Predavanje 11 - Numerično integriranje.ipynb

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   "source": [
    "<font size=\"2\"> [github.com/jankoslavic/pypinm](http://github.com/jankoslavic/pypinm)</font>\n",
    "\n",
    "<center>\n",
    "<img width=300 src=\"./fig/ladisk.png\">\n",
    "<br>\n",
    "<font size=\"7\" color=\"f00e0e\" face=\"garamond\"><b>Numerično integriranje</b></font>\n",
    "<br><br>\n",
    "<font size=\"4\">\n",
    "[izr. prof. dr. Janko Slavič](http://www.ladisk.si/~slavic)\n",
    "</font>\n",
    "<br><br>\n",
    "<font size=\"2\">November 2017 </font>\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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    },
    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#Uvod\" data-toc-modified-id=\"Uvod-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Uvod</a></div><div class=\"lev2 toc-item\"><a href=\"#Motivacijski-primer\" data-toc-modified-id=\"Motivacijski-primer-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Motivacijski primer</a></div><div class=\"lev1 toc-item\"><a href=\"#Newton-Cotesov-pristop\" data-toc-modified-id=\"Newton-Cotesov-pristop-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Newton-Cotesov pristop</a></div><div class=\"lev2 toc-item\"><a href=\"#Trapezno-pravilo\" data-toc-modified-id=\"Trapezno-pravilo-21\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Trapezno pravilo</a></div><div class=\"lev2 toc-item\"><a href=\"#Sestavljeno-trapezno-pravilo\" data-toc-modified-id=\"Sestavljeno-trapezno-pravilo-22\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Sestavljeno trapezno pravilo</a></div><div class=\"lev2 toc-item\"><a href=\"#Simpsonova-in-druge-metode\" data-toc-modified-id=\"Simpsonova-in-druge-metode-23\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Simpsonova in druge metode</a></div><div class=\"lev2 toc-item\"><a href=\"#Sestavljeno-Simpsonovo-pravilo\" data-toc-modified-id=\"Sestavljeno-Simpsonovo-pravilo-24\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>Sestavljeno Simpsonovo pravilo</a></div><div class=\"lev2 toc-item\"><a href=\"#Rombergova-metoda*\" data-toc-modified-id=\"Rombergova-metoda*-25\"><span class=\"toc-item-num\">2.5&nbsp;&nbsp;</span>Rombergova metoda*</a></div><div class=\"lev1 toc-item\"><a href=\"#Gaussov-integracijski-pristop\" data-toc-modified-id=\"Gaussov-integracijski-pristop-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Gaussov integracijski pristop</a></div><div class=\"lev2 toc-item\"><a href=\"#Gaussova-kvadratura-z-enim-vozliščem\" data-toc-modified-id=\"Gaussova-kvadratura-z-enim-vozliščem-31\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Gaussova kvadratura z enim vozliščem</a></div><div class=\"lev2 toc-item\"><a href=\"#Gaussova-integracijska-metoda-z-več-vozlišči\" data-toc-modified-id=\"Gaussova-integracijska-metoda-z-več-vozlišči-32\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Gaussova integracijska metoda z več vozlišči</a></div><div class=\"lev1 toc-item\"><a href=\"#scipy.integrate\" data-toc-modified-id=\"scipy.integrate-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span><code>scipy.integrate</code></a></div><div class=\"lev2 toc-item\"><a href=\"#Intergracijske-funkcije,-ki-zahtevajo-definicijsko-funkcijo:\" data-toc-modified-id=\"Intergracijske-funkcije,-ki-zahtevajo-definicijsko-funkcijo:-41\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Intergracijske funkcije, ki zahtevajo definicijsko <em>funkcijo</em>:</a></div><div class=\"lev2 toc-item\"><a href=\"#Intergracijske-funkcije,-ki-zahtevajo-tabelo-vrednosti:\" data-toc-modified-id=\"Intergracijske-funkcije,-ki-zahtevajo-tabelo-vrednosti:-42\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>Intergracijske funkcije, ki zahtevajo tabelo vrednosti:</a></div><div class=\"lev1 toc-item\"><a href=\"#Nekaj-vprašanj-za-razmislek!\" data-toc-modified-id=\"Nekaj-vprašanj-za-razmislek!-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Nekaj vprašanj za razmislek!</a></div><div class=\"lev1 toc-item\"><a href=\"#Dodatno\" data-toc-modified-id=\"Dodatno-6\"><span class=\"toc-item-num\">6&nbsp;&nbsp;</span>Dodatno</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Uvod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V okviru tega poglavja bomo za dano funkcijo $f(x)$ izračunali določen integral:\n",
    "$$\\int_a^b\\,f(x)\\,\\textrm{d}x$$\n",
    "kjer sta $a$ in $b$ meji integriranja, $f(x)$ pa so vrednosti funkcije, ki jih pridobimo iz tabele vrednosti ali s pomočjo analitične funkcije."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "subslide"
    }
   },
   "source": [
    "Pri numeričnem integriranju integral ocenimo z $I$ in velja\n",
    "$$\\int_a^b\\,f(x)\\,\\textrm{d}x= I + E,$$\n",
    "kjer je $E$ napaka ocene integrala."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Numerični integral bomo računali na podlagi diskretne vsote:\n",
    "$$I=\\sum_{i=0}^{m}A_i\\,f(x_i),$$\n",
    "kjer so $A_i$ uteži, $x_i$ pa vozlišča na intervalu $[a, b]$ in je $m+1$ število vozlišč."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "Ogledali si bomo dva različna pristopa k numerični integraciji:\n",
    "\n",
    "1. *Newton-Cotesov pristop*, ki temelji na ekvidistantnih vozliščih (konstanten korak integracije) in \n",
    "2. *Gaussov integracijski pristop*, kjer so vozlišča postavljena tako, da se doseže natančnost za polinome."
   ]
  },
  {
   "cell_type": "markdown",
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   },
   "source": [
    "## Motivacijski primer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pri numeričnem integriranju si bomo pomagali s konkretnim primerom:\n",
    "\n",
    "$$\\int_1^2 x\\,\\sin(x)\\,\\textrm{d}x$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "Pripravimo si vozlišča. Osnovni korak naj bo $h=0.25$, v tem primeru imamo štiri štiri podintervale in pet vozlišč, pri koraku $2h$ so tri vozliščne točke in pri koraku $4h$ samo dve (skrajni):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "xg, hg = np.linspace(1, 2, 100, retstep=True) # goste točke (za prikaz)\n",
    "x2v, h2v = np.linspace(1, 2, 2, retstep=True)  # korak h2v = 1 (2 vozlišči)\n",
    "x3v, h3v = np.linspace(1, 2, 3, retstep=True)  # korak h3v = 0.5 (3 vozlišča)\n",
    "x4v, h4v = np.linspace(1, 2, 4, retstep=True)  # korak h4v = 0.33.. (4 vozlišča)\n",
    "x5v, h5v = np.linspace(1, 2, 5, retstep=True)  # korak h5v = 0.25 (5 vozlišč)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo še funkcijske vrednosti:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "yg = xg * np.sin(xg)\n",
    "y2v = x2v * np.sin(x2v)\n",
    "y3v = x3v * np.sin(x3v)\n",
    "y4v = x4v * np.sin(x4v)\n",
    "y5v = x5v * np.sin(x5v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo prikaz podatkov:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import rc # to uvozimo, da so fonti na sliki latex ustrezni\n",
    "rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})\n",
    "rc('text', usetex=True)\n",
    "%matplotlib inline\n",
    "def fig_motivacija():\n",
    "    plt.fill_between(xg, yg, alpha=0.25, facecolor='r')\n",
    "    plt.annotate('$\\int_1^2\\,x\\,\\sin(x)\\,\\\\textrm{d}x$', (1.3, 0.5), fontsize=22)\n",
    "    plt.plot(xg, yg, lw=3, alpha=0.5, label='$x\\,\\sin(x)$')\n",
    "    plt.plot(x2v, y2v, 's', alpha=0.5, label=f'$h={h2v}$', markersize=14)\n",
    "    plt.plot(x3v, y3v, 'o', alpha=0.5, label=f'$h={h3v}$', markersize=10)\n",
    "    plt.legend(loc=(1.01, 0))\n",
    "    plt.ylim(0, 2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažimo podatke:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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hsqSbuyj8+bYf16fCxry8SAQIBIFYDFYBHnda8GSDzVx5pQEbky/hg8B/IZiKosnihF2q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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x27536b3c128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_motivacija()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Analitično izračunajmo točen rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAAPBAMAAAC/7vi3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdMrsi\niUTmZFZyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACt0lEQVQ4EbWUO2zTUBSGf8dpnnZqqWVhSUWR\nEJTBVKWIoYpVUhRY4qEVAxQFxENCRQ0qW4ZEHWABtUjdAOEBBqZkYKJIWEhIICHVA48KFKUrU0lT\nKFDAnHOvafqYucPv/+g79/f1ta+BjZGwyZKo47cd6bSxeXIvTz+lEo8CJ7A6/tYFBq+eBQ4vftxB\nHo/U6x6g9PJEGvGalKih3pNuAPofqBYaS1TfDZzA/RXtIGAjbOAajrqyp01mfN+3MDDaonk8znM0\nyTTwQbpjwB6ETYR6gEhf4AQeBkagVKAVYz3Qi7KnTc4AYYqIBNHaTYpmeQKkTeH2A3knNIvwT+BV\nCdIJvAY0oNegziYtKK3thDaDdmkjWo9SNMsqsGwIVzYpOtoS0V4J0jHuWgcmndS6qVc6LaSa2wmQ\nKmyKvsXRJOpXmltgRxAPTJJ4CzG3xGW8JfE+WvUS8t8XkbGR+raDIMndwYaoBcpi0X4DVU+UdHMq\n6E4eBiGilz2JG0C5At2fwoKHBDdtJVjkeUG0Dopm0WjVVU+UtEqLW+YAT0bPQeJoQWkY6P68srRg\ny+itRP3C84Lo3RzNIp9YlMAYd3TYUBwRTS7YrxtXGk7MwmRvsCFbCSLFdrTqcSoLv8Z0l3CI2Nxx\nERiCiCYnsEHXvBlykWjSa1SaomcTkWckWLWSzeb3nmMpzAMzh4TDLmguFBtvurPZlePCgbFJ0deR\nIc0la4i1thNkLGL/NgQI1agioS//lHR06iMun/AJqvvEWZ+QeNpN/AKtGs/oyHQUZY+YKAiq4nF5\nrzNFmtvJ0SRxQz0g3ev6uxPQ7tdHZ6leD5zAl80jFaSmECvgDvod2dMmSHN0x/BaTnwI+vNV+pxI\n1PcXHDo85Mq+/wNR+h9Q9Cc/J53A+hj9mPBwnHTo0smgZxN5YRD+f+Mv8rQ1O7ARp8wAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$1.4404224209802097$$"
      ],
      "text/plain": [
       "1.4404224209802097"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "x = sym.symbols('x')\n",
    "I_točno = float(sym.integrate(x*sym.sin(x), (x, 1, 2)).evalf())\n",
    "I_točno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Newton-Cotesov pristop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Na sliki je prikazan splošen primer, kjer je razdalja med vozlišči $x_i$ enaka $h$ (gre za **ekvidistantno delitev**).\n",
    "<img src='./fig/integriranje.png' width=400>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "V okviru tega poglavja si bomo najprej pogledali trapezno ter sestavljeno trapezno pravilo, pozneje pa si bomo pogledali Simpsonovo ter Romergovo metode."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Trapezno pravilo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trapezno pravilo vrednosti podintegralske funkcije $f(x)$ na (pod)intervalu $[x_0, x_1]$ interpolira z linearno funkcijo. Za dve vozliščni točki to pomeni, da površino pod grafom funkcije približno izračunamo kot:\n",
    "\n",
    "$$I_{\\textrm{trapezno}}=\\sum_{i=0}^{n=1}A_i\\,f(x_i)=\\frac{h}{2}\\cdot\\left(f(x_0)+f(x_1)\\right).$$\n",
    "\n",
    "In so uteži:\n",
    "\n",
    "$$A_0 = A_1 = \\frac{1}{2}\\,h.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerična implementacija"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Numerična implementacija je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def trapezno(y, h):\n",
    "    \"\"\"\n",
    "    Trapezno pravilo integriranja.\n",
    "    \n",
    "    :param y: funkcijske vrednosti\n",
    "    :param h: korak integriranja\n",
    "    \"\"\"\n",
    "    return (y[0] + y[-1])*h/2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerični zgled\n",
    "\n",
    "V konkretnem primeru to pomeni, da prvo in zadnjo funkcijsko vrednost utežimo s $h/2$. V našem primeru je $h=1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nol07Ldr/e/4C5w3bo+Z2+9UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.33003291923$$"
      ],
      "text/plain": [
       "1.33003291923"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_trapezno = trapezno(y2v, h=h2v)\n",
    "I_trapezno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo sliko:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fig_trapezno():\n",
    "    plt.fill_between(x2v, y2v, alpha=0.25, facecolor='r')\n",
    "    plt.vlines(x2v, 0, y2v, color='r', linestyles='dashed', lw=1)\n",
    "    plt.annotate('$I_{\\\\textrm{trapezno}}$', (1.4, 0.5), fontsize=22)\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.5, 1.4), xytext=(1.6, 1.8),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.plot(xg, yg, lw=3, alpha=0.5, label='$x\\,\\sin(x)$')\n",
    "    plt.plot(x2v, y2v, 'o', alpha=0.5, label=f'$h={h2v}$')\n",
    "    plt.legend(loc=(1.01, 0))\n",
    "    plt.ylim(0, 2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BrjH8iKLk+eefx+DgIL7xjW94t/3+97+HyWTyW4F9Oj39TlzscOBihx0dfcNTHrcoKxlF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\n7vx/OxfnC0nQ7ukAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x27538d87208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_trapezno()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Napaka trapeznega pravila"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Razlika med analitično vrednostjo integrala in numeričnim približkom $I$ je napaka metode:\n",
    "\n",
    "$$E = \\int_a^bf(x)\\,d x-I,$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če je funkcija $f(x)$ vsaj dvakrat odvedljiva, se lahko (glejte npr. vir: Burden, Faires, Burden: Numerical Analysis 10th Ed) izpelje ocena napake, ki velja samo za trapezni približek prek celega integracijskega intervala:\n",
    "$$E_{\\textrm{trapezno}}=-\\frac{h^3}{12}f''(\\xi),$$\n",
    "kjer je $h=b-a$ in $\\xi$ neznana vrednost na intervalu $[a,b]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sestavljeno trapezno pravilo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Če razdelimo interval $[a, b]$ na $n$ podintervalov in na vsakem uporabimo trapezno pravilo integriranja,  govorimo o *sestavljenem trapeznem pravilu* (angl. *composite trapezoidal rule*)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V tem primeru za vsak podinterval $i$ uporabimo trapezno pravilo in torej za meje podinterval $x_i$ in $x_{i+1}$ uporabimo uteži $A_i=A_{i+i}=h/2$. Ker so notranja vozlišča podvojena, sledi:\n",
    "$$A_0=A_{n}=\\frac{h}{2}\\quad\\textrm{in za ostala vozlišča:}\\quad A_i=h.$$\n",
    "Pri tem smo predpostavili podintervale enake širine:\n",
    "$$h=\\frac{x_{n}-x_0}{n}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sledi torej:\n",
    "$$I_{\\textrm{trapezno sest}}=\\sum_{i=0}^{n}A_i\\,f(x_i)=\\left(\\frac{y_0}{2} + y_1+y_2+\\cdots+y_{n-1}+\\frac{y_{n}}{2}\\right)\\,h.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerična implementacija"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Numerična implementacija je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def trapezno_sest(y, h):\n",
    "    \"\"\"\n",
    "    Sestavljeno trapezno pravilo integriranja.\n",
    "    \n",
    "    :param y: funkcijske vrednosti\n",
    "    :param h: korak integriranja\n",
    "    \"\"\"    \n",
    "    return (np.sum(y) - 0.5*y[0] - 0.5*y[-1])*h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerični zgled\n",
    "\n",
    "Zgoraj smo že pripravili podatke za dva podintervala (tri vozlišča):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1. ,  1.5,  2. ])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x3v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoklEQVQIHWNgEDIxZQCBPIGgAgaGdAaJCSDe\n+v+fGRhYFjAwJwA5DMnngILcBxjYP4J4B0AE/wEG3u8gxgEQIa/AwPsPxDjip8rAMN+AgRWonYFh\nI8P8CwzzFaA8oBENCJUMDEw/Qaawg0zhSWBg/svAvYGBBWQDVwID00eQ7UwJQB4TAwMHkG5nkApg\n/MbAqcDgBXSM8LujDAw9DAyTbBQZAHJpJhsPt5MdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0.5$$"
      ],
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h3v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunajmo oceno integrala s sestavljenim trapeznim pravilom:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAPBAMAAADe9tr1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nyjasWjLzv+Mf2U7EqeO3JO4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.41313769957$$"
      ],
      "text/plain": [
       "1.41313769957"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_trapezno_sest = trapezno_sest(y3v, h=h3v)\n",
    "I_trapezno_sest"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo sliko:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fig_trapezno_sest():\n",
    "    plt.fill_between(x3v, y3v, alpha=0.25, facecolor='r')\n",
    "    plt.vlines(x3v, 0, y3v, color='r', linestyles='dashed', lw=1)\n",
    "    plt.annotate('$I_{\\\\textrm{trapezno sestavljeno}}$', (1.2, 0.5), fontsize=22)\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.75, 1.68), xytext=(1.4, 1.8),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.2, 1.1), xytext=(1., 1.4),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.plot(xg, yg, lw=3, alpha=0.5, label='$x\\,\\sin(x)$')\n",
    "    plt.plot(x3v, y3v, 'o', alpha=0.5, label=f'$h={h3v}$')\n",
    "    plt.legend(loc=(1.01, 0))\n",
    "    plt.ylim(0, 2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ehG8TpifZJVsTZjR+/WtKfiKi5EdIknE4XegZnYRhyIqu4fFZZ1HJy0zFrUU5uLU4G7co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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x27538da77f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_trapezno_sest()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Napaka sestavljenega trapeznega pravila"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Napaka sestavljenega trapeznega pravila izhaja iz napake trapeznega pravila; pri tem tako napako naredimo $n$-krat."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ker velja $h\\cdot n=b-a$, izpeljemo napako sestavljenega trapeznega pravila kot:\n",
    "\n",
    "$$E_{\\textrm{trapezno sest}}=-\\frac{h^2(b-a)}{12}f''(\\eta),$$\n",
    "\n",
    "$\\eta$ je vrednost na intervalu $[a,b]$. Napaka je drugega reda $\\mathcal{O}(h^2)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Boljši približek integrala\n",
    "\n",
    "V nadaljevanju si bomo pogledali t. i. **Richardsonovo ekstrapolacijo**, pri kateri na podlagi ocene integrala s korakom $h$ in $2h$ izračunamo boljši približek."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V kolikor integral $I$ numerično izračunamo pri dveh različnih korakih $h$ in $2\\,h$, velja:\n",
    "\n",
    "$$\\int_a^b f(x)\\,\\textrm{d}x = I_h + E_h = I_{2h} + E_{2h},$$\n",
    "\n",
    "kjer sta $I_h$ in $I_{2h}$ približka integrala s korakom $h$ in $2h$ ter $E_h$ in $E_{2h}$ oceni napake pri koraku $h$ in $2h$. Izpeljemo $I_{2h}-I_{h} = E_{h}-E_{2h}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Naprej zapišemo:\n",
    "\n",
    "$$E_h=-\\frac{h^2(b-a)}{12}f''(\\eta)=h^2\\,K.$$\n",
    "\n",
    "Ob predpostavki, da je $f''\\left (\\eta \\right )$ pri koraku $h$ in $2h$ enak ($\\eta$ je pri koraku $h$ in $2h$ dejansko različen), zapišemo:\n",
    "$$E_{2h}=-\\frac{(2h)^2(b-a)}{12}f''(\\eta)=4\\,h^2\\,K$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sledi:\n",
    "\n",
    "$$I_{2h}-I_h=-3K\\,h^2.$$\n",
    "\n",
    "Sedaj lahko ocenimo napako pri koraku $h$:\n",
    "\n",
    "$$E_h=h^2\\,K=\\frac{I_h-I_{2h}}{3}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Na podlagi ocene napake lahko izračunamo boljši numerični približek  $I_{h}^*$:\n",
    "$$I_h^* = I_h + \\frac{1}{3}\\,(I_{h}-I_{2h})$$\n",
    "ali\n",
    "$$I_h^* = \\frac{4}{3}\\,I_h - \\frac{1}{3}\\,I_{2h}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerični zgled"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Predhodno smo s trapeznim pravilom že izračunali integral pri koraku $h=1$ in pri koraku $h=0,5$, rezultata sta bila:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASkAAAAUBAMAAAAjJNZ2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDFVqnfRBZVJteWLQxf/VutBj3r+Q98PVPrFcqgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ 1.33003291923, \\quad 1.41313769957\\right ]$$"
      ],
      "text/plain": [
       "[1.33003291923, 1.41313769957]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[I_trapezno, I_trapezno_sest]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "S pomočjo zgornje formule izračunamo boljši približek:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Točen rezultat:   1.4404224209802097\n",
      "Boljši približek: 1.4408392930139313\n"
     ]
    }
   ],
   "source": [
    "I_trapezno_boljši = 4/3*I_trapezno_sest - 1/3*I_trapezno\n",
    "print(f'Točen rezultat:   {I_točno}\\nBoljši približek: {I_trapezno_boljši}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### ``numpy`` implementacija sestavljenega trapeznega pravila"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sestavljeno trapezno pravilo je implementirano tudi v paketu ``numpy``, s funkcijo ``numpy.trapz``:\n",
    "\n",
    "```python\n",
    "trapz(y, x=None, dx=1.0, axis=-1)\n",
    "```\n",
    "\n",
    "* `y` predstavlja tabelo funkcijskih vrednosti, \n",
    "* `x` je opcijski parameter in definira vozlišča; če parameter ni definiran, se privzame ekvidistančna vozlišča na razdalji `dx`,\n",
    "* `dx` definira (konstanten) korak integracije, ima privzeto vrednost 1,\n",
    "* `axis` definira *os* po kateri se integrira (v primeru, da je `y` večdimenzijsko numerično polje).\n",
    "\n",
    "Funkcija vrne izračunani integral po sestavljenem trapeznem pravilu. Več informacij lahko najdete v [dokumentaciji](https://docs.scipy.org/doc/numpy/reference/generated/numpy.trapz.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAPBAMAAADe9tr1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdMrsi\niUTmZFZyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACJElEQVQoFaWSPWgTYRjHf5dLcvm65mZBjFZQ\nsMhRlOKUUBoUURrBTJo2KCqI0oAixcXgoB0EHRQHB7M6SLM4ODUUuilkqCLKkeDgXGm0tZacz3sf\n6e2+3Mf/ed73+d3/nveF0chUfJlzvlkRlXlecxxwbrc46ZwDvbbewbiaucCbsuN0QRsPEOm2H6yh\nbUdUynXdAVP9REM/ymqHyWZugpjrdnkqMyVOXNwMAFfafrBg8SeiEh0o8Y5sO14gXmEayhjrt+Ay\nxKU2GQByj8WBCr739a2IMsFsje3KynwDcwOZ63mFiP2vEYBpBACxL7/gc5WCH2QL8iq2Se7of2Gh\npb4sY6wuj9DBkz3AKWH7aaWgTf7+5zli4mDIEXHQjy9da8lEVu4QoNdHgI8v7CDtKcwmxUVilvg3\ntsU/D5pZOzeU2i8RgMkIgHk85CpFUa5NEuPcY/9PjLrWsyT7XnZ0IwLYFwFwyA6NieKG9O8ZqSGp\n62934NHNXksK5zskG3sAvTsCfIAVywP4ileQbpBS/Yx7WzZrn1aNQJ2c0Ks2MzN7uO6VuXYI8JU+\nkMKCcgBGSZXcVZ2ctyl6UbgLxISngmPwWuyNVOYXZKQHhdQEyx0edjK7VOAlLMvLryk2ROQDwBLa\nbz/tK00AnGfSSt7RF6Uj9lSTNYTGAQVITG+dIV2S47YyaHrBWPVS3097itxBWWZWz8Knal+pOcjV\nyjasWjLzv+Mf2U7EqeO3JO4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.41313769957$$"
      ],
      "text/plain": [
       "1.41313769957"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#%%timeit\n",
    "I_trapezno_np = np.trapz(y3v, dx=h3v)\n",
    "I_trapezno_np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Simpsonova in druge metode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zgoraj smo si pogledali trapezno pravilo, ki temelji na linearni interpolacijski funkciji na posameznem podintervalu. Z interpolacijo višjega reda lahko izpeljemo še druge integracijske metode."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Izračunati moramo:\n",
    "$$\\int_{a}^b f(x)\\,dx.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Tabeliramo podintegralsko funkcijo $f(x)$ in tabelo interpoliramo z Lagrangevim interpolacijskim polinomom $P_n(x)$ stopnje $n$:\n",
    "\n",
    "$$P_n(x)=\\sum_{i=0}^{n}\\,f(x_i)\\,l_i(x),$$\n",
    "\n",
    "kjer so $y_i=f(x_i)$ funkcijske vrednosti v vozliščih $x_i$ in je Lagrangev polinom $l_i$ definiran kot:\n",
    "\n",
    "$$l_i(x)=\\prod_{j=0, j\\ne i}^n \\frac{x-x_j}{x_i-x_j}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Za numerični izračun integrala $\\int_{a}^b f(x)\\,dx$ (meje so: $a=x_0$, $b=x_n$) namesto funkcije $f(x)$ vstavimo v integral interpolacijski polinom $P_n(x)$:\n",
    "\n",
    "$$I=\\int_{x_0}^{x_{n}} P_n(x)\\,dx=\\int_{x_0}^{x_{n}} \\sum_{i=0}^{n}\\,f(x_i)\\,l_i(x)\\,dx.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ker je integriranje linearna operacija, lahko zamenjamo integriranje in vsoto:\n",
    "\n",
    "$$I=\\sum_{i=0}^{n}\\,f(x_i)\\,\\underbrace{\\int_{x_0}^{x_{n}} l_i(x)\\,dx}_{A_i}.$$\n",
    "\n",
    "Lagrangev polinom integriramo in dobimo uteži $A_i$:\n",
    "\n",
    "$$A_i = \\int_{x_0}^{x_{n}} l_i(x)\\,dx$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Izpeljava trapeznega pravila z uporabo Lagrangevih polinomov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si kako z Lagrangevim interpolacijskim polinomom prve stopnje strojno izpeljemo uteži $A_i$ v primeru trapeznega pravila.\n",
    "\n",
    "Najprej v simbolni obliki definirajmo spremenljivko `x`, vozlišči `x0` in `x1` ter korak `h`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, x0, x1, h = sym.symbols('x x0, x1, h')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo Python funkcijo, ki v simbolni obliki vrne seznam $n$ koeficientov Lagrangevih polinomov $[l_0(x), l_1(x),\\dots, l_{n-1}(x)]$ stopnje $n-1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lagrange(n, x, vozlišča_predpona='x'):\n",
    "    if isinstance(vozlišča_predpona, str):\n",
    "        vozlišča = sym.symbols('{0:s}:{1:g}'.format(vozlišča_predpona, n)) \n",
    "    coeffs = []\n",
    "    for i in range(0, n):\n",
    "        numer = []\n",
    "        denom = []\n",
    "\n",
    "        for j in range(0, n):\n",
    "            if i == j:\n",
    "                continue\n",
    "\n",
    "            numer.append(x    - vozlišča[j])\n",
    "            denom.append(vozlišča[i] - vozlišča[j])\n",
    "        \n",
    "        numer = sym.Mul(*numer)\n",
    "        denom = sym.Mul(*denom)\n",
    "\n",
    "        coeffs.append(numer/denom)\n",
    "    return coeffs    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej poglejmo Lagrangeva polinoma za linearno interpolacijo ($n=2$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAyBAMAAAAASDLRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2qyJEZt3N\nu5lcG7FTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAC+ElEQVRYCe2Yv4sTQRTH3yW74cwqCZzYmsar\n5DwLf4BNwLMTs1icFsItgTu70+oQbK7VQq6yNVhqo4UHegemECyM4H+gCCJoE/RyahVndt7O7JuZ\nzO6eJlXS7Jvvvvf9ZGY38CVwfPgDxvspDYd1mFu6NF4K+EsX63B0zBBu700xxU954ofmr915OE+/\n5/LNe+2ISubKbLI4qd1chafRLnHxG9XtoAtel6jaApvSqnCqzDeUqDAn4FR4Xt1gVSksDcqb3tdn\nRNUWoomIwukxnFOqwoTwPpb9dzvss9sE8OHwByaVnRhsAjUGwukl1ELJURiAfakmRS1ilRsDEDcl\nE+LKnPw9mFmUagrj7Uk1KVpNVmVh4qZkIr5yJ+83BB2pKsyZSh9Wpc6LcngLKmEGBpvSg7GTN4Ag\nkqrEHPp5pD/bkTovHi1+hPtZu8Gm1KBwYruZiaQqMf7G6voTKcfFtbXlb50sDDalJoUTeza1LalK\njFSMIuvZGANCGPmmjegv10fccMvr9t/NqKHZ77+iUfdceunzF3U7x6Gp5oNXU8wBzm6yhzbM/Bhb\nqGWOJA39abIxTi+PUOAVWOnmMbT3FMAEL+wWedQCGKCBJI+77CmCacipwkUBjB/lcz9raSuAqVrG\nbdI/YmyWNs2FMZNw7GCJw4az3pNg0pZ4aJiE223NBIP1ZU0mSz0yI4YkcMSIJFxtlG8TBxBx+Mpd\nqtKVHpkRQxJ4sps4LgdbMGAWRhyGC9RYrG7wrL3zhkbmk73e817vLXdJJ3D5pvEkvNKEOd1wnwtW\njOpkPanIjLvBcC3il8TwJNzahFdqOK5EsHZjtMgsMakEjhiRhFshbFMMBmsnRo/MiCEJHDEiCbPd\nHCOYJFi7MEZkRgxJ4IgRSZg9mz8EkwRrF8aIzIghCVw+G26PbxohxQsXBrtVZEYM6torwNXKJ/13\ng72n8eq4qMh8nXSJBE52AwsLIenBxYPXGzaZaCQyqzuYwClG3f7PFcNM6I+uyfxt9xdHfCWiueuW\nQgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\frac{x - x_{1}}{x_{0} - x_{1}}, \\quad \\frac{x - x_{0}}{- x_{0} + x_{1}}\\right ]$$"
      ],
      "text/plain": [
       "⎡ x - x₁   x - x₀ ⎤\n",
       "⎢───────, ────────⎥\n",
       "⎣x₀ - x₁  -x₀ + x₁⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lag = lagrange(2, x)\n",
    "lag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj Lagrangev polinom $l_0(x)$ integriramo čez celotni interval:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaYAAAAyBAMAAADl32LRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n36IC/gfr/0IjYIAGiwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{x_{0}^{2}}{2 x_{0} - 2 x_{1}} + \\frac{x_{0} x_{1}}{x_{0} - x_{1}} + \\frac{x_{1}^{2}}{2 x_{0} - 2 x_{1}} - \\frac{x_{1}^{2}}{x_{0} - x_{1}}$$"
      ],
      "text/plain": [
       "        2                       2           2  \n",
       "      x₀         x₀⋅x₁        x₁          x₁   \n",
       "- ─────────── + ─────── + ─────────── - ───────\n",
       "  2⋅x₀ - 2⋅x₁   x₀ - x₁   2⋅x₀ - 2⋅x₁   x₀ - x₁"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int0 = sym.integrate(lag[0], (x, x0, x1))\n",
    "int0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izraz poenostavimo in dobimo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAqBAMAAACQMGKTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny6+knyDgaxDGyaVVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{1}{2} \\left(x_{0} - x_{1}\\right)$$"
      ],
      "text/plain": [
       "-(x₀ - x₁) \n",
       "───────────\n",
       "     2     "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int1 = int0.factor()\n",
    "int1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ker je širina podintervala konstantna je $x_1=h_0+h$, izvedemo zamenjavo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAArBAMAAAC+8nRaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAELvv3c2ZVESJZnYi\nqzKqLvLVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAo0lEQVQYGWNgEDJhAIMwCPUdTHF8BFOcDmCK\nNX21DpAhH8NwF0jVP2BQAlKpDAzWQMqTgecfAwPjDwZOAwYGrg8MzBMWMvAGMPA3TGBgWsDAORuo\nhDLwHwyIMoMnTekBUKEsA+cPIHWRgcEYSAG9dx8oGi8ApoAC/gJAgoHjJ4hkYLkAptLAJHcCmCpj\n4DnAwMCXwMANpI7MnHYFKBr///8XBgChCihQyOTicgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{h}{2}$$"
      ],
      "text/plain": [
       "h\n",
       "─\n",
       "2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zamenjave = {x1: x0+h}\n",
    "int1.subs(zamenjave)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Zgornje korake za Lagrangev polinom  $l_0(x)$ lahko posplošimo za seznam Lagrangevih polinomov:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAyBAMAAAAEkSCpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\left [ \\frac{h}{2}, \\quad \\frac{h}{2}\\right ]$$"
      ],
      "text/plain": [
       "⎡h  h⎤\n",
       "⎢─, ─⎥\n",
       "⎣2  2⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, x0, x1, h = sym.symbols('x, x0, x1, h')\n",
    "zamenjave = {x1: x0+h}\n",
    "A_trapez = [sym.integrate(li, (x, x0, x1)).factor().subs(zamenjave) \n",
    "            for li in lagrange(2, x)]  # za vsak lagrangev polimom `li` v seznamu lagrange(2,x)\n",
    "A_trapez"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Izpeljali smo uteži, ki smo jih uporabili pri trapezni metodi:\n",
    "\n",
    "$$A_0=h/2\\qquad A_1=h/2.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Trapezno pravilo je:\n",
    "\n",
    "$$I_{\\textrm{trapezno}}=\\frac{h}{2}\\left(y_0+y_1\\right)$$\n",
    "\n",
    "Ocena napake je (vir: Burden, Faires, Burden: Numerical Analysis 10th Ed):\n",
    "$$E_{\\textrm{trapezno}}= -\\frac{h^3}{12}f''(\\xi).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Izračun uteži za Simpsonovo pravilo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potem ko smo zgoraj pokazali strnjen izračun za trapezno pravilo, lahko podobno izvedemo za kvadratno interpolacijo čez tri točke ($n=3$).\n",
    "\n",
    "Izračun uteži je analogen zgornjemu:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAAyBAMAAACAOwXCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXcVD5tiPb499+Ab6B+LhpMYDQNtlAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\frac{h}{3}, \\quad \\frac{4 h}{3}, \\quad \\frac{h}{3}\\right ]$$"
      ],
      "text/plain": [
       "⎡h  4⋅h  h⎤\n",
       "⎢─, ───, ─⎥\n",
       "⎣3   3   3⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, x0, x1, x2, h = sym.symbols('x, x0, x1, x2, h')\n",
    "zamenjave = {x1: x0+h, x2: x0+2*h}\n",
    "A_Simpson1_3 = [sym.integrate(li, (x, x0, x2)).factor().subs(zamenjave).factor() \n",
    "             for li in lagrange(3, x)]\n",
    "A_Simpson1_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Simpsonovo pravilo je:\n",
    "\n",
    "$$I_{\\textrm{Simpsonovo}}=\\frac{h}{3}\\left(y_0+4\\,y_1+y_2\\right)$$\n",
    "\n",
    "Ocena napake je (vir: Burden, Faires, Burden: Numerical Analysis 10th Ed):\n",
    "$$E_{\\textrm{Simpsonovo}}= -\\frac{h^5}{90}f^{(4)}(\\xi).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Primer uporabe:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAPBAMAAADQc2bfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1.44083929301$$"
      ],
      "text/plain": [
       "1.44083929301"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_Simps = h3v/3 * np.sum(y3v * [1, 4, 1])\n",
    "I_Simps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo sliko. Ker Simsonovo pravilo temelji na kvadratni interpolaciji, moramo naprej pripraviti interpolacijski polinom (pomagamo si s ``scipy.interpolate``):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import interpolate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def fig_Simpsonovo():\n",
    "    y_interpolate = interpolate.lagrange(x3v, y3v)\n",
    "    plt.fill_between(xg, y_interpolate(xg), alpha=0.25, facecolor='r')\n",
    "    plt.vlines(x3v, 0, y3v, color='r', linestyles='dashed', lw=1)\n",
    "    plt.annotate('$I_{\\\\textrm{Simpsonovo}}$', (1.2, 0.5), fontsize=22)\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.75, 1.7), xytext=(1.4, 1.8),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.2, 1.1), xytext=(1., 1.4),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.plot(xg, yg, lw=3, alpha=0.5, label='$x\\,\\sin(x)$')\n",
    "    plt.plot(x3v, y3v, 'o', alpha=0.5, label=f'$h={h3v}$')\n",
    "    plt.legend(loc=(1.01, 0))\n",
    "    plt.ylim(0, 2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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rkLSYdBOEY4IGzrkBALyBxxir9BsKJTEyODgIo9GI7373uwAAt9sNo9GIhx9+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\njr9Sf2lCAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2753a4d8f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_Simpsonovo()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### `scipy.integrate.newton_cotes`\n",
    "\n",
    "Koeficiente integracijskega pristopa Newton-Cotes pridobimo tudi s pomočjo `scipy.integrate.newton_cotes()` [dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.newton_cotes.html#scipy.integrate.newton_cotes):\n",
    "\n",
    "```python\n",
    "newton_cotes(rn, equal=0)\n",
    "```\n",
    "\n",
    "kjer sta parametra:\n",
    "\n",
    "* `rn`, ki definira število podintervalov (mogoč je tudi nekonstanten korak, glejte [dokumentacijo](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.newton_cotes.html#scipy.integrate.newton_cotes)),\n",
    "* `equal`, ki definira ali se zahteva konstantno široke podintervale.\n",
    "\n",
    "Funkcija vrne terko, pri čemer prvi element predstavlja numerično polje uteži in drugi člen oceno napake."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 0.375,  1.125,  1.125,  0.375]), -0.0375)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import integrate\n",
    "integrate.newton_cotes(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Izračun uteži za Simpsonovo 3/8 pravilo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nadaljujemo lahko s kubično interpolacijo čez štiri točke ($n=4$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAyBAMAAAA0MfPNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\frac{3 h}{8}, \\quad \\frac{9 h}{8}, \\quad \\frac{9 h}{8}, \\quad \\frac{3 h}{8}\\right ]$$"
      ],
      "text/plain": [
       "⎡3⋅h  9⋅h  9⋅h  3⋅h⎤\n",
       "⎢───, ───, ───, ───⎥\n",
       "⎣ 8    8    8    8 ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, x0, x1, x2, x3, h = sym.symbols('x, x0, x1, x2, x3, h')\n",
    "zamenjave = {x1: x0+h, x2: x0+2*h, x3: x0+3*h}\n",
    "A_Simpson3_8 = [sym.integrate(li, (x, x0, x3)).factor().subs(zamenjave).factor() \n",
    "                for li in lagrange(4, x)]\n",
    "A_Simpson3_8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Simpsonovo 3/8 pravilo je:\n",
    "\n",
    "$$I_{\\textrm{Simpsonovo 3/8}}=\\frac{3h}{8}\\left(y_0+3\\,y_1+3\\,y_2+y_3\\right)$$\n",
    "\n",
    "Ocena napake je (vir: Burden, Faires, Burden: Numerical Analysis 10th Ed):\n",
    "$$E_{\\textrm{Simpsonovo 3/8}}= -\\frac{3\\,h^5}{80}f^{(4)}(\\xi).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer uporabe. Uporabimo pripravljeno tabelo vrednosti funkcije v štirih točkah:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.84147098,  1.2959172 ,  1.65901326,  1.81859485])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y4v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1.44060715408$$"
      ],
      "text/plain": [
       "1.44060715408"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_Simps38 = 3*h4v/8 * np.sum(y4v * [1, 3, 3, 1])\n",
    "I_Simps38"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo še prikaz:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def fig_Simpsonovo38():\n",
    "    y_interpolate = interpolate.lagrange(x4v, y4v)\n",
    "    plt.fill_between(xg, y_interpolate(xg), alpha=0.25, facecolor='r')\n",
    "    plt.vlines(x4v, 0, y4v, color='r', linestyles='dashed', lw=1)\n",
    "    plt.annotate('$I_{\\\\textrm{Simpsonovo 3/8}}$', (1.2, 0.5), fontsize=22)\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.75, 1.7), xytext=(1.4, 1.8),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.5, 1.47), xytext=(1.1, 1.6),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.2, 1.1), xytext=(1., 1.4),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))    \n",
    "    plt.plot(xg, yg, lw=3, alpha=0.5, label='$x\\,\\sin(x)$')\n",
    "    plt.plot(x4v, y4v, 'o', alpha=0.5, label=f'$h={h4v:.6f}$')\n",
    "    plt.legend(loc=(1.01, 0))\n",
    "    plt.ylim(0, 2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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i1TmZMme7xMRErF27FlevXh0xi5LH40FLSwsuXbqES5dMaO9ox+pVa/HA5q14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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2753a5364a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_Simpsonovo38()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sestavljeno Simpsonovo pravilo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interval $[a, b]$ razdelimo na sodo število $n$ podintervalov enake širine $h=(b - a)/n$, s čimer so definirana vozlišča $x_i=a+i\\,h$ za $i=0,1,\\dots,n$. V tem primeru zapišemo sestavljeno Simpsonovo pravilo:\n",
    "\n",
    "$$\\int_{a}^{b}f(x)\\,\\textrm{d}x=\n",
    "\\frac{h}{3}\\left(\n",
    "f(x_0)\n",
    "+4\\sum_{i=1}^{n/2}f(x_{2i-1})\n",
    "+2\\sum_{i=1}^{n/2-1}f(x_{2i})\n",
    "+f(x_n)\n",
    "\\right)\n",
    "\\underbrace{\n",
    "-\\frac{b-a}{180}h^4\\,f^{(4)}(\\eta)\n",
    "}_{E_{\\textrm{Sestavljeno Simpsonovo 1/3}}}\n",
    ",$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "kjer je $\\eta$ neznana vrednost na intervalu $[a, b]$. Napaka je četrtega reda $\\mathcal{O}(h^4)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Numerična implementacija:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simpsonovo_sest(y, h):\n",
    "    \"\"\"\n",
    "    Sestavljeno Simpsonovo pravilo integriranja.\n",
    "\n",
    "    :param y: funkcijske vrednosti\n",
    "    :param h: korak integriranja\n",
    "    \"\"\"    \n",
    "    return h/3 * (y[0] + 4*np.sum(y[1:-1:2]) + 2*np.sum(y[2:-1:2]) + y[-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gEUv86r1Hx4HwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.44044796026$$"
      ],
      "text/plain": [
       "1.44044796026"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_Simps_sest = simpsonovo_sest(y5v, h=h5v)\n",
    "I_Simps_sest"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo sliko:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fig_Simpsonovo_sest():\n",
    "    y_interpolate = interpolate.lagrange(x5v, y5v)\n",
    "    plt.fill_between(xg, y_interpolate(xg), alpha=0.25, facecolor='r')       \n",
    "    plt.vlines(x5v, 0, y5v, color='r', linestyles='dashed', lw=1)\n",
    "    plt.annotate('$I_{\\\\textrm{Simpsonovo sestavljeno}}$', (1.2, 0.5), fontsize=22)\n",
    "    plt.annotate('Napaka', fontsize=20, xy=(1.70, 1.68), xytext=(1.4, 1.8),\n",
    "            arrowprops=dict(facecolor='gray', shrink=0.05))\n",
    "    plt.plot(xg, yg, lw=3, alpha=0.5, label='$x\\,\\sin(x)$')\n",
    "    plt.plot(x5v, y5v, 'o', alpha=0.5, label=f'$h={h5v}$')\n",
    "    plt.legend(loc=(1.01, 0))\n",
    "    plt.ylim(0, 2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kZyMnOw8ZGRlYns6wOlOF1RkMy9KZtJGb27cDn366cCeWaKg7NC5QS5CQBPMvL7+C74bG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\nkrQoCRJCCEla/xd/x01zZLh3xwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x27538dc6fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_Simpsonovo_sest()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Boljša ocena integrala"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Napaka sestavljene Simpsonove metode je definirana z:\n",
    "$$E_{\\textrm{Sestavljeno Simpsonovo 1/3}}= \n",
    "-\\frac{b-a}{180}h^4\\,f^{(4)}(\\eta),$$\n",
    "kjer je $\\eta$ neznana vrednost na intervalu $[a, b]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izboljšano oceno integrala določimo na podoben način kakor pri sestavljeni trapezni metodi; integral $I$ ocenjujemo pri dveh različnih korakih $h$ in $2\\,h$, velja natančno:\n",
    "\n",
    "$$\\int_a^b f(x)\\,\\textrm{d}x = I_h + E_h = I_{2h} + E_{2h},$$\n",
    "\n",
    "kjer je $I_h$ približek integrala s korakom $h$ in $E_h$ ocena napake pri koraku $h$; analogno velja za $I_{2h}$ in $E_{2h}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če predpostavimo, da je $f^{(4)}\\left (\\eta \\right )$ v obeh primerih enak, lahko določimo razliko $I_{2h}-I_{h} = E_{h}-E_{2h}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Naprej zapišemo:\n",
    "\n",
    "$$E_h=-\\frac{b-a}{180}h^4\\,f^{(4)}(\\eta)=h^4\\,K.$$\n",
    "\n",
    "Ob predpostavki, da je $f^{(4)}\\left (\\eta \\right )$ pri koraku $h$ in $2h$ enak ($\\eta$ je pri koraku $h$ in $2h$ dejansko različen), zapišemo:\n",
    "\n",
    "$$E_{2h}=-\\frac{(b-a)}{180}(2h)^4\\,f^{(4)}(\\eta)=16\\,h^2\\,K$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sledi:\n",
    "\n",
    "$$I_{2h}-I_h=-15K\\,h^4.$$\n",
    "\n",
    "Sedaj lahko ocenimo napako pri koraku $h$:\n",
    "\n",
    "$$E_h=h^4\\,K=\\frac{I_h-I_{2h}}{15}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Na podlagi ocene napake lahko izračunamo boljši približek  $I_{h}^*$:\n",
    "\n",
    "$$I_h^* = I_h + \\frac{1}{15}\\,(I_{h}-I_{2h})$$\n",
    "\n",
    "ali\n",
    "\n",
    "$$I_h^* = \\frac{16}{15}\\,I_h - \\frac{1}{15}\\,I_{2h}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerični zgled"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Predhodno smo s Simpsonovim pravilom že izračunali integral pri koraku $h=0,5$ in pri koraku $h=0,25$, rezultata sta bila:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASkAAAAUBAMAAAAjJNZ2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nNiaVtZZ9nzpTbb69z97/rC1yq/MPHoHHIcZ8twcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ 1.44083929301, \\quad 1.44044796026\\right ]$$"
      ],
      "text/plain": [
       "[1.44083929301, 1.44044796026]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[I_Simps, I_Simps_sest]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "S pomočjo zgornje formule izračunamo boljši približek:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Točen rezultat:   1.4404224209802097\n",
      "Boljši približek: 1.4404218714139077\n"
     ]
    }
   ],
   "source": [
    "I_Simps_boljši = 16/15*I_Simps_sest - 1/15*I_Simps\n",
    "print(f'Točen rezultat:   {I_točno}\\nBoljši približek: {I_Simps_boljši}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pridobimo boljši numerični približek, izgubimo pa oceno napake!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Simpsonova metoda v ``scipy.integrate.simps``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "V ``scipy`` je implementirano sestavljeno Simpsonovo pravilo v `scipy.integrate.simps()` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simps.html)):\n",
    "\n",
    "```python\n",
    "simps(y, x=None, dx=1, axis=-1, even='avg')\n",
    "```\n",
    "\n",
    "kjer so parametri:\n",
    "\n",
    "* `y` tabela funkcijskih vrednosti, ki jih integriramo,\n",
    "* `x` vozlišča, gre za opcijski parameter, če je `x=None`, se predpostavi ekvidistantne podintervale širine `dx`,\n",
    "* `dx` širina ekvidistantnih podintervalov oz korak integriranja,\n",
    "* `axis` os integriranja (pomembno v primeru večdimenzijskega numeričnega polja)\n",
    "* `even` je lahko {`'avg'`, `'first'`, `'last'`} in definira način integriranja v primeru lihega števila podintervalov."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer, najprej uvozimo funkcijo `simps`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import simps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gEUv86r1Hx4HwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.44044796026$$"
      ],
      "text/plain": [
       "1.44044796026"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#%%timeit\n",
    "simps(y5v, dx=h5v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Rombergova metoda*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rombergova metoda temelji na Richardsonovi ekstrapolaciji. Predpostavimo, da integral $\\int_a^b f(x)\\textrm{d}x$ integriramo na intervalu $[a,b]$, ki ga razdelimo na $n=1,2,4,8, \\dots$ podintervalov.\n",
    "\n",
    "Rezultat trapeznega pravila pri $n=1$ označimo z $R_{\\underbrace{1}_{n},\\underbrace{1}_{j}}$, pri čemer $j$ označuje natančnost pridobljenega rezultata $\\mathcal{O}(h^{2j})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če uporabimo trapezno integracijsko pravilo pri $n=1,2,4 \\dots$ podintervalih, izračunamo:\n",
    "\n",
    "* $R_{1,1}$, korak $h_1=h$, red natančnosti $\\mathcal{O}(h_1^2)$,\n",
    "* $R_{2,1}$, korak $h_2=h/2$, red natančnosti $\\mathcal{O}(h_2^2)$,\n",
    "* $R_{3,1}$, korak $h_3=h/4$, red natančnosti $\\mathcal{O}(h_3^2)$,\n",
    "* $R_{4,1}$, korak $h_4=h/8$, red natančnosti $\\mathcal{O}(h_4^2)$,\n",
    "* $\\dots$\n",
    "* $R_{n,1}$, korak $h_4=h/(2^{n-1})$, red natančnosti $\\mathcal{O}(h_n^2)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Na podlagi Richardsonove ekstrapolacije izračunamo boljši približek\n",
    "\n",
    "* $R_{2,2} = R_{2,1} + \\frac{1}{3}\\left(R_{2,1}-R_{1,1}\\right)$, korak $h_2=h/2$, red natančnosti $\\mathcal{O}(h_2^4)$,\n",
    "* $R_{3,2} = R_{3,1} + \\frac{1}{3}\\left(R_{3,1}-R_{2,1}\\right)$, korak $h_3=h/4$, red natančnosti $\\mathcal{O}(h_3^4)$,\n",
    "* $\\dots$\n",
    "* $R_{n,2} = R_{n,1} + \\frac{1}{3}\\left(R_{n,1}-R_{n-1,1}\\right)$, korak $h_n=h/(2^{n-1})$, red natančnosti $\\mathcal{O}(h_n^4)$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato nadaljujemo z Richardsonovo ekstrapolacijo:\n",
    "\n",
    "* $R_{3,3} = R_{3,2} + \\frac{1}{15}\\left(R_{3,2}-R_{2,2}\\right)$, korak $h_3=h/4$, red natančnosti $\\mathcal{O}(h_3^6)$,\n",
    "* $\\dots$\n",
    "* $R_{n,3} = R_{n,2} + \\frac{1}{15}\\left(R_{n,2}-R_{n-1,2}\\right)$, korak $h_n=h/(2^{n-1})$, red natančnosti $\\mathcal{O}(h_n^6)$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Richardsonovo extrapolacijo lahko posplošimo:\n",
    "\n",
    "* $R_{n,j} = R_{n,j-1} + \\frac{1}{4^{j-1}-1}\\left(R_{n,j-1}-R_{n-1,j-1}\\right)$, korak $h_n=h/(2^{n-1})$, red natančnosti $\\mathcal{O}(h_n^{2j})$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pri tem je boljši približek $R_{2,2}$ pri koraku $h/2$ enak rezultatu, ki ga dobimo po Simpsonovi metodi pri koraku $h/2$. Podobno je boljši približek $R_{3,2}$ pri koraku $h/4$ enak numeričnemu integralu Simpsonove metode pri koraku $h/4$. In je dalje $R_{3,3}$ enak popravljenemu približku Simpsonove metode pri koraku $h/4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo numerične podatke:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "x9v, h9v = np.linspace(1, 2, 9, retstep=True)  # korak h9v = 0.125 (9 vozlišč)\n",
    "y9v = x9v * np.sin(x9v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer od zgoraj. Najprej s sestavljeno trapezno metodo izračunamo integral pri različnih korakih (drugi red napake):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAmoAAAAUBAMAAADPSwzUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ 1.33003291923, \\quad 1.41313769957, \\quad 1.43362039509, \\quad 1.43872310575\\right ]$$"
      ],
      "text/plain": [
       "[1.33003291923, 1.41313769957, 1.43362039509, 1.43872310575]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# O(h^2)\n",
    "R1_1 = trapezno_sest(y9v[::8], 8*h9v) # h=1.0\n",
    "R2_1 = trapezno_sest(y9v[::4], 4*h9v) # h=0.5\n",
    "R3_1 = trapezno_sest(y9v[::2], 2*h9v) # h=0.25\n",
    "R4_1 = trapezno_sest(y9v, h9v) # h=0.125\n",
    "[R1_1, R2_1, R3_1, R4_1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato izračunamo boljše približke (dobimo četrti red napake):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAckAAAAUBAMAAAAUxtsHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndxX+AUspmx49UYfTAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 1.44083929301, \\quad 1.44044796026, \\quad 1.44042400931\\right ]$$"
      ],
      "text/plain": [
       "[1.44083929301, 1.44044796026, 1.44042400931]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# O(h^4)\n",
    "R2_2 = R2_1 + 1/3 * (R2_1 - R1_1)\n",
    "R3_2 = R3_1 + 1/3 * (R3_1 - R2_1)\n",
    "R4_2 = R4_1 + 1/3 * (R4_1 - R3_1)\n",
    "[R2_2, R3_2, R4_2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rezultati predstavljajo rezultat Simpsonove metode pri koraku $h=0,5$, $h=0,25$ in $h=0.125$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAckAAAAUBAMAAAAUxtsHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndxX+AUspmx49UYfTAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 1.44083929301, \\quad 1.44044796026, \\quad 1.44042400931\\right ]$$"
      ],
      "text/plain": [
       "[1.44083929301, 1.44044796026, 1.44042400931]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[simpsonovo_sest(y9v[::4], 4*h9v), simpsonovo_sest(y9v[::2], 2*h9v), simpsonovo_sest(y9v, h9v)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ponovno izračunamo boljše približke (dobimo šesti red napake):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASkAAAAUBAMAAAAjJNZ2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ 1.44042187141, \\quad 1.44042241258\\right ]$$"
      ],
      "text/plain": [
       "[1.44042187141, 1.44042241258]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# O(h6)\n",
    "R3_3 = R3_2 + 1/15 * (R3_2 - R2_2)\n",
    "R4_3 = R4_2 + 1/15 * (R4_2 - R3_2)\n",
    "\n",
    "[R3_3, R4_3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rezultat predstavlja boljši rezultat Simpsonove pri koraku $h=0,25$ in $h=0.125$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASkAAAAUBAMAAAAjJNZ2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ 1.44042187141, \\quad 1.44042241258\\right ]$$"
      ],
      "text/plain": [
       "[1.44042187141, 1.44042241258]"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a4h, a2h, ah = [simpsonovo_sest(y9v[::4], 4*h9v), simpsonovo_sest(y9v[::2], 2*h9v), simpsonovo_sest(y9v, h9v)]\n",
    "[16/15*a2h-1/15*a4h, 16/15*ah-1/15*a2h]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ponovno izračunamo boljše približke (dobimo osmi red napake):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAPBAMAAADe9tr1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrr/ULsAhtbGX6gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.44042242117$$"
      ],
      "text/plain": [
       "1.44042242117"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# O(h8)\n",
    "R4_4 = R4_3 + 1/63 * (R4_3 - R3_3)\n",
    "R4_4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rombergova metoda nam torej ponuja visoko natančnost rezultata za relativno majhno numerično ceno. Napako pa ocenimo:\n",
    "\n",
    "$$E = \\left|R_{n,n}-R_{n-1,n-1}\\right|.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Rombergova metoda v ``scipy.integrate.romb``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "V ``scipy`` je implementirana Rombergova metoda v `scipy.integrate.romb()` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.romb.html#scipy.integrate.romb)):\n",
    "\n",
    "```python\n",
    "romb(y, dx=1.0, axis=-1, show=False)\n",
    "```\n",
    "\n",
    "kjer so parametri:\n",
    "\n",
    "* `y` tabela funkcijskih vrednosti, ki jih integriramo,\n",
    "* `dx` širina ekvidistantnih podintervalov oz korak integriranja,\n",
    "* `axis` or integriranja (pomembno v primeru večdimenzijskega numeričnega polja),\n",
    "* `show` če je `True` prikaže elemente Richardsonove ekstrapolacije."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si primer od zgoraj:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import romb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.84147098,  1.01505104,  1.18623077,  1.34872795,  1.49624248,\n",
       "        1.62261343,  1.72197541,  1.78891084,  1.81859485])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y9v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "       Richardson Extrapolation Table for Romberg Integration       \n",
      "==================================================================== 1.33003 \n",
      " 1.41314  1.44084 \n",
      " 1.43362  1.44045  1.44042 \n",
      " 1.43872  1.44042  1.44042  1.44042 \n",
      "====================================================================\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAPBAMAAADe9tr1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdMrsi\niUTmZFZyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB5ElEQVQoFaWSv09TURiG3/aUWyu32ERdXGrE\nRVhK0rgRGq0GJzpAHJCkGt0Y3DtgXFwwMrABoQsDUxmcZLAxIcHE4S7+iKbR/0BiMQSMXt/3O5d6\nd8/w5vnuud9zz7n5gME62yAy3MLztqdwbpe0N/+KJbYGpEpr+1avFwGZUV+isENg5Etu3VMVxT9w\nNXz9xnrtlKxB8SKO4xqqs33uaj2QgPEU+OjpJnAFuQqyl4FgPCFrsLgH5PhikAjCZxQoXgLlitE1\nYKadXUHuBNhvISFrUPD4n1OCYp4CxSHwvWS0VKEg3zdB1EJCAwEw0kwJliVguJ8UNEXcxGaFUejj\nTLelkuSP7M89rGce4ZrsUIS/gU5kJT/Bgr4I12ECUlrwSbuJoAgKFCFP0Ims5BdremVV9zUBKSVw\nB9pNBJckUPgrWAnM6Y2hBjJtE5DSguCxtr3ARepV6CeWzxshUAMeAZMwASktsMlJBJl6febqfUVz\nlyMyYYSLCLvINPDuQr3+47ZRWjBVGwgF2R0fHKS7njjXQVczvMh63KaZZEe26NgBhVO6zDkJGIWS\nG/P0tvf+DsKN3uwK61+n9E9QlmDoxtG0/ezi68MnULgPD9scKdJSHB8jz4mn4Es87ckaLPCmRMH/\nrr/ULsAhtbGX6gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.44042242117$$"
      ],
      "text/plain": [
       "1.44042242117"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "romb(y9v, dx=h9v, show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Gaussov integracijski pristop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Newton-Cotesov pristop temelji na polinomu $n$-te stopnje in napaka je $n+1$ stopnje. To pomeni, da integracija daje točen rezultat, če je integrirana funkcija polinom stopnje $n$ ali manj."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Gaussov integracijski pristop je v principu drugačen: cilj je integral funkcije $f(x)$ nadomestiti z uteženo vsoto vrednosti funkcije pri diskretnih vrednostih $f(x_i)$:\n",
    "$$\\int_a^bf(x)\\,\\textrm{d}x\\approx \\sum_{i=0}^{n-1} w_i\\, f(x_i).$$\n",
    "Pri tem je neznana utež $w_i$ in tudi lega vozlišča $x_i$. Za stopnje polinoma $n$ bomo potrebovali tudi več  točk $(x_i, f(x_i))$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V nadaljevanju bomo spoznali, da lahko zelo učinkovito izračunamo numerično točen integral. Prednost Gaussove integracije je tudi, da lahko izračuna integral funkcij s singularnostmi (npr: $\\int_0^1\\sin(x)/\\sqrt{(x)}\\,dx$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Gaussova kvadratura z enim vozliščem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Predpostavimo, da želimo integrirati polinom stopnje $n=1$ (linearna funkcija):\n",
    "\n",
    "$$f(x)=P_1(x)=A_0+A_1\\,x.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunajmo: \n",
    "\n",
    "$$\\int_a^bP_1(x)\\,\\textrm{d}x=\\left(A_0\\,x+A_1\\,\\frac{x^2}{2}\\right)_a^b=-A_0\\,a+A_0\\,b-\\frac{A_1\\,a^2}{2}+\\frac{A_1\\,b^2}{2}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Po drugi strani pa želimo integral izračunati glede na ustrezno uteženo $w_0$ vrednost funkcije $f(x_0)$ v neznanem vozlišču $x_0$ (samo eno vozlišče!):\n",
    "\n",
    "$$\\int_a^bP_1(x)\\,\\textrm{d}x = w_0\\,P_1(x_0)= w_0\\,A_0+w_0\\,A_1\\,x_0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$A_0$ in $A_1$ sta koeficienta linearne funkcije, ki lahko zavzame poljubne vrednosti. Z enačenjem zgornjih izrazov izpeljemo:\n",
    "\n",
    "$$-A_0\\,a+A_0\\,b-\\frac{A_1\\,a^2}{2}+\\frac{A_1\\,b^2}{2}=w_0\\,A_0+w_0\\,A_1\\,x_0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Gre za sistem linearnih enačb z rešitvijo:\n",
    "\n",
    "$$w_0= b-a, \\qquad x_0=\\frac{b+a}{2}.$$\n",
    "\n",
    "Če je integrirana funkcija linearna, bomo samo na podlagi vrednosti v eni točki izračunali pravo vrednost!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Da je Gaussov integracijski pristop neodvisen od mej integriranja $a$, $b$, pa uvedemo standardne meje."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Standardne meje: $a=-1$ in $b=1$\n",
    "\n",
    "Zaradi splošnosti meje $x\\in[a,b]$ transformiramo v $\\xi\\in[-1, +1]$ s pomočjo:\n",
    "\n",
    "$$x=\\frac{b+a}{2}+\\frac{b-a}{2}\\xi$$\n",
    "\n",
    "in \n",
    "$$\\textrm{d}x=\\frac{b-a}{2}\\textrm{d}\\xi.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Velja:\n",
    "\n",
    "$$\\int_a^bf(x)\\,\\textrm{d}x=\\int_{-1}^1 g\\left(\\xi\\right)\\,\\textrm{d}\\xi,$$\n",
    "\n",
    "kjer je:\n",
    "\n",
    "$$g(\\xi)=\\frac{b-a}{2}\\,f\\left(\\frac{b+a}{2}+\\frac{b-a}{2}\\xi\\right).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V primeru standardnih mej je pri eni Gaussovi točki utež $w_0=2$ in $x_0=0$ vrednost, pri kateri moramo izračunati funkcijo $f(x_0)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Strojno izpeljevanje uteži in Gaussove točke"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definirajmo simbole in nastavimo enačbo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAakAAAAvBAMAAACrnCgZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDG71r29V/wGJlsTAviV36QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- A_{0} a + A_{0} b - \\frac{A_{1} a^{2}}{2} + \\frac{A_{1} b^{2}}{2} = w_{0} \\left(A_{0} + A_{1} x_{0}\\right)$$"
      ],
      "text/plain": [
       "                   2       2                  \n",
       "               A₁⋅a    A₁⋅b                   \n",
       "-A₀⋅a + A₀⋅b - ───── + ───── = w₀⋅(A₀ + A₁⋅x₀)\n",
       "                 2       2                    "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A0, A1, x, a, b, w0, x0 = sym.symbols('A0, A1, x, a, b, w0, x0') # simboli\n",
    "P1 = A0 + A1*x                                                   # linearni polinom\n",
    "eq = sym.Eq(P1.integrate((x, a, b)).expand(), w0*P1.subs(x, x0)) # teoretični integral = ocen z utežmi\n",
    "eq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo dve enačbi (za prvo predpostavimo $A_0=0, A_1=1$, za drugo predpostavimo $A_0=1, A_1=0$) ter rešimo sistem za `w0` in `x0`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANYAAAAyBAMAAADSGDiJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhBEmau73WYi\nds1/9lIHAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAELklEQVRYCa1YzU8TQRR/tGxLSz8w+IWa2Hgh\nMVHgZGI0lARvJpTExJqo3f+AGg9ojFIPXiQEPBk91ZMJ0cjNaIz0ZoIX9MLJ0INGLgKKcPCDOm/m\nzXY/ZtpBOoeZ937v47dv93Vmu3C09gP4CNlibfGcp3x9te/QPXxOaL0tJqF0kaoQLgwPwF7CQiUS\nWr28lQmHHK6BnMSM1/A+E9dYhrwcLmu/SZzHJ39j06NrFKubDA5XrKhxbQAnjbhgwM81uPNbCIZc\n8Ywgc+p62uD6dSZDrvAfL1cn6bq0StyQC6gOWVd8RpmtMWjKNS+ej+QasRunVVpNuQqLPFxyXVMm\nawKacqWqHq6DTdIqzaZcSdEMVFf7b2WyJqApV0JkJy5ibpLbbzblAnHXiCvW5c9joEfvbk8Kt/c6\n7zfC8JgvxNXWpOUvHdJlQzyhirYelAE6KmiHPj4TV6HINf20ojcBpCoqa6oEYFW5Zd7GhbhGFjkW\nnM4StBE01ZGXdZFJMiTdz5QT3CTSE9cg4qpBgZGGO/ozT6TkGssxeJSbxG0jrvmsx72uUGCoXIcC\nUsJrlFzH0TFl45wu4UxcyxVUnPGu9wNeFBsUmJpa0lUOkCwyx0AIrHy6DhBCG6RncCauvgoqckSO\nwBrJxFX4YukPgtgcQDDE2s4VKtBZxURtfCauiRxCcozOwU2SiYudpPRmJ31ca7yfPZdASHgT0hkI\nl9Gxjc/ENY0IXJ3F8RkY83PUT66uPlldxXa/A1aAK13DwYxtNihColUoZKB9nTlAvAtnDxcCOHrA\n+ikk+by+QkTf9cgVDEkVYaxfxeW5h6yEML8cxkf38DdE9V3P7qEiJD2HxUbKeNFxPqt6w9qAWPUM\n+jhcW/hAdIP1hiIkbVuHQdUby1l3om/wupQRANU1CQ/B2nb7uGTs62BIxxy+BkZ5HrHdUl1iw5Lx\n+Y/np2yhENfFJVbWbE46eFe8T8EQ6/Yt5pbKom+asTq9MWijohjExS3hitchMf4oyxH1HsVNC3xm\nDckG1aXdey9zXzGFXDKKeQhtcci797pDVHuvKNKXy69e8QH3APZwKFbxWRyVzhSxtVNd8apj1gvs\nkXkGOz+ns4goz0ruSlchtkDiiq5z086mtRxxQbN3ALEtEVen/pfakP+FpjV9QT1cJ652/Q7kC/Oo\nkcAm6TFLJfGXS8RFb1XSaLrGS0ae3ndRmDAK8juN+wG1HitzXNY1ZqvdGqL8QG7oIYy+/w5xvosY\nxLldTkGi4tZ18rLoIFnX/zRiexGSRlz0x0RywQHdRenxhWO99/XWusX/HxYGzX4p9QwAa7XaL7eu\nk9kJzYdTlwR0AbvA5Vcahysh3zB2kVQd6nylcbjgtNpz96jzlabOFS3tPq0ywyuJDtW/6U1JrLWr\n+5ue860yareWhLLlqcHZt8p//cj2RNRKH1AAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( - a + b, \\quad \\frac{1}{2} \\left(a + b\\right)\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛        a + b⎞⎤\n",
       "⎢⎜-a + b, ─────⎟⎥\n",
       "⎣⎝          2  ⎠⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solve([eq.subs(A0, 0).subs(A1, 1), eq.subs(A1, 0).subs(A0, 1)], [w0, x0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Za dodatno razlago priporočam [video posnetek](https://www.youtube.com/watch?v=iQ5-4hx25Rw)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Gaussova integracijska metoda z več vozlišči"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Izpeljati želimo Gaussovo integracijsko metodo, ki bo upoštevala na intervalu $[a,b]$ $v$ vozlišč in bo točno izračunala integral polinomov do stopnje $n=2v-1$. Veljati mora:\n",
    "\n",
    "$$\\int_a^bP_{2v-1}(x)\\,\\textrm{d}x = \\sum_{i=0}^{v-1} w_i\\,P_{2v-1}(x_i).$$\n",
    "\n",
    "Pri izpeljavi se bomo omejili na standardne meje $a=-1$, $b=1$,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "kjer je polinom stopnje $n=2v-1$ definiran kot:\n",
    "$$P_{2v-1}(x)=\\sum_{i=0}^{2v-1} A_i\\,x^i.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Z dvema Gaussovima točkama/vozliščema točno izračunamo integral polinoma do tretjega reda, s tremi Gausovimi vozlišči pa točno izračunamo integral polinoma do petega reda!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Strojno izpeljevanje\n",
    "Pripravimo si najprej simbolni zapis polinoma in ustreznih spremenljivk:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "def P_etc(n=1, A='A', x='x'):                       # n je stopnja polinoma\n",
    "    Ai = sym.symbols('{0:s}:{1:g}'.format(A, n+1))  # seznam A_i\n",
    "    x = sym.symbols(x)                              # spremenljivka x\n",
    "    return Ai, x, sum([Ai[i]*x**i for i in range(n+1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj pa poiščimo uteži $w_i$ in vozlišča $x_i$ za primer dveh Gaussovih vozlišč; polinom je torej tretje stopnje."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vozlišča: (x0, x1)\n",
      "Uteži:    (w0, w1)\n"
     ]
    }
   ],
   "source": [
    "v = 2 # število vozlišč\n",
    "Ai, x, P = P_etc(n=2*v-1)\n",
    "xi = sym.symbols('x:{0:g}'.format(v)) # seznam x_i\n",
    "wi = sym.symbols('w:{0:g}'.format(v)) # seznam w_i\n",
    "print(f'Vozlišča: {xi}\\nUteži:    {wi}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Polinom:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAAXBAMAAADpf/4JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs1m3RAiVJl2RKu7\n74l6DlUiAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACm0lEQVRIDa2Vz2sTQRiG3202u0lsY+g/kBAQ\nEQUDevBYFIQK4p6EqtUFD4Ig7UWLIDRgPVUwl4JFSOPRixYPgvRgLoK39iLiySAoCR6MaBvUQ5yZ\nb2Z/THYdAl1IdmeeZ95vdyd8AfbvKD58YAzrrXpGZwzhK16a7Fwt1zY54/CzKPsGP9fJ9A3KWPgc\nFlrGBfv7lMCyeaPmS8a7GkuYNdr5I1KZTlNTAS3QcLGekKM56sUeT1DFVCqgBRo+lRSjOc5ASJkL\n0T24Gy6MA5pPx4VOQW5UqjNfcn6KmBuP/LAMIn4cjJTU8PTSPXnrqRG2n38vYmobreSScTBSUsOf\nhkMZEykZd6zeN587BW+5Jl1+Cv0QZNdfPeuRY8BCSnLmntxsNigCuIQyu2425Tj0CVgV4CpONk4T\n13H3nRfFQtId1lmzncyszZLoaGGhjkzH3aRh6Atgrc4AX/Ddu5OIi/6BShQLSYvgnbXoFXddnyJg\nbW29acNuYZdNfK5Wz1erRwUiAHcG8HBMTI1i27f2Apzi8M6axURbYP5VAKYquFbCB5oKbpGAKAn8\nVr6G7Zbzh6EAC01zAN4ADgY7iRpg7/A/gTOUGvgEqKT1g2Dk1yUxXPZ2QhwvqRzeWcvsQ4ezCEz0\nUfYgO6QqKYEoeTvfxxr5OobdQAQLSXdYZ3W9DeQ9irjFTpN7/Ck/xjMl4CULg8m+U0vG6MawkFRJ\nFcFe7IvFbVynhCvDQ3C3f9XZXvI9YYf0FeAls6/XHp8gOoIz9RhOjIAzuLw+d1/etAxSv9iwpAK0\nl8FIlQwmnuNpcK0u1FPKcdBZFadzvu5u0tXFOIC7E5vQsHV46W2M84HmBJ1VE7tdT5uhobPyt5EI\naHJqOOz/BwukOit5/wAp+9RiFAIUnQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$A_{0} + A_{1} x + A_{2} x^{2} + A_{3} x^{3}$$"
      ],
      "text/plain": [
       "                2       3\n",
       "A₀ + A₁⋅x + A₂⋅x  + A₃⋅x "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Podobno kakor zgoraj za eno vozlišče, tukaj definirajmo enačbe:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAqkAAAAyBAMAAACNNl9dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXXRxUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ 2 = w_{0} + w_{1}, \\quad 0 = w_{0} x_{0} + w_{1} x_{1}, \\quad \\frac{2}{3} = w_{0} x_{0}^{2} + w_{1} x_{1}^{2}, \\quad 0 = w_{0} x_{0}^{3} + w_{1} x_{1}^{3}\\right ]$$"
      ],
      "text/plain": [
       "⎡                                           2        2           3        3⎤\n",
       "⎣2 = w₀ + w₁, 0 = w₀⋅x₀ + w₁⋅x₁, 2/3 = w₀⋅x₀  + w₁⋅x₁ , 0 = w₀⋅x₀  + w₁⋅x₁ ⎦"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqs = [sym.Eq(P.integrate((x, -1, 1)).coeff(A_),\\\n",
    "              sum([wi[i]*P.subs(x, xi[i]) \\\n",
    "                   for i in range(v)]).expand().coeff(A_)) \\\n",
    "       for A_ in Ai]\n",
    "eqs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešimo jih za za neznane $x_i$ in $w_i$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfEAAAA/BAMAAAACiew+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARM1UEHbdIu8yibtm\nq5lTG27OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIHUlEQVRoBe1bz28bRRR+dhxv4jg/UC89NRZC\nUA6okSIBJ7wCNeKCkgsSB6RYqKpQD62FQEU9kLSqaBFIzX8QCyRUFaQEIc7xiQNCarhxgMai3FCb\n9AcUStswMzsz+2bnzc56Y++JkeJ58703732fZ7273nFgdn+/AaKNLkX90F5LvfTUQycARySBrf39\nOZidn5+Jxi+m8xqA97v0HMMnMCEX99B8kylXbIJPlDW0/nX5HtMFCiBQOa9KjyHl1WEf7AClS6ow\n1RdAAA63ZWWs/HOCzco+b4QjggLh3nP6LcdvFoIATOCDbJn7JjDVkAWR8uAOIqHMn5RB95OpRy8x\nZ3GOACVkEPAUVln6JlC5J6ci5RObKl3cB43Ypqz3KDANq626vZiAr7DK0jcBuCqnIuXNNZUu7qdS\nlohH3YxDs1mVP91xmICvsMrSNwFYXIrmIuVPq2yoX4Dg2ldu9ZUWnLv2LYr3mxuhMwYTWIBMmXMQ\nqN2PCMTKxx8RlG7CRzBCOaLYqbmgB80uMdEJrTijDQI3IciSOQeBQMqJlU/u2WzZW7o8B3/ZDoks\nwOg9mG45/YSj3CBAAWECrHCmzDkIgDy0YuWL2zalWhcOz1Tu2g6JfA2l87DYcPoJx+gDAhQQJsAK\nZ8qcgwAst0W5WPn1GZsS+7RBytHOVoa1/o72ykM+h2qYgCjsz5yHACx2RPVY+RbB5m2OLXQIj4BG\nu6yrnHK5afwoDQNgAqKwP3MuAuVVwSBWnjimS/wQaLG/c6dD9mq1pxhSZX/Bl79bvlRgY83hlgR0\nYU/m3ARq/wgCWnkpcRS+1QCIvjTW9E0+ojx6gQ3eFMBV4mOCIpPm7FISicaKQFwY0jLnJ1CKbim0\n8todg1BwnI35mrJ2ORSd8fL+MtPbEtDYRcPjGyxu0hGSAC6cljk/gcq/EW31Xa0sL/CKFvf/wAZn\n2QdwToGoZwLqHXawt2FC3Qkjb4pZXaWdioAq7MucmwC8KwjoNbeuyrttOMFC9kNa+dQDqIYA0/f7\nVT7puKxpArKwL3NuArDT5tK18pXkQbjS42sKnwKcEZE8GrXgiTgkxjah6pCCgrFZc9wSagKysC9z\nbgKw0eV8tPJmjw9Rm7zP1xR+hBH6S8ZOyA+JeheaS2iW36wnTqVqhiYgC3sz5yUA1zu8pla+nOQ/\n/vAVHlC6/ccM7622/kWPY0duf2O5UoHSY9qtCcjC3sx5CcDsNmegle92k4SOPpNEjHH5QmiMsw6C\nxI2DmhcT8BRWE/ISgJUGT6GVb1krqw8/VcrsR56YY2M0ctEYioHC/rZdHIkJeAqr6X0TgJdDMXd6\nlXda+bG2QNFL2UKQk5nPmUM8euNW4hrJnBpzPNWLCfgKq0p9Eqjc2AnF1Ok93mnlVyJUuA7+Mmor\nB4U5KjngnFxUMWP6ciiGY4KcVu5YCWNm9gFVWGHkRZLdOGTPniFSFTNCpfLoMvy/8gLecrUMx9aM\nhVCD4ta8LL6sqTWvDL+wUh6fxJVo3hdAAOTRPiG+nGnl0bXm1ZO8Pc8+dUbDFIU9bbjFudKIUSox\nqDCHcnmxMxJby1F5lvN7Zw3yEEhVjpkexFYqcQ6FpSvHMw5gq2JGCnrNrbfXmNPvgCqsMIfy4ROg\n17zIM9wM+T4Wp7xsfM49ylOePAsZySfPfH0pjAdH5/ak10XAiiPfNrKYNVUe7ea53VVY1TkZKovs\nqycMeOrnu78AhfGoM+KqlvA672SsOKOQHiTCKAJw68oLXT6halzVgLp5RHtqI21dRBvIDS9pNDYo\njHnlPVzSiwn4MqsaaFMvmY6HUBiDE/dwZ0Ieaza0p1YzPWKE3NAh/BTGwq5EoUkvJuDLHGWAAG29\nJdPxEApj8Ji4CqvrOWyJY5DHxw3tqR2PUW0hd2VJo9qgMO6MTmWWFxPwZFYl0NablY7FUBifmviu\ntjun8sU92lPrxqi2kLuuwdigMOatRBuUlhcT8GRWNdDWm5WOxVAYn7rY4q96zfXDII7qlrKDzGM8\nbp0GG+PR424MCdskkDVzs2sl8gLrmzxEK292qBkLJKojPW4dhw3XoxSTQMbM/q03XFra0TMfrXx9\nm4hx7KmpSI9bhZn9qOOps0EgY2bP1ptZWI+Wu9zUyq3Lvggk99SEJ4s7DkTWlLiaIkCaJgFPYT09\nbetNByWM3TUOaOXVvYQ/Gl4OSViBHrcKw32ZLgQJAhkzp2294arYPiYGWjm1FGcde2oyjceNiyF7\nuoUGyMQEMmb2bb2h7NiMLi5aOXUy3Xfsqck0Hjcuhuz1HhogExPImNm39YayI3M8elyulQePkVOa\nrj21bG47n0DYXQrZMAFPYTXft/Wm4sye3f/wppXDx6afj5x7alGox23nE8iOA8cEMmau972px2vL\n32DFyjfaFiXnnloU6XFb6SKArSfdEIGsmfve1OOV2ZEiOvXLAVjpCGDYL/K3GkSZggjA7JIoHq95\nuUWwGTw0Sey+RFUKIgA7oagXK6877jAGLH6650pYEAGQO6GxcnjoojRQ/PqaM10xBNQbjJRvhE5O\nA3TgHzQn0hZDoNyIyiLlY50Ek2EMSymfqUIIwKw86JDyuvPcM8C3oNpzJyuEAJyWBJByOOUmNTBP\nM0xJVQSBEbW+WPl6O4XVgFyfpeUpgsBYVzLAyuutNFoD8U1up6UpgAD8qghg5XBDoUPrv0/PPHwC\n8ZvLlev/yC3PpRM7sDe4lJ5i6ATgQ/mJ3uL/kfua/o/cylo6sQN72aOE1DZ0AvwHzKIdmp9v/wfD\nOmf6T3xWPwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left ( - \\frac{\\sqrt{3}}{3}, \\quad \\frac{\\sqrt{3}}{3}, \\quad 1, \\quad 1\\right ), \\quad \\left ( \\frac{\\sqrt{3}}{3}, \\quad - \\frac{\\sqrt{3}}{3}, \\quad 1, \\quad 1\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛-√3   √3      ⎞  ⎛√3  -√3       ⎞⎤\n",
       "⎢⎜────, ──, 1, 1⎟, ⎜──, ────, 1, 1⎟⎥\n",
       "⎣⎝ 3    3       ⎠  ⎝3    3        ⎠⎦"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = sym.solve(eqs, sym.flatten((xi, wi)))\n",
    "sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Določili smo seznam dveh (enakih) rešitev. \n",
    "\n",
    "Najprej sta definirani vozlišči: $x_0=-\\sqrt{3}/3$ in $x_1=\\sqrt{3}/3$, katerima pripadata uteži $w_0=w_1=1$.\n",
    "\n",
    "Koda zgoraj je izpeljana v splošnem - število vozlišč `v` lahko povečate ter izračunate vozlišča ter pripadajoče uteži."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Tukaj je podana tabela vozlišč in uteži za eno, dve in tri vozlišča (za meje $a=-1$, $b=1$):\n",
    "\n",
    "| Število točk | Vozlišče  $x_i$ | Utež $w_i$ |\n",
    "|:------------:|:------------:|:------------:|\n",
    "||||\n",
    "|1 | 0 | 2 |\n",
    "||--||\n",
    "|2 | $-\\frac{\\sqrt{3}}{3}$ | 1 |\n",
    "| | $+\\frac{\\sqrt{3}}{3}$ | 1 |\n",
    "||--||\n",
    "| | $-\\frac{\\sqrt{15}}{5}$ | $\\frac{5}{9}$ |\n",
    "|3 | $0$ | $\\frac{8}{9}$ |\n",
    "| | $\\frac{\\sqrt{15}}{5}$ | $\\frac{5}{9}$ |\n",
    "\n",
    "Za več vozlišč in tudi oceno napake, glejte [Mathworld  Legendre-Gauss Quadrature](http://mathworld.wolfram.com/Legendre-GaussQuadrature.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Za primer treh Gaussovih točk numerični integral izračunamo (standardne meje):\n",
    "\n",
    "$$I_{\\textrm{Gauss3}}= \\frac{5}{9}\\,f\\left(-\\frac{\\sqrt{15}}{5}\\right) + \\frac{8}{9}\\,f\\left(0\\right)+\\frac{5}{9}\\,f\\left(\\frac{\\sqrt{15}}{5}\\right).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Numerična implementacija"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Numerična implementacija (vključno s transformacijo mej) za eno, dve ali tri vozlišča:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Gaussova(fun, a, b, vozlišč=1):\n",
    "    \"\"\" \n",
    "    Gaussova integracijska metoda.\n",
    "\n",
    "    :param fun: objekt funkcije, ki jo integriramo\n",
    "    :param a:   spodnja meja\n",
    "    :param b:   zgornja meja\n",
    "    :param vozlišča: število vozlišč (1, 2 ali 3)\n",
    "    \"\"\"\n",
    "    def g(xi): # funkcija za transformacijo mej\n",
    "        return (b-a)/2 * fun((b+a +xi * (b-a))/2)\n",
    "    \n",
    "    if vozlišč == 1:\n",
    "        return 2*g(0.)\n",
    "    elif vozlišč == 2:\n",
    "        return 1. * g(-np.sqrt(3)/3) + 1. * g(np.sqrt(3)/3)        \n",
    "    elif vozlišč == 3:\n",
    "        return 5/9 * g(-np.sqrt(15)/5) +8/9 * g(0.) + 5/9 * g(np.sqrt(15)/5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si zgled. Najprej definirajmo funkcijo, ki jo želimo integrirati:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x*np.sin(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj pa funkcijo (v konkretnem primeru `f`) in ne vrednosti (npr. `f(0.)`) posredujemo funkciji `Gaussova`. Najprej za eno vozlišče, nato dve in tri:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAPBAMAAADQc2bfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$1.49624247991$$"
      ],
      "text/plain": [
       "1.49624247991"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gaussova(fun=f, a=1., b=2., vozlišč=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1.44014401845$$"
      ],
      "text/plain": [
       "1.44014401845"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gaussova(fun=f, a=1., b=2., vozlišč=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\np2QafvUc71W6NGb1/9YfngDI7MQT4mwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.44042294912$$"
      ],
      "text/plain": [
       "1.44042294912"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gaussova(fun=f, a=1., b=2., vozlišč=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### ``scipy.integrate.fixed_quad``\n",
    "\n",
    "Znotraj ``scipy`` je Gaussova integracijska metoda implementirana v okviru funkcije `scipy.integrate.fixed_quad()` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.fixed_quad.html)):\n",
    "\n",
    "```python\n",
    "fixed_quad(func, a, b, args=(), n=5)\n",
    "```\n",
    "\n",
    "kjer so parametri:\n",
    "\n",
    "* `func` je ime funkcije, ki jo kličemo,\n",
    "* `a` je spodnja meja,\n",
    "* `b` je zgornja meja,\n",
    "* `args` je terka morebitnih dodatnih argumentov funkcije `func`,\n",
    "* `n` je število vozlišč Gaussove integracije, privzeto `n=5`.\n",
    "\n",
    "Funkcija vrne terko z rezultatom integriranja `val` in vrednost `None`: `(val, None)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo primer od zgoraj:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\np2QafvUc71W6NGb1/9YfngDI7MQT4mwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.44042294912$$"
      ],
      "text/plain": [
       "1.44042294912"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.integrate import fixed_quad\n",
    "fixed_quad(f, a=1, b=2, n=3)[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rezultat je enak predhodnemu:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\np2QafvUc71W6NGb1/9YfngDI7MQT4mwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.44042294912$$"
      ],
      "text/plain": [
       "1.44042294912"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gaussova(fun=f, a=1., b=2., vozlišč=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# ``scipy.integrate``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "``scipy.integrate`` je močno orodje za numerično integriranje (glejte [dokumentacijo](https://docs.scipy.org/doc/scipy/reference/integrate.html)). V nadaljevanju si bomo pogledali nekatere funkcije.\n",
    "\n",
    "## Intergracijske funkcije, ki zahtevajo definicijsko *funkcijo*:\n",
    "\n",
    "* `quad(func, a, b[, args, full_output, ...])`\tizračuna določeni integral `func(x)` v mejah [`a`, `b`],\n",
    "* `dblquad(func, a, b, gfun, hfun[, args, ...])`\tizračuna določeni integral `func(x,y)`,\n",
    "* `tplquad(func, a, b, gfun, hfun, qfun, rfun)`\tizračuna določeni integral `func(x,y,z)`,\n",
    "* `nquad(func, ranges[, args, opts, full_output])`\tizračuna določeni integral $n$ dimenzijske funkcije `func( ...)`,\n",
    "* `romberg(function, a, b[, args, tol, rtol, ...])`\tintegracija Romberg za funkcijo `function`,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### `scipy.integrate.quad`\n",
    "\n",
    "Poglejmo si zelo uporabno funkcijo za integriranje, `scipy.integrate.quad()` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html#scipy.integrate.quad)):\n",
    "\n",
    "```python\n",
    "quad(func, a, b, args=(), full_output=0, epsabs=1.49e-08, epsrel=1.49e-08, limit=50, \n",
    "     points=None, weight=None, wvar=None, wopts=None, maxp1=50, limlst=50)\n",
    "```\n",
    "\n",
    "Izbrani parametri so:\n",
    "\n",
    "* `func` Python funkcija, ki jo integriramo,\n",
    "* `a` spodnja meja integriranja (lahko se uporabi `np.inf` za mejo v neskončnosti),\n",
    "* `b` zgornja meja integriranja (lahko se uporabi `np.inf` za mejo v neskončnosti),\n",
    "* `full_output` za prikaz vseh rezultatov, privzeto 0,\n",
    "* `epsabs` dovoljena absolutna napaka,\n",
    "* `epsrel` dovoljena relativna napaka."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo primer od zgoraj:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import integrate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAc0AAAAUBAMAAAAdLXt9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left ( 1.4404224209802095, \\quad 1.5991901369585387e-14\\right )$$"
      ],
      "text/plain": [
       "(1.4404224209802095, 1.5991901369585387e-14)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate.quad(f, a=1, b=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### `scipy.integrate.dblquad`\n",
    "\n",
    "Gre za podobno funkcijo kot `quad`, vendar za integriranje po dveh spremenljivkah ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.dblquad.html#scipy.integrate.dblquad)).\n",
    "Funkcija `scipy.integrate.dblquad()` izračuna dvojni integral `func(y, x)` v mejah od `x = [a, b]` in `y = [gfun(x), hfun(x)]`.\n",
    "\n",
    "```python\n",
    "dblquad(func, a, b, gfun, hfun, args=(), epsabs=1.49e-08, epsrel=1.49e-08)\n",
    "```\n",
    "\n",
    "Izbrani parametri so:\n",
    "\n",
    "* `func` je Python funkcija, ki jo integriramo,\n",
    "* `a` je spodnja meja integriranja `x` (lahko se uporabi `np.inf` za mejo v neskončnosti),\n",
    "* `b` je zgornja meja integriranja `x` (lahko se uporabi `np.inf` za mejo v neskončnosti),\n",
    "* `gfun` je Python funkcija, ki definira spodnjo mejo `y` v odvisnosti od `x`,\n",
    "* `hfun` je Python funkcija, ki definira zgornjo mejo `y` v odvisnosti od `x`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo primer izračuna površine polkroga s polmerom 1:\n",
    "\n",
    "$$\\int_{-1}^{1}\\left(\\int_0^{\\sqrt{1-x^2}}1\\,\\textrm{d}y\\right)\\,\\textrm{d}x=\\frac{\\pi}{2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo ustrezne Python funkcije in izračunajmo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAc0AAAAUBAMAAAAdLXt9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left ( 1.5707963267948986, \\quad 1.0002356720661965e-09\\right )$$"
      ],
      "text/plain": [
       "(1.5707963267948986, 1.0002356720661965e-09)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def func(y, x): # integracijska funkcija je enostavna, konstanta = 1!\n",
    "    return 1.\n",
    "def gfun(x):    # spodnja meja = 0\n",
    "    return 0.\n",
    "def hfun(x):    # zgornja meja\n",
    "    return np.sqrt(1-x**2)\n",
    "integrate.dblquad(func=func, a=-1, b=1, gfun=gfun, hfun=hfun)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Intergracijske funkcije, ki zahtevajo tabelo vrednosti:\n",
    "\n",
    "* `trapz(y[, x, dx, axis])` sestavljeno trapezno pravilo,\n",
    "* `cumtrapz(y[, x, dx, axis, initial])` kumulativni integral podintegralske funkcije (vrne rezultat v vsakem vozlišču),\n",
    "* `simps(y[, x, dx, axis, even])`\tSimpsonova metoda,\n",
    "* `romb(y[, dx, axis, show])`\tRombergova metoda."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Tukaj si bomo na primeru ogledali funkcijo `cumtrapz`. Kot primer si, ko na maso $m=1$ (enote izpustimo) deluje sila $F=\\sin(t)$. Iz 2. Newtonovega zakona sledi: $F=m\\,\\ddot x$. Pospešek integriramo, da izračunamo hitrost; nato integriramo še enkrat za pomik. Definirajmo najprej funkcijo za pospešek:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pospešek(t):\n",
    "    F = np.sin(t)\n",
    "    m = 1\n",
    "    return F/m"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Zanima nas dogajanje v času 3 sekund:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, h = np.linspace(0, 3, 100, retstep=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunajmo tabelo pospeškov ter nato integrirajmo za hitrost (pri tem je pomembno, da definiramo začetno vrednost):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = pospešek(t)\n",
    "v = integrate.cumtrapz(y=a, dx=h, initial=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Hitrost sedaj še enkrat integrirajmo, da izračunamo pot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = integrate.cumtrapz(y=v, dx=h, initial=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažimo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EWQfgzI7/fehPWzZqENTN3NIO7QvtekNIL3Pr244mAaszoVt6tSzWNG0O5pb4FMum9UCs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\nvAubEsxLwY14CkbNBDf7foP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      "text/plain": [
       "<matplotlib.figure.Figure at 0x2753a6321d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t, a, label='Pospesek')\n",
    "plt.plot(t, v, label='Hitrost')\n",
    "plt.plot(t, s, label='Pot')\n",
    "plt.xlabel('$t$ [s]')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Nekaj vprašanj za razmislek!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Na sliki (vir: J. Slavič: Dinamika, meh. nihanja ..., 2014) je prikazan trikotnik s stranicami dolžine $a$, $b$, z debelino $h$ in gostoto $\\rho$.\n",
    "<img src=\"./fig/Slavic_2014_str_242_slika_5.4.png\" width=200>\n",
    "V simbolni obliki določite masni vztrajnostni moment glede na prikazano os $y$:\n",
    "$$J_{yy}=\\int_0^b y^2\\,\\rho\\,h\\,(a-a/b\\,y)\\,\\textrm{d}y.$$\n",
    "Upoštevajte tudi: $m=a\\,b\\,h\\,\\rho/2$. Za izmišljene vrednosti izračunajte numerični rezultat.\n",
    "2. Izračunajte integral tudi numerično. Uporabite ``scipy.integrate`` in integrirajte glede na pravila: trapezno, Simpsonovo 1/3. Rezultat primerjajte tudi z Gaussovo kvadraturo.\n",
    "Raziščite natančnost in hitrost metod.\n",
    "3. Preštudirajte ``scipy.special.legendre``, ki vam vrne objekt ``orthopoly1d``. Ta objekt ima metodo ``weights``, ki vrne seznam ``[x, w, mu0]`` vrednosti, ki jih uporabimo pri Gaussovi kvadraturi. (Če vsega ne razumete, ne skrbite preveč, bo asistent pokazal/komentiral). Opazite lahko, da smo vrednosti izpeljali na predavanjih!\n",
    "4. S pomočjo zgoraj pridobljenih uteži in vozlišč izračunajte integral s pomočjo Gaussove kvadrature: $\\sum_iw_i\\,f(x_i)$. Pazite na transformacijo mej.\n",
    "5. Preprost integral $\\int_0^2x^2\\,dx$ izrabite za prikaz trapeznega in Simpsonovega 1/3 pravila (osnovno pravilo, ne sestavljeno). Uteži izračunajte z uporabo ``scipy``.\n",
    "6. Integral predhodne točke razširite za sestavljeno trapezno pravilo (lastna koda). Prikažite vpliv števila podintervalov, primerjajte napako izračuna s predhodnim številom podintervalov in prikažite konvergenco.\n",
    "7. Integral predhodne točke razširite za sestavljeno Simpsonovo 1/3 pravilo (lastna koda). Prikažite vpliv števila podintervalov, primerjajte napako izračuna s predhodnim številom podintervalov in prikažite konvergenco.\n",
    "8. Z različnimi metodami izračunajte integrala (namig: vse metode niso primerne):\n",
    "$$\\int_1^2\\frac{\\sin(x)}{\\sqrt{x}}.$$\n",
    "9. S pomočjo numeričnega integriranja določite ekvivalentno silo porazdeljene obremenitve ($Q$) ter njeno prijemališče vzdolž nosilca ($x_Q$) dolžine $L = 2\\,$m. Konstanta obremenitve: $q_0 = 5\\,$kN/m.\n",
    "<img src=\"./fig/nal-num_integriranje_1-1.jpg\" width=350>\n",
    "Rešitev: \n",
    "ekvivalentna obremenitev $Q = \\int_0^L{q(x) \\mathrm{d}x}$, pozicija (težišče) $x_Q = \\frac{ \\int_0^L{x\\,q(x) \\mathrm{d}x} }{ \\int_0^L{q(x) \\mathrm{d}x} }$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dodatno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Obravnavajte prikazan enoosni primer, obremenjen s porazdeljeno obremenitvijo $n(x)$ ter točkovno silo $F=10\\,$kN. Dolžina palice je $L = 2\\,$m, konstanta $n_0 = 15\\,$kN/m in $EA = 200000\\,\\mathrm{MPa} \\times 50×50\\, \\mathrm{mm^2}$.\n",
    "<img src=\"./fig/nal-num_integriranje_2-2.jpg\" width=350>\n",
    "Naloga: \n",
    "    1. S pomočjo simbolnega integriranja določite funkcijo notranje osne sile $N(x)$.\n",
    "    2. S pomočjo numeričnega integriranja izračunajte pomik prostega konca palice $u_0$."
   ]
  }
 ],
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