{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "fd4dff47",
   "metadata": {},
   "source": [
    "# Universelle Funktionen (`ufunc`)\n",
    "\n",
    "Eine universelle Funktion, oder `ufunc`, ist eine Funktion, die elementweise Operationen auf Daten in `ndarrays` durchführt. Man kann sie sich als schnelle vektorisierte Wrapper für einfache Funktionen vorstellen, die einen oder mehrere skalare Werte annehmen und ein oder mehrere skalare Ergebnisse erzeugen.\n",
    "\n",
    "Viele `ufuncs` sind einfache elementweise Transformationen, wie [sqrt](https://numpy.org/doc/stable/reference/generated/numpy.sqrt.html) oder [exp](https://numpy.org/doc/stable/reference/generated/numpy.exp.html):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "f559f67e",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.107739Z",
     "iopub.status.busy": "2026-05-21T16:51:21.107473Z",
     "iopub.status.idle": "2026-05-21T16:51:21.139867Z",
     "shell.execute_reply": "2026-05-21T16:51:21.139511Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.107719Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "data = np.arange(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "26e18ead",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.140405Z",
     "iopub.status.busy": "2026-05-21T16:51:21.140285Z",
     "iopub.status.idle": "2026-05-21T16:51:21.143616Z",
     "shell.execute_reply": "2026-05-21T16:51:21.143292Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.140396Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0a3b82aa",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.143985Z",
     "iopub.status.busy": "2026-05-21T16:51:21.143912Z",
     "iopub.status.idle": "2026-05-21T16:51:21.146328Z",
     "shell.execute_reply": "2026-05-21T16:51:21.145991Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.143978Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 1.        , 1.41421356, 1.73205081, 2.        ,\n",
       "       2.23606798, 2.44948974, 2.64575131, 2.82842712, 3.        ])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8da698e7",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.146850Z",
     "iopub.status.busy": "2026-05-21T16:51:21.146758Z",
     "iopub.status.idle": "2026-05-21T16:51:21.149169Z",
     "shell.execute_reply": "2026-05-21T16:51:21.148945Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.146842Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.00000000e+00, 2.71828183e+00, 7.38905610e+00, 2.00855369e+01,\n",
       "       5.45981500e+01, 1.48413159e+02, 4.03428793e+02, 1.09663316e+03,\n",
       "       2.98095799e+03, 8.10308393e+03])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c16e2f2",
   "metadata": {},
   "source": [
    "Diese werden als einstellige ufuncs bezeichnet. Andere, wie [add](https://numpy.org/doc/stable/reference/generated/numpy.add.html) oder [maximum](https://numpy.org/doc/stable/reference/generated/numpy.maximum.html), nehmen zwei Arrays (also binäre ufuncs) und geben ein einziges Array als Ergebnis zurück:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "ee94de95",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.149586Z",
     "iopub.status.busy": "2026-05-21T16:51:21.149510Z",
     "iopub.status.idle": "2026-05-21T16:51:21.158466Z",
     "shell.execute_reply": "2026-05-21T16:51:21.158152Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.149578Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.44268441,  0.91852841, -3.0452678 , -0.19302094,  1.04784189,\n",
       "         0.78246093, -1.24320392, -0.85376618]),\n",
       " array([ 0.00448541, -1.31346391,  0.29722027,  2.53364962,  0.06792141,\n",
       "        -1.52562906, -0.25359309,  0.45988813]))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "rng = np.random.default_rng()\n",
    "x = rng.normal(size=8)\n",
    "y = rng.normal(size=8)\n",
    "x, y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "583063e6",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.160094Z",
     "iopub.status.busy": "2026-05-21T16:51:21.159940Z",
     "iopub.status.idle": "2026-05-21T16:51:21.162342Z",
     "shell.execute_reply": "2026-05-21T16:51:21.162093Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.160086Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00448541,  0.91852841,  0.29722027,  2.53364962,  1.04784189,\n",
       "        0.78246093, -0.25359309,  0.45988813])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.maximum(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce9fc716",
   "metadata": {},
   "source": [
    "Hier berechnete `numpy.maximum` das elementweise Maximum der Elemente in `x` und `y`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "901af41f",
   "metadata": {},
   "source": [
    "Manche `ufunc`, wie z.B. [modf](https://numpy.org/doc/stable/reference/generated/numpy.modf.html) , eine vektorisierte Version des eingebauten Python [divmod](https://docs.python.org/3/library/functions.html#divmod), geben mehrere Arrays zurückgeben: die Bruch- und Integralteile eines Gleitkomma-Arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a4b665ed",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.162787Z",
     "iopub.status.busy": "2026-05-21T16:51:21.162723Z",
     "iopub.status.idle": "2026-05-21T16:51:21.164539Z",
     "shell.execute_reply": "2026-05-21T16:51:21.164315Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.162780Z"
    }
   },
   "outputs": [],
   "source": [
    "data = x * 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "34c8f475",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.164922Z",
     "iopub.status.busy": "2026-05-21T16:51:21.164807Z",
     "iopub.status.idle": "2026-05-21T16:51:21.167187Z",
     "shell.execute_reply": "2026-05-21T16:51:21.166771Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.164911Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ -2.21342203,   4.59264206, -15.22633899,  -0.96510468,\n",
       "         5.23920946,   3.91230463,  -6.2160196 ,  -4.26883088])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "1daabbb7",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.167574Z",
     "iopub.status.busy": "2026-05-21T16:51:21.167490Z",
     "iopub.status.idle": "2026-05-21T16:51:21.169119Z",
     "shell.execute_reply": "2026-05-21T16:51:21.168873Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.167567Z"
    }
   },
   "outputs": [],
   "source": [
    "remainder, whole_part = np.modf(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "9d827342",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.169691Z",
     "iopub.status.busy": "2026-05-21T16:51:21.169536Z",
     "iopub.status.idle": "2026-05-21T16:51:21.171791Z",
     "shell.execute_reply": "2026-05-21T16:51:21.171575Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.169681Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.44268441,  0.91852841, -0.0452678 , -0.19302094,  0.04784189,\n",
       "        0.78246093, -0.24320392, -0.85376618])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "remainder"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "9846dd84",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.172358Z",
     "iopub.status.busy": "2026-05-21T16:51:21.172225Z",
     "iopub.status.idle": "2026-05-21T16:51:21.174707Z",
     "shell.execute_reply": "2026-05-21T16:51:21.174426Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.172350Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.,  0., -3., -0.,  1.,  0., -1., -0.])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "whole_part"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2248cfbd",
   "metadata": {},
   "source": [
    "Ufuncs akzeptieren ein optionales `out`-Argument, das es euch erlaubt, eure Ergebnisse an ein bestehendes Array zu übertragen, anstatt ein neues zu erstellen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "28726c4d",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.175112Z",
     "iopub.status.busy": "2026-05-21T16:51:21.175034Z",
     "iopub.status.idle": "2026-05-21T16:51:21.176819Z",
     "shell.execute_reply": "2026-05-21T16:51:21.176485Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.175106Z"
    }
   },
   "outputs": [],
   "source": [
    "out = np.zeros_like(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "a725be5a",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.177222Z",
     "iopub.status.busy": "2026-05-21T16:51:21.177124Z",
     "iopub.status.idle": "2026-05-21T16:51:21.179356Z",
     "shell.execute_reply": "2026-05-21T16:51:21.179141Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.177179Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ -1.21342203,   5.59264206, -14.22633899,   0.03489532,\n",
       "         6.23920946,   4.91230463,  -5.2160196 ,  -3.26883088])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.add(data, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "1046df1e",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.180065Z",
     "iopub.status.busy": "2026-05-21T16:51:21.179744Z",
     "iopub.status.idle": "2026-05-21T16:51:21.182359Z",
     "shell.execute_reply": "2026-05-21T16:51:21.182063Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.180041Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ -1.21342203,   5.59264206, -14.22633899,   0.03489532,\n",
       "         6.23920946,   4.91230463,  -5.2160196 ,  -3.26883088])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.add(data, 1, out=out)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "9a00a932",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-21T16:51:21.182992Z",
     "iopub.status.busy": "2026-05-21T16:51:21.182873Z",
     "iopub.status.idle": "2026-05-21T16:51:21.185072Z",
     "shell.execute_reply": "2026-05-21T16:51:21.184796Z",
     "shell.execute_reply.started": "2026-05-21T16:51:21.182985Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ -1.21342203,   5.59264206, -14.22633899,   0.03489532,\n",
       "         6.23920946,   4.91230463,  -5.2160196 ,  -3.26883088])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "out"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6d35a55",
   "metadata": {},
   "source": [
    "Einige einstellige ufuncs:\n",
    "\n",
    "Funktion | Beschreibung\n",
    ":------- | :-----------\n",
    "`abs`, `fabs` | berechnet den absoluten Wert elementweise für Ganzzahl-, Gleitkomma- oder komplexe Werte\n",
    "`sqrt` | berechnet die Quadratwurzel aus jedem Element (entspricht `data ** 0,5`)\n",
    "`square` | berechnet das Quadrat eines jeden Elements (entspricht `data ** 2`)\n",
    "`exp` | berechnet die Exponentialfunktion aller Elemente in einem Eingabearray\n",
    "`log`, `log10`, `log2`, `log1p` | Natürlicher Logarithmus (Basis e), log Basis 10, log Basis 2 bzw. log(1 + x)\n",
    "`sign` | berechnet das Vorzeichen jedes Elements: `1` (positiv), `0` (Null), oder `-1` (negativ)\n",
    "`ceil` | berechnet die Obergrenze jedes Elements (d.h. die kleinste ganze Zahl, die größer oder gleich dieser Zahl ist)\n",
    "`floor` | berechnet die Untergrenze jedes Elements (d.h. die größte ganze Zahl, die kleiner oder gleich jedem Element ist)\n",
    "`rint` | rundet Elemente auf die nächste Ganzzahl, wobei der `dtype` erhalten bleibt\n",
    "`modf` | gibt den gebrochenen und ganzzahligen Teile des Arrays als separate Arrays zurück\n",
    "`isnan` | gibt ein boolesches Array zurück, das angibt, ob jeder Wert `NaN` (Not a Number) ist\n",
    "`isfinite`, `isinf` | gibt ein boolesches Array zurück, das angibt, ob jedes Element endlich (non-`inf`, not-`NaN`) bzw. unendlich ist\n",
    "`cos`, `cosh`, `sin`, `sinh`, `tan`, `tanh` | reguläre und hyperbolische trigonometrische Funktionen\n",
    "`arccos`, `arccosh`, `arcsin`, `arcsinh`, `arctan`, `arctanh` | Inverse trigonometrische Funktionen\n",
    "`logical_not` | berechnet den Wahrheitswert von `not x` elementweise (entspricht `~data`)\n",
    "\n",
    "Einige binäre universelle Funktionen:\n",
    "\n",
    "Funktion | Beschreibung\n",
    ":------- | :-----------\n",
    "`add` | hinzufügen entsprechender Elemente in Arrays\n",
    "`subtract` | subtrahiert Elemente im zweiten Array vom ersten Array\n",
    "`multiply` | Array-Elemente multiplizieren\n",
    "`divide`, `floor_divide` | Dividieren oder Abschneiden des Rests\n",
    "`power` | erhöht Elemente im ersten Array auf die im zweiten Array angegebenen Potenzen\n",
    "`maximum`, `fmax` | elementweises Maximum; `fmax` ignoriert `NaN`\n",
    "`minimum`, `fmin` | elementweises Minimum; `fmin` ignoriert `NaN`\n",
    "`mod` | Elementweiser Modulus (Rest der Division)\n",
    "`copysign` | kopiert das Vorzeichen der Werte im zweiten Argument auf die Werte im ersten Argument\n",
    "`greater`, `greater_equal`, `less`, `less_equal`, `equal`, `not_equal` | Elementweise Vergleiche durchführen, die ein boolesches Array ergeben (entspricht den Infix-Operatoren `>`, `>=`, `<`, `<=`, `==`, `!=`)\n",
    "`logical_and` |berechnet den elementweisen Wahrheitswert der logischen Operation AND (`&`).\n",
    "`logical_or` | berechnet den elementweisen Wahrheitswert der logischen Operation OR (`|`).\n",
    "`logical_xor` | berechnet den elementweisen Wahrheitswert der logischen Operation XOR (`^`).\n",
    "\n",
    "<div class=\"alert alert-block alert-info\">\n",
    "\n",
    "**Hinweis:**\n",
    "\n",
    "Eine vollständige Übersicht über binäre universelle Funktionen findet ihr in [Universal functions (ufunc)](https://numpy.org/doc/stable/reference/ufuncs.html).\n",
    "</div>"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.13 Kernel",
   "language": "python",
   "name": "python313"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.13.0"
  },
  "widgets": {
   "application/vnd.jupyter.widget-state+json": {
    "state": {},
    "version_major": 2,
    "version_minor": 0
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}