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    "latex_envs": {
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        "T9WFkaVbbgyJ",
        "tvRMCe_abgzC"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/klajosw/python/blob/master/kl_py_modul_diagram.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9EBhrOofbgxf",
        "colab_type": "text"
      },
      "source": [
        "A `Python`ban, mint minden más modern nyelvben, szokás előre megírt, hasznos függvényeket és függvénycsomagokat használni. \n",
        "A Python szintaxisában az `import` parancs segítségével tudunk külső függvényeket, csomagokat vagy modulokat betölteni. \n",
        "\n",
        "Modulnak nevezünk egy `Python` nyelven írt file-t, ami `Python`-függvények, esetleg osztályok definícióit tartalmazza. \n",
        "\n",
        "Rendszerint ez egy `.py` kiterjeszésű file. \n",
        "\n",
        "Több ilyen definíciókat tartalmazó file-ok összességét hívjuk csomagnak. Nagy, sok függvénydefiníciót tartalmazó, csomagokat szokás alcsomagokra osztani. \n",
        "\n",
        "Ebben a notebookban a `numpy` a `matplotlib` és a `csv` csomagok néhány hasznos függvényével és adatstruktúrájával fogunk megismerkedni.  \n",
        "\n",
        "A modulokat a környezetünkbe installálni kell.\n",
        "\n",
        "Konkrét python programunknál a használatba vétel első lépése, hogy be kell őket tölteni (Import utasítás). \n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wtekkaaAbgxh",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fZbJeKIKbgxk",
        "colab_type": "text"
      },
      "source": [
        "A fenti parancs betöltötte a `numpy` modult és ezzel a modulban definiált függvények és adatstruktúrák rendelkezésünkre állnak! A `numpy` csomag sok gyakran használt matematikai függvény implementációját is tartalmazza Határozzuk meg például, hogy mennyi $\\sin(3)$ :"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jGVDaCZibgxk",
        "colab_type": "code",
        "colab": {},
        "outputId": "a6129088-ada6-4f89-bd8a-c52cf1af142f"
      },
      "source": [
        "numpy.sin(3)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.14112000805986721"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 2
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4qbuEJambgxo",
        "colab_type": "text"
      },
      "source": [
        "Amint a fenti példa is illusztrálja egy betöltött modul függvényeit az alábbi szintaxis szerint kell használni\n",
        ">```python\n",
        "modul_neve.fuggveny_neve(...)\n",
        "```\n",
        "\n",
        "Egy modul nem csak függvényeket hanem előre definiált változókat is tartalmazhat. Például a `numpy` modul definiálja a matematikában és a fizikában is gyakran használt $e$ és $\\pi$ számokat. Ezekre  - talán nem túl meglepő módon - így hivatkozunk:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BqHafIuGbgxo",
        "colab_type": "code",
        "colab": {},
        "outputId": "7382c7fa-1bea-4bb0-bef0-fcddee2baca5"
      },
      "source": [
        "numpy.e"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.718281828459045"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "EIgAYfS3bgxq",
        "colab_type": "code",
        "colab": {},
        "outputId": "c81d3794-fffe-4960-984d-a0291905bbc2"
      },
      "source": [
        "numpy.pi"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3.141592653589793"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 4
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5auPWdmRbgxs",
        "colab_type": "text"
      },
      "source": [
        "Nagy moduloknak sok almodulja is van. Az almodulokra az alábbi szintaxis alapján hivatkozunk:\n",
        ">```python\n",
        "modul_neve.almodul_neve.fuggveny_neve(...)\n",
        "```\n",
        "\n",
        "Például a `numpy` véletlenszámokat generáló függvényei a `random` almodul függvényeiként vannak definiálva. Az alábbi kódcella $0$ és $99$ között generál egy véletlen számot, ehhez a `numpy` modul  `random` almoduljának a `randint` fügvényét használja."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "07mzkJgubgxt",
        "colab_type": "code",
        "colab": {},
        "outputId": "5e16443e-c0a8-4963-9b1a-c359b421bb9b"
      },
      "source": [
        "numpy.random.randint(100)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "82"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GXqZU8YYbgxv",
        "colab_type": "text"
      },
      "source": [
        "Előfordulhat, hogy egy modulból csak bizonyos függvényeket szeretnénk betölteni. Erre ad lehetőséget a \n",
        ">```python\n",
        "from modul_neve import egyik_fuggveny_neve, masik_fuggveny_neve\n",
        "```\n",
        "\n",
        "konstrukció. Például ha a numpy csomagból csak a $\\sin$ és $\\cos$ függvényeket szeretnénk használni, akkor ezt az alábbi módon tehetjük meg:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "l5wKw-JNbgxv",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "from numpy import sin,cos"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rePHGv6qbgxx",
        "colab_type": "text"
      },
      "source": [
        "Ezután a modulnév használata nélkül rendelkezésünkre állnak a $\\sin()$ és $\\cos()$ függvények!"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mqitVlA3bgxy",
        "colab_type": "code",
        "colab": {},
        "outputId": "cb643c8f-90c2-4946-efe2-c34c0f046bea"
      },
      "source": [
        "sin(23)**2+cos(23)**2"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "t8M7zfl-bgx2",
        "colab_type": "text"
      },
      "source": [
        "Egy modulból akár betölthetjük az összes függvényt és változót is:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dR876UlObgx2",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "from numpy import *"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0grEIT9mbgx4",
        "colab_type": "text"
      },
      "source": [
        "Ekkor már például hivatkozhatunk a $\\pi$ értékét tároló `pi` változóra a `numpy` kiírása nélkül."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "b8NBVqArbgx5",
        "colab_type": "code",
        "colab": {},
        "outputId": "8c999e78-0815-4275-c0b3-570fc581aeae"
      },
      "source": [
        "pi"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3.141592653589793"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pD4Af8JHbgx7",
        "colab_type": "text"
      },
      "source": [
        "A kurzus további részében - eltekintve néhány speciális esettől -  célszerű lesz minden notebookot néhány sokszor használt modul betöltésével kezdeni. Ezt a `jupyter` notebookokban az alábbi paranccsal ( cell magic-el ) tehetjük meg:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "rIsuBLgLbgx7",
        "colab_type": "code",
        "colab": {},
        "outputId": "d649a0ac-f4cf-4216-8c07-9a9d5da46735"
      },
      "source": [
        "%pylab inline"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Populating the interactive namespace from numpy and matplotlib\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vnpoSer4bgx9",
        "colab_type": "text"
      },
      "source": [
        "A fenti utasítás a következő importálásokat tartalmazza:\n",
        "\n",
        ">```python\n",
        ">import numpy\n",
        ">import matplotlib\n",
        ">from matplotlib import pylab, mlab, pyplot\n",
        ">np = numpy\n",
        ">plt = pyplot\n",
        ">\n",
        ">from IPython.display import display\n",
        ">from IPython.core.pylabtools import figsize, getfigs\n",
        ">\n",
        ">from pylab import *\n",
        ">from numpy import *\n",
        ">```\n",
        "\n",
        "Azaz a `numpy` numerikus csomagon túl többek közt betölti a `matplotlib` ábra készítő modult is. "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bWxdlKlAbgx-",
        "colab_type": "text"
      },
      "source": [
        "# adatstruktúra tipus: `array`"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "collapsed": true,
        "id": "u6J1AVHubgx-",
        "colab_type": "text"
      },
      "source": [
        "A numpy csomag egy a listekhez hasonló adatstruktúrával gazdagítja a repertoárunkat. Ennek az új adatstruktúrának a neve: array. Tekintsük át az array-k viselkedésének  néhány alapvető tulajdonságát! (Részletesebb leírások [angol](https://scipy-lectures.github.io/intro/numpy/array_object.html) nyelven. )\n",
        "\n",
        "A listek hez hasonlóan számok vagy más objektumok listáinak segítségével tudjuk őket definiálni:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "N0Q0AjAxbgx_",
        "colab_type": "code",
        "colab": {},
        "outputId": "750d689d-7f5e-49b9-a773-570fabaeb0eb"
      },
      "source": [
        "vec=array([1,2,3])\n",
        "vec"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([1, 2, 3])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "GXClNsLrbgyB",
        "colab_type": "code",
        "colab": {},
        "outputId": "2e842cc8-2d29-4fe0-9420-c51536828133"
      },
      "source": [
        "matr=array([[1,1,3],[4,3,5],[6,2,3]])\n",
        "matr"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[1, 1, 3],\n",
              "       [4, 3, 5],\n",
              "       [6, 2, 3]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "h5I4R5s4bgyD",
        "colab_type": "code",
        "colab": {},
        "outputId": "cd9a9818-f04c-4998-8385-5f1cdecad975"
      },
      "source": [
        "b=array(['a','b','cd'])\n",
        "b"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array(['a', 'b', 'cd'], \n",
              "      dtype='<U2')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qWBXp9ffbgyF",
        "colab_type": "code",
        "colab": {},
        "outputId": "034a210d-49bd-498e-cd68-8fb2731f5d6b"
      },
      "source": [
        "c=array([1,2,3,'sd',])\n",
        "c"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array(['1', '2', '3', 'sd'], \n",
              "      dtype='<U21')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kfDE1MMebgyH",
        "colab_type": "code",
        "colab": {},
        "outputId": "73a2721f-2cd4-483b-d550-de0730897b2a"
      },
      "source": [
        "d=array([[1,2],[1,3,4],23])\n",
        "d"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[1, 2], [1, 3, 4], 23], dtype=object)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 15
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xYA5A8udbgyJ",
        "colab_type": "text"
      },
      "source": [
        "Az utolsó pár példában a kiírásban megjelent `dtype=` azt jelzi, hogy az adott array-ben milyen jellegű dolgok vannak csoportosítva. Egy `array`-ban mindig a legáltalánosabb adattípus érvényesül. A `vec` és `matr` változókban skalárok vannak, a `b` és a `c` válltozóban karakterláncok. \n",
        "A `d` változóban pedig, mivel skalár (az utolsó szám) is van benne, meg listák is (az első két vektor), ezért általános \"objektumok\" halmazának csoportjaként jelenik meg."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "T9WFkaVbbgyJ",
        "colab_type": "text"
      },
      "source": [
        "## Egy pár hasznos array művelet:"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Z3GJm1AnbgyK",
        "colab_type": "text"
      },
      "source": [
        "Egy array alakjáról a `shape` attribútum segítségével érdeklődhetünk:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Fq-wvWfFbgyK",
        "colab_type": "code",
        "colab": {},
        "outputId": "21532f6d-58e2-463a-a283-51c42fefc4f3"
      },
      "source": [
        "vec.shape"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(3,)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "piaww_b2bgyN",
        "colab_type": "code",
        "colab": {},
        "outputId": "010faa52-857d-4aa4-8725-c9daf131b75d"
      },
      "source": [
        "matr.shape"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(3, 3)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "poHTfOy-bgyR",
        "colab_type": "code",
        "colab": {},
        "outputId": "eb4c3ca3-e4bd-4274-8f19-5110f7dd872e"
      },
      "source": [
        "d.shape"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(3,)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rZ2R972WbgyS",
        "colab_type": "text"
      },
      "source": [
        "A `size` attribútum az elemek összes számáról ad tájékoztatást:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Lnr73tc4bgyT",
        "colab_type": "code",
        "colab": {},
        "outputId": "3b243929-0a7b-4c2b-979b-ff580f99b994"
      },
      "source": [
        "matr.size"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "9"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 19
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pxvg0dRrbgyU",
        "colab_type": "text"
      },
      "source": [
        "Számosorok statisztikai vizsgálatában segít a `min` és a `max` (legkisebb és legnagyobb elem értéke), illetve a `mean` (átlag) és az `std` (szórás) függvények"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YKP86-DLbgyV",
        "colab_type": "code",
        "colab": {},
        "outputId": "53a1ebc3-daa5-42e1-8beb-c88551ed2a1a"
      },
      "source": [
        "vec.min()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 20
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Z7QJbC5wbgyX",
        "colab_type": "code",
        "colab": {},
        "outputId": "a4fa8722-cfc1-4767-c3a6-4f1aa4eeb3b1"
      },
      "source": [
        "vec.max()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9rDe7ektbgyZ",
        "colab_type": "code",
        "colab": {},
        "outputId": "cba0d1ce-8008-4754-8c5f-a37e7691a4c1"
      },
      "source": [
        "vec.mean()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PwgQJNrybgya",
        "colab_type": "code",
        "colab": {},
        "outputId": "6e2d8203-c980-4115-c0c6-30c08927d910"
      },
      "source": [
        "vec.std()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.81649658092772603"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 23
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l_VxPpw9bgyc",
        "colab_type": "text"
      },
      "source": [
        "Ha az array alakja nem egydimenziós, mint például a `matr` változó esetében, akkor a fenti függvényeket soronként vagy oszloponként is hattathatjuk a változóra:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "htGmp5febgyd",
        "colab_type": "code",
        "colab": {},
        "outputId": "f68c7a4b-144b-4ed4-83ac-43d4768e7ea2"
      },
      "source": [
        "matr.mean(axis=0) #oszlop szerinti átlag"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 3.66666667,  2.        ,  3.66666667])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "iDyEtVOwbgye",
        "colab_type": "code",
        "colab": {},
        "outputId": "b4f3e638-815b-4a62-ed3c-a801efd534b1"
      },
      "source": [
        "matr.mean(axis=1) #sor szerint"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 1.66666667,  4.        ,  3.66666667])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 25
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NcfGUy0Sbgyg",
        "colab_type": "text"
      },
      "source": [
        "A `matr` mátrix transzponáltját a `T` attribútum segítségével kapjuk."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Xv45hBzPbgyh",
        "colab_type": "code",
        "colab": {},
        "outputId": "4f47ac88-2aca-454e-f54e-b9a2067dff5d"
      },
      "source": [
        "matr.T"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[1, 4, 6],\n",
              "       [1, 3, 2],\n",
              "       [3, 5, 3]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 26
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ky_EOV6Rbgyj",
        "colab_type": "text"
      },
      "source": [
        "Egy array-ban jelenlévő elemek szorzata és összege a `sum`, illetve a `prod` segítségével kapható:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "c1wSQO9ibgyk",
        "colab_type": "code",
        "colab": {},
        "outputId": "51898317-79c8-4bc1-cf58-5b8d71ccb921"
      },
      "source": [
        "vec.sum() # A vec változó összes elemének az összege"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "6"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 27
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CVRSYbhDbgyl",
        "colab_type": "code",
        "colab": {},
        "outputId": "8eeca081-51ee-4589-f9e2-0c4d9fef5280"
      },
      "source": [
        "matr.prod() # A matr változó összes elemének szorzata"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "6480"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_fD9X3cJbgyn",
        "colab_type": "text"
      },
      "source": [
        "Azonban vigyázni kell, mert nem minden arrayfüggvény alkalmazható minden array-re! Ha karaktereket tartalmazó mátrix produktumára vagyunk kíváncsiak akkor hibát fogunk kapni!"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "collapsed": true,
        "id": "tG8t-t5Tbgyn",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "matrS=array([['a','b'],['c','d']])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "QnevjSe8bgyp",
        "colab_type": "code",
        "colab": {},
        "outputId": "cc7fca73-ca88-4c8d-ffc4-f2abe2294836"
      },
      "source": [
        "matrS.prod()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "error",
          "ename": "TypeError",
          "evalue": "cannot perform reduce with flexible type",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-30-9768fae74e07>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mmatrS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mprod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
            "\u001b[0;32m/opt/conda/lib/python3.5/site-packages/numpy/core/_methods.py\u001b[0m in \u001b[0;36m_prod\u001b[0;34m(a, axis, dtype, out, keepdims)\u001b[0m\n\u001b[1;32m     33\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     34\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_prod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0maxis\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mout\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkeepdims\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 35\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mumr_prod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0maxis\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mout\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkeepdims\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     36\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     37\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_any\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0maxis\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mout\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkeepdims\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mTypeError\u001b[0m: cannot perform reduce with flexible type"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nh6GpkTIbgyr",
        "colab_type": "text"
      },
      "source": [
        "Néhány függvény segíti, hogy valamilyen előre megadott struktúrával rendelkező array-ek generáljunk. Lássunk erre néhány példát!\n",
        "A `linspace()` függvény egy megadott kezdő- és végérték között megadott számú egyenletesen mintavételezett számot ad:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CH-1QS-jbgys",
        "colab_type": "code",
        "colab": {},
        "outputId": "e1b107f6-1535-4c81-cb66-8eac588435bd"
      },
      "source": [
        "linspace(0,pi,10) #10 szám 0 és pi között.. "
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 0.        ,  0.34906585,  0.6981317 ,  1.04719755,  1.3962634 ,\n",
              "        1.74532925,  2.0943951 ,  2.44346095,  2.7925268 ,  3.14159265])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9WapNxs3bgyv",
        "colab_type": "text"
      },
      "source": [
        "A `rand()` függvény és a hozzá hasonló `randn()`, illetve `randint()` véletlen számokat tartalmazó array-ket adnak."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jvU3j8C2bgyv",
        "colab_type": "code",
        "colab": {},
        "outputId": "eb6d4a25-8c4b-445d-a711-f1abf0e872b4"
      },
      "source": [
        "rand() #0 és 1 között egy véletlen szám"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.1262308349286887"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 32
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "lbyF3QpBbgyx",
        "colab_type": "code",
        "colab": {},
        "outputId": "7fe9445b-fd41-45e4-954d-6e9c8409003e"
      },
      "source": [
        "randn(3) #3 darab véletlen Normális elpszlású szám "
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 0.41822238, -0.74672338, -0.76571842])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 33
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OMYaAHRebgyz",
        "colab_type": "code",
        "colab": {},
        "outputId": "7583f0db-874c-47e3-9ace-9e94b8df06bc"
      },
      "source": [
        "randint(0,9,(2,3)) # egy 2x3 as véletlen mátrix amely 0 és 9 közötti egész számokat tartalmaz"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[5, 1, 0],\n",
              "       [7, 3, 1]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 34
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gnTMmuw3bgy0",
        "colab_type": "text"
      },
      "source": [
        "Fontos megjegyezni, hogy bizonyos alapműveletek (összeadás, kivonás, szorzás és osztás) és alapvető matematikai függvények (például a sin, cos és exp függvények) array típusú változókra elemenként hatnak!"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xDxotbqgbgy1",
        "colab_type": "code",
        "colab": {},
        "outputId": "a92ff53e-a161-4ebb-e8cb-09064a825e3c"
      },
      "source": [
        "v1=array([1,2,3])\n",
        "v2=array([2,3,3])\n",
        "v1*v2"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([2, 6, 9])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "74Qqw-4Jbgy2",
        "colab_type": "code",
        "colab": {},
        "outputId": "454e2199-0f0a-4cb5-8f53-b07143938768"
      },
      "source": [
        "sin(v1)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 0.84147098,  0.90929743,  0.14112001])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 36
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zGkO3hxWbgy7",
        "colab_type": "text"
      },
      "source": [
        "Bool típusú változók array-énak összege elemenkénti `or`,  szorzata elemenkénti `and` műveletnek felel meg:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "collapsed": true,
        "id": "bOfl-B-8bgy8",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "b1=array([True,False,True,False])\n",
        "b2=array([False,False,True,True])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "J1mVJRWEbgzA",
        "colab_type": "code",
        "colab": {},
        "outputId": "af937876-682a-4ee0-e3a9-0d533be5a385"
      },
      "source": [
        "b1+b2 "
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ True, False,  True,  True], dtype=bool)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 38
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "bFkl4ytHbgzB",
        "colab_type": "code",
        "colab": {},
        "outputId": "295d6919-6331-42e5-b87f-6f0283964f7e"
      },
      "source": [
        "b1*b2"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([False, False,  True, False], dtype=bool)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 39
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tvRMCe_abgzC",
        "colab_type": "text"
      },
      "source": [
        "## Array típusú változók indexelései"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "A2zga2N1bgzD",
        "colab_type": "text"
      },
      "source": [
        "Az array típusú változók legfontosabb tulajdonsága, hogy a list-eknél jóval gazdagabb indexelési módszerekkel rendelkeznek. Nézzünk ezekre néhány példát! Előszöris definiáljunk néhány változót:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "collapsed": true,
        "id": "8pbI7O3ybgzD",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "proba1=linspace(0,10,10) #1-től 10-ig 10 db egyenletes szám\n",
        "proba2=rand(10) #10 véleteln szám\n",
        "proba3=randint(0,10,(5,5)) #5x5 ös véletlen mátrix"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jBXYI5AYbgzF",
        "colab_type": "text"
      },
      "source": [
        "A list-eknél megszokott szeletelések itt is működnek:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ic0EIWEpbgzG",
        "colab_type": "code",
        "colab": {},
        "outputId": "6ea08ec5-8775-4818-d107-9bd71bf59edd"
      },
      "source": [
        "proba1[0:3]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 0.        ,  1.11111111,  2.22222222])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 41
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3mqelD58bgzH",
        "colab_type": "code",
        "colab": {},
        "outputId": "d8d35a34-f68f-49f3-ec90-7c7055f71455"
      },
      "source": [
        "proba1[-4:-1]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 6.66666667,  7.77777778,  8.88888889])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 42
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rR_DnEYZbgzJ",
        "colab_type": "text"
      },
      "source": [
        "A list-ekkel ellentétben itt egy tetszőleges indexlistát is megadhatunk indexelésként:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "On0-ZAOBbgzJ",
        "colab_type": "code",
        "colab": {},
        "outputId": "18853410-67a7-4d68-a6ef-edc263d91edb"
      },
      "source": [
        "proba1[[3,5,2]]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 3.33333333,  5.55555556,  2.22222222])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 43
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ewP8acClbgzL",
        "colab_type": "text"
      },
      "source": [
        "Egy másik hasznos dolog, hogy egy array-ből Bool típusú array segítségével valamilyen kritériumokat teljesítő elemeket választhatunk ki:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "loGCGSk1bgzL",
        "colab_type": "code",
        "colab": {},
        "outputId": "ae730996-b3f3-461a-9a9f-9372335337d6"
      },
      "source": [
        "proba1>5 # Ez a kifejezés egy bool típusú array-t ad vissza"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([False, False, False, False, False,  True,  True,  True,  True,  True], dtype=bool)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 44
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "C8pA7WEAbgzM",
        "colab_type": "text"
      },
      "source": [
        "Ha a fenti Bool típusú array-t mint indexet használjuk a proba1 hasában, akkor egy olyan array-t kapunk, amely a `proba1`-nek csak azon elemeit tartalmazza, ahol a Bool érték `True` volt, azaz az utolsó 5 elemet!"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Gwf38tFkbgzN",
        "colab_type": "code",
        "colab": {},
        "outputId": "4f9ed06a-1db2-4dd3-d2ef-c959dc84ec51"
      },
      "source": [
        "proba1[proba1>5]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([  5.55555556,   6.66666667,   7.77777778,   8.88888889,  10.        ])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 45
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pQv8pV4RbgzP",
        "colab_type": "text"
      },
      "source": [
        "Adatbázisok elemzésénél egy igen hasznos művelet valamilyen tulajdonság szerint válogatni az adatbázisban. Ezen feladatok  sokszor megfogalmazhatóak úgy, mint egy array elemeinek egy másik array elemei szerinti szelektálása! Vizsgáljuk meg például a `proba1` azon elemeit amelyeknek megfelelő elemek a `proba2`-ben 0.25-nél nagyobbak:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Q7ClHeR0bgzP",
        "colab_type": "code",
        "colab": {},
        "outputId": "a4b9ba7d-f43b-451d-d992-733b576b7e10"
      },
      "source": [
        "proba1[proba2>0.25]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 0.        ,  1.11111111,  2.22222222,  4.44444444,  5.55555556,\n",
              "        6.66666667,  7.77777778,  8.88888889])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 46
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RnCbcgFJbgzR",
        "colab_type": "text"
      },
      "source": [
        "Végül áljon itt egy pár grafikus példa magasabb dimmenziójú array változók indexeléseire:\n",
        "<table style=\"width:100%;border: 0px solid #ffffff\">\n",
        "<tr style=\"border: 0px solid #ffffff\">\n",
        "<td style=\"border: 0px solid #ffffff\">\n",
        "<img src=\"https://scipy-lectures.github.io/_images/numpy_indexing.png\" width=400> \n",
        "</td>\n",
        "<td style=\"border: 0px solid #ffffff\">\n",
        "<img src=\"http://scipy-lectures.github.io/_images/numpy_fancy_indexing.png\" width=400>\n",
        "</td>\n",
        "</tr>\n",
        "</table>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "H8-Hw11MbgzR",
        "colab_type": "text"
      },
      "source": [
        "# Diagramok alapjai\n",
        "\n",
        "Igen sokszor egy program futása végén a kiszámolt eredményeket ábrákban foglaljuk össze. A `Python` nyelvben sok modul van, ami diagram, ábra készítésére alkalmas, ezek közül talán a legelterjedtebb a `matplotlib`. Az alábbiakban néhány egyszerűbb ábrakészítési feladatot tekintünk át. \n",
        "A `plot()` függvény, amit a későbbiekben is igen gyakran fogunk használni, a legalapvetőbb ábrakészítő függvény.\n",
        "Ezt a függvény a `matplotlib` modul `pyplot` almoduljában található. Tehát ha a `matplotlib` modult betöltöttük akkor az alábbi módon használható:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5l-Yf9ajbgzS",
        "colab_type": "code",
        "colab": {},
        "outputId": "62259791-ba0d-497e-93c2-e243c69a8fc5"
      },
      "source": [
        "x=[1,2,3,4,5]\n",
        "y=[1,4,13,10,25]\n",
        "matplotlib.pyplot.plot(x,y)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f8d21942828>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 47
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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utqxfny9qcdZZ+bqUNnSpXDZ1tWzdunzF+b/5G7jiirKrkQQ2dbVo7dq8h/6lL8Gll5Zd\njaTtzCeoaatX5/n5lVfCOeeUXY2kiWzqasozz8App+SLXJxxRtnVSJrMpq6GPf44zJ+fL0M3b17Z\n1UgqYlNXQ1auzOdyWbIkf8FIUjV5oFTTuv9+OPNMuPNOG7pUdTZ17dDSpfCZz8CyZTntIqnabOqa\n0pIlcNFF+VzoRx5ZdjWSGmFTV6HFi3MG/cEH4Ygjyq5GUqM8UKrfcc01cNVVMDYGhxxSdjWSmmFT\n17ssWgQ33ACPPAKzZ5ddjaRm2dQFQEowMgK33w4PPwyzZpVdkaRW2NRFSnDZZbBiRW7o++1XdkWS\nWmVTH3Lj43DxxfD00/DQQ7D33mVXJKkdNvUhtm0bnHcePP98TrnssUfZFUlql019SG3dCmefnS9D\nt3w57LZb2RVJ6gSb+hB68818Hpe33oJ774X3vKfsiiR1il8+GjJbtsBpp8HMmTA6akOXBo1NfYi8\n/jqcfDLsuy/cdhvsskvZFUnqNJv6kNi8OZ9h8UMfgptuynvqkgaPTX0IvPIKHHsszJ0L118PM/yv\nLg0s/3oPuA0boFbL1xS9+mqIKLsiSd3U1ofwiHgBeA0YB7amlOZ2oih1xvr1cNxx+eLQV1xRdjWS\neqHdyeo4UEsp/WcnilHnrFuXG/rFF8Oll5ZdjaReaXf8Eh14D3XY2rX5KkVf+pINXRo27TbkBKyI\niKcj4rOdKEjtWb06HxT9+tfhggvKrkZSr7U7fjkqpbQhIvYlN/dnU0qPTX7RyMjI2/drtRq1Wq3N\nzarIM8/AKafki1yccUbZ1UhqxtjYGGNjY22/T6SU2q8GiIiFwOsppW9Oejx1ahua2uOPw/z5+TJ0\n8+aVXY2kdkUEKaWm82otj18iYteI2L1+fzfgE8DPWn0/tW7lSjj9dPj+923o0rBrZ/yyP3BXRKT6\n+yxJKT3QmbLUqPvvh3PPhTvvzAdHJQ23jo1fptyA45euWboULrwQ7rkHjjyy7GokdVLPxy8q15Il\ncNFF+VzoNnRJ29nU+9DixTmD/uCDcMQRZVcjqUo8V1+fueYauOoqGBuDQw4puxpJVWNT7xMvvwz/\n/M9w663wyCMwe3bZFUmqIscvFbZ+fd4zr9XyedB/+Ut4+GEbuqSpmX6pmOeey5eZu+uufFKuefNg\nwQL48z/30nPSMGk1/WJTL1lKsGpVbuSjo/kKRfPn50b+8Y97hSJpWNnU+8j4OPz7v7/TyHfaCf7i\nL3IznzvXKxNJar2pux/YI1u35sTK6CjcfTfst1/eG1+2DD7yEa9IJKkzbOpdtGULPPBAno/fe28+\n2LlgATz6KHzwg2VXJ2kQOX7psNdeg/vuy3vkK1bAn/xJbuSnnw6zZpVdnaR+4Uy9RC+/nMcoo6Pw\n2GM5gjh/Ppx6KuyzT9nVSepHNvUeW78+j1VGR3N65cQT8x75SSfBe99bdnWS+p1NvQfMkEvqFZt6\nF5ghl1QWm3qHFGXIFyzINzPkknrFnHobijLk8+fni08cdpgZckn9Y2ibuhlySYNoqMYvZsgl9Qtn\n6lOYnCE/5pjcyM2QS6oym/oEkzPkJ5yQG/nJJ5shl9Qfhr6pmyGXNEiGrqmbIZc0yIaiqU/OkM+Y\nkc9DboZc0qAZ2Jz65Az5vvvmJm6GXJJ+VyWb+uQM+SGH5Eb+yCP5viSpWGXGL2bIJekdfTlTN0Mu\nScX6pqlPlSE/6STYY4+uliJJfaPSTX3t2vSuDPmpp+ZGfvzxZsglqUilm/r735/elSHfeeeublKS\n+l6lm/q2bckMuSQ1odWm3pNWa0OXpN6w3UrSALGpS9IAaaupR8SJEbE2Iv5PRHy5U0VJklrTclOP\niBnAt4ETgEOBsyLiw50qrNfGxsbKLqEh1tk5/VAjWGen9UudrWpnT30u8MuU0osppa3AbcBpnSmr\n9/rlP7R1dk4/1AjW2Wn9Umer2mnqs4D1E35+qf6YJKkkHiiVpAHS8pePIuJIYCSldGL9568AKaW0\naNLryrvqtCT1sZ5+ozQidgKeA44DNgBPAWellJ5t6Q0lSW1r+SIZKaVtEXER8AB5jHOjDV2SytX1\nc79IknqnIwdKI+LGiNgUEat38JprIuKXEbEqIuZ0YrvNmq7OiDgmIjZHxI/rt6+VUOOBEbEyIn4e\nEWsi4m+neF2p69lInRVZz9+LiCcj4if1OhdO8bqy13PaOquwnvU6ZtS3v2yK50v/u16vY8o6q7KW\n9VpeiIif1v/bPzXFaxpf05RS2zfgaGAOsHqK508C7qvf/yjwRCe224U6jwGWlVHbhBoOAObU7+9O\nPm7x4aqtZ4N1lr6e9Tp2rf+5E/AEMLdq69lgnVVZz0uA7xfVUpW1bKDOSqxlvZZ1wF47eL6pNe3I\nnnpK6THgP3fwktOAm+uvfRLYMyL278S2m9FAnQBNH23upJTSxpTSqvr9N4Bn+d38f+nr2WCdUPJ6\nAqSUttTv/h75ONLkmWPp61nf9nR1QsnrGREHAicDi6d4SSXWsoE6oQL/b9YFO56aNLWmvcqpT/6i\n0n9Q3S8q/Y/6R5z7IuK/lVlIRBxM/mTx5KSnKrWeO6gTKrCe9Y/hPwE2AitSSk9Pekkl1rOBOqH8\n9fyfwGUU/4MDFVlLpq8Tyl/L7RKwIiKejojPFjzf1Jr65aN3+xHw+ymlOeTz2txdViERsTtwJ/CF\n+p5wJU1TZyXWM6U0nlI6AjgQ+GjZ/1hPpYE6S13PiPgksKn+CS2ozp7uuzRYZyX+36w7KqX0R+RP\nFp+PiKPbebNeNfX/AA6a8POB9ccqJaX0xvaPwCmlHwI7R8Teva4jImaSG+X3Ukr3FLykEus5XZ1V\nWc8J9fxf4CHgxElPVWI9t5uqzgqs51HAvIhYB9wK/FlE3DzpNVVYy2nrrMBaTqxlQ/3PXwN3kc+r\nNVFTa9rJpr6jf7mXAX8Nb38TdXNKaVMHt92MKeucOKeKiLnkyOervSpsgu8Av0gpfWuK56uynjus\nswrrGRH7RMSe9fv/BTgeWDvpZaWvZyN1lr2eKaXLU0q/n1L6AHAmsDKl9NeTXlb6WjZSZ9lrOWHb\nu9Y/7RIRuwGfAH426WVNrWnLXz6aVNgtQA14X0T8ClgI7EI+bcD/SindHxEnR8TzwP8DPtOJ7Xa6\nTuBTEXEBsBX4LfDpEmo8CvgrYE19vpqAy4HZVGg9G6mTCqwn8H7gpsinip4B/Gt9/c6nQuvZSJ1U\nYz1/RwXXslBF13J/4K7Ip1OZCSxJKT3Qzpr65SNJGiAeKJWkAWJTl6QBYlOXpAFiU5ekAWJTl6QB\nYlOXpAFiU5ekAWJTl6QB8v8BDxgIIoGJlhkAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d21cfc3c8>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mTrwAfJ9bgzT",
        "colab_type": "text"
      },
      "source": [
        "A fenti parancs tehát az `y` tömböt ábrázolja az `x` függvényében. "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FMROwS9XbgzT",
        "colab_type": "text"
      },
      "source": [
        "Mivel a notebook elején lefuttatuk a  `%pylab inline` parancsot, ami többek között a `matplotlib.pyplot` modulból is betöltötte az összes függvényt, ezért elegendő a `plot()` függvényt magában hívni!"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3fg01IBrbgzT",
        "colab_type": "code",
        "colab": {},
        "outputId": "9a4625fd-259f-474a-ac67-fcde8b10ad14"
      },
      "source": [
        "plot(x,y)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f8d20eb8828>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 48
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d21cfc320>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BTFqOXmLbgzU",
        "colab_type": "text"
      },
      "source": [
        "A `Python`-ban konkrét matematikai függvények ábrázolásának az a legegyszerűbb módja, ha két tömböt generálunk, egyet, amely azokat a pontokat tartalmazza, ahol a függvényt ki szeretnénk értékelni, illetve egyet, amely a függvény értékét tartalmazza a kiértékelendő pontokban. Ezeket a `plot()` függvény segítségével ábrázoljuk.\n",
        "Az alábbi példán a $\\sin(t)$ függvényt ábrázoljuk a $[0,2\\pi]$ intervallumon tíz darab mintavételezési pontot használva."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wwqn9eaObgzV",
        "colab_type": "code",
        "colab": {},
        "outputId": "641c40f5-bae0-48b6-c31b-a84e7471e53e"
      },
      "source": [
        "t=linspace(0,2*pi,10)\n",
        "plot(t,sin(t))"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f8d2175ccf8>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 49
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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wqJk1BFYC3YHzK3uSZEKLiEjq0m3LPMPMlgLHAc+b2QuJj+9vZs8DOOe2AH2B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\n4AAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d221899b0>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "B1CHNJD_bgzW",
        "colab_type": "text"
      },
      "source": [
        "Ha több mintavételezési pontot használunk akkor simább függvényt kapunk. Ezt úgy érhetjük el, ha a `linspace()` függvény harmadik változójának segítségével megnöveljük a legenerált értékek számát. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PYsm0IMZbgzW",
        "colab_type": "code",
        "colab": {},
        "outputId": "949c28e4-dc7d-4d1b-cbaf-fa5fd8e8b002"
      },
      "source": [
        "t=linspace(0,2*pi,100) \n",
        "plot(t,sin(t))"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f8d200f0860>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 50
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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DF8kz++0H77wDNWrAEUf4HSiz5bXXoFkzfzbvG2+o2EdVSgXfzM4zswVmVmRm\nzcq4rq2ZLTGzpWZ2bSr3FJHS7bKLP0TkgQf8qVl9+8J332Xufl9+6e9z2WV+y4ShQ2GHHTJ3P0lN\nqi38D4GzgTdLu8DMqgAjgdOBQ4AuZnZgiveNpMLCwtARUqL8YaUzf/v2MH++323zwAPhzjth48a0\nPTxr18LgwXD44X7HywULoEaNwvTdIIC4//5UREoF3zn3kXNuGVBW31ELYJlzbqVz7mdgPHBWKveN\nqrj/wih/WOnOv8cefrHTzJkwe7Y/ROWaa2D58so/5pIlMGCAP3t31Sr/2EOG+JlC+vlHXzamZe4N\nfL7N56vwTwIikgVNmsBzz/ktlkeNgmOPhaZNoXVrf4Riixb+PNmSrF3rt0WYPt3vgbN2re/C+eAD\naNAgu38PSV25Bd/MpgJ1t/0S4IDrnXMvZyqYiKTXAQfA3XfDrbfCm2/CtGnQv7/v+qldG+rX9+83\nbvRv33wD69b5bpsWLeCRR/yTRRVN9YittEzLNLNpQH/n3Psl/NkxwGDnXNvE5wMB55y7q5TH0pxM\nEZEkVWRaZjq7dEq72WygsZk1BL4EOgNdSnuQioQWEZHkpTots6OZfQ4cA0w2s1cTX69nZpMBnHNF\nQB9gCrAQGO+cW5xabBERSVbkVtqKiEhmRGb4Jc6Ls8zscTP72szmh85SGWZW38zeMLOFZvahmV0Z\nOlMyzKyGmc0ys7mJ/INCZ0qWmVUxs/fNbFLoLMkys0/NbF7i5/9u6DzJMrNaZjbBzBYn/g+0DJ2p\nosysSeLn/n7i/bqy/v9GooWfWJy1FDgV+ALf79/ZObckaLAKMrPjgfXAGOfc4aHzJMvM9gL2cs59\nYGa7AO8BZ8Xl5w9gZjs55zaYWVXgLeBK51xsio+Z/RFoDuzmnOsQOk8yzOwToLlz7l+hs1SGmT0J\nvOmce8LMqgE7Oed+CBwraYk6ugpo6Zz7vKRrotLCj/XiLOfcDCCWv+wAzrmvnHMfJD5eDyzGr5+I\nDefchsSHNfCTEcK3ZCrIzOoDZwB/CZ2lkozo1JKkmNluwAnOuScAnHNb4ljsE1oDH5dW7CE6/0gl\nLc6KVcHJFWa2L3AkMCtskuQkukTmAl8BU51zs0NnSsK9wABi9CS1HQdMNbPZZtYzdJgk7Qd8Z2ZP\nJLpFRplZzdChKul84K9lXRCVgi8RkOjOmQhclWjpx4Zzrtg5dxRQH2hpZhE7/6lkZnYm8HXiFZZR\n9jYlUXXzgDPRAAABiElEQVScc64Z/lXKFYkuzrioBjQDHkz8HTYAA8NGSp6Z7QB0ACaUdV1UCv5q\nYJ9tPq+f+JpkSaLvciIw1jn3Uug8lZV4OT4NaBs6SwUdB3RI9IP/FTjZzMYEzpQU59yXifffAi8Q\nr61TVgGfO+fmJD6fiH8CiJt2wHuJf4NSRaXg/2dxlplVxy/Oittshbi2zrYaDSxyzt0fOkiyzGwP\nM6uV+LgmcBoQiwFn59x1zrl9nHON8L/3bzjneoTOVVFmtlPilSFmtjPQBlgQNlXFOee+Bj43syaJ\nL50KLAoYqbK6UE53DkTkTFvnXJGZbV2cVQV4PE6Ls8zsGaAA+LWZfQYM2joIFAdmdhzQDfgw0Q/u\ngOucc38Pm6zC6gFPJWYpVAGedc69EjhTvqgLvJDYEqUaMM45NyVwpmRdCYxLdIt8AlwUOE9SzGwn\n/IBtr3KvjcK0TBERybyodOmIiEiGqeCLiOQJFXwRkTyhgi8ikidU8EVE8oQKvohInlDBFxHJEyr4\nIiJ54v8A42wJvv9FA/kAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d21772860>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_pcm8Jw_bgzZ",
        "colab_type": "text"
      },
      "source": [
        "Ha egyszerre több függvényt szeretnénk ábrázolni, akkor egy kódcellán belül több `plot()` parancsot is kiadhatunk:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wLHab-zBbgzZ",
        "colab_type": "code",
        "colab": {},
        "outputId": "38655cc2-6053-4ed5-93b5-cc67cba32d7b"
      },
      "source": [
        "plot(t,sin(t))\n",
        "plot(t,cos(t))"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f8d200feba8>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 51
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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WURYCV0yebAdyKKXeUEolAOoAi/5xzSKgUWSwEsCN5xV74Thp0XTcW29BgQLS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\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d200febe0>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6LSi7WGLbgza",
        "colab_type": "text"
      },
      "source": [
        "Ha kétváltozós függvényt szeretnénk ábrázolni, akkor ahhoz a mintavételezést a `numpy` csomag `meshgrid()` függvényével tehetjük meg az alábbi szintaxis szerint:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1zckCVUYbgzb",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "xrange=linspace(-3,3,100) # határok és pontok száma az x irányba\n",
        "yrange=linspace(-3,3,100) # határok és pontok száma az y irányba\n",
        "x,y=meshgrid(xrange,yrange) # mintavételezés az x és y síkban"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NhcQ8i4zbgzc",
        "colab_type": "text"
      },
      "source": [
        "Két változós függvényt a `pcolor()` `matplotlib` függvény segítségével tudunk ábrázolni. A fent definiált `x` és `y` tömbök segítségével például az $$f(x,y)=\\mathrm{e}^{-(x^2+y^2)}$$ kétdimenziós Gauss-görbét az alábbi módon ábrázolhatjuk:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zen_qJdjbgzc",
        "colab_type": "code",
        "colab": {},
        "outputId": "d1d3b0d0-a2f3-464a-a24d-6fe02a506eed"
      },
      "source": [
        "pcolor(x,y,exp(-(x**2+y**2)))"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.collections.PolyCollection at 0x7f8d21475f60>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 53
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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DUJlPhei2992FKFPlFdbDmTqv0jIBRSsn0rB0q8x9zu5c/nEfim8DgBKXQ6++7bFX7bau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\nDcDJnBBCGoCTOSGENMD/B4QQRzpDbIoGAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d2172cd68>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BZTrBcHsbgzd",
        "colab_type": "text"
      },
      "source": [
        "Végül a statisztikus problémák vizsgálatakor számtalanszor használt hisztogramkészítéssel ismerkedjünk. Erre a `matplotlib` modul a `hist()` függvényét fogjuk használni. Előszöris gyártsunk a `numpy` modul `random` almoduljának `randn()` függvényével ezer darab véletlen számot. A  `randn()` függvény Gauss-eloszlás szerint generál véletlen számokat. A `numpy` `random` modulja sok egyéb más eloszlás szerint is tud véletlenszámokat generálni, a rendelkezésre álló generátorokról [itt](http://docs.scipy.org/doc/numpy/reference/routines.random.html) található információ."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FJLdka1Zbgze",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "gauss_eloszlas=random.randn(1000)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "anUKDZCDbgzf",
        "colab_type": "text"
      },
      "source": [
        "Egy adatsor hisztogramját egyszerűen a `hist()` függvénnyel lehet legyártani:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "35ohJxyObgzg",
        "colab_type": "code",
        "colab": {},
        "outputId": "bfc03cf0-1f4e-440e-eb70-38faddac0e82"
      },
      "source": [
        "hist(gauss_eloszlas)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(array([   1.,    6.,   14.,   79.,  197.,  271.,  273.,  117.,   34.,    8.]),\n",
              " array([-4.29185613, -3.54341105, -2.79496596, -2.04652088, -1.29807579,\n",
              "        -0.54963071,  0.19881438,  0.94725946,  1.69570454,  2.44414963,\n",
              "         3.19259471]),\n",
              " <a list of 10 Patch objects>)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 55
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f8d217317b8>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ICOdCXICbgzh",
        "colab_type": "text"
      },
      "source": [
        "A fenti ábrán az ábrázolt intervallumot a hist parancs, az alapértelmezésnek megfelelően 10 alintervallumra osztja, és azt jeleníti meg, hogy az adott alintervallumban hány érték található a bemeneti `gauss_eloszlas` tömbben. "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5tCRO52Pbgzi",
        "colab_type": "text"
      },
      "source": [
        "Fontos megjegyezni hogy a `plot()`, `pcolor()` és`hist()` parancsoknak a fent tárgyalt legegyszerűbb használatán túl számos alapértelmezett értékekkel ellátott kulcsszavas argumentuma van, melyekről egy rövid leírás a megfelelő függvény dokumentációjában ( a docstringjében ) található."
      ]
    }
  ]
}