Predavanje 10 - Numerično odvajanje.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "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>Numerično odvajanje</b></font>\n",
    "<img src=\"\">\n",
    "<font size=\"2\">September 2021</font>\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "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=\"#Aproksimacija-prvega-odvoda-po-metodi-končnih-razlik\" data-toc-modified-id=\"Aproksimacija-prvega-odvoda-po-metodi-končnih-razlik-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Aproksimacija prvega odvoda po metodi končnih razlik</a></span></li><li><span><a href=\"#Centralna-diferenčna-shema\" data-toc-modified-id=\"Centralna-diferenčna-shema-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Centralna diferenčna shema</a></span><ul class=\"toc-item\"><li><span><a href=\"#Odvod-$f'(x)$\" data-toc-modified-id=\"Odvod-$f'(x)$-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Odvod $f'(x)$</a></span></li><li><span><a href=\"#Zgled:-$\\exp(-x)$\" data-toc-modified-id=\"Zgled:-$\\exp(-x)$-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Zgled: $\\exp(-x)$</a></span></li><li><span><a href=\"#Odvod-$f''(x)$\" data-toc-modified-id=\"Odvod-$f''(x)$-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>Odvod $f''(x)$</a></span></li><li><span><a href=\"#Odvod-$f'''(x)$\" data-toc-modified-id=\"Odvod-$f'''(x)$-3.4\"><span class=\"toc-item-num\">3.4&nbsp;&nbsp;</span>Odvod $f'''(x)$</a></span></li><li><span><a href=\"#Odvod-$f^{(4)}(x)$\" data-toc-modified-id=\"Odvod-$f^{(4)}(x)$-3.5\"><span class=\"toc-item-num\">3.5&nbsp;&nbsp;</span>Odvod $f^{(4)}(x)$</a></span></li><li><span><a href=\"#Povzetek-centralne-diferenčne-sheme*\" data-toc-modified-id=\"Povzetek-centralne-diferenčne-sheme*-3.6\"><span class=\"toc-item-num\">3.6&nbsp;&nbsp;</span>Povzetek centralne diferenčne sheme*</a></span></li><li><span><a href=\"#Uporaba-scipy.misc.central_diff_weights\" data-toc-modified-id=\"Uporaba-scipy.misc.central_diff_weights-3.7\"><span class=\"toc-item-num\">3.7&nbsp;&nbsp;</span>Uporaba <code>scipy.misc.central_diff_weights</code></a></span></li><li><span><a href=\"#Izboljšan-približek---Richardsonova-ekstrapolacija\" data-toc-modified-id=\"Izboljšan-približek---Richardsonova-ekstrapolacija-3.8\"><span class=\"toc-item-num\">3.8&nbsp;&nbsp;</span>Izboljšan približek - Richardsonova ekstrapolacija</a></span><ul class=\"toc-item\"><li><span><a href=\"#Zgled\" data-toc-modified-id=\"Zgled-3.8.1\"><span class=\"toc-item-num\">3.8.1&nbsp;&nbsp;</span>Zgled</a></span></li></ul></li></ul></li><li><span><a href=\"#Necentralna-diferenčna-shema\" data-toc-modified-id=\"Necentralna-diferenčna-shema-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Necentralna diferenčna shema</a></span><ul class=\"toc-item\"><li><span><a href=\"#Diferenčna-shema-naprej\" data-toc-modified-id=\"Diferenčna-shema-naprej-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Diferenčna shema naprej</a></span></li><li><span><a href=\"#Diferenčna-shema-nazaj\" data-toc-modified-id=\"Diferenčna-shema-nazaj-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>Diferenčna shema nazaj</a></span></li></ul></li><li><span><a href=\"#Uporaba-numpy.gradient\" data-toc-modified-id=\"Uporaba-numpy.gradient-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Uporaba <code>numpy.gradient</code></a></span><ul class=\"toc-item\"><li><span><a href=\"#Zgled\" data-toc-modified-id=\"Zgled-5.1\"><span class=\"toc-item-num\">5.1&nbsp;&nbsp;</span>Zgled</a></span></li></ul></li><li><span><a href=\"#Zaokrožitvena-napaka-pri-numeričnem-odvajanju*\" data-toc-modified-id=\"Zaokrožitvena-napaka-pri-numeričnem-odvajanju*-6\"><span class=\"toc-item-num\">6&nbsp;&nbsp;</span>Zaokrožitvena napaka pri numeričnem odvajanju*</a></span><ul class=\"toc-item\"><li><span><a href=\"#Zgled\" data-toc-modified-id=\"Zgled-6.1\"><span class=\"toc-item-num\">6.1&nbsp;&nbsp;</span>Zgled</a></span></li></ul></li><li><span><a href=\"#Aproksimacija-odvoda-s-kompleksnim-korakom*\" data-toc-modified-id=\"Aproksimacija-odvoda-s-kompleksnim-korakom*-7\"><span class=\"toc-item-num\">7&nbsp;&nbsp;</span>Aproksimacija odvoda s kompleksnim korakom*</a></span></li><li><span><a href=\"#Nekaj-vprašanj-za-razmislek!\" data-toc-modified-id=\"Nekaj-vprašanj-za-razmislek!-8\"><span class=\"toc-item-num\">8&nbsp;&nbsp;</span>Nekaj vprašanj za razmislek!</a></span></li><li><span><a href=\"#Dodatno\" data-toc-modified-id=\"Dodatno-9\"><span class=\"toc-item-num\">9&nbsp;&nbsp;</span>Dodatno</a></span></li></ul></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Uvod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vsako elementarno funkcijo lahko analitično odvajamo. Definicija odvoda je:\n",
    "$$f'(x)=\\lim_{\\Delta x \\rightarrow 0}\\frac{f(x+\\Delta x)-f(x)}{\\Delta x}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Neposredna uporaba zgornje enačbe vodi v odštevanje zelo podobnih funkcijskih vrednostih ($f(x+\\Delta x)$, $f(x)$), obremenjenih z zaokrožitveno napako, ki jih delimo z majhno vrednostjo $\\Delta x$; posledično ima odvod bistveno manj signifikantnih števk kakor pa funkcijske vrednosti. Numeričnemu odvajanju se izognemo, če imamo to možnost; je pa v nekaterih primerih (npr. reševanje diferencialnih enačb) nepogrešljivo orodje!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pri numeričnem odvajanju imamo dva, v principu različna, pristopa:\n",
    "\n",
    "1. najprej izvedemo **interpolacijo/aproksimacijo**, nato pa na podlagi znanih interpolacijskih/aproksimacijskih funkcij izračunamo odvod (o tej temi smo že govorili pri interpolaciji oz. aproksimaciji) in \n",
    "2. računanje odvoda **neposredno iz vrednosti iz tabele**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V okviru tega poglavja se bomo seznanili s tem, kako numerično izračunamo odvod funkcije $f(x)$; pri tem so vrednosti funkcije $f(x)$ podane tabelarično (pari $x_i$, $y_i$), kakor je prikazano na sliki:\n",
    "<img src='./fig/odvajanje_tabela.png' width=400>\n",
    "Najprej se bomo osredotočili na ekvidistantno, s korakom $h$, razporejene vrednosti $x_i$; vrednosti funkcije pa bodo $y_i=f(x_i)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Glede na zgornjo definicijo odvoda, bi prvi odvod (za mesto $i$) lahko zapisali:\n",
    "$$y_i'=\\frac{y_{i+1}-y_{i}}{h},$$\n",
    "kjer je $h=x_{i+1}-x_{i}$. S preoblikovanjem enačbe:\n",
    "$$y_i'=-\\frac{y_{i}}{h}+\\frac{y_{i+1}}{h},$$\n",
    "lahko tudi rečemo, da za prvi odvod funkcije na mestu $i$, **utežimo** funkcijsko vrednost pri $i$ z $-1/h$ in funkcijsko vrednost pri $i+1$ z $+1/h$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V nadaljevanju si bomo pogledali teoretično ozadje kako določimo ustrezne **uteži** za različne stopnje odvodov, katere možnosti pri tem imamo in kako to vpliva na red natančnosti."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Aproksimacija prvega odvoda po metodi končnih razlik"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Za uvod si oglejmo spodnji video:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/jpeg": "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\n",
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"800\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/YYuGL-VP2BE\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x272aa2fb2e0>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('YYuGL-VP2BE', width=800, height=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Odvod $f'(x)$ lahko aproksimiramo na podlagi razvoja Taylorjeve vrste. To metodo imenujemo **metoda končnih razlik** ali tudi **diferenčna metoda**.\n",
    "\n",
    "Razvijmo **Taylorjevo vrsto naprej** (naprej, zaradi člena $+h$):\n",
    "$$f{\\left (x + h \\right )} =\\sum_{n=0}^{\\infty}\\frac{h^n}{n!}\\frac{d^n}{dx^n}f(x)= f{\\left (x \\right )} + h\\, f'\\left (x \\right ) + \\underbrace{\\frac{h^2}{2}\\,f''(x)+\\cdots}_{\\mathcal{O}\\left(h^{2}\\right)}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Člen $\\mathcal{O}\\left(h^{2}\\right)$ označuje napako drugega reda. Če iz enačbe izrazimo prvi odvod:\n",
    "\n",
    "$$f'{\\left (x \\right )}=\\frac{1}{h}\\left(f{\\left (x + h \\right )} - f{\\left (x \\right )}\\right) - \\underbrace{\\frac{h}{2}\\,f''(x)+\\cdots}_{\\mathcal{O}\\left(h^{1}\\right)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ugotovimo, da lahko ocenimo prvi odvod v točki $x_i$ (to je: $f_o'(x_i)$) na podlagi dveh zaporednih funkcijskih vrednosti:\n",
    "$$f_o'(x_i)=\\frac{1}{h}\\left(y_{i+1}-y_i\\right)$$\n",
    "in pri tem naredimo **napako metode**, ki je prvega reda $\\mathcal{O}\\left(h^{1}\\right)$. \n",
    "\n",
    "Uporabili smo $y_i=f(x_i)$ (glejte sliko zgoraj)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Napaka je:\n",
    "\n",
    "$$e=-\\frac{h}{2}\\,f''(\\xi),$$\n",
    "\n",
    "kjer je $\\xi$ neznana vrednost na intervalu $[x_i, x_{i+1}]$ in smo zanemarili višje člene."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Velja torej izraz:\n",
    "$$f'(x_i)=f_o'(x_i)+e$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj si poglejmo, kako pridemo do istega rezultata s strojno izpeljavo; najprej uvozimo ``sympy``:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo simbole:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sym.Function('f')\n",
    "x, h = sym.symbols('x, h')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato nadaljujemo z razvojem **Taylorjeve vrste naprej** (angl. *forward Taylor series*):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} + h \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=x }} + O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       "         ⎛ d        ⎞│        ⎛ 2⎞\n",
       "f(x) + h⋅⎜───(f(ξ₁))⎟│     + O⎝h ⎠\n",
       "         ⎝dξ₁       ⎠│ξ₁=x        "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x+h).series(h, n=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Člen $\\mathcal{O}\\left(h^{2}\\right)$ vsebuje člene drugega in višjega reda. V zgornji enačbi je uporabljena začasna spremenljivko za odvajanje $\\xi_1$; izvedmo odvajanje in vstavimo $\\xi_1=x$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} + h \\frac{d}{d x} f{\\left(x \\right)} + O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       "         d           ⎛ 2⎞\n",
       "f(x) + h⋅──(f(x)) + O⎝h ⎠\n",
       "         dx              "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x+h).series(h, n=2).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Zapišemo enačbo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(h + x \\right)} = f{\\left(x \\right)} + h \\frac{d}{d x} f{\\left(x \\right)} + O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       "                    d           ⎛ 2⎞\n",
       "f(h + x) = f(x) + h⋅──(f(x)) + O⎝h ⎠\n",
       "                    dx              "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enačba = sym.Eq(f(x+h), f(x+h).series(h, n=2).doit())\n",
    "enačba"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešimo jo za prvi odvod $f'(x)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{f{\\left(h + x \\right)} - f{\\left(x \\right)} + O\\left(h^{2}\\right)}{h}$"
      ],
      "text/plain": [
       "                   ⎛ 2⎞\n",
       "f(h + x) - f(x) + O⎝h ⎠\n",
       "───────────────────────\n",
       "           h           "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_naprej_točno = sym.solve(enačba, f(x).diff(x))[0]\n",
    "f1_naprej_točno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V kolikor odvoda drugega in višjih redov ne upoštevamo, smo naredili torej napako:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle O\\left(h\\right)$"
      ],
      "text/plain": [
       "O(h)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_naprej_O = f1_naprej_točno.expand().getO()\n",
    "f1_naprej_O"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Napaka $\\mathcal{O}\\left(h^{1}\\right)$ je torej prvega reda in če ta člen zanemarimo, naredimo *napako metode* in dobimo oceno odvoda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{f{\\left(x \\right)}}{h} + \\frac{f{\\left(h + x \\right)}}{h}$"
      ],
      "text/plain": [
       "  f(x)   f(h + x)\n",
       "- ──── + ────────\n",
       "   h        h    "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_naprej_ocena = f1_naprej_točno.expand().removeO()\n",
    "f1_naprej_ocena"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ugotovimo, da gre za isti izraz, kakor smo ga izpeljali zgoraj, torej je:\n",
    "\n",
    "$$y_i'=\\frac{1}{h}\\left(-y_i+y_{i+1}\\right).$$\n",
    "\n",
    "Uteži torej so:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   | $y_{i}$   |$y_{i+1}$|\n",
    "|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{h}\\cdot$| -1 | 1 |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Centralna diferenčna shema"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Odvod $f'(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej si poglejmo razvoj **Taylorjeve vrste nazaj** (angl. *backward Taylor series*):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} - h \\frac{d}{d x} f{\\left(x \\right)} + \\frac{h^{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)}}{2} + O\\left(h^{3}\\right)$"
      ],
      "text/plain": [
       "                         2              \n",
       "                     2  d               \n",
       "                    h ⋅───(f(x))        \n",
       "                         2              \n",
       "         d             dx           ⎛ 3⎞\n",
       "f(x) - h⋅──(f(x)) + ──────────── + O⎝h ⎠\n",
       "         dx              2              "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x-h).series(h, n=3).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ugotovimo, da se pri razliki vrste naprej in nazaj odštevajo členi sodega reda; definirajmo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 h \\frac{d}{d x} f{\\left(x \\right)} + O\\left(h^{3}\\right)$"
      ],
      "text/plain": [
       "    d           ⎛ 3⎞\n",
       "2⋅h⋅──(f(x)) + O⎝h ⎠\n",
       "    dx              "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def razlika(n=3):\n",
    "    return f(x+h).series(h, n=n).doit()-f(x-h).series(h, n=n).doit()\n",
    "razlika(n=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izvedemo sledeče korake:\n",
    "\n",
    "1. Taylorjevo vrsto nazaj odštejemo od vrste naprej, sodi odvodi se odštejejo,\n",
    "2. rešimo enačbo za prvi odvod,\n",
    "3. določimo napako metode,\n",
    "4. določimo oceno odvoda.\n",
    "\n",
    "Izvedimo zgornje korake:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "f1_cent_točno = sym.solve(\n",
    "           sym.Eq(f(x+h) - f(x-h), razlika(n=3)), # 1 korak\n",
    "           f(x).diff(x))[0]                       # 2.korak\n",
    "f1_cent_O = f1_cent_točno.expand().getO()         # 3.korak\n",
    "f1_cent_ocena = f1_cent_točno.expand().removeO()  # 4.korak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ocena 1. odvoda torej je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{f{\\left(- h + x \\right)}}{2 h} + \\frac{f{\\left(h + x \\right)}}{2 h}$"
      ],
      "text/plain": [
       "  f(-h + x)   f(h + x)\n",
       "- ───────── + ────────\n",
       "     2⋅h        2⋅h   "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_cent_ocena"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ali:\n",
    "\n",
    "$$y_i'=\\frac{1}{2h}\\left(-y_{i-1}+y_{i+1}\\right)$$\n",
    "\n",
    "Uteži torej so:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   |$y_{i-1}$ | $y_{i}$   |$y_{i+1}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{2h}\\cdot$| -1 | 0 | 1 |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Napaka metode pa je torej drugega reda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       " ⎛ 2⎞\n",
       "O⎝h ⎠"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_cent_O"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Zgled: $\\exp(-x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Poglejmo si zgled eksponentne funkcije $f(x)=\\exp(-x)$ in za točko $x=1,0$ izračunajmo prvi odvod $f'(x)=-\\exp(-x)$ pri koraku $h_0=1$ in $h_1=0,1$. \n",
    "\n",
    "Najprej pripravimo tabelo numeričnih vrednosti in točen rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "x0 = np.array([0., 1., 2.]) # korak h=1\n",
    "y0 = np.exp(-x0)\n",
    "h0 = x0[1]-x0[0]\n",
    "\n",
    "x1 = np.array([0.9, 1.0, 1.1]) # korak h=0.1\n",
    "y1 = np.exp(-x1)\n",
    "h1 = x1[1]-x1[0]\n",
    "\n",
    "f1_točno0 = - np.exp(-1) # točen rezultat\n",
    "f1_točno1 = - np.exp(-1) # točen rezultat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Potem uporabimo shemo naprej:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{f{\\left(x \\right)}}{h} + \\frac{f{\\left(h + x \\right)}}{h}$"
      ],
      "text/plain": [
       "  f(x)   f(h + x)\n",
       "- ──── + ────────\n",
       "   h        h    "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_naprej_ocena # da se spomnimo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1_naprej0 = (y0[1:]-y0[:-1])/h0   # korak h_0\n",
    "f1_naprej1 = (y1[1:]-y1[:-1])/h1   # korak h_1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunajmo napako pri $x=1,0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.135335283236613$"
      ],
      "text/plain": [
       "-0.1353352832366127"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_točno0 - f1_naprej0[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.0177958664378144$"
      ],
      "text/plain": [
       "-0.017795866437814445"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_točno1 - f1_naprej1[1] # korak h1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potrdimo lahko, da je napaka pri koraku $h/10$ res približno 1/10 tiste pri koraku $h$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo sedaj še napako za centralno diferenčno shemo, ki je drugega reda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{f{\\left(- h + x \\right)}}{2 h} + \\frac{f{\\left(h + x \\right)}}{2 h}$"
      ],
      "text/plain": [
       "  f(-h + x)   f(h + x)\n",
       "- ───────── + ────────\n",
       "     2⋅h        2⋅h   "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_cent_ocena"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1_cent0 = (y0[2:]-y0[:-2])/(2*h0) # korak h_0\n",
    "f1_cent1 = (y1[2:]-y1[:-2])/(2*h1) # korak h_1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Analizirajmo napako:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0644529172102513$"
      ],
      "text/plain": [
       "0.06445291721025131"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_točno0 - f1_cent0[0] # korak h0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.000613439041155517$"
      ],
      "text/plain": [
       "0.0006134390411555168"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_točno1 - f1_cent1[0] # korak h1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potrdimo lahko, da je napaka pri koraku $h/10$ res približno 1/100 tiste pri koraku $h$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Odvod $f''(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Če Taylorjevo vrsto naprej in nazaj seštejemo, se odštejejo lihi odvodi:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 f{\\left(x \\right)} + h^{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + O\\left(h^{4}\\right)$"
      ],
      "text/plain": [
       "              2              \n",
       "          2  d           ⎛ 4⎞\n",
       "2⋅f(x) + h ⋅───(f(x)) + O⎝h ⎠\n",
       "              2              \n",
       "            dx               "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vsota(n=3):\n",
    "    return f(x+h).series(h, n=n).doit() + f(x-h).series(h, n=n).doit()\n",
    "vsota(n=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Določimo drugi odvod:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "f2_cent_točno = sym.solve(\n",
    "           sym.Eq(f(x+h) + f(x-h), vsota(n=4)),   # 1 korak\n",
    "           f(x).diff(x,2))[0]                     # 2.korak\n",
    "f2_cent_O = f2_cent_točno.expand().getO()         # 3.korak\n",
    "f2_cent_ocena = f2_cent_točno.expand().removeO()  # 4.korak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ocena drugega odvoda je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{2 f{\\left(x \\right)}}{h^{2}} + \\frac{f{\\left(- h + x \\right)}}{h^{2}} + \\frac{f{\\left(h + x \\right)}}{h^{2}}$"
      ],
      "text/plain": [
       "  2⋅f(x)   f(-h + x)   f(h + x)\n",
       "- ────── + ───────── + ────────\n",
       "     2          2          2   \n",
       "    h          h          h    "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2_cent_ocena"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ali:\n",
    "\n",
    "$$y_i''=\\frac{1}{h^2}\\left(y_{i-1}-2\\,y_{i}+y_{i+1}\\right)$$\n",
    "\n",
    "Uteži torej so:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   |$y_{i-1}$ | $y_{i}$   |$y_{i+1}$ |\n",
    "|:--------:|:-------------------:|:----------:|:----------:|\n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$ | 1 | -2 | 1 |\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Napaka metode pa je ponovno drugega reda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       " ⎛ 2⎞\n",
       "O⎝h ⎠"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2_cent_O"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Odvod $f'''(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Če želimo določiti tretji odvod, moramo Taylorjevo vrsto razviti do stopnje 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - f{\\left(- h + x \\right)} + f{\\left(h + x \\right)} = 2 h \\frac{d}{d x} f{\\left(x \\right)} + \\frac{h^{3} \\frac{d^{3}}{d x^{3}} f{\\left(x \\right)}}{3} + O\\left(h^{5}\\right)$"
      ],
      "text/plain": [
       "                                            3              \n",
       "                                        3  d               \n",
       "                                       h ⋅───(f(x))        \n",
       "                                            3              \n",
       "                            d             dx           ⎛ 5⎞\n",
       "-f(-h + x) + f(h + x) = 2⋅h⋅──(f(x)) + ──────────── + O⎝h ⎠\n",
       "                            dx              3              "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_h = sym.Eq(f(x+h)-f(x-h), razlika(n=5))\n",
    "eq_h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uporaba 1. odvoda, ki smo ga izpeljali zgoraj, nam ne bi koristila, saj je red napake $\\mathcal{O}\\left(h^{2}\\right)$, kar pomeni, da bi v zgornji pri deljenju s $h^3$ dobili $\\mathcal{O}\\left(h^{-1}\\right)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uporabimo trik: ponovimo razvoj, vendar na podlagi dodatnih točk, ki sta od $x$ oddaljeni za $2h$ in $-2h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - f{\\left(- 2 h + x \\right)} + f{\\left(2 h + x \\right)} = 4 h \\frac{d}{d x} f{\\left(x \\right)} + \\frac{8 h^{3} \\frac{d^{3}}{d x^{3}} f{\\left(x \\right)}}{3} + O\\left(h^{5}\\right)$"
      ],
      "text/plain": [
       "                                                  3              \n",
       "                                              3  d               \n",
       "                                           8⋅h ⋅───(f(x))        \n",
       "                                                  3              \n",
       "                                d               dx           ⎛ 5⎞\n",
       "-f(-2⋅h + x) + f(2⋅h + x) = 4⋅h⋅──(f(x)) + ────────────── + O⎝h ⎠\n",
       "                                dx               3               "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_2h = eq_h.subs(h, 2*h)\n",
    "eq_2h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj imamo dve enačbi in dve neznanki; sistem bomo rešili po korakih:\n",
    "\n",
    "1. enačbo `eq_h` rešimo za prvi odvod,\n",
    "2. enačbo `eq_2h` rešimo za prvi odvod,\n",
    "3. enačimo rezultata prvih dveh korakov in rešimo za tretji odvod,\n",
    "4. določimo napako metode,\n",
    "5. določimo oceno odvoda.\n",
    "\n",
    "Izvedimo navedene korake:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "f3_cent_točno = sym.solve(\n",
    "        sym.Eq(sym.solve(eq_h, f(x).diff(x))[0],  # 1. korak\n",
    "        sym.solve(eq_2h, f(x).diff(x))[0]),       # 2. korak\n",
    "        f(x).diff(x,3))[0]                        # 3. korak\n",
    "f3_cent_O = f3_cent_točno.expand().getO()         # 4.korak\n",
    "f3_cent_ocena = f3_cent_točno.expand().removeO()  # 5.korak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ocena 3. odvoda je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{f{\\left(- 2 h + x \\right)}}{2 h^{3}} + \\frac{f{\\left(- h + x \\right)}}{h^{3}} - \\frac{f{\\left(h + x \\right)}}{h^{3}} + \\frac{f{\\left(2 h + x \\right)}}{2 h^{3}}$"
      ],
      "text/plain": [
       "  f(-2⋅h + x)   f(-h + x)   f(h + x)   f(2⋅h + x)\n",
       "- ─────────── + ───────── - ──────── + ──────────\n",
       "         3           3          3            3   \n",
       "      2⋅h           h          h          2⋅h    "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f3_cent_ocena"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ali:\n",
    "\n",
    "$$y_i'''=\\frac{1}{h^3}\\left(-y_{i-2}/2+y_{i-1}-y_{i+1}+y_{i+2}/2\\right)$$\n",
    "\n",
    "Uteži torej so:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   |$y_{i-2}$|$y_{i-1}$ | $y_{i}$   |$y_{i+1}$ |$y_{i+2}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'''=\\frac{1}{h^3}\\cdot$| -0.5 | 1 | 0 | -1 | 0.5|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potrdimo, da je napaka metode drugega reda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       " ⎛ 2⎞\n",
       "O⎝h ⎠"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f3_cent_O"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Odvod $f^{(4)}(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ponovimo podoben postopek kot za 3. odvod, vendar za 4. odvod seštevamo Taylorjevo vrsto (do stopnje 6) naprej in nazaj:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(- h + x \\right)} + f{\\left(h + x \\right)} = 2 f{\\left(x \\right)} + h^{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + \\frac{h^{4} \\frac{d^{4}}{d x^{4}} f{\\left(x \\right)}}{12} + O\\left(h^{6}\\right)$"
      ],
      "text/plain": [
       "                                                    4              \n",
       "                                                4  d               \n",
       "                                               h ⋅───(f(x))        \n",
       "                                     2              4              \n",
       "                                 2  d             dx           ⎛ 6⎞\n",
       "f(-h + x) + f(h + x) = 2⋅f(x) + h ⋅───(f(x)) + ──────────── + O⎝h ⎠\n",
       "                                     2              12             \n",
       "                                   dx                              "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_h = sym.Eq(f(x+h)+f(x-h), vsota(n=6))\n",
    "eq_h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo dodatno enačbo na podlagi točk, ki sta od $x$ oddaljeni za $2h$ in $-2h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(- 2 h + x \\right)} + f{\\left(2 h + x \\right)} = 2 f{\\left(x \\right)} + 4 h^{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + \\frac{4 h^{4} \\frac{d^{4}}{d x^{4}} f{\\left(x \\right)}}{3} + O\\left(h^{6}\\right)$"
      ],
      "text/plain": [
       "                                                            4              \n",
       "                                                        4  d               \n",
       "                                                     4⋅h ⋅───(f(x))        \n",
       "                                           2                4              \n",
       "                                       2  d               dx           ⎛ 6⎞\n",
       "f(-2⋅h + x) + f(2⋅h + x) = 2⋅f(x) + 4⋅h ⋅───(f(x)) + ────────────── + O⎝h ⎠\n",
       "                                           2               3               \n",
       "                                         dx                                "
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_2h = eq_h.subs(h, 2*h)\n",
    "eq_2h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Iz dveh enačb določimo 4. odvod:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "f4_cent_točno = sym.solve(\n",
    "        sym.Eq(sym.solve(eq_h, f(x).diff(x,2))[0],  # 1. korak\n",
    "        sym.solve(eq_2h, f(x).diff(x,2))[0]),       # 2. korak\n",
    "        f(x).diff(x,4))[0]                          # 3. korak\n",
    "f4_cent_O = f4_cent_točno.expand().getO()           # 4.korak\n",
    "f4_cent_ocena = f4_cent_točno.expand().removeO()    # 5.korak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ocena 4. odvoda je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAi4AAAAsCAYAAABCHTP9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAOQUlEQVR4Ae2d75XVthbFB9YUwJsSoIMkVPBIBw+oAOggWe/T8C1r0kGSCnihg0AFEDogHWQyHfD2z7FubF3Z1rVkW9Y9WktjW3+Oj/Y+ks6VbM/Fly9fLtaO19fXD+feM6Xu3HuuWa/29i2JpWH3d182HI7HNMOkTtuondfa2zc0H0y1+/7FyuH169ff6ZZfJdz2YSsjQUSZVTNgU2bD1tOqWtuIhdBsaBAps430sXcQ3A0zquX1zPvyKK/38Hi6QWDddK91/kZpn7y0o0uV+UmJd23GD7p254eySvuPLh7r+P0hccaJ6uP83On484zqm1SRrqP4KD8LNn7jJPeh0hze3+j8lmul9zjV9QOl/6rYlNH1I53vLkjvXdiG9MR5f6XjqxiQVW7UfpChMkvZkNlGDEmZyojH1WxD99o9t2rDLvq8Mw/pO7svq+7ZjOdjvB5WXABE8XeB+5uOTGxMdoDEZDYaVBZn50rxjeJLxSeKvaAydJD/tnJ7eadeSMaPqsOgj8zig/QcxadtRxZsumBILvz9pGMzQer4ta5xWH7XeY8jXeMIfqu8PxTfKe4yqA17sQ36FX1mMqhNo/aDAJXJ1r98hSTbbMMHZdnr1WyjBm7Vhr30+QvpOrsvq+5ZjedjvB4cF/VDOgurK91Ji8GQiWwq4KzgtDAQf1TsytBlEyAMTzNXQBYy9xCm8MmNjcMEub1f9OIXh/ROccgh5dfeb4pZgu73RJH2rxmKtg3hwS/EU8KU/SBrKRvq6mm20UVjgfMNbSMbt9bnRw0jpS+f43geHMsbx0WGxhIzhtvbelH6t8QxGjBS5ePgvNM5kTpMjH54pvSefL/AKdetLGRy72KD9IvBJys2HTC49+cARjiWD5SOB38Ira5chxzPQ7kTT+BnVY7UDuysSNuQbvQz+gdxMrScgN+q/ctXzGzDRyT/9Va2sQC32CtxtaA2FNvnHQgtzil9+ezG8yFe3YoLv8pZDr5zIJ9wbLYXxuoqD8coZuXmhNs2RZH57NRKK5cfxWdBbGgmDsgfI9z4g8uorivjlnq7Um3jufg4xYGf5GRhG3I8TOrhCu7gaLbRJ6kWbkvl1aE9ifNEXz7X8fyI18sWUR7IZILj1+BzxT8VeTjzV6UFf30rnSUviMALvNU1Ww8ce1sTSiNQLiiHTNVhAnXbCY91/kKR1QB0IXxQmbd/n/b+IhPZp0wEPQFLXUjfWHxGsUnRTzo8HagPzxfK7z2gqyS45Jc9juaVIjZAWZ6RwXhWD7pvNbahtrBFFLVdqrKx9gMni9lQh3CzjQ4YuU83to2iuK2pz2Mnufqy5Ox6PM/Jq1txcb+8v5FwHsz9UREHBMeFSewoKP1nRYCkLg+APlUMOS3UxTH6zMlAuFFd7slDVh8Uf1Hk2Qiex0A+e3uhgEwcnOKCdI/FZwqbrG2TXjgiYAa2h6B0cCYPfT61+lPmTnEIf2UtHqqwDeEJ5qxqRjmALf65+lcSSdLFbCMJwfHKW9pGodxW0ecd60v2Zcne03iejdf7reGC8VcA7MBuj//T8ZdOmV620p3TMLia0lZg4OM13KMgGfyy7E6MTJQ4S06XKy9fl4eATKfDIbGUk0h8BrFZqB2sjL2VbjiJ3cCvLsIL5fmTKzquHiqzDVatnE1HYanyzrZn96+oG00XMtuYxiilxJa2URS3lfX5g00s2Jd3MZ7n5vXygGz4GRRej8ax4Fd4aPDE27uQUv6WA8ndgPOBQxIKH1W/O1E2r+wqrSmv49DyGLKot8mkGmpIIC0Gn0Fs1Hba9v7ENrLyFeRD6WxTsCUYwpTtBlZa/Lq0IbRNp+R/QivbDYL/ZLSv/Co/tBrH/UK6uPpV2IbaSB+K2iJyDW+PMfZD0aVtyGzDIybX5Za20bZhNrfSHZu2Ph9nDMl92b9Ni/9exvOsY/mlGs/yNZgMORbkuV9+nHfDY110nY5uXtS57u1PlDxs+0NU5ZEBO7L+0sWS8IEbKYgjlxwki8nzSkcGqlBgAOo5KCoL78Q3oQrdNJUNOSYXSmf1jG8E+Ss83erBc9XZvW2oDeDHG1xz+kmS/QCq7pvDhsw2ghaalri1bbTaz+ZW+lufjzeB5L7cvZWw39V4Ln2zjuWXLRispgw5JxQZGnTxIn2FKO+HWyWwejAa1Dg6EeUOE6jSmno6MgD7gTxklxpi8InCJqWBwg7n4ZGOT50cnTd864jHDo5c+99vYWDCsY3h2Ile5Cgd9mob4MrXolnS7QZsA4eOdDjoPXPUFoyxH4ouZkPSy2yjJWOBw6a2UTq30m+vfT5kKtn6snDZ9Xieg1fnuLDk5w+sgM+vfSau0DYR+WwhxayO4Pg0EyWVXJBcBkXuy0M73MNNlF1HaeyLsleq0y2ry6JCDD5BbHK1QrjSYUL/ZgHjd89cNK+UB3jGq2/KKI9z3jhaBW/dpwrbaDE96j9K/6vF8+BM6toPMfZDnSVtyGzDZyXTdQG2URS3tfT5AfPI0peF0e7G8yV4bRwXCeZhTSYlHIjml5+OTBwY9osQEcrHEaFMzK9xyrBU5gc8aiJvMiGLX46HoDTyPhwSjk/cJ+yPczZOOQGfIWySW9DqgGMIt/4zFry15bZv2D46mlyVBidum4gVG+foKHnxUK1ttMiBLTEYWu7IT+lfQdknJpptnAhYhuJr2UZp3FbZ53P15VbOHsfz7Lxeuk4mUPjiLY6Lm+BYzfi3rocGTjy/C+WHJjwn1h2Z/ADcD9RlMqRhyOLpev5XEjrwYDDfhTlsG1HGC9QL7rN65ba4jMVnCJscOrP1g4PJaokfurwyUIb4wYnlg2k4OaGtDF9mzusqbaO1bbf6CK7gzneKnBPpMIy1H8ovaUNmG46RhY8b2EZp3FbZ52U2ufryXsfz7Lwe/Xfo2L7ZdjIe9hxb6j6IUzm+uTL4xsuhYOSJ5DH44+QU+V+MpRfOVxQ+ubGJhHCVYmobW1KzHs6dq6DuWbRtxLTrFPtBnspn7V8xOqaWkc5mGzNALN02jNc+qaXz1dd2/GptbnW/4Fh+f1zNfi5KK7pf5uzZuW2EfsHw1Y2Sc66OsAKAzGJCAj65sSkGEyly18Y1dSrONmIan2A/iN+jDZltxBiGyuzMNs6e153xFWmFTbG1uQ2O5SetuIgMftXxICCrLO91fdKruirPUlfy5+MlBy+Mr/WyR5scJIctJ1YFkp7hUP3Z+KhuFmySwdi5AOGY1TbWhCPFftDTbGicLbON9LF3HOFtcnPzKnnJ84FkzJ4LQFH1z34+EAaDY/nliabGr7oHihyjtog8+dRhxSbV4WAbJufqDW0ipoYUfHJhk9qGvdfPbRtr4pFiP+hpNjTOltlG+tg7jvA2ubl5zTEfWF9Ot4VBXk9acUnXo/Ek8aLYcvIfRowSr3rfqSBvQWV7LVeyVt9rDzVWeiRhE5J5TmlL2Mbe8DMbCjNmtpE+9oaR3TZ1CV4l0+aDbWm9mOJ1dcdlYzyCty/FUIPKWaIhYAgYAobAagjYfLAa1LNvdH92TatoCBgChoAhYAgYAobAygiY47Iy4HY7Q8AQMAQMAUPAEJiPwL3r6+sv86uXVVNLfPfGNFI+D/vwxLgfrtqEWz9D11P/wbipItnV4BjAoIqkMftQHg/kvVc85SHtye8SmV2UbzpjdoH2Zhvlczik4Ri3yltkPpBcmwuGCMmUbs+4CEgZWhEPY2Xi1MQYAoaAIWAIzETA5oOZwK1YzbaKVgTbbmUIGAKGgCFgCBgCaQiY45KGn9U2BAwBQ8AQMAQMgRUR2Nxx0bIc/0zRwk4QML52QtQGapptbAB6Abc03gsgYSEVSuV2U8dFoPAxueY/Zy6Eu4nNiIDxlRHMykSZbVRGaGRzjPdIoHZYrGRuT/3kfxB+NZA3MfiUP/948VbXk/+xWWV4uyfb128ly0IkAsZXJFBnWMxs4wxJV5ON93p5r5HbLCsuAuZOkf8/hCPybsoEWiBZaZksOyUrU/6d5BDPIlTA11nwtEUjzTa2QH37exrvPQ6qmg9q5DaL49KhHGeE/2o5FV4KzFn/q2hK8Jx86fJOMek/Q8+5bwF1dslXAbidgwpmG+fA8nEbz573iueDarjN5riIbPdht9FVFJXjmylvj/uLpayJgPG1Jtr7upfZxr74yqWt8Z4LyfLk1MZtNsdFVDVbRQJocMtFeQ9V7kpHe7Zle9s2vrbnoFQNzDZKZWZZvYz3ZfHdUnpV3Gb7cq6cEV5r/qjIVtGVIg/osjT1yjkqOnL9XNGFBzp5qci20Qfl20qMQ2bho/G1MMA7Fm+2sWPyElQ33hPAK7xqbdxmcVwECg7IX4qfFPn/Lc2Kio68aXSh41OOflA6KzCfFf+l88GVGr+eXachIKyNrzQIq61ttlEttaMNM95H4dl1Zo3c5toqcs+3vBBI/jYQk+RRUDlWWm7ajBtd8+yLhXUQML7WwXmPdzHb2CNr6Tob7+kYliqhOm4vMyHN/hn/RZkVl25gayi4/aOyvMVzjm/ydPHZ6tz42gr58u9rtlE+R0toaLwvgWoZMqvjNueKS+9tIjkmbAMR35TBnWnRQQAP3PjqAGKnBwTMNg5QnNWJ8V4v3dVxm+y4yEFhKwgHxf9+yyul8WE6fxWmXvPYQcuMrx2QtJGKZhsbAb/xbY33jQlY8Pa1cpvsuAjzZ+AugHq/4JXEMyzNVpDy+OAczo2F7REwvrbnoFQNzDZKZWZZvYz3ZfHdUnqV3OZwXNg/850WiGIlxm0TPZLj4j+0SxkL6yNgfK2P+V7uaLaxF6by6mm858WzJGlVcnuZAWEclOa1Z0/W97p+LofliSLnFspAwPgqg4cStTDbKJGV5XUy3pfHeKs7VMnt/wFw3P7LnupyogAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\frac{6 f{\\left(x \\right)}}{h^{4}} + \\frac{f{\\left(- 2 h + x \\right)}}{h^{4}} - \\frac{4 f{\\left(- h + x \\right)}}{h^{4}} - \\frac{4 f{\\left(h + x \\right)}}{h^{4}} + \\frac{f{\\left(2 h + x \\right)}}{h^{4}}$"
      ],
      "text/plain": [
       "6⋅f(x)   f(-2⋅h + x)   4⋅f(-h + x)   4⋅f(h + x)   f(2⋅h + x)\n",
       "────── + ─────────── - ─────────── - ────────── + ──────────\n",
       "   4           4             4            4            4    \n",
       "  h           h             h            h            h     "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f4_cent_ocena"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ali:\n",
    "\n",
    "$$y_i^{(4)}=\\frac{1}{h^4}\\left(y_{i-2}-4\\,y_{i-1}+6\\,y_i-4\\,y_{i+1}+y_{i+2}\\right)$$\n",
    "\n",
    "Uteži torej so:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   |$y_{i-2}$|$y_{i-1}$ | $y_{i}$   |$y_{i+1}$ |$y_{i+2}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| 1 | -4 | 6 | -4 | 1|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potrdimo, da je napaka metode drugega reda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle O\\left(h^{2}\\right)$"
      ],
      "text/plain": [
       " ⎛ 2⎞\n",
       "O⎝h ⎠"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f4_cent_O"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Povzetek centralne diferenčne sheme*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zgoraj smo izpeljali prve štiri odvode z napako metode 2. reda. Bistvo zgornjih izpeljav je, da nam dajo **uteži**, s katerimi moramo množiti funkcijske vrednosti, da izračunamo približek določenega odvoda. Iz tega razloga bomo tukaj te uteži zbrali. Če ste neučakani, lahko skočite na tabelo spodaj. Z branjem nadaljujte, če pa želite spoznati, kako predhodno izpeljane izraze strojno uredimo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej zberimo vse ocene odvodov v seznam:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - \\frac{f{\\left(- h + x \\right)}}{2 h} + \\frac{f{\\left(h + x \\right)}}{2 h}, \\  - \\frac{2 f{\\left(x \\right)}}{h^{2}} + \\frac{f{\\left(- h + x \\right)}}{h^{2}} + \\frac{f{\\left(h + x \\right)}}{h^{2}}, \\  - \\frac{f{\\left(- 2 h + x \\right)}}{2 h^{3}} + \\frac{f{\\left(- h + x \\right)}}{h^{3}} - \\frac{f{\\left(h + x \\right)}}{h^{3}} + \\frac{f{\\left(2 h + x \\right)}}{2 h^{3}}, \\  \\frac{6 f{\\left(x \\right)}}{h^{4}} + \\frac{f{\\left(- 2 h + x \\right)}}{h^{4}} - \\frac{4 f{\\left(- h + x \\right)}}{h^{4}} - \\frac{4 f{\\left(h + x \\right)}}{h^{4}} + \\frac{f{\\left(2 h + x \\right)}}{h^{4}}\\right]$"
      ],
      "text/plain": [
       "⎡  f(-h + x)   f(h + x)    2⋅f(x)   f(-h + x)   f(h + x)    f(-2⋅h + x)   f(-h\n",
       "⎢- ───────── + ────────, - ────── + ───────── + ────────, - ─────────── + ────\n",
       "⎢     2⋅h        2⋅h          2          2          2              3          \n",
       "⎣                            h          h          h            2⋅h           \n",
       "\n",
       " + x)   f(h + x)   f(2⋅h + x)  6⋅f(x)   f(-2⋅h + x)   4⋅f(-h + x)   4⋅f(h + x)\n",
       "───── - ──────── + ──────────, ────── + ─────────── - ─────────── - ──────────\n",
       " 3          3            3        4           4             4            4    \n",
       "h          h          2⋅h        h           h             h            h     \n",
       "\n",
       "   f(2⋅h + x)⎤\n",
       " + ──────────⎥\n",
       "        4    ⎥\n",
       "       h     ⎦"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "odvodi = [f1_cent_ocena, f2_cent_ocena, f3_cent_ocena, f4_cent_ocena]\n",
    "odvodi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Na razpolago imamo 5 funkcijskih vrednosti (pri legah $x-2h, x-h, x, x+h, x+2h$), ki jih damo v seznam:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "funkcijske_vrednosti = [f(x-2*h), f(x-h), f(x), f(x+h), f(x+2*h)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Utež prvega odvoda za funkcijsko vrednosti $f(x-h)$ izračunamo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{1}{2 h}$"
      ],
      "text/plain": [
       "-1 \n",
       "───\n",
       "2⋅h"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_cent_ocena.expand().coeff(funkcijske_vrednosti[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj posplošimo in izračunajmo uteži za vse funkcijske vrednosti in za vse ocene odvodov:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left[ 0, \\  - \\frac{1}{2 h}, \\  0, \\  \\frac{1}{2 h}, \\  0\\right], \\  \\left[ 0, \\  \\frac{1}{h^{2}}, \\  - \\frac{2}{h^{2}}, \\  \\frac{1}{h^{2}}, \\  0\\right], \\  \\left[ - \\frac{1}{2 h^{3}}, \\  \\frac{1}{h^{3}}, \\  0, \\  - \\frac{1}{h^{3}}, \\  \\frac{1}{2 h^{3}}\\right], \\  \\left[ \\frac{1}{h^{4}}, \\  - \\frac{4}{h^{4}}, \\  \\frac{6}{h^{4}}, \\  - \\frac{4}{h^{4}}, \\  \\frac{1}{h^{4}}\\right]\\right]$"
      ],
      "text/plain": [
       "⎡⎡   -1       1    ⎤  ⎡   1   -2   1    ⎤  ⎡-1    1      -1    1  ⎤  ⎡1   -4  \n",
       "⎢⎢0, ───, 0, ───, 0⎥, ⎢0, ──, ───, ──, 0⎥, ⎢────, ──, 0, ───, ────⎥, ⎢──, ───,\n",
       "⎢⎣   2⋅h     2⋅h   ⎦  ⎢    2    2   2   ⎥  ⎢   3   3       3     3⎥  ⎢ 4    4 \n",
       "⎣                     ⎣   h    h   h    ⎦  ⎣2⋅h   h       h   2⋅h ⎦  ⎣h    h  \n",
       "\n",
       " 6   -4   1 ⎤⎤\n",
       " ──, ───, ──⎥⎥\n",
       "  4    4   4⎥⎥\n",
       " h    h   h ⎦⎦"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "centralna_diff_shema = [[odvod.expand().coeff(fv) for fv in funkcijske_vrednosti] \\\n",
    "                        for odvod in odvodi]\n",
    "centralna_diff_shema"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Zgornje povzetke lahko tudi zapišemo v tabelarični obliki:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   |$y_{i-2}$|$y_{i-1}$ | $y_{i}$   |$y_{i+1}$ |$y_{i+2}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{h}\\cdot$| 0     | -0.5 | 0 | 0.5 | 0|\n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$| 0 | 1 | -2 | 1 | 0|\n",
    "|$y_i'''=\\frac{1}{h^3}\\cdot$| -0.5 | 1 | 0 | -1 | 0.5|\n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| 1 | -4 | 6 | -4 | 1|\n",
    "\n",
    "Prikazana centralna diferenčna shema ima napako 2. reda $\\mathcal{O}(h^{2})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Uporaba ``scipy.misc.central_diff_weights``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uteži za centralno diferenčno metodo lahko izračunamo tudi z uporabo ``scipy.misc.central_diff_weights()`` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.misc.central_diff_weights.html)):\n",
    "\n",
    "```python\n",
    "central_diff_weights(Np, ndiv=1)\n",
    "```\n",
    "\n",
    "`Np` predstavlja število točk, čez katere želimo izračunati odvod, `ndiv` pa stopnjo odvoda (privzeto 1).\n",
    "Za več informacij glejte [dokumentacijo](https://docs.scipy.org/doc/scipy/reference/generated/scipy.misc.central_diff_weights.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uvozimo funkcijo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.misc import central_diff_weights"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3$.$ odvod (čez pet točk), napaka reda  $\\mathcal{O}(h^{2})$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.5,  1. ,  0. , -1. ,  0.5])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "central_diff_weights(Np=5, ndiv=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Opomba: ``SymPy.finite_diff_weights`` omogoča izračun uteži tudi pri nekonstantnem koraku; v podrobnosti ne bomo šli in radovednega bralca napotimo na [dokumentacijo](http://docs.sympy.org/latest/modules/calculus/index.html#sympy.calculus.finite_diff.finite_diff_weights)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Izboljšan približek - Richardsonova ekstrapolacija"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Če je točen odvod je izračunan kot:\n",
    "\n",
    "$$f'(x_i)=f_o'(x_i)+e,$$\n",
    "\n",
    "kjer je $f_o'(x_i)$ numerično izračunan odvod v točki $x_i$ in $e$ ocena napake."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Za metodo reda točnosti $n$: $\\mathcal{O}(h^{n})$ pri koraku $h$ velja:\n",
    "$$f'(x_i)=f_o'(x_i, h)+K\\,h^n,$$\n",
    "kjer je $K$ neznana konstanta. \n",
    "\n",
    "Če korak razpolovimo in predpostavimo, da se $K$ ne spremeni, velja:\n",
    "$$f'(x_i)=f_o'\\left(x_i, \\frac{h}{2}\\right)+K\\,\\left(\\frac{h}{2}\\right)^n.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Iz obeh enačb izločimo konstanto $K$ in določimo izboljšan približek:\n",
    "$$\\overline{f}'(x_i)=\\frac{2^n\\,f_o'\\left(x_i, \\frac{h}{2}\\right)-f_o'(x_i, h)}{2^n-1}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Zgled \n",
    "\n",
    "Poglejmo si zgled $f(x)=\\sin(x)$ (analitični odvod je: $f'(x)=\\cos(x)$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(0, 2*np.pi, 9)\n",
    "y = np.sin(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pri koraku $h$ imamo funkcijske vrednosti definirane pri:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.78539816, 1.57079633, 2.35619449, 3.14159265,\n",
       "       3.92699082, 4.71238898, 5.49778714, 6.28318531])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Numerični odvod pri $x=\\pi$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.14159265358979$"
      ],
      "text/plain": [
       "3.141592653589793"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x[4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "in koraku $h$ je"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALoAAAAPCAYAAABTCdU3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGDElEQVRoBe2a23EVORCGDy4HACaCZTPAkAFkYHYjADJYat/8RkEGQARgMgAiWHAGsBHgPRl4/0+otZoZXVrnzKnaB7pKlqbV6l990WXG58b19fXmJ/30QM0D5+fnd1Uu834939Tziepv8FXfUfVA5Z3aW3j/Nzpea0Iy8EXU9V31ryovxAuO8GBI9lUmd6L245LTvDgeOckQoKcRl+DxzLw/Rt6iUt9dMS9UTtWuBlV95g/T8Va8ecIM4a+JbZNy6PwkGXxjc6cNnf6owl98QvxeSTYwZn+24t+a8TbidX0pGfOjK68yeYMLfl8l0aX8i7Q+V/0e7apxxhfVD1WayW6yksdJL+N4HPC3nkmmNF5tF45HTjLM8ZlqS/SN2mfifVD9SCXYEueD7BuVK5V7KiRokTSOPhYCusOCUc0zhQ0gkHgu/Ch3CGyXTk0WmyFiQizwC7Heqja6rwa2plhZh2p2+mf2PGiPK97olt6m3/dOdAE8Ec5N1XlisIJ5ZpU/VGkRDucYDEmOoNqXKp/VTOP17MLxykk3+p5InsS2udtO/qf6jLdRP0F9pEL7D1UEvUYkNLuI6UKOpJ4ngQv/ENiDOolFsB1DaiSZRZzFCxuC6mFfaowr3tl8mn4/ygR3beIEO9ZyHX/p4YEmTJBbxC46TwLk0ZmP9+J45dBPAlMCaa6pbbyRWuOxhUXwOh8nPifbPBFWxR/Ezqe3RptYl4hrYDoxSwINnjeOG4/te+/omihH0ySwcfKWvPSnFR37QqUJ2iK4yvmxzZ0M4qrA7ujFcckJG52Te2N0mNjhJKEeJYLKadZdMAfAd2OPGtWTly2L+IrH3fp5b2yj3xXHOL5r+3EDqNslYyxRW7IntU6NJynoLsncjuPueHG8clHvpNJYHEtwnqpdWrgT+coDi/KbxrOr/67CYuVefiFefpURa0or4O+MPZ3J8klzC9cI9RATriPc0TmRiqQ+ZPhak+7mRcEKU+NG86pr+16JrnlagrZ2sN6k2Q1IsjmRLBDjvTheuaCYP3IqOODfVyF4n1V2JbP1Xh5ktf9R4StSaedbC38Y22kketNnQ9lAEn9VzXWstnjZMOxriRNmIjYax67tRxP1h3mwnbmm/TEdclpKdrUJ/jYOsCtQfKxWPRwbOJETFi9bL1W4E75V4WvRmQl7a40xZ7OTzU+Ed9LzJpNJasXbGz/TO4SdJtFoSDcJbbHYqE08SHA+FCxI/SwE3q1qi2AxZkdGiKNwXH4/joKfBGYDPLh8fmP3K92tbbytSrtrG39SSw/Xl1/E5MWFBMcAXm4oJByO9eJ45aRyScJ/z3zUw1XjVmwvBduc0sLkMxnHP0dsNQFWwN8Zu23Sohcckplr5RyT+/Kct1DQYewSxxJm8juJTmBPO8DFbsaq0FdaJMYrTWCiL84BByUSz44+7rwuHK8cIJJlUVFfUmfE1YXThbK4amRyk6b02BzxZ43Y7QKtiT+KbXPo1dL7QTJ8+q3lh8U4V2WbU84bamf2lPQbL+RVJtv0+9HQDMrC7FApgJmI7ejVHSyTLTVJxI8YEju9OF45VjvXFHNcaQ6jPLBb+vJFvzb+CLbXLk6gkj0htvLdZJOIviQXLGZenJKcN46M7dq+RqJfCAiHzIldgPtn02j1n6nwspYcGtvsqPlbuxfHK8e87KqSz91swXmjxL21tOjxBTt+rnNt/BFsr12vNef039xsELHJbbEu813r6mGyvdobR/R0bd870eUIXryuVJ/ZzNUmaX9TCS+a8OGpXKuwk+VEYswdg5F85ks7htpeHJec9LOIOJoTCQMbmDvYJGKJbkemnVhJRmO46nAK2bVro/bCF3HALvhrYac5q1HVqT5+ljF56dQz/xmGeHmfE7ZCNd/96P3vbxVbON44biTb9fuNNX69KCAMJLgY+F2FT3WLb62S+yo+iVC7j5ujit+dB3C882FnygPGouOleLFbicfigxiDfhYhVxF+QkBQEukZX5gtLIiFLxCWnAtfcofAdukUNj6xkxVb2JT4Hc9W9YSiLBtZ8VOqCXvtkZwrjpneqt//BXnwuQojBYQ3AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle -0.900316316157106$"
      ],
      "text/plain": [
       "-0.9003163161571061"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = x[1] - x[0]\n",
    "f_ocena_h = (-0.5*y[3] + 0.5*y[5])/h\n",
    "f_ocena_h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračun pri koraku $2h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.636619772367581$"
      ],
      "text/plain": [
       "-0.6366197723675814"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h2 = x[2] - x[0]\n",
    "f_ocena_2h = (-0.5*y[2] + 0.5*y[6])/h2\n",
    "f_ocena_2h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunajmo izboljšano oceno za $x=\\pi$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.988215164086948$"
      ],
      "text/plain": [
       "-0.9882151640869475"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_ocena_izboljšana = (2**2 * f_ocena_h - f_ocena_2h)/(2**2-1)\n",
    "f_ocena_izboljšana"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vidimo, da je izboljšana ocena najbližje teoretični vrednosti $\\cos(\\pi)=-1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Necentralna diferenčna shema"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Centralna diferenčna shema, ki smo jo spoznali zgoraj, je zelo uporabna in relativno natančna. Ker pa je ne moremo vedno uporabiti (recimo na začetku ali koncu tabele), si moramo pomagati z **necentralnimi diferenčnimi shemami** za računanje odvodov."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poznamo:\n",
    "\n",
    "* **diferenčno shemo naprej**, ki odvod točke aproksimira z vrednostmi funkcije v naslednjih  točkah in \n",
    "* **diferenčno shemo nazaj**, ki odvod točke aproksimira z vrednostmi v predhodnih točkah.\n",
    "\n",
    "Izpeljave so podobne, kakor smo prikazali za centralno diferenčno shemo, zato jih tukaj ne bomo obravnavali in bomo prikazali samo končni rezultat."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Diferenčna shema naprej"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Diferenčna shema naprej z redom napake $\\mathcal{O}(h^{1})$:\n",
    "\n",
    "|  Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$        |$y_{i}$|$y_{i+1}$ | $y_{i+2}$   |$y_{i+3}$ |$y_{i+4}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{h}\\cdot$| -1     | 1 | 0 | 0 | 0|\n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$| 1 | -2 | 1 | 0 | 0|\n",
    "|$y_i'''=\\frac{1}{h^3}\\cdot$| -1 | 3 | -3| 1 | 0|\n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| 1 | -4 | 6 | -4 | 1|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Diferenčna shema naprej z redom napake $\\mathcal{O}(h^{2})$:\n",
    "\n",
    "|  Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$        |$y_{i}$|$y_{i+1}$ | $y_{i+2}$   |$y_{i+3}$ |$y_{i+4}$|$y_{i+5}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{2h}\\cdot$| -3 | 4 | -1| 0 | 0|  0| \n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$| 2 | -5 | 4 | -1| 0| 0| \n",
    "|$y_i'''=\\frac{1}{2h^3}\\cdot$| -5 | 18| -24| 14| -3| 0| \n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| 3 | -14 | 26 | -24 | 11| -2| "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Diferenčna shema nazaj"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Diferenčna shema nazaj z redom napake $\\mathcal{O}(h^{1})$:\n",
    "\n",
    "| Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$         |$y_{i-4}$|$y_{i-3}$ | $y_{i-2}$   |$y_{i-1}$ |$y_{i}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{h}\\cdot$| 0 | 0| 0 | -1| 1|\n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$| 0 | 0 | 1 | -2| 1|\n",
    "|$y_i'''=\\frac{1}{h^3}\\cdot$| 0 | -1| 3| -3| 1|\n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| 1 | -4 | 6 | -4 | 1|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Diferenčna shema nazaj z redom napake $\\mathcal{O}(h^{2})$:\n",
    "\n",
    "| Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$         |$y_{i-5}$|$y_{i-4}$|$y_{i-3}$ | $y_{i-2}$   |$y_{i-1}$ |$y_{i}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{2h}\\cdot$| 0 | 0| 0| 1 |-4| 3| \n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$| 0 | 0 | -1| 4|-5| 2| \n",
    "|$y_i'''=\\frac{1}{2h^3}\\cdot$| 0 | 3| -14| 24|-18| 5| \n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| -2| 11| -24| 26 | -14| 3| "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Uporaba ``numpy.gradient``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Za izračun numeričnih odvodov (centralna diferenčna shema 2. reda) lahko uporabimo tudi ``numpy.gradient()`` ([dokumentacija](http://docs.scipy.org/doc/numpy/reference/generated/numpy.gradient.html)):\n",
    "\n",
    "```python\n",
    "gradient(f, *varargs, **kwargs)\n",
    "```\n",
    "\n",
    "kjer `f` predstavlja tabelo vrednosti (v obliki numeričnega polja) funkcije, katere odvod iščemo. `f` je lahko ene ali več dimenzij. Pozicijski parametri `varargs` definirajo razdaljo med vrednostmi argumenta funkcije `f`; privzeta vrednost je 1. Ta vrednost je lahko skalar, lahko pa tudi seznam vrednosti neodvisne spremenljivke (ali tudi kombinacija obojega). Gradientna metoda na robovih uporabi shemo naprej oziroma nazaj; parameter `edge_order` definira red sheme, ki se uporabi na robovih (izbiramo lahko med 1 ali 2, privzeta vrednost je 1). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rezultat funkcije `gradient` je numerični seznam (ali seznam numeričnih seznamov) z izračunanimi odvodi.\n",
    "\n",
    "Za podrobnosti glejte [dokumentacijo](http://docs.scipy.org/doc/numpy/reference/generated/numpy.gradient.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Zgled"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pogledali si bomo zgled, kako uporabimo **uteži**, funkcijo gradient in posebnosti na robovih. Najprej pripravimo tabelo podatkov:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, h = np.linspace(0, 1, 20, retstep=True)\n",
    "y = np.sin(2*np.pi*x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uteži diferenčih shem:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "centralna = np.array([-0.5, 0, 0.5]) # bi lahko tudi pridobili prek central_diff_weights(3,1)\n",
    "naprej = np.array([-3/2, 2, -1/2])\n",
    "nazaj = np.array([1/2, -2, 3/2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj izvedemo odvod notranjih točk (prvi način je z izpeljevanjem seznamov, drugi je vektoriziran):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "odvod_notranje = np.array([y[i-1:i+2] @ centralna/h for i in range(1, len(x)-1)]) # izpeljevanje seznamov\n",
    "odvod_notranje = np.convolve(y, centralna[::-1], mode='valid') / h # vektoriziran"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Na robovih uporabimo diferenčno shemo naprej oziroma nazaj:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "odvod_prva = y[:len(naprej)] @ naprej / h  # naprej\n",
    "odvod_zadnja = y[-len(nazaj):] @ nazaj / h # nazaj"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sestavimo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "odvod_cel = np.hstack([odvod_prva, odvod_notranje, odvod_zadnja])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prikažemo rezultat skupaj z rezultatom funkcije `np.gradient`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "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": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.plot(x, odvod_cel, 'ko', lw=3, label='lastna implementacija')\n",
    "plt.plot(x, np.gradient(y, h), 'g.', label='np.gradient, edge_order=1')\n",
    "plt.plot(x, np.gradient(y, h, edge_order=2), 'r.', label='np.gradient, edge_order=2')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Zaokrožitvena napaka pri numeričnem odvajanju*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zgoraj smo se osredotočili na napako metode. Pri numeričnem odvajanju pa moramo biti zelo pozorni tudi na **zaokrožitveno** (ali tudi *upodobitveno*) napako! Poglejmo si prvi odvod (po centralni diferenčni shemi) zapisan z napako metode ($k\\,h^2$) in zaokrožitveno napako $\\varepsilon$:\n",
    "\n",
    "$$y_i'=\\frac{1}{2h}\\left((-y_{i-1}\\pm\\varepsilon)+(y_{i+1}\\pm\\varepsilon)\\right) + k\\,h^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V najslabšem primeru se zaokrožitvena napaka sešteje in je skupna napaka:\n",
    "\n",
    "$$n=\\frac{\\varepsilon}{h}+k\\,h^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ko je $h$ velik prevladuje napaka metode $k\\,h^2$; ko pa je $h$ majhen, pa prevladuje zaokrožitvena napaka. Napaka ima minimum, ko velja:\n",
    "\n",
    "$$n'=-\\frac{\\varepsilon}{h^2}+2\\,k\\,h=0.$$\n",
    "\n",
    "Sledi:\n",
    "\n",
    "$$h=\\sqrt[3]{\\frac{\\varepsilon}{2\\,k}}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Zgled"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Spodaj si bomo pogledali primer, kjer bomo natančnost spreminjali v treh korakih:\n",
    "\n",
    "1. ``float16`` -\n",
    "    16-bitni zapis: predznak 1 bit, 5 bitov eksponent, 10 bitov mantisa\n",
    "2. ``float32`` -\n",
    "    32-bitni zapis: predznak 1 bit, 8 bitov eksponent, 23 bitov mantisa\n",
    "3. ``float64`` -\n",
    "    64-bitni zapis: predznak 1 bit, 11 bitov eksponent, 52 bitov mantisa (to je privzeta natančnost).\n",
    "\n",
    "Za več o tipih v ``numpy`` glejte [dokumentacijo](http://docs.scipy.org/doc/numpy/user/basics.types.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Določimo sedaj osnovno zaokrožitveno napako za posamezni tip:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Osnovna zaokrožitvena napaka za tipe `float16`, `float32` in `float64` je:\n",
      "[0.000977, 1.1920929e-07, 2.220446049250313e-16]\n"
     ]
    }
   ],
   "source": [
    "eps16 = np.finfo(np.float16).eps\n",
    "eps32 = np.finfo(np.float32).eps\n",
    "eps64 = np.finfo(np.float64).eps\n",
    "print(f'Osnovna zaokrožitvena napaka za tipe \\\n",
    "`float16`, `float32` in `float64` je:\\n{[eps16, eps32, eps64]}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Kot primer si poglejmo seštevanje: k številu `1.` prištejemo polovico osnovne zaokrožitvene napake `eps16` in pretvorimo v tip `float16`, ugotovimo, da je nova vrednost še vedno enaka vrednosti `1.`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1.+eps16/2).astype('float16')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo najprej funkcijo $\\exp(x)$, ki bo dala rezultat natančnosti, ki jo definira parameter `dtype`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-58-1f75dd94d9da>:1: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.\n",
      "Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations\n",
      "  def fun(x, dtype=np.float): # funkcija\n"
     ]
    }
   ],
   "source": [
    "def fun(x, dtype=np.float): # funkcija\n",
    "    return np.exp(-dtype(x)).astype(dtype)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definirajmo še funkcijo za analitično določljiv odvod (to bomo pozneje potrebovali za določitev relativne napake):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def f1_fun(x): # \"točen\" odvod funkcije\n",
    "    return -np.exp(-x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Podobno kakor zgoraj pri seštevanju, lahko tudi pri vrednosti funkcije ugotovimo, da sprememba vrednosti $x$, ki je manjša od $\\epsilon$, vodi v isti rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.368"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fun(1., dtype=np.float16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.368"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fun(1+eps16/2, dtype=np.float16)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uporabimo sedaj centralno diferenčno shemo za prvi odvod. Pri tem naj bodo števila zapisana z natančnostjo `dtype`, s pomočjo točnega odvoda pa se izračuna še relativna napaka:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f1_CDS(fun, x, h, dtype=np.float64):\n",
    "    f1_ocena = (fun(x+h, dtype=dtype)-fun(x-h,dtype=dtype))/(2*dtype(h))\n",
    "    f1_točno = f1_fun(x)\n",
    "    relativna_napaka = (f1_točno - f1_ocena) / f1_točno\n",
    "    return f1_ocena, relativna_napaka"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo primer odvoda pri $x=1,0$ (Python funkcija vrne vrednost in relativno napako):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( -0.366132723112128, \\  0.00474807196007504\\right)$"
      ],
      "text/plain": [
       "(-0.36613272311212813, 0.00474807196007504)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_CDS(fun, x=1., h=.01, dtype=np.float16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcUAAAAaCAYAAADG3kfKAAAACXBIWXMAAA7EAAAOxAGVKw4bAAANzklEQVR4Ae2d7ZEUtxaGmy0CwOsILs4AlgiMM4DrCMAZ2MUv+EfZGRhHYOMMsCPAkIF9I2A9GXDfR6PTpVaru496ZnZ2d6Qqoe6jV+dLH6el7lnufP78uWupeaB5oHmgeaB54DZ44NWrV/dlx3fK76I93F+K/nu8ny3O8lo1fK78OKe3++aB5oHmgeaB5oEb4AGC4HNlguLPyl+VAqJo1I3S3ZQi0BPdP1T5JqW36+aB5oHmgeaB5oEb5IGvFcc+Luj7ozDvlL9JcXfs+FQV91Txp8qHKaBdNw80DzQPNA80D9wUDyiGcdJ5qXyuTFz7Q7SNylESnR3lPZU/WWV6fPpWxOJ20sC3oZTx93GEMs668iS5D3Kh6KLMlr9PHhxtlPdui0d2r2i7aB6o8EDtmD3Vseixu9aXFd10G6AvZMQ/yn8o/yJfcQo6SqJzKvqdyj4ehONTnKuKC5WDbeSIww4E8f4xNv+k8itltq4ovZiifrw4JaE8+tIeg0cpkWV1v4pmW2mCEsH/Z9GsPi03on8BQSWycK4l7t+KPpDrxUUm7MbhY/pwTcp36B6c2xYESC5++4FrpQtlnqZ+EN10gU7yyA7ACp6Gt3EQ7vVP2jdGQ1ds40GN4/xNX1G48GCFSR/4zsXm2RLfgqgrJXnsmlNI7b2+vkpc1ZiVfTVj0exYXGMS38xiPThhvPOqq8B67K71JfKvbB5I1uIcFsbdZ3NjPa0TT9bnfo3W/Wvdf1C+k+KSa3zyi/JTaPZOkYXyYO8RpRQKvVYZvv5RSSD4oPIb5dnAGLEs3BYUO10T9TkLfqrcf1GkawYnCyn44BSV3JMJxKRHytSV5LLttqChyxB4e7kQ4KfMzq6XW4GDBYGIxIBBB/jgm3zh9+Dctog/vuFBoH/w0TUD0vqhH0SieWR3NTwjdrZvhGFcMDiRT9BG52LyYiOO8Yft4YhEJb7/n0oCbmkcFGVeBVH6uH0wpY94eOZBdySce8xG+7xj0b3GyG4X1oOLPnTNqxqsbPfY7falZIc1V3wPOg+iHO8cXuwH8WOuWuCMQ6JYsNsLc1nlE+V0fTb6Y9HTdc4YEfvYZLGubywocq7aL5aG3EcpIXZm2yuJYGXuidBLcmnPESFB0HiYYS9UZzRdhuDHzsPqoTEYBguf6kcyRQsLsMlQiVw6LU/PRPhTOcj14hImH9XmaXI/denCideiLVEAAysP8Dw8YCfBKuyOI9YlW9gansiY7RvpshEm+EbX3+uaCVFMFVgm6Lnw/TsDXWPfX6J7xl9R/qGIFXbNqbDo69j4KDjZ6B2zqLk4FsXPvcZ4sV6c9KuZAzXYRbtxjvT0+vJK5oH0cc1hr3+F+ygzRzZi+0xi48IDL21J59tiGAcirROOeAT2v8pvznRjCw+LxCESi5wpl/J/rxsiN0FrLtEWR5NDwgi7tlI0do/YMtjxis5uNHUqckuJJ4U0aLCzTNuV2kDz4qba70L32oIMdsF/y8bc3zxA8IQ0uSuj8URy8RRvb99MiNmJjOzBQ1HkxrjyjL+dhF91Y6+vj4WTP2rGrNd9NWuMF+vFueZANKQG67G9xpfXbR54/evxQ44hBqRzHtt/15hPaXkb4h86dXeV6ah/1GAUaADsIcF/EKgiT1OQ+rDrKsmSXiza6S6mEw0jSTzpWyKgEfFn7VD9SJZoPMG9NkaxZMB9r7p3KjmmNb5gU7leXGS7v6LCFoTixweJHbkiebDM60v3Xp6uvikJ2IUmW82mywKfT5F2oRI7bkvy+voouMox6+2TmjXGi/XivHMAW2qwi7Z7fXlN54HXv4t+KAB4BfZCdjPHv6Re1yHgcT2R/hadnWIIio9U2oIPbW8p6Yw5nra1ncP0deKJM8MxhK7TYMviRnBnt/itMg5hFzf6MEa0PgnPDolgkb5L7HTPkwUBlAD8b6yHX3qM68apXZ/EKxz3iECHIZ93iqPdtBdnjIUv2kK96qYGRTgpUP1Avu4XdazguapvzK61pfTjIYnmpTEWJovq8NltSl5fHws38LX6Z3LMGlCYybGoOnvwMXipDP3vxXpxCBLWPa9qsJH3pN0lI8W/6EvRr9U8kD7uPivZuUTDXmEG6/lSG9Wz/nFidu+MC+XSk7TIOydbjFByKnkc1ElZAhfvmXjCxQC2u2kyPnxFy7uyn5TBEhQJbFOJAEseJbVjwFvgBUNAHgQPGnlxYJXQ8ze1QT86jszHLvBOkxeXtpm0JQXZtWQSEJlI+QBaIzuwneC5tm9M1V1KHmywMU/YTjLdtnc3/1+zZ2keHAuXe3hpzKLn3HypWWO8WC8utyXcT8yBWuyS3SV+c768TvNgJ/+WDN8DzWLgfYIiCm72wHQtC3tin22vgcaLZwIJgepXZQJJCHYqbYITOC2IGb/fdMHvVAxj9E40FsvHKovHZ6LDH9+wQwQDnvdygyDrxaltJyzvOHt/65pjZHinR7JuHDxJ4jNryxY1+vetKOyI+49QQOjepeOI25Yw4Cle5veqvpngvYb8jEbSo3/o0PUDkawP7Bh/De9r1cbr62PhcmdJj8UxK8wuY9FEutaYCPZi53CDOWBKTJRFbK3dDl/etHkw598JV+5EtvXg/K7YsGhZlBxxlbOp52tLSm/iHRw7qkm+qrOnBXu34+XdiTcLOUawC0zfN5YWuA/CcQxxoZwHP3aSpTbIoA1fMIEhMTkJhgxigmz4KwleHAxmEjo8Fi9+jF/UJ7adw03aUpIrOQRhjpunjn/yZnOyA3aBZ8muub7J5a+6l04cHf1HjfmQimDIZHsfM/1Z0kvkdUkydpkv64SOW5VsSn1tpyxXjXPPv7FJAwp6h/mismaN8WK9uIFS3Kj/3fOqBhsF9Xarbd53s/Nf+J3mgdozrv9VZnPyMOqztljt37UCHe16nQiKBBcLUKO2OFPEVU6IHQFPHJono+WdO8CJBwtZp/LjoGJ7fMrTP5ODIEk1uk4lnkrzNLcochTBYtqnKIddIy9lkc2RhBfXqT0f7ZyrnPJn8IkXJ15pmrMlxaEHAR89Rl/XrpQ9yVP87H1Gbd8MdN7lBh3U3h5uAivR6DfS7PjbQvz/RllT/etntALp9bVwPNAhYbZP9o0rmDQ7ZiV/cb4Iw4MdrMPcyWQYLfSxcDYWjZ7CjRY+OvTyTBmozeS8SnFcz2FVt2h3zk/3s76MMjcqV80D6YTv8KM9UMFyVYq8aGs+T/kYba/zMhUwcW0x8PJsArBPsh075jxNifzpMcfxhMtRqTkrr7d7+MxhBk6O/AiUG2NgZay7R+cZzUrR4EMwJKggbxFnbVVegE/u7TL4Qvws8HtxoX3Uo2iLCbBSWCYPfzW+3yHqmh2qPTRUyYav2i7xrOob0/XAJQ9bk38T8cCyD8ne6+tj4YLtGjPMg6Ux6x2L2GLjN/VtaY3xYr04s2dpDvR6OeaL126T7fFlLz+7cM8D6c26MQiqGa+a2yr/1jBeibV1eXMmBizyRljJb7YZx410cp54mmYrvskrsnvq7bg0rTKeOJfEsUVpYiCHpxzDgSVZ+37bvCWHRR6ZtCnxA4a/bEFdxNEgpjfiyU4zT+w6U/28OOMzaYsBKCWbCfBIZf5hzRPRzQ9Vsp08a/sGdfeSpB9/3YKvh+mzkOI1Ph/4IcVE6E0svL4+Fs586hmz3rFYs8Z4sV5cp3HjmVfBbifWa7fbl5LrngfG9MCl278H1sPY2wNU2Cm+F9UGqAH2Vqoz+PDlkk4xprpmgeI3IeHlL3Royp+V2RmmiYWL44Q+CQMvePCnfTZUqGT3RqCyYzFoIzlgY6KOFNpvLwf/spPinaXhQqXuOSLhZxm28/TiaM+fWGIx6pPu+aKW1O/cdO3FhYb6x3ScsqWTHAI8AxE/B/5Witb7Uddu2V6ewtX2jdTY/r5IpQ1WaFPpy1hRwmK3BXxrjx+w2Xbmna7xIcEzH3/W5hjlpF3oqzyaL6K5fH0sXOLExTErrGssyhbXGoNsL7YC551XyPZiXXZX+tI1DxKe+7icG7/uPtuHIg4e+IcNzubOy5cveWIm6HwBwdG4GiK+TACCFfw/KT9SHv02Tzje1RHYBlt03aNjGjQwgA8n0t2VSGHQI8cmHIvkSE7EwYMF8Jn4sJCMkugPRHyhnC6qo989enEIEBa5tkNBP3jzE5KB7704tTWeS7bgW2SX0uDluVe2cG6eCBV+sW+EIWCR6HP6kcDFAwgPIoMvi73YKFcs+nEx6kMqhcMe0uIfId/CDvNvhV3F+YJW4rHo6yPjFudf1M87Xxgri2tM5OnCyoeLOGHcc6AS67I78dHs/I84/EPCLlJxHmyr1v8rO11z2OPf9VrUtZQubFZ4JfY0/H+KuuCrIr4YHQWZOtYN3Txwsz2gOcAphB2N32xjmvbNA80DLg9o3vNQwQ79zVlswW/5Rl8iurg1UPPA7fIA71w3t8ukZk3zQPPAlAc039k5cypIHOwsKLJ15F1ZS80DJ+uBODk+nawDmuHNA6fpAb5v6T/mDEFRiwHvbTgy6j+GOU3fNKtP3APPNQcGf93nxP3RzG8eOAUP8I2HfefR7xQxnC9B+aikpeaBk/RAC4gn2e3N6BP2gOY8J6S8S7RfE3ThQxvziSrYKZZ+x2aQVjYPNA80DzQPNA/ceA8o3vGFLwFx8D2NvVMMBqqSnybwB6/5HL6l5oHmgeaB5oHmgdvqAY5M05/6BTv/D0NFUO6ygltvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left( -0.367885572526119, \\  -1.66667500022579 \\cdot 10^{-5}\\right)$"
      ],
      "text/plain": [
       "(-0.3678855725261193, -1.6666750002257943e-05)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_CDS(fun, x=1., h=.01, dtype=np.float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo sedaj korak:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.00000000e+00, 2.50000000e-01, 6.25000000e-02, 1.56250000e-02,\n",
       "       3.90625000e-03, 9.76562500e-04, 2.44140625e-04, 6.10351562e-05,\n",
       "       1.52587891e-05, 3.81469727e-06])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h=0.25**np.arange(30)\n",
    "h[:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Izračunamo oceno odvodov za različne natančnosti zapisa (zaradi deljenja z 0 dobimo opozorilo):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-62-a9f3bf704e2c>:2: RuntimeWarning: invalid value encountered in true_divide\n",
      "  f1_ocena = (fun(x+h, dtype=dtype)-fun(x-h,dtype=dtype))/(2*dtype(h))\n"
     ]
    }
   ],
   "source": [
    "f1_16 = f1_CDS(fun, x=1., h=h, dtype=np.float16)\n",
    "f1_32 = f1_CDS(fun, x=1., h=h, dtype=np.float32)\n",
    "f1_64 = f1_CDS(fun, x=1., h=h, dtype=np.float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izrišemo različne tipe v odvisnosti od velikosti koraka $h$. Najprej uvozimo potrebne knjižnice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo sliko:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fig_ocena():\n",
    "    plt.semilogx(h, f1_16[0], 'b', lw=3, alpha=0.5, label='float16')\n",
    "    plt.semilogx(h, f1_32[0], 'r', lw=3, alpha=0.5, label='float32')\n",
    "    plt.semilogx(h, f1_64[0], 'g', lw=3, alpha=0.5, label='float64=float')\n",
    "    plt.title('Ocena odvoda za različne tipe natančnosti')\n",
    "    plt.xlabel('$h$')\n",
    "    plt.ylabel('Ocena odvoda: $-\\exp(-1)=−0.3678794$')\n",
    "    plt.ylim(-0.37, -0.365)\n",
    "    plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažimo jo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "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": [
    "fig_ocena()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pri relativno velikem koraku $h$ prevladuje napaka metode, pri majhnem koraku pa zaokrožitvnea napaka; optimalni korak lahko ocenimo glede na:\n",
    "\n",
    "$$h=\\sqrt[3]{\\frac{\\varepsilon}{2\\,k}}.$$\n",
    "\n",
    "V konkretnem primeru velja:\n",
    "\n",
    "$$k=-\\frac{f'''(x)}{6}=-\\frac{-e^{-x}}{6}=\\frac{1}{6\\,e}\\qquad(x=1)$$\n",
    "\n",
    "Sledi:\n",
    "\n",
    "$$h=\\sqrt[3]{3\\,\\varepsilon\\,e}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunamo primeren korak za 16, 32 in 64-bitni zapis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAPCAYAAAB0p1TfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFb0lEQVRoBe2a73FUNxTF1xkX4JgKcDqITQfQQUgqIHQAX/2NsTsAKghOBzYVENxBSAVhtgPn/LS6z9q3etLRxjN84c5o9e/o6F7pviu9Zx/c3d2tvsv9Cpyfn5+o9lTpg8rr+579Sy6niwtNhP9Z6Tbq5KofKTtW/qVs/1Zl6WHp2MMdlgYIfJHr/yr/Seli1GAm1LgrpVOVqxut9rfqDzlW4UUN6+IgEjZ0D94/1La1iRnXmxv9wbzVeIbMZa32H8tG1Xs2u5wuLqb/qLlxzLCTMnK6ybZ/DT1XwsQ6dn3AxLo6NnGTo2rSzzLrjfI/MU85Rn9W/kyp+XRm7HvhvyqdKRGVdiQ41YETXAJQzub8oxzHTvMoT3P3cHk8c/FgvNa4m9xGncTDlmSA84kGwFOzmUj7GsLM17UZrMTiHMAlUv2w3ghriL7sHXu4Vp5kRE9hbR8YwHZ1zKo2cclRNenvAh8pT07KQJWJHNSJLs9oWxKw6ntOv8qvlLFwNWFjOZaSkwJQ+VbpLxXLeVwcFDgk0TM5KQ0SHH3uaDanuHbsVVt6+JSnNVLu2pwUcjgBurhEqkgqfFr3XN/J1G/pKZztAyNYKdTV0bHlhwzC2NtcLrNPqjyVYmz8Q8gvIpk7ELzMXc5j4aQXOB6Kd0qTqJ1TYO5sFqdIsLkmXINe1jqMNpfTxRlTDkNGfGAEO6xIbUA4KkdahN4SF05F//8SbXI4e20e7kPImYvbwFc4DpGfqLEoI5zCTqdKEKqNe9ubqI/mLqeLG53fxI/4wAjWnL4NO9TihAO1kMetTqdP8+BQQGtcjzLHiTA3Di7jz5R/EZ6o+ptSvABcwZMxK5WtuQNf5hp7ojpvruluWvbtW3Y5HZww6ciWLqwhunJHrZ2Oi+oKb/vACDYmdHVs4Yio4TitqOQYEnq1cqIVizkXHA2JeVxc4InEvExdKhFlcVSO+lJcznIMZaJpvAnP+/atu5w9HPbzGQ27eZBIvACPnoAjPjCCZX1cHZs4HNWRiHgOtoV5QWe5kCrjpOs8KK4aXZzGYRhCtNu6o6rtg9L7AgOuywmoFI3noeLuPEXnsn+fssvp4IThLh5rt1KZ9UPX8hPcPmrWxoz4wIR1dezhcNTanTEUjacn7pDRvlcuZVjUx0rPVX6lRMTAGeIlIjmqi9M4JJx7U9v88pkFR+ZqkGSQM4YRnWv80b9P7nK6uLkO6MsVinV1ZcQHRrBL87s6TjjuqHF/iwhVkkfbg21Wdhg2YRK1xdE6zdPDFXrj/EuytVk9zgrJ0peCCtRucjmbONlyrRn51He6MHPs3UL3fXOxlrUx0Zb2ZhBr6ejYQkRFOC62NjW13t9fH+zoy7zzjOOfl6iW0zFmjkOvWEj65zI5/ryjqM85U5d0gZc16elUULWLLqeJ47So2Z5OQXEMvVCJa8QHXKyrYxd3mJf2SnlEtdyUMp5WPtg+yGaJhyjBh/fHwamcxebyP0UGF6cx3MXQfS5wcVJMD9gAZ3CxeEjrqNsg/F+X08G9k021LxGs5WS3r1paR9cHXH9xdeziUkSVwbyMfM2bmWxTGQf6VSm9hNBIm9KdEnfAJXmUO9KTPQMRoeYbj9EvxVlGAAunMbzJE4mnBVZ5R++sg8WZsWTwIOtN1vxt2VwOdDkdHH+G3nppUp2/CiJLf61a1FNjLR+AfADr6tjFHcR/T2lyFocNZ2N4eXqitPNNTri/1Y5zzO+ZEdl4ouHC8Th6r7NhKiYjw6liM7a+eSbQAK7gDD4ekB29CxzFwFbnzlgcmweSf5jhgdgRtVs2x0Dhu5xgB3ERVbGbIMBnujU8IapbegrHunR9AF4XKxw2Ozo2cf8BuaOQQAbcsKEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0.199697177510055$"
      ],
      "text/plain": [
       "0.19969717751005522"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.power(3*eps16*np.exp(1),1/3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.00990623467670444$"
      ],
      "text/plain": [
       "0.009906234676704437"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.power(3*eps32*np.exp(1),1/3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANgAAAATCAYAAADyD8OaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHNUlEQVR4Ae2b0XEUORCG15QDMJABZGBDBOfLAO4iADKA8pvfXJAB5wjuTAZABBzOAGeA2Qx8/6dVqzQzGk2Pdvx001WyNK3uv1tSt6QZloO7u7vNSusMrDPQPgPn5+dPpP1G5XNE4flW/E8PImOt1hlYZ6B9Bkio1yok2EeVpySX6s0hf1ZaZ2Cdgb1n4Dcl1XUfZU2w/oysz+sMNM6AEuxUqkcqX9TeAvO/uSJqwMcMOCfxjlQ43ldaZ2DfGTgTwI3KF5VLxdULADsnWAzCK/FP1A4ZiNAUxSB9F+Weqb5VeSf+4MhERnyCfdSO+tkFcNiI5yvxcT6RVy4qfI3y5hOY0MmuKv+VDr6+Uc1LbKKZtpMejQomyW528I/n95LvjFs8MOhzzblk36Mj+qnyVAVMgmFA4lfXxhQ8mJJx+2i4+9Qe3z1+z/VBmKxPWiM9X+j5u8rBoR5YyEsVkoLkYFLcJH3kP6r+3ZTUZkG/w1MJhlXPsUMAWKAFWD2TYJw44eUx2vLKIc74IAKI4ALnQnhb1TViI7CkzOXm2M71aA8wGZv4bEpp3GqzC35W/VIljVtt15xjSLIsNOMM+qqxY2sTkizy3DHgxHT7iJ+tNMd3p9/Eh21INbfYdG3+Xqid1kdKxj8lwQiwlyCp/VYVBuYQzqSgQFE4BApfVQikh5HnshP1CIo+vRLjq4oFCviTchnItbDDODNetSl55mNAXh8HimKMYaqL8bxWPwlli2W7Iqe58dQMAeCZczA7m5KwtxGfr11hU4SntisGJOvC9Pooub3I67vXb8mxmabDwukcmz+3PtuIH0W9mwdOgJoYL3Y/BM7OmBPB0fKOwxXGM0CvXO6Tu63xsNEQeJQ+NdmewGRxOvYkX7KNL945J2ls0dEz+qbGaWHNrL9WezG9PtZsLdnn9bvF5l9SCqdWVObm8Unzu0iCkUg3lWA4ika9FYvPkctOnutyUrLrGnnlTH5u/afsM3ElarU9iilbfHl6SG0G1WahoHzcPHvnnCC/RaFHFgz0zyUvptfHufZb5b1+t+DzLnym9XqrQpw+Vh1uBIctaLmOAeW82A5XTfWXdtCC+I4leTKf6xDB9UttnOfEyK9OG6+c9BJJJ1xvxHiswjsC7yYD/8TjatgParF21Gi7imnYVssGARGu32p3El3PYfFMNqvTnEsm35wykU7TrjId5tjDHEyPj2N2lubP8bvFtvC5adgHpw7EElfEDiAPMshCE8BFo8jUSPoEkAUVQUawlRLBJRdtEXD/CPuDCn5ReNkHO5Ge8Zv3FNvlU1/eUL/bthcTfMkeq5CMvGMx5n9VJgk9CeVzbsnD4o+RJwlz3b0wCz7m2PfZ3svvfRy7lwSTQ1cqnEQfWpyTHqcXgcHJxVWDwOE9z65MegzB6JKLsnzRTMGmNgkEdv+k4uuQJTeqRYq+TPoYlV2YyAqXjzFsAiTw3ypsAp1xI1egljnnJF+aapgtPi7t3xheze8xnUn+4gmmYCBgeScjQGaT9LjGkQx8iQSHDx6GdannsOuqdslNOECSPREWCbyJmP2EG0DMse3FHBjZ+cNVmSQO/0RRkoEnG6U5vx2TF9929J8VmVJXM+aIjyUbgSd5PpDdqZS+FI/qjXQ0+z2C52YvmmCaDIL+kWrPV8AxJ7kSdq6WwiPQOM1ILrvSeeU20uf9rbZQ9rWTuno1lH3IZVtYJK4LU7JcDY8DevePXRFt3J1e6RTnXHwSEwob0q6Z/hrPM9ak1Io55mMCLjSiLfyz8RekfKxWv33odanDere/V4PgGsOviO202ahtJ4NrISXPwhOQFhzJAfE4zUg0Etgll5R3/4Be2sXCTi48rmUE8HPVXGNyIug55eAzjguVSR8lAzH+SUxhh/dBFNTmS+Jg/PT1SXJTc27X676qnWD0z6VZmA4fR+1Ll011KZrl91JGF0kwTQRBSCB1Th7xCIDJ9xkbjPS3sRDQpaQkscIPKT1yhosPku/7RjdJFYJM/dShTYeR+L/Uxma+ceBn1Uf0JePGlDhJlX4kin6kZ7Hu+CZsz5yzKXDa9ulEDDYVVyL3lN2YTh978Pf26PZ7SQ/6V0R70bMdLtnSZLFrD+7E4rFL4zz9/GQqFfF4uS8t4qgd6RDIg3cO4XAV4qpnieeVk1r4KVfn3Uo49iuNlDgIFoikpuQ0x3auZ+0SJhuA/Ye9ICcf2aCQ7cyj+K45lxyb223EMUzw/lB5FRjDP7W12XgxJefycWh+L86o716/97JeUD7gfzTLOAkCsaOzANcqBDIBnU4gtX+Ixy6bfqYTeUxmidglT6xDba8dduczlfxaR9KVdvFJOexLFx/tFGMDAZsPKVvVAxKfhESHOYG4nn4TP3wZVe3yMWjGPw5MbOUJj31+89gfN+vgnXPWk1OMcfJR47nKhTBZ40R6dq0NCpKdxJSM28fkRGPD67vH70YXRtX+A93RaAcTmYNzAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1.21885484320102 \\cdot 10^{-5}$"
      ],
      "text/plain": [
       "1.2188548432010213e-05"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.power(3*eps64*np.exp(1),1/3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Najbolje se izkaže 64-bitni zapis, vendar pa tudi pri tem korak manjši od cca `1e-5` ni priporočen!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Aproksimacija odvoda s kompleksnim korakom*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Aproksimacija odvoda s kompleksnim korakom je uporabna takrat, ko imamo definirano analitično funkcijo, vendar pa nimamo na voljo analitičnega izraza za prvi odvod, za podrobnosti glejte [vir](https://nhigham.com/2020/10/06/what-is-the-complex-step-approximation/). Ideja izhaja iz razvoja funkcije $f(x)$ v Taylorjevo vrsto, vendar se izvede korak v smeri imaginarne osi $i\\,h$ ($i=\\sqrt(-1)$):\n",
    "\n",
    "$$f{\\left(x + i\\,h \\right)} = f{\\left(x \\right)} + i\\,h\\,\\frac{d}{d x} f{\\left(x \\right)} - \\frac{h^{2}}{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + O\\left(h^{3}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če sedaj predpostavimo, da funkcija $f(x)$ realne vrednosti slika na realno os in da sta $x$ in $h$ realni vrednosti, potem lahko izpeljemo:\n",
    "\n",
    "$$\\mathbf{Im} \\left( f{\\left(x + i\\,h \\right)}\\right)= h\\,\\frac{d}{d x} f{\\left(x \\right)}+ O\\left(h^{3}\\right)$$\n",
    "\n",
    "$$\\mathbf{Re} \\left( f{\\left(x + i\\,h \\right)}\\right) = f{\\left(x \\right)} - \\frac{h^{2}}{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + O\\left(h^{3}\\right)\\rightarrow\n",
    "\\mathbf{Re} \\left( f{\\left(x + i\\,h \\right)}\\right) = f{\\left(x \\right)} + O\\left(h^{2}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Iz prve enačbe (zgoraj) potem izpeljemo izraz za prvi odvod:\n",
    "\n",
    "$$f'{\\left(x \\right)}=\\frac{\\mathbf{Im} \\left( f{\\left(x + i\\,h \\right)}\\right)}{h}+ O\\left(h^{2}\\right)$$\n",
    "\n",
    "Rezultat je na nek način zelo presenetljiv. Zgoraj smo z eno koračno shemo naprej uspeli pridobiti natančnost $O\\left(h^{1}\\right)$, tukaj pa $O\\left(h^{2}\\right)$!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če pogledamo sedaj uporabo metoda na primeru od zgoraj $\\exp(-x)$, spomnimo se najprej točnega rezultata pri vrednosti $x=1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.367879441171442$"
      ],
      "text/plain": [
       "-0.36787944117144233"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_točno0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pri klicu s korakom $h=0.1$ dobimo točni dve (tri) števki (podobno kakor pri centralni diferenčni shemi), zmanjšanjem koraka na desetino pa bi število točnih števk približno podvojili:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.367266615262709$"
      ],
      "text/plain": [
       "-0.3672666152627091"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h1_kompleksno = 0.1\n",
    "y1_kompleksno = np.exp(-(1.+1.j*h1_kompleksno))\n",
    "f1_kompleksno = np.imag(y1_kompleksno)/h1_kompleksno\n",
    "f1_kompleksno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Nekaj vprašanj za razmislek!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Za batni mehanizem na spodnji sliki strojno izpeljite kinematiko gibanja bata, če se kolenasta gred giblje po zakonu $\\varphi(t)=\\omega\\,t$.\n",
    "<img src='./fig/batni_mehanizem.png' width=300>\n",
    "2. Za kotno hitrosti $\\omega=2\\,\\pi\\,50\\,$rad/s izrišite lego bata v treh obratih gredi. Uporabite: $r=0,03\\,$m in $l=0,1\\,$m.\n",
    "3. Simbolno odvajajte lego $x(t)$, da pridobite pospešek $\\ddot x(t)$.\n",
    "4. Pripravite funkcijo za klicanje simbolnih izrazov za lego $x(t)$ in pospešek $\\ddot x(t)$ iz ``numpy``.\n",
    "5. S pomočjo ``scipy`` pripravite centralno diferenčno shemo za 2. odvod čez 3, 5, in 7 točk.\n",
    "6. Raziščite funkcijo ``numpy.convolve`` in z njo na podlagi numeričnih vrednosti za $x$ numerično izračunajte pospešek $\\ddot x$. Kje je odvod pravilen?\n",
    "7. S centralno diferenčno shemo 2. odvoda čez tri točke ste izračunali notranje točke, nastavite diferenčno shemo naprej za izračun prve točke z natančnostjo $\\mathcal{O}(h^{2})$.\n",
    "8. Dodajte podatkom lege, določeno mero šuma in preverite, zakaj ni primerna uporaba numeričnega odvajanja na šumnih podatkih.\n",
    "9. S centralno diferenčno shemo 2. odvoda čez tri točke ste doslej izračunali notranje točke, nastavite diferenčno shemo nazaj za izračun zadnje točke z natančnostjo $\\mathcal{O}(h^{2})$.\n",
    "10. Raziščite vpliv časovnega koraka na izračun 2. odvoda.\n",
    "11. Izmerjene imamo sledeče pozicije (gibanja) avtomobila:\n",
    "\n",
    "    $t = [ 0. ,  0.1,  0.2,  0.3,  0.4,  0.5,  0.6,  0.7,  0.8,  0.9,  1. ]$ [h]\n",
    "\n",
    "    $s = [0, 3, 10, 13, 17, 25, 33, 46, 58, 75]$ [km]\n",
    "\n",
    "    Izračunajte hitrost in pospešek avtomobila pri času 0,5 h. Hitrost in pospešek prikažite tudi v grafični obliki.\n",
    "    \n",
    "Opomba:\n",
    "Dodatek k domačim nalogam: 6. vprašanje bi lahko nadaljevali in zašumljene podatke gladili ter nato izvedli odvajanje. Glajenje izvedite s konvolucijo med [0,21194156,  0,57611688,  0,21194156] in $x$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dodatno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Video predavanja na temo numeričnega odvajanja](https://www.youtube.com/watch?v=ZJkGI5DZQv8&list=PLYdroRCLMg5OvLx1EtY1ByvveJeTEXQd_&index=18)."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "pinm",
   "language": "python",
   "name": "pinm"
  },
  "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": "371px",
    "width": "252px"
   },
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Kazalo",
   "title_sidebar": "Kazalo",
   "toc_cell": true,
   "toc_position": {
    "height": "calc(100% - 180px)",
    "left": "10px",
    "top": "150px",
    "width": "235.599px"
   },
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}