuucud2VkptUUotquS9Br9e1YzLlOullEpSSu0oO2eFVYvMuocJ+2WrS2R+ArwFzK3i/auBiLJXb2BW2Z9mxwWwWms9ogFiKa8EeFBrvVkp5QVsUkr9qrUuvyC8GdesOnFBw1+zQuAyrXWeUsoV+EMptURrva7cPncAmVrrDkqpscBLwM02EBfAl1rre+s5lsrcB+wGvCt5z4zrVZ24wLzrNURrXVXxE7PuYcJO2eSTtdZ6FXChdVdHAXO1YR3gq5RqYwNxmUJrnaq13lz291yMG1fQebs1+DWrZlwNruwanFmv1rXsdf5Iy1HAnLK/LwAuV0rVa1nHasZlCqVUMHAN8EEVuzT49apmXLbKlHuYsF82mayrIQhIKff1YWwgCZTpW9aMuUQp1aWhT17W/NgdWH/eW6ZeswvEBSZcs7Km063AceBXrXWV10trXQJkAy1sIC6A0WVNpwuUUiH1HVOZN4BHqGKRAUy6XtWIC8y5Xhr4RSm1SSk1tZL3bfkeJmyQvSZrW7UZo3ZvN+B/wHcNeXKlVDPgG+B+rXVOQ577Qi4SlynXTGtdqrW+BAgGeimlohvivBdTjbh+BMK11l2BX/n7abbeKKVGAMe11pvq+1w1Uc24Gvx6lRmgte6B0dx9j1JqUAOdVzRS9pqsjwDlPyEHl20zldY650wzptb6J8BVKeXfEOcu6+P8BpintV5YyS6mXLOLxWXmNSs7ZxawHBh23ltnr5dSygXwAU6aHZfW+qTWurDsyw+A2AYIpz8wUimVBMwHLlNKfXbePmZcr4vGZdL1Qmt9pOzP48C3QK/zdrHJe5iwXfaarH8AJpWNqOwDZGutU80OSinV+kw/nVKqF8b1rfcbfNk5PwR2a61fq2K3Br9m1YnLjGumlApQSvmW/b0JMBRIPG+3H4Bby/5+I/C7rucKQtWJ67x+zZEY4wDqldb6Ua11sNY6HBiLcS1uOW+3Br9e1YnLjOullGpaNqASpVRT4EoqLntok/cwYbtscjS4UuoLYDDgr5Q6DDyBMdgGrfW7wE/AcGA/kA9MtpG4bgSmK6VKgNPA2Pq+YZXpD0wEdpT1dwL8GwgtF5sZ16w6cZlxzdoAc5RSzhgfDr7SWi9SSj0NxGutf8D4kPGpUmo/xqDCsfUcU3Xj+qdSaiTGSPsM4LYGiKtSNnC9qhOXGderFfBt2WdQF+BzrfVSpdQ0MPceJuyXlBsVQgghbJy9NoMLIYQQDkOStRBCCGHjJFkLIYQQNk6StRBCCGHjJFkLIYQQNk6StRBCCGHjJFkLIYQQNu7/A7JKnPhRAnUdAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "spl1 = spl.derivative(1)\n",
    "#spl2 = spl.derivative(2)\n",
    "#spl3 = spl.derivative(3)\n",
    "plt.plot(xint, spl(xint), lw=3, label='Zlepek')\n",
    "plt.plot(xint, spl1(xint), label='Prvi odvod')\n",
    "#plt.plot(xint, spl2(xint), label='Drugi odvod')\n",
    "#plt.plot(xint, spl3(xint), label='Tretji odvod')\n",
    "plt.plot(x, y, 'ro', label='Znane vrednosti')\n",
    "#plt.plot(spl.get_knots(), spl(spl.get_knots()), 'k.', label='Vozli B-zlepka')\n",
    "plt.legend(loc=(1.01, 0));"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "pinm2021",
   "language": "python",
   "name": "pinm2021"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  },
  "latex_envs": {
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 0
  },
  "livereveal": {
   "theme": "sky",
   "transition": "default",
   "width": 1200
  },
  "nav_menu": {},
  "toc": {
   "base_numbering": 1,
   "nav_menu": {
    "height": "299px",
    "width": "252px"
   },
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Kazalo",
   "title_sidebar": "Kazalo",
   "toc_cell": true,
   "toc_position": {},
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}