{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Bevezetés\n",
"Sykora Henrik - BME-GPK Műszaki Mechanikai Tanszék (sykora@mm.bme.hu)\n",
"### Python programozási nyelv\n",
"\n",
"A Python egy open-source (OS), interpretált, általános célú programozási nyelv (vagy script-nyelv).\n",
"\n",
"**Tulajdonságai:**\n",
"- Objektum orientált\n",
"- Interpretált\n",
" - Nem szükséges fordítani (mint a pl a *C++*-t), elég csak beírni a parancsot, és már futtatható is a kód\n",
" - Alkalmas teszi számítások gyors-prototipizálására\n",
" - Cserébe lassú\n",
"- Open-source:\n",
" - Ingyenes\n",
" - Folyamatosan karban tartott \n",
" - Széles körben felhasznált iparban és akadémiában is\n",
" - Nagy \"Community\", sok segédlettel, fórummal (pl.: [stackoverflow](https://stackoverflow.com/questions/tagged/python)) \n",
"- Moduláris:\n",
" - Rengetek feladatra létezik \"*package*\" (pl.: numerikus számításokra *numpy*/*scipy*, szimbolikus számításokra *sympy*, táblázatfájl-kezelésre *CSV*)\n",
" - Csak azt kell behívni, amire szükségünk van\n",
" - Ismerni kell a *package* ekoszisztémát, mik léteznek, mi mire jó, stb...\n",
"- Sok IDE (*Integrated Development Environment*) létezik:\n",
" - Alapvetően shell (terminál) alapú\n",
" - Notebook: **_jupyter notebook_**, *jupyter lab*\n",
" - Szövegszerkesztő: *Spyder*, *VS Code* (ingyenes/open source - ezek tartalmaznak *Debugger*-t is)\n",
" - Fizetős szövegszerkeszők (lista nem teljes): *Visual Studio*, *PyCharm*, stb...\n",
" \n",
"### Jupyter notebook működése (+ Python kernel):\n",
"\n",
"Legfontosabb tudnivalók:\n",
"\n",
"- Csak egy *front-end*, ami kommunikál egy *kernel*-lel (ez a kernel menüben választható).\n",
"- Két mód létezik:\n",
" - Command mode (cellaműveleteket lehet végezni)\n",
" - Edit mode (szövegbevitel cellába)\n",
"- Command mode (`ESC` billentyű lenyomásával érhető el, kék csík a cella kijelölése esetén):\n",
" - Notebook mentése: `s`\n",
" - Cellák hozzáadása: `b` az aktuális cella alá, `a` az aktuális cella fölé\n",
" - Cella törlése: kétszer egymás után a `d` billentyű lenyomása\n",
" - Cella törlésének visszavonása: `z`\n",
" - Cella másolás: `c`, kivágás: `x`, beillesztés az aktuális cella alá: `v`\n",
" - Számozás bekapcsolása a cella soraira: `l` (kis L), vagy `Shift + l` az összes cellára\n",
" - Cellamódok: futtatandó kód: `y`, nyers kód (nem futtatható): `r`, markdown (formázott szöveg): `m` \n",
"- Edit mode (`Enter` billenytű lenyomásával érhető el, zöld szín):\n",
" - Sor \"kikommentelése\"/\"visszaállítása\": `Ctrl + /`\n",
" - Több kurzor lehelyezése: `Ctrl + Bal egérgomb` \n",
" - Téglalap-szerű kijelölés (rectangular selection): `Alt + Bal egérgomb` \"húzása\" (dragging)\n",
"- Közös\n",
" - Cella futtatása, majd cellaléptetés: `Shift + Enter` (ez létrehoz egy új cellát, ha nincs hova lépnie)\n",
" - Cella futtatása cellaléptetés nélkül: `Ctrl + Enter` \n",
"\n",
"**Jupyter notebook help-jének előhozása**: *Edit mode*-ban `h` lenyomásával \n",
"**Python help**: Kurzorral a függvény nevén állva `Shift + Tab` vagy egy cellába `?\"fv_név\"` beírása és futtatása\n",
"\n",
"### A bevezető felépítése:\n",
"- [Alapműveletek](#Alapmuveletek)\n",
"- [Összetett függvények](#Osszetettfuggvenyek)\n",
"- [Saját függvények](#Sajatfuggvenyek)\n",
"- [Listák és vezérlési szerkezetek](#Vezerlesiszerkezetek)\n",
"- [Külső függvénykönyvtárak](#Kulsofuggvenykonyvtarak)\n",
" - [Szimbolikus matematikai műveletek](#Szimbolikus)\n",
" - [Deriválás/Integrálás](#DerivalIntegral)\n",
" - [Vektor és mátrixszámítások Sympy-ban](#SzimVektorMatrix)\n",
" - [Vektor és mátrixszámítások Numpy-ban](#NumVektorMatrix)\n",
" - [Egyenletek megoldása Scipy-ban](#Egyenletek)\n",
" - [Szimbolikus függvényekből numerikus függvények](#SymToNum)\n",
"- [Egyszerű ábrák készítése](#Egyszeruabrak)\n",
"- [+ Osztályok](#Osztalyok)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Python bevezető "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Alapműveletek (Shift/Ctrl + Enter-rel futtassuk)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.030566Z",
"start_time": "2020-02-24T14:30:27.011636Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"24"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"17 + 7 #Összeadás"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.048421Z",
"start_time": "2020-02-24T14:30:27.032860Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"2"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1+1"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.064491Z",
"start_time": "2020-02-24T14:30:27.054480Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"326"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"333 - 7 #Kivonás"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.079706Z",
"start_time": "2020-02-24T14:30:27.066825Z"
},
"scrolled": false
},
"outputs": [
{
"data": {
"text/plain": [
"242"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"11 * 22 #Szorzás"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.094382Z",
"start_time": "2020-02-24T14:30:27.082300Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"1.2857142857142858"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"9/7 #Osztás (ez nem egész (int) lesz: külön típus float)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.108835Z",
"start_time": "2020-02-24T14:30:27.096313Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"1"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"9//7"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.122691Z",
"start_time": "2020-02-24T14:30:27.110438Z"
},
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"-2.7755575615628914e-17"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"0.3-0.1-0.2 # float: számábrázolási hiba lehet!!"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.136661Z",
"start_time": "2020-02-24T14:30:27.124273Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"81"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"9**2 # Hatványozás (** és NEM ^!)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.151890Z",
"start_time": "2020-02-24T14:30:27.138916Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"1.4142135623730951"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2**(0.5) # Gyökvönás hatványozás segítségével"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.166327Z",
"start_time": "2020-02-24T14:30:27.153901Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"0.005"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"5e-3 #normálalak e segítségével (vagy 5E-3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Néhány alapművelet működik szövegre is"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.181132Z",
"start_time": "2020-02-24T14:30:27.168589Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"'str1_str2_'"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'str1_' + 'str2_' #Összeadás"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.194650Z",
"start_time": "2020-02-24T14:30:27.183558Z"
},
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"'str2_str2_'"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 * 'str2_' #Szorzás"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Összetettebb függvények "
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:27.239794Z",
"start_time": "2020-02-24T14:30:27.197061Z"
},
"scrolled": true
},
"outputs": [
{
"ename": "NameError",
"evalue": "name 'sin' is not defined",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m#szinusz\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mNameError\u001b[0m: name 'sin' is not defined"
]
}
],
"source": [
"sin(2) #szinusz"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Összetettebb függvények már nincsenek a python alapnyelvben - ilyenkor szükséges behívni külső csomagokat, pl a **math** csomagot"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:37.494160Z",
"start_time": "2020-02-24T14:30:37.490940Z"
}
},
"outputs": [],
"source": [
"import math"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:37.654331Z",
"start_time": "2020-02-24T14:30:37.640304Z"
}
},
"outputs": [
{
"ename": "NameError",
"evalue": "name 'sin' is not defined",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# ez így továbbra sem létezik\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mNameError\u001b[0m: name 'sin' is not defined"
]
}
],
"source": [
"sin(2) # ez így továbbra sem létezik"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:38.278324Z",
"start_time": "2020-02-24T14:30:37.811454Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAAPCAYAAABJNTW5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHMklEQVRoBe2a4ZEWNwyGF+YKIFAB0AEcHUAHECogdECGf/xjoAOgggAdABUQ6ABSQeA6uDyPb2W8u95vtcePzGSiGZ+9svxKlmR594MLp6enw//0wwNPnjy5xtNt2mvGJz9m/vsj9nuD9rndKc+XeL5M/7Xl/1tj7EjF56g1kEXPxue/6a/Tnu3ZELIvGrzLjB/AWyRHVk8GDxk3+vuo95j+m8/wJwEa54cE5g1k3ccLZGNZ25/A/6VlxBi+ax/SPwxe9PD22hmxCIg/wJjsiWeT7nEI0Pv8Bv77hjfwnNX9YcQMPeJJN8+65V/kD9o54m3aGMjI68M3tJuMF7kDPxWfmtiAfGLRU/q39AO9m/pEf4d28LSGLPImw/NxvQb8xbMG1vWMN/UgU3Rv4SFnwNR5R50SYx0ddtcAw09hsvYWzXXVZsZBVvI4RMFrewMSSVH56N5jp7LieDiL/fQ+2yw2LVl4JodIWdolWsQxrRvgbyO4sXP/YpgTiwSDl7UzY6OxeUVTv8VJ7DVKxackNkb+Bkp1hohuhubGrF41cZzrkEZ5XZWkdp7xZ9qfDOt6nrN6Unhgm8TzwJoQ6jER2sqaxRxYv9gvvOJs+pIwYE8I/qMJY/qwx07ttjrXQ8mzgZ8cNObdo0ViTg9gfKCFnXt0G7N7c8CV5007szYi58EpehnrRw/WKiGzGZ+L42pBF5UG3kfabYB07CG6y+TE8aOwmO36rJ4snhX0S8c+k8KDWpJxtCWL6Z57tKg8IYQeA2FwbD1K2QmONor1sgWB7605D6bVe85rl8U4pTuEM/0OO7M2ZtSGTCo+kdhuPq6hALCPZHW+S2wykr633nd16fisKx9lPbmqZydeeWVgzVpCFdv2YCIblW40udw+Vr2nlbEc3GfdJBlnIik7WePt4025tp8W1gDfRfYdLWLgvLZ6SwZldYd8ps/ambUxo7PIsNdUfI5mTllTcHltgvUGwumezJVx3bWsniyeuMiuXZtWPee9MexTNio7J9Za9f21oPtuDd+rs02kOcSATMpOFloAviKv/fdpFgarXu+D8C1yBtkq/52x9ilrotfgM87qZmnxVXldZGjs3Lvv2MWPjINSdmrHaMtBGwP0PD343fhYsSMhD1WJtiL09OtIFcypJBhM1+/Rk8Gb6yrPbFSd2jJPxPNiWgFtCxqd6qGJG2chs8ZYsTP8fMy83wrPaVZHE9vkmBA8kzZuCm30Zp0n4WSND6xb85H6/ZlTvfrP5of4/MZO28nac9mI3ix142NiZygq75qsHyxD6wDGOu9kXJANfOj5Gbw36LRS1A/Z0YbdmGB4QG7Te533yJ/2IrF684d4EzvBiWTxdphjvgboVSNTcHk22fWxlVobtddvjsUhgN/SRHdMsM53+YjZwNi4iVtvJHi77ET+vDaGWas92Kvxuciq3jtvgEWVjXfl4E96FOiMq7R7jB/RPEUq9R1L0kFpPUm8Atz+YZ0B8CpfXL/nxLRadg8leF7ZNeCtHVvjQ3au6PPXDxPqOLBH/Saild09+yEZ+14cgmbdqo9CZta7f18ljWdLPb9M7GSNPtptY6tkY7wanyOUx/tnnMQWK3i9TbRygzgwVFQJXlzhOn6Xni28qmQcIK8T/cnRAHdpLyYgVpvF3sExyJfoF3NdxQ2TNV07tY2mpH5coza59K3FpBLrvams3l9ovj7Ud22FmOvqHufe0eu/mz53qOQC83vs3G1jR+8hVjc+LjgaV8U1NgeJir12Fc/l58++jrzXGePEz+qZ4xVY8N3gdfqoWAPjkgT0W8m3hmkgxei9s8q/BbZXektiWd3ke5gn7/k8b9m55p/QUfYCjrZ5sMKvMT/AU68JfbkyGcDb0n2MWO9WLTisb/2waSfyu21s7d0aj/hr8amJbSCiuraYnl5/tF84sBViXqf5DyBXQ3ZUbNVoK0BKzw68AVmTySSbJBE8barvqnswWScZaGkRbLAMrG1C8L/D8CDXAxYC8DJ2+powPyxC6EMrZdFJ79jmIeodXJOq2odMRvdL5OY+VLcxrFgyoD12pmw8g931dzU+olz0DxsyAb7RmwyFGOucX2nlo0umPNopzXepljw58wQwQH5c1ZPOOKWHdVk85dSjXf7Tem2j7vZApjBZF+T+pRbjjLP+1zWxrkphV8pO5Ky0HoxaZBiLN4nDCOzhKf98Pj6XDnlfN/zJL6p7Sjdriu9mWP6UKU0OKthZO1M2nqmof6+Mo3JTVO5yEH4+WU4Nw4X4332jA3Wogn4s3qItfsNEzvc3nb/2Ph0KF7+9sm7YoSeCu4o32mLgeuRN094W6t7EDCBkxfUAP2A8eVcNmeiZt4Ipb3WTlP8IP/7fjD7ba2fs2wAv4gBvAN9K/JjWFpWJ35FJ60ZWG6Nqq1dcP07NiQXB158H7URm00aBkYubSh+KaUH0cHpI683LcyF4B+PzDz5Vplwn5OHdAAAAAElFTkSuQmCC\n",
"text/latex": [
"$\\displaystyle 0.9092974268256817$"
],
"text/plain": [
"0.9092974268256817"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"math.sin(2)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:38.282905Z",
"start_time": "2020-02-24T14:30:38.280034Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1.4142135623730951\n",
"-2.185039863261519\n",
"1.1071487177940904\n"
]
}
],
"source": [
"# Több parancs együttes beírásakor nem látszik, csak az utolsó sor kimenete: print függvény alkalmazása!\n",
"print(math.sqrt(2))\n",
"print(math.tan(2))\n",
"print(math.atan(2))"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:38.298398Z",
"start_time": "2020-02-24T14:30:38.284168Z"
}
},
"outputs": [],
"source": [
"# Kimenet el is rejthető a ; segítségével (\"suppress output\")\n",
"1+1;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Amennyiben szükséges, definiálhatunk mi is saját változókat az `=` jellel. \n",
"Megjegyzés: a `=` értékadó függvénynek nincs kimenete"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:38.919260Z",
"start_time": "2020-02-24T14:30:38.914779Z"
}
},
"outputs": [],
"source": [
"a=2\n",
"b=3\n",
"c=4.0 # automatikus típusadás"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:39.669381Z",
"start_time": "2020-02-24T14:30:39.224038Z"
},
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 196.0$"
],
"text/plain": [
"196.0"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(a+b*c)**a # a legáltalánosabb típus lesz a kimenet (int < float)"
]
},
{
"cell_type": "raw",
"metadata": {
"ExecuteTime": {
"end_time": "2019-09-09T09:30:55.312952Z",
"start_time": "2019-09-09T09:30:55.302551Z"
}
},
"source": [
"# Fontos, hogy igyekezzük kerülni védett változó neveket! ILYET NE!\n",
"math.sqrt = 1\n",
"math.sqrt(2)\n",
"\n",
"# KERNEL RESTART SZÜKSÉGES"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Ha véletlenül ilyet teszünk, akkor érdemes újraindítani a *kernel* a fent látható körkörös nyíllal, vagy a *Kernel* $\\rightarrow$ *Restart* segítségével**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Saját függvények \n",
"\n",
"Szerkezet:\n",
"```python\n",
"def function(*arguments):\n",
" instruction1\n",
" instruction2\n",
" ...\n",
" return result\n",
"```\n",
"\n",
"A függvény alá tartozó utasításokat tabulátoros behúzással (indent) kell beírni (nincs `{}` zárójel, vagy `end`). A függvény neve után jönnek az argumentumok majd kettősponttal `:` jelezzük, hogy hol kezdődik a függvény."
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:39.971178Z",
"start_time": "2020-02-24T14:30:39.964602Z"
}
},
"outputs": [],
"source": [
"def foo(x):\n",
" return 3*x\n",
"\n",
"def bar(x,y):\n",
" a = x+y**2\n",
" return 2*a + 4"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:40.165605Z",
"start_time": "2020-02-24T14:30:40.157459Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"9\n",
"9.0\n",
"szöveg_szöveg_szöveg_\n",
"42.0\n"
]
}
],
"source": [
"print(foo(3))\n",
"print(foo(3.))\n",
"print(foo('szöveg_'))\n",
"\n",
"print(bar(3,4.))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lehetséges úgynevezett anonim függvényeket (*anonymous function* vagy *lambda function*) is létrehozni, amely gyors módja az egyszerű, egysoros függvények létrehozására:\n",
"\n",
"```python\n",
"lambda arguments: instruction\n",
"```\n",
"\n",
"Ez akár egy változóhoz is hozzárendelhető, mint egy szám vagy string."
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:40.765556Z",
"start_time": "2020-02-24T14:30:40.761230Z"
}
},
"outputs": [],
"source": [
"double = lambda x : x*2\n",
"multiply = lambda x,y : x*y"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:40.963430Z",
"start_time": "2020-02-24T14:30:40.958127Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"6\n",
"30\n"
]
}
],
"source": [
"print(double(3))\n",
"print(multiply(10,3))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Listák és vezérlési szerkezetek (Control Flow) - csak a legfontosabbak "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Listák"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:41.588956Z",
"start_time": "2020-02-24T14:30:41.580914Z"
}
},
"outputs": [],
"source": [
"lista = [1,2,3,4,\"valami\",[1.0,4]]"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:41.808134Z",
"start_time": "2020-02-24T14:30:41.800693Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1\n",
"4\n",
"[1.0, 4]\n",
"valami\n"
]
}
],
"source": [
"print(lista[0]) # lista 1. eleme\n",
"print(lista[3]) # lista 4. eleme\n",
"print(lista[-1]) # negatív számokkal hátulról indexeljük a listát, és (-1)-től indul\n",
"print(lista[-2]) # lista utolsó előtti eleme"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:42.018213Z",
"start_time": "2020-02-24T14:30:42.009461Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[2, 3, 4, 'valami']\n",
"[2]\n",
"[3, 4, 'valami', [1.0, 4]]\n"
]
}
],
"source": [
"print(lista[1:-1]) # egyszerre több elem [inkluzív:exklúzív módon]\n",
"print(lista[1:2]) # egyszerre több elem [inkluzív:exklúzív módon]\n",
"print(lista[2:]) # lista utolsó elemét is figyelembe vesszük"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:42.709853Z",
"start_time": "2020-02-24T14:30:42.226340Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ 0, \\ 1, \\ 1, \\ 2, \\ 3, \\ 4, \\ 9, \\ 64, \\ 89\\right]$"
],
"text/plain": [
"[0, 1, 1, 2, 3, 4, 9, 64, 89]"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lista = [2,3,64,89,1,4,9,0,1]\n",
"\n",
"lista.sort()\n",
"lista"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### if-then-else\n",
"\n",
"```python\n",
"if condition:\n",
" instruction1\n",
"elif condition2:\n",
" instruction2\n",
"else:\n",
" intsturction3\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:42.715958Z",
"start_time": "2020-02-24T14:30:42.711849Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\"a\" nagyobb mint 3, de kisebb mint 10\n"
]
}
],
"source": [
"a=4\n",
"if a<=3:\n",
" print('\"a\" nem nagyobb, mint 3')\n",
"elif a>=10:\n",
" print('\"a\" nem kisebb, mint 10')\n",
"else:\n",
" print('\"a\" nagyobb mint 3, de kisebb mint 10')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### for ciklus (for loop)\n",
"```python\n",
"for i in array:\n",
" instruction\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:42.986576Z",
"start_time": "2020-02-24T14:30:42.971779Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0\n",
"1\n",
"2\n",
"\n",
"lista 0. eleme: 0\n",
"lista 1. eleme: 1\n",
"lista 2. eleme: 1\n",
"lista 3. eleme: 2\n",
"lista 4. eleme: 3\n",
"lista 5. eleme: 4\n",
"lista 6. eleme: 9\n",
"lista 7. eleme: 64\n",
"lista 8. eleme: 89\n"
]
}
],
"source": [
"for i in range(3):\n",
" print(i)\n",
" \n",
"print()\n",
"\n",
"for (i,elem) in enumerate(lista):\n",
" print('lista ',i,'. eleme: ',elem,sep='') # több elem printelése egyszerr, szeparátor = ''\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Listák gyors létrehozása (List comprehension)"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:43.781146Z",
"start_time": "2020-02-24T14:30:43.359439Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ 12, \\ 27, \\ 48\\right]$"
],
"text/plain": [
"[12, 27, 48]"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lista2 = [3*i**2 for i in range(2,5)] # range: 2,3,4\n",
"lista2"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:43.787406Z",
"start_time": "2020-02-24T14:30:43.783334Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"range(0, 10000)"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"range(10000)"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:44.326285Z",
"start_time": "2020-02-24T14:30:43.789447Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ 0, \\ 1, \\ 2, \\ 3, \\ 4, \\ 5, \\ 6, \\ 7, \\ 8, \\ 9\\right]$"
],
"text/plain": [
"[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lista3 = list(range(10))\n",
"lista3"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:44.828780Z",
"start_time": "2020-02-24T14:30:44.329304Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ 12, \\ 48, \\ 75, \\ 147, \\ 192\\right]$"
],
"text/plain": [
"[12, 48, 75, 147, 192]"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"myfun = lambda x: 3*x**2\n",
"\n",
"lista4 = [myfun(i) for i in range(2,10) if i%3 != 0] # ha i nem osztható 3-al\n",
"lista4"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Külső függvénykönyvtárak: "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A mérnöki gyakorlatban előforduló alapszámításokhoz 2 fő csomag importálása ajánlott:\n",
"- `sympy`: szimbolikus számításokhoz\n",
"- `scipy`/`numpy`: numerikus számításokhoz (pl mátrix algebra)\n",
"- `matplotlib.pyplot`: Függvények ábrázolása\n",
"A `module`-ok hívása az alábbi módokon történhet\n",
"- `import` *`modulename`* : importálja az adott `module`-t. Ekkor a `module`-ban lévő függvények hívása a `module.functionname(arguments)` segítségével történik (Az *1_Alapok.ipynb*-ben láthattunk erre példát a `math` `module` esetében). \n",
"- `import` *`modulename`* `as` *`alias`* : hasonló az előzőhöz, de megválasztjuk, hogy milyen *alias*-ként hivatkozunk a `module`-unkra\n",
"- `from` *`modulename`* ` import` *`function1, function2, ...`* : csak bizonyos függvények importálása (nem szükséges a `module`-ra hivatkozni a függvények hívása során)\n",
"- `from` *`modulename`* ` import *` : a `module` összes függvényének importálása (nem szükséges a `module`-ra hivatkozni a függvények hívása során)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Szimbolikus matematikai műveletek "
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:44.834533Z",
"start_time": "2020-02-24T14:30:44.830880Z"
}
},
"outputs": [],
"source": [
"import math\n",
"import sympy as sp\n",
"import scipy as sc\n",
"sp.init_printing()"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:44.960902Z",
"start_time": "2020-02-24T14:30:44.956953Z"
}
},
"outputs": [],
"source": [
"F, m, a, b, c, x = sp.symbols(\"F m a b c x\")"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:45.152461Z",
"start_time": "2020-02-24T14:30:45.147206Z"
}
},
"outputs": [],
"source": [
"F=m*a"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:45.827638Z",
"start_time": "2020-02-24T14:30:45.343873Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left( a, \\ 7\\right)$"
],
"text/plain": [
"(a, 7)"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(a,7)"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:46.299119Z",
"start_time": "2020-02-24T14:30:45.829733Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 7 m$"
],
"text/plain": [
"7⋅m"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"F.subs(a,7)"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:46.797595Z",
"start_time": "2020-02-24T14:30:46.301353Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAPCAYAAAD6Ud/mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDUlEQVQ4EZ2U7RGCMAyGi+cAugIj+LGBbqAzMII/4S+rqBvoBnqM4AjCBvgGKxdiWoO9C9Ckz5ukFJI8z1Pn3AZ2LIqiwX3UAGPiEyTaQfkUUW8gNg/FETPxUwisYVfYQxGjTg+Kn7tMPCVyqGrLSe+jLaHYWcbk3MJTopsE/byEwD4Q424b37atk4b3VsIW0m+da7yWJMXCi1VUrgOr8hO+B58tw71U/FYXsd88r8hXU3PfmOcYLzvKUI12zK3dBHmZiD6+p1VVWRfk+0Q4yjOAKWz0b4gS/uL7RFi7IgDj346ifPdneOs76oiG2pGvuEa8wvOyWzm8RHneUeWTqF86xKkAOij3oX4/i/IvLjgmApqttJ4AAAAASUVORK5CYII=\n",
"text/latex": [
"$\\displaystyle 7.7$"
],
"text/plain": [
"7.70000000000000"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"F.subs([(a,7),(m,1.1)])"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:47.363940Z",
"start_time": "2020-02-24T14:30:46.801139Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$"
],
"text/plain": [
" 3 2 2 3\n",
"a + 3⋅a ⋅b + 3⋅a⋅b + b "
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"((a+b)**3).expand()"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:47.877779Z",
"start_time": "2020-02-24T14:30:47.366027Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle a^{7} + 7 a^{6} b + 21 a^{5} b^{2} + 35 a^{4} b^{3} + 35 a^{3} b^{4} - 8 a^{3} + 21 a^{2} b^{5} - 12 a^{2} b + 7 a b^{6} - 6 a b^{2} + b^{7} - b^{3}$"
],
"text/plain": [
" 7 6 5 2 4 3 3 4 3 2 5 2 \n",
"a + 7⋅a ⋅b + 21⋅a ⋅b + 35⋅a ⋅b + 35⋅a ⋅b - 8⋅a + 21⋅a ⋅b - 12⋅a ⋅b + 7⋅a\n",
"\n",
" 6 2 7 3\n",
"⋅b - 6⋅a⋅b + b - b "
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"((a+b)**7 - (b+2*a)**3).expand()"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:48.368238Z",
"start_time": "2020-02-24T14:30:47.880176Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left(a + b\\right)^{2}$"
],
"text/plain": [
" 2\n",
"(a + b) "
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(a**2+b**2+2*a*b).factor()"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:48.856643Z",
"start_time": "2020-02-24T14:30:48.370976Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAaCAYAAADsS+FMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADXklEQVRYCeWY7VEbMRCGDZMCnKQD04EJFQQ6IHQAdIAnv+y/pINABQQ6cFJBCB04HYS4A+d9NFqNTqc739lwscc7s5a0q9N+aD/uvLdYLHq7BJPJpC97P3ubB348F33+Zpcc4W29luGXZrfmXzX/JTzYN+IOjRdywHFk77XmA9GGu+gMouIxckaY7u1azQiW+4kigsg41VhOExGtqKTPbcW6jf7aO8QRwkOMK6SJmFeisWGbgfzHjlrwTiMqDjWfszmkiQh46EjjCMYmgnSjLd4LPwiftT7I6Sk6zphrvKngE/0j8V1X0eiywTlDC4T80OjCJXfAJtGkJ63wUWNokal+fs9Hje7Wja81htNO40vnnJGlCeHChk5ACh0LL9YQRipPlzyPPdiVAo6ktTIa0m7DS9eZFpVeTk97gTWRCLYGHOkf+l73sPbdCP8KSYcQHZq/rXpuX0xqxe+qDRtIP0Hf2MAaHbHrrIZfYPE6zuHLvEyeEXIc/kf4IKQ3f9HYNRAZT5JtHeNI6zut0SkF7MK+bCFNN1MzqMyzlGFrCbH8ItxAHODyUfOVQt3Objt6edQLLoc0QJdz4b3m0FLArhw93efWOAODnnNcL5xWhhPiVGLeNFRzR69Ks3oRuoT0oh6ApHsK2NXYGaTJOyGH5YDU6EtgGmYolQvLwhl6jggyA2IeMnvi54o2KfAp3hzNCXn4qb590d9H+2zKpcFrBDijDig+hXoiRTgcby9rbT3tzRkLnVvkTbFtzSldgs4wZ//UmSnUXXS6172OE0ol70kINDA12lVn8QtOKp38OgQuITXaoiinD/pnS0BOPWoGoVSXV3Gt4AwXqkzkEDpK3bNse2kI+kg2xnI5l5rPM4KIjLA/wy+QcMaTkPZUAH843g7GikZ4D4X2f8CJaI2FFQSstqBO0f0MKO7fpENa04zP5wX2NQJqxp2QQ3NACN5KGD2dAsVeIoO/zqB19govWQBtFH34QCMqaKlVjhDbFe9s3YJZAv7cGY/HM+GQeRcoWafCq9eUpfMHwlkbGaQJQAtt7kH3yFo/5Hcux9c6NHmYr1Lsag7mOXlxijdtvc2jj4ppWxssMvAe9aHrGtD81trtxI7WkR7+6UKWihGd4399gKHC2iAbKOwPGlt3uX+qgpYbzIC0HAAAAABJRU5ErkJggg==\n",
"text/latex": [
"$\\displaystyle \\left(a + b\\right)^{2}$"
],
"text/plain": [
" 2\n",
"(a + b) "
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.factor(a**2+b**2+2*a*b)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:49.357643Z",
"start_time": "2020-02-24T14:30:48.858711Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left(a + b\\right)^{3}$"
],
"text/plain": [
" 3\n",
"(a + b) "
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.factor(b**3 + 3*a*b**2 + 3*a**2*b + a**3)"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:49.882404Z",
"start_time": "2020-02-24T14:30:49.361845Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{a}{b} + \\frac{c}{b} + \\frac{7}{b}$"
],
"text/plain": [
"a c 7\n",
"─ + ─ + ─\n",
"b b b"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a/b+c/b+7/b"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:50.346024Z",
"start_time": "2020-02-24T14:30:49.883959Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{a + c + 7}{b}$"
],
"text/plain": [
"a + c + 7\n",
"─────────\n",
" b "
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.ratsimp(a/b+c/b+7/b)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:50.858971Z",
"start_time": "2020-02-24T14:30:50.347747Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{a + c + 7}{b}$"
],
"text/plain": [
"a + c + 7\n",
"─────────\n",
" b "
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(a/b+c/b+7/b).ratsimp()"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:51.406560Z",
"start_time": "2020-02-24T14:30:50.863638Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZElEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUACBj0chKHhJQQLN0ZQZAGDGBRBIOACxKC4OQfE94B4NwDm+hiAOyllRAAAAABJRU5ErkJggg==\n",
"text/latex": [
"$\\displaystyle 1$"
],
"text/plain": [
"1"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(sp.sin(x)**2 + sp.cos(x)**2).simplify().expand()"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:51.911217Z",
"start_time": "2020-02-24T14:30:51.408590Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\cos{\\left(2 x \\right)}$"
],
"text/plain": [
"cos(2⋅x)"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(sp.cos(2*x)).expand()"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.437696Z",
"start_time": "2020-02-24T14:30:51.913785Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAYCAYAAAArrNkGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFJUlEQVRoBe2Z63EUORCA11sEYCADyADsCDgy4BEBJoO78i/73xVkABcCZOAjAjgygAwwzsD3fUKt0uw8dpYd9uG7rtKqp7vVavVD0sweXF9fz/YBzs/PD7HzNNt6L/cvoF/tg/2bsvHWpiaaYJ5XBO9l6AF/A/4P7X7Q/u9ns/keOeGEIP5W2fsK/B60BxXtP4/uU0Ctzk/rRIzgx1a9spp1xq482RoD9iagOPQtrT4vDfBXaJ/HrB+535Fbp5rdDdSx03DgpQhDzdw/sqVH9Jc+Qx/lrDxuYx12GZh3tIfgdZA7bUDmCYxj+lhjp9wyIuMN6BX922Wym+BjR8sPB2dnZwbzDczHYQS455PGPwb/O+i70GNPshdbnoKPCeYhsh+QfTiF/ejxIvZozNxTzLeog3ldz180i87iM6i3wx5vuQav3B7BZzCtzhNQq+C2tF0AbEo7CX1Kvvw8o/86YJ/r80Y8Fair5bOplC/Tw1pN4qfKgbeOkTl0b45fYBr5GqzMQ+g6ceuQ7dCZ7iYPbOBuoWbqEDxDdrItMutS56K/hmzYGM8KNXA6yMh3wa4Y7lanLfYFsLuxuxQGCDzPzqHqrcVXwdX5jDZZoqwy+ZDsLRadyrdDyArQKb0XI3g62O2nToaP0N87VqhkvvygpA8B76CXsxncudwp1KNO4Tn0cu6B/8zW79Zc5klaqx90OpdHi3BMe0FzR3pOExpr+UFKv+pU9+4FtDKyoNnB6bwqxAUEGflWi5/fUgDpDYrBSq8T9AbKc9jLVakU8At5tNc0nepXoHQugifgubfyQmZE76Vh6PwsX5+Yz/PIy4ZB9A4RZ2VJzmo+k9O17hzMeywyCO9Z1OsevmRlDFy94Kiuqzwu9JRgZrpnn8404Dr9CDzGZpHBQITMsl6dnWcs81mZ7i4B2uwWHVV3B7zmh5y9Ok3onQPP0AawUDPTQPVtxTN4OspgNAIO3eBGtcp30R9pDUDuM02a26rVoIO+07uVXdCWJRMio8CgGKgu+MR8daK5vWtXkqfvXT9yjtMHg4AOZT7QlspWinwd6z3mKrlOtBFQFJm1d+gb21/HyMjObx28IIVMclAQF3qDLujMU5oVYlVYvX4ZmmLbRV0b0L3oNC85f7YlOylDiVIGMIdrL/eAwviFyDx0M7nOvE9fMhPcz10RmBC1j8we+qcjZIay050g6ae3Up3/AP0G8gQ8Au6cPwNW/tD8SSfzeB4qV44PaL6y9Y2Vru6dgxTQ7LiuT2MGuWU48lfQzfCjrhXBf0KTr1yr2uHpQMEz1qDFTVPaDL7nmM7t1K/MSDCpWgmJfoPlxSzsMIH8pBdJqPpTnrW/C6zQWrZLZiu0OUa7YB3rIn1pLw3ay4FFWclW8GIw3DINpvCI5kv4YqUp4w03Xil0nllfg8/Br+mr4Npx3DHAQNou87yXtQw0ea2zv5JJ5231vC30bp7YBEvgt1yv4K0sznwvCb1nQHZG3ATjPdMLTclecHV7q41s99mkScGi13lWcX0ea6gVtFZAGW8i+RrVOBp4Nlm0O9nMs8nlbqQdvooZ6LL98twAeI4z2deyr6F0hQfmtQAFfedaTFx9fpH+bQG5sZCdv9bNsXYO+kxIk62RJLXMNvH5Niff0NxW4pS35fQOvSHbV57mxgeUSvKC1XdbX8lhuTrVFR8fVhq/CeEbH9DsRC9wQ58Ax/paHVNW+9h5R8vd+DM0PJGry9epxtet4C/rGee33saFb9mYbfD/BasE5Yb2Eji6AAAAAElFTkSuQmCC\n",
"text/latex": [
"$\\displaystyle 2 \\cos^{2}{\\left(x \\right)} - 1$"
],
"text/plain": [
" 2 \n",
"2⋅cos (x) - 1"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.expand_trig(sp.cos(2*x))"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.443179Z",
"start_time": "2020-02-24T14:30:52.440376Z"
}
},
"outputs": [],
"source": [
"import scipy.constants"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.953415Z",
"start_time": "2020-02-24T14:30:52.445167Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 1.618033988749895$"
],
"text/plain": [
"1.618033988749895"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sc.constants.golden"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.975935Z",
"start_time": "2020-02-24T14:30:52.957853Z"
}
},
"outputs": [
{
"ename": "TypeError",
"evalue": "can't convert complex to float",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mmath\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msqrt\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m0j\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# A math csomagban nincsenek implementálva a komplex számok!\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m: can't convert complex to float"
]
}
],
"source": [
"math.sqrt(-1+0j) # A math csomagban nincsenek implementálva a komplex számok!"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.980622Z",
"start_time": "2020-02-24T14:30:51.148Z"
}
},
"outputs": [],
"source": [
"sc.sqrt(-1+0j)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.982560Z",
"start_time": "2020-02-24T14:30:51.655Z"
}
},
"outputs": [],
"source": [
"sp.limit(sp.sin(x)/x,x,0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"Taylor-sor megadása. Első paraméter a függvény, második a változó, harmadik az érték ami körül akarjuk a sort kifejteni, negyedik pedig a fokszám:\n",
"\n",
"$$f\\left(x\\right) \\approx \\sum\\limits_{i=0}^{N} \\dfrac{\\left(x - x_0\\right)^i}{i!} \\left.\\dfrac{\\mathrm{d}^i f}{\\mathrm{d} x^i}\\right|_{x = x_0}$$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:52.984881Z",
"start_time": "2020-02-24T14:30:52.400Z"
}
},
"outputs": [],
"source": [
"sp.series(sp.sin(x),x,0,20)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Deriválás/Integrálás "
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:53.092158Z",
"start_time": "2020-02-24T14:30:53.088452Z"
}
},
"outputs": [],
"source": [
"a,Δt,x = sp.symbols('a,Δt,x')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Deriválás"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:54.021312Z",
"start_time": "2020-02-24T14:30:53.658992Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 3 x^{2} \\cos{\\left(x^{3} \\right)}$"
],
"text/plain": [
" 2 ⎛ 3⎞\n",
"3⋅x ⋅cos⎝x ⎠"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.diff(sp.sin(x**3),x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Többszöri deriválás"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:54.725092Z",
"start_time": "2020-02-24T14:30:54.405133Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAZMAAAAaCAYAAABy1LHFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAMbklEQVR4Ae2d2ZUUNxSGmzkTwAARGGfAEoEhAy8RABnYhzfeOHYG2BFgyMAmAjAZ2I6AYTLA/yd0NapqVZW6S6pWu+ueo9FSWq7+u2ipmpkbnz9/3pwyPX/+/GfN/6PH4Er5X08JD833jub7UOFSgfQDhRcq/6B4pR4Cp47Xqc+/pw5Hla0tu7MaaIjpJwo4qKZJPP4lBl8p/kXxnwovm2Z4gLmZeLOYfq0+3ngc3in/Nh5K5UeJSzyHgulJvAqOVaWrVV+qwFq905lyg79J3c2x9aE6xRcTDfStmL6nGOfcLIk/gH2v2O3AffyoWYYHGBPfc/F+oa5fRd3fVppTSkw/a5w/4oITTufg1Sw8q740K5pRxgrIjf5zdDfH1pN1bpS85tKEL8TwW8X3RpFp4KF4/CQ2vlP4R+Guwp8qu1J8NFQDb/X5twBAWTrXfco/UfmFYk5xK3kEhvBqESDxWtw+h+av8lVfCilBDbnB2hzZpeRb+mTyWjw2fyXihYNhcRIh5hSFA8UAjomK4a25/6jAtd9LxZ2FBEB82VPF4HXyJBxG8WoUoFVfGhXMBFvF5MY4U7qr59j/qK2n6oSTiR7y8vUpg4lwGORxsFnXVb79X4pv0kFpUr/xInVL/T9W2dU+43he2YEDmHOcipkzpxXeH3BaaZr8HIri7TFAcf9QeusEorIf9eyBYk50ByfxwYkSfrlW3dIFlSHTZxGj5F+rPEuno3bJpO9/EK9ko4FC9YW9sZj/pPTWYj7QLLvY+ldczD7Vl8NTTDStL37uP3mw7ivmGhecm//IxPNe1M7BQf3Olp366PgDdzLxHQOuBRYVnDdKwp18DiGsGkbA1QqO/2/FOH944+7vX6UxwJ1J7WyxeG+NVXbl081/OOD5LI63x4B+2UTgqPuEfL/VMxTxIMTYCiwI6OdvCmM6wDxMp4nRHXQoV6dH56h+0JkxvEbb9x4yD3CtdUV8qvoCrpy2ne9QDL4sIjjoY7D14nLT3Dea+5Tu5th6p84ZHYu43uELrNjIbPcW7+xc5YEf9FHjJS0O45Z4CztlpVEGFgIcyr7EgsIJp0+20PTLW8vPxls44pg/KcQLB0oGsYPrkOrxDOy/7zxYMAMPCt8psDDEHw50uNBz8GGn36fHKsjV6U5b9bkTXp3GExn1jb3d9POaqL3X46PVF2HyUAH+9yE+tEFXAqkvHDS6zKmydZotNyaoOe+ku6o/aev9OraY4CBoTHDkK1p2NFZdc0Zhpz/aYLeHLHApBw/PKNnFbt2F2ihZuK7xc/hHsS2ioWJriVJ4qx/kfakQ48tujfLfFVKEjANuqQqNlH0tPop+nbcnXtlw+P6z6+dW/B/oCza+r52jz9xq9Ntj5zjYO7k4Ll2vlNzgW31h0zVsPfiDcz+Q2xWRNtLgdkrJ2f0jMBwxDBcj9WcKAAh9+ugL2EHvvACo718VeInKokJfOJ9aVwzquiiVxJuF4ZlwMDwfKJ98B+FnwJXjwU4mnoec6J0qIV9Oy5xkTDeRd45OD42xK15D/SxZfsr6gm+4G8m/j7v5mH55C/mScmM+u+pujq2HOucpxAQ8k3DHQ6Vz3oPggMxYU13uVYYCKNA2dR1123cadhaqi2LYcRieuNLg+Q8K0DvVefMl6VbrcHVmZTmxHwd84jn3+4YX6gA2xGLVefmrfjjRgTX9UB/6QeVTi9oo3p6/LBxUlxMeIZeoy46OEM8/t/0i9cQbv4SJrNkUcZXH1QYy4D1g0AHlN8qDfS28GIIxBmWtZ+goVy7EfKKO0Vsbey/ESRF9sk0eOoDO5ejwyeqLYSms+uRuU/R8UPf1DL1o1s6ZkOexlu7m2Hqo01lMxJgpPMpHJY4wOQTolzkV96iD4eNw++SUQYWMbcQLV3c/qpgvDTBEDI6Xr+xGUYyOI1F+J1I/GDx38XxN5vpSDH8sFJzOPijAG87hEWWKHSmNI2OX9IuCU1TFnasY5Tv3u9a2F0/hXRMHkzM4oCPNkrDkRILcMTZkjyyI+1QTr414GJW1nsMXp0H0KpDy4Es5G5JbCk53qKAydIyXyCyaQcd4lqBVXyJQPHboLxuMJKnOMdg5vNfU3RxbD3XOYiQFII4QR8fO6JUCymo7obhqP42iX/ULC+U5XWzER1hQlMaQbDxnSCozh2HD8hze7WQFjylHYvVzYxYJFo14UcJYIeOJOikjR3kRPvzfJyhtbZV1lHMFM4j3AjjYHOGhaRIWyB9+OZFw3YGD4P486PQCeGnIbFmbYdImJnT8oXgNOqc0Cw0U7OJLNvlz1ZcuLGafY6e6pu2c6Syguzm2HuqcdzG+zqG4ClRkx81XJtboutJ16kLJpCGoHc/eKhDnEjtKZyyMq/CVGpoT5nrrnQ84BduVvVc9S6vYvf9gcXR8K2aBnEXqgzmwEHSUUOUYuZ1SeI7TgscOqR78UMZVFqclMOMKBkfH3T6Yd/pWWYrgI4m3ymvjMDRuis+N5gOve8s/2WlGocZlc8Gu3k56nBLRF5zEb0rbXzyojdfGj7WvrG227y3Ri8F3io5CX4QTG6nU4ug2Lnpusozni01l27Yfg83gYBs9A6/W7RwMautujq2HOm4xEXgAt1H8gTgiFBjhEpyzjJ7FyStlnMDjQtLqk2dT7wD6zTp530dHkVRmpww7mfR55yXxi05H8zMsEpC9rP6S6/60Osx7iBzeegguzxRwcsyHBZMPAzpzVXmfxvCujYPJOShRn7k472U3S/5xfzukwZNNSCDxwmLNKYVrI6fTytfGy8bfV9bWfk58FPoiWST1XuXYxx3FORutQZzUng0Gv2bQuVpONDAbbtbO4VnzqK27ObYe6px5ILmr5UrrwuePIcIh2+6yw6/mgaNgLmEBZG4F5mcnHxzSEFmdMSzZGTmFVcwJhd+6v6EOMaYnSjO32aR+auBg87qazWClDjRveEx+IKBnyAe9MCMIXFTCa6N+q8s6TGJGotL8m9AXzY0FCTsLJxLkQkhAZjZ8FHYO/5rHoWw9yNcWExwDuzbimO77zNSnt4BvncbtZ6fFE79xzVVQ6N+nAc+9QCOvwMttyiCcMtdjphSU8flrf36UZ5Nvz27AcOm01XN45TnjbO1+9Mz446qFBYOdUiA95/0Oji7Zf6j45Wov4GHlar8EDuaEL23c1mLhAP7IP+UoYBfs2IgsgRfjzZE17edS0j4Xmv/B9UXzBP8Hivsv3FlgtvRY9dCfZu0cZRCPS+hujuxCHVtMAJk7+0BiFqAxOv4MAeCOEe8HphzgWPuxZziEvsBxxvCFwCGcNOESkIkVAqmMZ/BYgtjZsKPpLwRcqxg/3yj9veqgxDFRhw8cbHFmgYPfmMjb87g8Tg/hvQQOyANHPaUTMb+10rd9x6bQ8TjIifd9HXyVR25sPHCwS+BlPOXKusOvb5yan/WbE5+svkjO6Cv+Auf7Mg4qG/NtLds5Ml9Cd3NsPdSJ/9AjzIUjoNJU4g5/yrFtVIe2LEZTL+pVZXdS/zhhyAwNJxH4Uppy6nAXvlEeh81iyOmAKzwWmXDlpfwsUl82Hv24MRVzsgsnIaXBj0X6SgEijzI7vhWDGfzF97I4RxxdmJvyW+TbbuGtcuOrGg4ag5ek3DvHurLFY80CjY1zgMCQObOIgz3Y2dd7G6VZzHknFW8ugu7oeXW8NDZ8jMo64pN6EPJ/rIDO2Ds19IjyF76c0zf1Xbn6GJSHH/9o9UX8Y8t7vTNRW2wBHFPEy/vB93l6ZvpB22bsHGYi3g5q6+Ij+IOwmMDgHFKn7v+DKB51hHPGWNteI3AovDUuizOLYnDa11ytqVYROGZ9Ee97LyatyiOXr0PJDf409qStx3XOcieVUe931dl6T5DRbq2yHwKL4y3FYafGbp+xVzouBI5ZXzh9EU6RFpcbIOfYer/OeUHpcNx5q9B/yVVwiLWrCIFD4M3n1qkPNSK21mSjCBytvshpnfJtxyHkhgrn2HqnTrGTiQTOvTVfyHAkXakyAgfCm43CulmoLNsa3a/6UgPV+n0eSG5MLMfWO3WKLSYeVl4aPvPpNaqPwGJ4S6n5Cop3JeEjg/rTW0cojMCqL4UBXai7xeTGfHJsPVWn2At4A1WDcDJJfdNtVda4IAJL4K0x+BqGheRRQdbXrg6AwKovBwC9wJBLyA02Nc6krQ/VKX0ygRk+weWP6dlnjgWgXLsYQmAhvDnODn56OsTbWt4eAqu+tCeTHI4Wkhus5Nh6ss5/3t68JJx68bwAAAAASUVORK5CYII=\n",
"text/latex": [
"$\\displaystyle 3 \\left(- 9 x^{6} \\cos{\\left(x^{3} \\right)} - 18 x^{3} \\sin{\\left(x^{3} \\right)} + 2 \\cos{\\left(x^{3} \\right)}\\right)$"
],
"text/plain": [
" ⎛ 6 ⎛ 3⎞ 3 ⎛ 3⎞ ⎛ 3⎞⎞\n",
"3⋅⎝- 9⋅x ⋅cos⎝x ⎠ - 18⋅x ⋅sin⎝x ⎠ + 2⋅cos⎝x ⎠⎠"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.diff(sp.sin(x**3),x,3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Integrálás"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:55.457551Z",
"start_time": "2020-02-24T14:30:55.046178Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\log{\\left(x + 1 \\right)}$"
],
"text/plain": [
"log(x + 1)"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.integrate(1/(1+x),x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Határozott integrál"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:56.048298Z",
"start_time": "2020-02-24T14:30:55.489243Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle - \\log{\\left(2 \\right)} + \\log{\\left(3 \\right)}$"
],
"text/plain": [
"-log(2) + log(3)"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.integrate(1/(1+x),(x,1,2))"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:56.055047Z",
"start_time": "2020-02-24T14:30:56.051454Z"
}
},
"outputs": [],
"source": [
"a = sp.Symbol('a')"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:56.684318Z",
"start_time": "2020-02-24T14:30:56.057707Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle - \\frac{\\sqrt{- \\frac{1}{a}} \\log{\\left(- a \\sqrt{- \\frac{1}{a}} + x \\right)}}{2} + \\frac{\\sqrt{- \\frac{1}{a}} \\log{\\left(a \\sqrt{- \\frac{1}{a}} + x \\right)}}{2}$"
],
"text/plain": [
" _____ ⎛ _____ ⎞ _____ ⎛ _____ ⎞\n",
" ╱ -1 ⎜ ╱ -1 ⎟ ╱ -1 ⎜ ╱ -1 ⎟\n",
" ╱ ─── ⋅log⎜- a⋅ ╱ ─── + x⎟ ╱ ─── ⋅log⎜a⋅ ╱ ─── + x⎟\n",
" ╲╱ a ⎝ ╲╱ a ⎠ ╲╱ a ⎝ ╲╱ a ⎠\n",
"- ──────────────────────────────── + ──────────────────────────────\n",
" 2 2 "
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.integrate(1/(x**2 + a),x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Láncszabályt is ismeri a szoftver"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:56.796990Z",
"start_time": "2020-02-24T14:30:56.792744Z"
}
},
"outputs": [],
"source": [
"y = sp.Symbol('y')\n",
"def f(x):\n",
" return x**2\n",
"def g(y):\n",
" return sp.sin(y)"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:58.130704Z",
"start_time": "2020-02-24T14:30:57.777512Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\sin^{2}{\\left(y \\right)}$"
],
"text/plain": [
" 2 \n",
"sin (y)"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(g(y))"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:58.638096Z",
"start_time": "2020-02-24T14:30:58.132064Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 2 \\sin{\\left(y \\right)} \\cos{\\left(y \\right)}$"
],
"text/plain": [
"2⋅sin(y)⋅cos(y)"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.diff(f(g(y)),y)"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:30:59.154720Z",
"start_time": "2020-02-24T14:30:58.640047Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 2 x \\cos{\\left(x^{2} \\right)}$"
],
"text/plain": [
" ⎛ 2⎞\n",
"2⋅x⋅cos⎝x ⎠"
]
},
"execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.diff(g(f(x)),x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Sok esetben nem létezik zárt alakú kifejezés a határozatlan integrálhoz. Ebben az esetben használhatjuk a [numerikus integrálást](https://en.wikipedia.org/wiki/Numerical_integration) a határozott integrál számításához:"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:00.188255Z",
"start_time": "2020-02-24T14:30:59.156424Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\int \\sin{\\left(\\cos{\\left(x \\right)} \\right)}\\, dx$"
],
"text/plain": [
"⌠ \n",
"⎮ sin(cos(x)) dx\n",
"⌡ "
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.integrate(sp.sin(sp.cos(x)),x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[numerikus intágráls SymPy-vel](https://docs.sympy.org/latest/modules/integrals/integrals.html#numeric-integrals)\n",
"\n",
"nem triviális"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Vektor- és mátrixszámítások *sympy*-ban \n",
"Kisebb méretek esetén, amelyek akár szimbolikus számokat is tartalmaznak"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:00.200099Z",
"start_time": "2020-02-24T14:31:00.190218Z"
}
},
"outputs": [],
"source": [
"v1= sp.Matrix([2.,3.,4.]) # oszlopvektor\n",
"v2= sp.Matrix([[3.,-2.,-7.]]) # sorvektor (még 1 szögletes zárójel)\n",
"mx1 = sp.Matrix([[1.,2.,3.],[2.,0.,4.],[3.,4.,1.]])\n",
"mx2 = sp.Matrix([[1.,2.,3.],[4.,5.,6.],[7.,8.,9.]])\n",
"EM = sp.eye(3) # egységmátrix"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:00.936217Z",
"start_time": "2020-02-24T14:31:00.508876Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}2.0\\\\3.0\\\\4.0\\end{matrix}\\right]$"
],
"text/plain": [
"⎡2.0⎤\n",
"⎢ ⎥\n",
"⎢3.0⎥\n",
"⎢ ⎥\n",
"⎣4.0⎦"
]
},
"execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:01.396324Z",
"start_time": "2020-02-24T14:31:00.938194Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}3.0 & -2.0 & -7.0\\end{matrix}\\right]$"
],
"text/plain": [
"[3.0 -2.0 -7.0]"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v2"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:01.844378Z",
"start_time": "2020-02-24T14:31:01.404834Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}-28.0\\end{matrix}\\right]$"
],
"text/plain": [
"[-28.0]"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v2.multiply(v1)"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:02.333778Z",
"start_time": "2020-02-24T14:31:01.846774Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}20.0\\\\47.0\\\\74.0\\end{matrix}\\right]$"
],
"text/plain": [
"⎡20.0⎤\n",
"⎢ ⎥\n",
"⎢47.0⎥\n",
"⎢ ⎥\n",
"⎣74.0⎦"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mx2.multiply(v1)"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:02.819298Z",
"start_time": "2020-02-24T14:31:02.336141Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}-54.0 & -60.0 & -66.0\\end{matrix}\\right]$"
],
"text/plain": [
"[-54.0 -60.0 -66.0]"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v2.multiply(mx2)"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:03.315668Z",
"start_time": "2020-02-24T14:31:02.824350Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
],
"text/plain": [
"⎡1 0 0⎤\n",
"⎢ ⎥\n",
"⎢0 1 0⎥\n",
"⎢ ⎥\n",
"⎣0 0 1⎦"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"EM"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:03.381187Z",
"start_time": "2020-02-24T14:31:03.320039Z"
}
},
"outputs": [],
"source": [
"EVs=mx1.eigenvals() # sajátértékek és multiplicitásuk (racionális számokkal)"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:03.823589Z",
"start_time": "2020-02-24T14:31:03.383153Z"
}
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/latex": [
"$\\displaystyle \\left\\{ -3.6088213318864 : 1, \\ -1.14824478501694 : 1, \\ 6.75706611690333 : 1\\right\\}$"
],
"text/plain": [
"{-3.6088213318864: 1, -1.14824478501694: 1, 6.75706611690333: 1}"
]
},
"execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mx1.eigenvals(rational=False) # sajátértékek numerikusan"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:04.217569Z",
"start_time": "2020-02-24T14:31:03.825447Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAiEAAAAVCAYAAACQYz4VAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAO50lEQVR4Ae2d75UctRLFBx8HsNgRABlgOwOTAX5EYMgADp/sbz6QwcMR8CcDIAIeZAAvAsxm8N799ah61T3q1q3ZmcXsts7RSF26qipVldRqTc/uOy9evPh8t9t9pkx69vLly9/21e1zs8Bmgc0CmwU2C2wW2CxwOgtojzHZc7yjTchXYv+jGn46nZiN02aBzQKbBa4soPXlfV09Vf5O9curluNr4vOh8uShSdcX4vhA5R/Hc9569iwg+9r+FLbrpwy/nm5b+z/DAvL5sPe4fy51S1DFCQsLA0H7leipzU5RtFbzW9HmCw+DIf2p/IEyciaLkK7R4UvlSFx/L/pEHxcHE2EZ0xfUlR4rv1H+QvSJfjSSRP9QxffKj1S/hFYn0eBn2SyDRYbwPdmWfQovG1vw4R8uSQc+3JOvPou+n6kMe1w1viW1nk17amb7L9lEdNsfwqZiNsawJDvao1zBEX//JgsT8Lq8FP1dCCrd8fxcsDHf6Ed6tC+mn8JGHC6uE9FD2NX5EjjKim+QD+JbGMYe6YEqz0W7DEKrVDs6nH0OOPo39LP9qb6OnzL8dpXOji/D7zGMA/9EQ8U3SE2scOi7uJZHZ0oX68gWxo6jit+ijYSx7jkurozX4hk2OssmRAqzGHAzHm8gqn8sGicufOXzQyiwVArDQHAyfIaNgkquyWw0hiTar6q8UjnwVInsX1V+pFxvRL7S9agPnXXNJuRCudbHwqkP+rGgfgQvkuoEe8gOndHntTIbFDYq9DtI6mvbzMUWXFd2UcYadwYr+ZYPD4yxJ+DnuLksQG6enLTpgYLX7L9kE8t3xR/dmD1Qek9Ykj2HL+GeCMicqOdk9OWEJDbz0KzxCMecInFDgC/zmLXgUuUkidZdJ7K+KfZkvItrVOGJbOz+NUqpRN//quRhpGUPYKQlW+5br/kp2deZnxl/On6y+Unvri8xTWZ8DlYYay0vsjPYrh+KbDuOhO/aqPDs3qddXDXuLk+wkc6yCRHzT8lSnk1H3ODjxIHTiKCFHq2SCfit+kc/MDh2nLRqQ85kEyEaT1TwZ8c4bBAKDqfM03MRflYe9HFxhQkbjvmmBuOjE7rHUx0L4jPlndr4LowFqJUyNrOwkmfJLjp37YPSGazgXR+2DCEZ2OmtTNLNsumS8sf2X7KJ6MSC5TvhrJid674kO4sTn3HDHn1FGzblKo+Zg7+p3zC3gl+rFAYbOetE1rdOfPMQwNdDwwYE/VRH7/+oOq5R0Ouk9puYA47+tVqTunTs+rN0cP3U5SeZli+L3Mz4uljJtuMjg5WuXdnC2HGUsJF1H5FsF4fZM1jwu3vD5+k/eILFYeQhFafE5WopLKcm3Ky/qYGic7pRByoLUOtp+RfRnwp7UfpzclL3K+SDwsXRkae33ysZwYxNEwvesLgG0SgzNstgDdHDyZJjH3hZNtL4XR9O9FM//D6JnQngDl50bGL5o5gtHbMd2aM3DBxzspXmpx6Z8bT4tWjuOtHq26RpvG58gxsfnCpmzOF6jRqbDFuO2GMrCf2XRLj+XOo/p7v8LF9mxpfBzpW+7nVCdiaOLBtJd/c+4uIwRwY7mO8sJyEyLDfi4SRgkKKPYmwu6++0onlecsLAica4iZkDyjWL6mSjUugx6WnnCYsA/1z8flTJ10HBlyfDWh8Xp27D0TIvXAUvaHW6qC96dfGxbZbB9uSW9sy4Xazrw7mKn2h8k+O8OeAOXq/ZxPUHZiPGsjG7Jrt2xSpOPh1OOuoOojH/XtU01TPjmXVdvHTXiUUGjYZufGt8sQa8afT/s9Aeq8QvdVq1ZQ28Rr2r/xrvhD/X2IxtCX6uLzPjy2BHnU9U6co+Io4sG4mvdc9xcdgjgw373Y/KOUsphlGGo2DVW5uGuXgm5h/C8lT8iTITliek8UVStcUEF3kxPaBF2B/IqrKb/Et1vn+GX/11kY1TP7DsNlsJnWlnR3h0Un/bZhlsSyH1t+xD3wS268O5LuLNEXS9KZxD7tx1zyYJf+yETcVsT3Y4w8UFnlJ9OClkQ1S/CwLdjsXCZzieV/2hMjx5J2Sce6rb6wT8Eqkb35LNgxQsh3Voxht9SZMTU+Fvag509d+r531K76Y/o7faV/0UuChb/ETL+DIzvgw2VDxV2ZWtcdtxlLTRZAzqa91zXBzMHez9iRYnvpAC3JAZ2BNlFga+B3VSBNtj8RgXKdXZQPBWORuKmNiXKwyDz059OAHhBsdkYEPEaQnlJLm4Sadyob6Ml8k46tzCrdEKD8tmGeyaTNrEy7JPAhu2X/PhqJbkYzcmW5xijW13teLaJOO7uS3VtxmzCdnH+o25dzD/0C8xHmJs/Mlv0ZmvSPnaNk4XUusE8s3kxjdrFfN5nrA7Kfjsiv43NQdCrjU/96qufi76s4yx56c58xa/jC8z48tg53pe99qV7cZRxkaD7oo7YrF7z3FxMM1g7w1anOlDivBC0tfKPIF9q8wvRz5eE6f2cApPSfNTk+/U93WFWWNFWzxt7NQHuZfKnICwQLF4smBN9HFx6ttK34vIk9z4EloLtEZTX9tmGeyaTNrEy7KPgxWvY3zITxHn/u6pfdvbLZtkfNcw2FLMWrLFz8WNoqUvc++pytgojG1U3PEIx2aDOT0k1dnAwjN7mjauE3tO65+Sk4nv53BTn3EjojqLfuhdb7rTtlzXtN2a1L/NpKKKX8+fKT/1+FWiW9WHmfFlsC1h16ElZWfiqKfWJN6lh3XPcXEIz2Dvt7QVAyYZvxqJydaCzWk8SY/HoPNGtXFzZuLxlcq7pT6H1df15Aw6vwLgJOOx8tqpSuwGh+9dJYs+/ByO799ITApuuizAbGp+UuYJxMLBYJ7Ul4WPr5CWjrznXbrX4mXbLIOdC1Zfe9wOtuLf8+FwEyo8szeOSoxflayTx7Yv3Ue6NnH8IQzz7iCJ3ozZpOxj/MY8bMXGrsjuztWDwVwR4MsG531l6m+umg5qk3XioLVPaI2hXqNiXXlPrHgBl80HNwDeeyGzBg08yriPsaVYHJ26+pucF/250h/ZtZ9q6BK/rC8z48tga11PUe/KVnxwf+rGkZTJ2miiv2RY9xwXB/MedmkTwqL1aKJd4kJCmWwIn29K2DjwREDmeOkgqQ/Ght5cOEsHFhgmOJfcVOYpaOFcjvZw4JjUF2NzKvK7cujj4kY+VMSHm/gDle4vTCb9Cw/bZpJjYw8EtQmZcXex0g/bImnVhwCE4ynqQmX4CvLZkuSg09GxfTbFKsZJm3T9IdYHc00ymjHrynZx1bDq6njzrYmlbo1H8nnJnDm35MthDVB7rCexJtQig5aKvYonsbSUiOshgVclHoCCxjhJPLjc+ByQTGRb+gPspEV/So7lpxn/Jj/sWPQOv9XdgoY9A9cdXwZbCztFPSsbvOQuxhE6VTzDHrWqQRviXVjrPuLiinyLZ63U/frihHWeBjCIc+LREhtfl7TaoMWisYR7UDqyUcHw3ORw4CSJRsCyQLOYWbgJA12oHxPmA5XjCYjqwwKkMvScd2tdZ2yWwbZkjTTpaI87g5WAJd+E7LANtnoi3pxK1YlgZrMJHT8d/Z5NzfQfUrdsorG8Uu7G9nzMsuVizArryubmkvabZBNvyJg/oOxKmzuex+LReuob5r541fyXYnFcJ8Qrm5Z4Bp+I77iel8R3nJQwlrQt5wyT19fVfxC35s+iT8ZPEQPN+Cj8lvSe+3IJF2aq/ZPBRv9TldeVPcZRpdASz7mN3PuIi0OFDHZQ+VybEG74wwQbpFx9EJAkjLSWOJac35TA89TDLjf6g4knCtojgeN7LvQgsOkTx7OBiZJFMRaDLi46UYonAcDiMb9Bfix69v2GjM0yWFRdTNKdMVvjzmAl0PKheOLL8Oeop+h/QVc5bu5o1PWF8uDXEXzLKhqfbRNsobwa27V5hF2NWbWzOLv+cHG1CrEGHGwgyljc8Xwj/HzeIedpQ39rnaiVNOpufLMWvFZ+j/HBV+WFCvQcTnF0bfub/iR4KF9nHlj676Wtfi76s/TK+IkuPX6uLzPjy2BXjXFEoyVbvu7GUSXbtRHx49ynXRwqZLCDyveGz9N/sDjwpDSmYsQLEXj5CkV3KplI/1Medk8B1jWnExhn3GCoTt9/KQ8v6IAVjRv9G5U4aEgtnBq4kQ1/on2P2n8Ky5E0P9ONXbGL26kPu3WczRj4k8xjFm0co+p1elguYkdat1k2Kx0y2JCxJtset5hZWNnD8mEo1yjxN3lM4sk1v5CaxMsIuPnKok3RVfkgtmcqLvaf4eLywCZqsPwBA+lzTMyuyY62umzpOG/n+rImVnV3PMN8q/oxvvgro/AYk+juOjH2UWXVN+Lpxjc2n2+4WDdYI+rTmlp21Ju2VD/o15oHrv7IUl6LY3QhLfnT9tOezTjnm/yki+VLd3zIzGCLjhSr8VHhVrEJ2XYciadlIynm3kdcHGPNYMHvzvZfdGUIdvr1YoAReTlr8vSka97JYMMx+a5LtJ1obEIiyLlxvxJtMnF1TTs4gvZP5SfKLRxPgF8q1wvC+HdHRB+S+Lk49GZMrcQpzKNoUJ1Fh4RN0JcxsPFhA0TADEl1y2aAXaxwrmxr3EV2Btv14TD48iF9eTLArtiCxGL/i+jxfzewO4mXF/H5jaeETZux7faPgQnfs4nlD/GxY9aVfQQO37KJfC598O1BEt0dD7xY9EgPlJnbX6j/QVyIxrxz1glrvojXkMS3G98FAx4dSAfrzp68/xR+1d+ghDnJPCi6hV7YsLV2NuO46OH4M+Mnh5/ly6Jf1z/gSKYt7PgQvwy2q2fRD1XDX4txJKxlI+Gse46LK3Z0eTLmH8+2CUGZLW0WOIcFNCE4+WodI55D3MZzs8BbaYFtHryVbtmUMi2g+B02IfdM/AbbLPA2WYD3cC7fJoU2XTYL/A0W2ObB32D0TeRpLbBtQk5rz43bmS2gzQfHjHzttqXNAnfWAts8uLOuv3UD3zYht86lt35An2oBPvov0t5662wDvCsW2ObBXfH0LR9nvBMSvy55pgV+8uLnLR//NrzNApsFNgtsFtgssFnghiygPQa/YuOHKLwI/ez/VubiAQ45sl0AAAAASUVORK5CYII=\n",
"text/latex": [
"$\\displaystyle \\left[ -3.6088213318864, \\ -1.14824478501694, \\ 6.75706611690333\\right]$"
],
"text/plain": [
"[-3.6088213318864, -1.14824478501694, 6.75706611690333]"
]
},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"list(mx1.eigenvals(rational=False)) # sajátértékek numerikusan, listaként (multiplicitás nélkül)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:04.694469Z",
"start_time": "2020-02-24T14:31:04.218939Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 28.0$"
],
"text/plain": [
"28.0000000000000"
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mx1.det() # mx1 determinánsa"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:05.892297Z",
"start_time": "2020-02-24T14:31:04.697112Z"
},
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}Ix & - Ixy\\\\- Ixy & Iy\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ Ix -Ixy⎤\n",
"⎢ ⎥\n",
"⎣-Ixy Iy ⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" Sajátértékek, vektorok: \n",
"\n"
]
},
{
"data": {
"image/png": 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\n",
"text/latex": [
"$\\displaystyle \\left[ \\left( \\frac{Ix}{2} + \\frac{Iy}{2} - \\frac{\\sqrt{Ix^{2} - 2 Ix Iy + 4 Ixy^{2} + Iy^{2}}}{2}, \\ 1, \\ \\left[ \\left[\\begin{matrix}\\frac{Ixy}{\\frac{Ix}{2} - \\frac{Iy}{2} + \\frac{\\sqrt{Ix^{2} - 2 Ix Iy + 4 Ixy^{2} + Iy^{2}}}{2}}\\\\1\\end{matrix}\\right]\\right]\\right), \\ \\left( \\frac{Ix}{2} + \\frac{Iy}{2} + \\frac{\\sqrt{Ix^{2} - 2 Ix Iy + 4 Ixy^{2} + Iy^{2}}}{2}, \\ 1, \\ \\left[ \\left[\\begin{matrix}\\frac{Ixy}{\\frac{Ix}{2} - \\frac{Iy}{2} - \\frac{\\sqrt{Ix^{2} - 2 Ix Iy + 4 Ixy^{2} + Iy^{2}}}{2}}\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$"
],
"text/plain": [
"⎡⎛ ⎡⎡ Ixy \n",
"⎢⎜ ______________________________ ⎢⎢──────────────────────────\n",
"⎢⎜ ╱ 2 2 2 ⎢⎢ _____________\n",
"⎢⎜Ix Iy ╲╱ Ix - 2⋅Ix⋅Iy + 4⋅Ixy + Iy ⎢⎢ ╱ 2 \n",
"⎢⎜── + ── - ─────────────────────────────────, 1, ⎢⎢Ix Iy ╲╱ Ix - 2⋅Ix⋅I\n",
"⎢⎜2 2 2 ⎢⎢── - ── + ────────────────\n",
"⎢⎜ ⎢⎢2 2 \n",
"⎢⎜ ⎢⎢ \n",
"⎣⎝ ⎣⎣ 1 \n",
"\n",
" ⎤⎤⎞ ⎛ ⎡⎡ \n",
"─────────────────⎥⎥⎟ ⎜ ______________________________ ⎢⎢─────\n",
"_________________⎥⎥⎟ ⎜ ╱ 2 2 2 ⎢⎢ \n",
" 2 2 ⎥⎥⎟ ⎜Ix Iy ╲╱ Ix - 2⋅Ix⋅Iy + 4⋅Ixy + Iy ⎢⎢ \n",
"y + 4⋅Ixy + Iy ⎥⎥⎟, ⎜── + ── + ─────────────────────────────────, 1, ⎢⎢Ix \n",
"─────────────────⎥⎥⎟ ⎜2 2 2 ⎢⎢── - \n",
"2 ⎥⎥⎟ ⎜ ⎢⎢2 \n",
" ⎥⎥⎟ ⎜ ⎢⎢ \n",
" ⎦⎦⎠ ⎝ ⎣⎣ \n",
"\n",
" Ixy ⎤⎤⎞⎤\n",
"──────────────────────────────────────⎥⎥⎟⎥\n",
" ______________________________⎥⎥⎟⎥\n",
" ╱ 2 2 2 ⎥⎥⎟⎥\n",
"Iy ╲╱ Ix - 2⋅Ix⋅Iy + 4⋅Ixy + Iy ⎥⎥⎟⎥\n",
"── - ─────────────────────────────────⎥⎥⎟⎥\n",
"2 2 ⎥⎥⎟⎥\n",
" ⎥⎥⎟⎥\n",
" 1 ⎦⎦⎠⎦"
]
},
"execution_count": 98,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Ix,Iy,Ixy = sp.symbols('Ix,Iy,Ixy')\n",
"mxSP=sp.Matrix([[Ix,-Ixy],[-Ixy,Iy]])\n",
"display(mxSP)\n",
"\n",
"print('\\n Sajátértékek, vektorok: \\n')\n",
"mxSP.eigenvects()"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:05.920073Z",
"start_time": "2020-02-24T14:31:05.893627Z"
}
},
"outputs": [],
"source": [
"ev=mxSP.eigenvals()"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:06.908101Z",
"start_time": "2020-02-24T14:31:05.921607Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}Ix & 0 & 0\\\\0 & Ix & 0\\\\0 & 0 & Iy\\end{matrix}\\right]$"
],
"text/plain": [
"⎡Ix 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢0 Ix 0 ⎥\n",
"⎢ ⎥\n",
"⎣0 0 Iy⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" Sajátértékek, vektorok: \n",
"\n"
]
},
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ \\left( Ix, \\ 2, \\ \\left[ \\left[\\begin{matrix}1\\\\0\\\\0\\end{matrix}\\right], \\ \\left[\\begin{matrix}0\\\\1\\\\0\\end{matrix}\\right]\\right]\\right), \\ \\left( Iy, \\ 1, \\ \\left[ \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$"
],
"text/plain": [
"⎡⎛ ⎡⎡1⎤ ⎡0⎤⎤⎞ ⎛ ⎡⎡0⎤⎤⎞⎤\n",
"⎢⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥\n",
"⎢⎜Ix, 2, ⎢⎢0⎥, ⎢1⎥⎥⎟, ⎜Iy, 1, ⎢⎢0⎥⎥⎟⎥\n",
"⎢⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥\n",
"⎣⎝ ⎣⎣0⎦ ⎣0⎦⎦⎠ ⎝ ⎣⎣1⎦⎦⎠⎦"
]
},
"execution_count": 100,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mxSP=sp.Matrix([[Ix,0,0],[0,Ix,0],[0,0,Iy]])\n",
"display(mxSP)\n",
"\n",
"print('\\n Sajátértékek, vektorok: \\n')\n",
"mxSP.eigenvects()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Vektor- és mátrixszámítások *numpy*-ban \n",
"Nagy mátrixok és vektorok esetén érdemes ezt használni, vagy ha sok numerikus adattal dolgozik az ember. Továbbá a sajátérték-sajátvektor számítások is jobban megoldott ebben a csomagban"
]
},
{
"cell_type": "code",
"execution_count": 101,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:06.912380Z",
"start_time": "2020-02-24T14:31:06.910193Z"
}
},
"outputs": [],
"source": [
"import sympy as sp\n",
"import numpy as np\n",
"import numpy.linalg"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:06.942061Z",
"start_time": "2020-02-24T14:31:06.913892Z"
}
},
"outputs": [],
"source": [
"v1= np.array([2.,3.,4.])\n",
"v2= np.array([3.,-2.,-7.])\n",
"mx1 = np.array([[1.,2.,3.],[2.,0.,4.],[3.,4.,1.]])\n",
"mx2 = np.array([[1.,2.,3.],[4.,5.,6.],[7.,8.,9.]])"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:06.958540Z",
"start_time": "2020-02-24T14:31:06.943692Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[20. 47. 74.]\n",
"[42. 51. 60.]\n",
"[-13. 26. -13.]\n"
]
}
],
"source": [
"print( np.dot(mx2,v1) ) # skalár szorat mx2*v1\n",
"print( np.dot(v1,mx2) ) # skalár szorat transpose(v1)*mx2\n",
"print( np.cross(v1,v2) ) # keresztszorzat v1×v2"
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:06.980852Z",
"start_time": "2020-02-24T14:31:06.961098Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1. sajátérték és -vektor:\n",
"λ = 6.76; v = [0.529, 0.834, -0.157]\n",
"\n",
"2. sajátérték és -vektor:\n",
"λ = -1.15; v = [0.543, -0.475, -0.693]\n",
"\n",
"3. sajátérték és -vektor:\n",
"λ = -3.61; v = [0.653, -0.281, 0.704]\n",
"\n"
]
}
],
"source": [
"(λ,V) = np.linalg.eig(mx1) # Mátrix sajátértékei, sajátvektorai a λ és a V változókban\n",
"\n",
"# kiíratás 2 értékes tizedes jeggyel és for ciklusra példa\n",
"for (i,v) in enumerate(V):\n",
" print(i+1, '. sajátérték és -vektor:',sep='')\n",
" print('λ = ', sp.N(λ[i],3), '; v = ', [sp.N(e,3) for e in v], sep='', end='\\n\\n')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Figyeljük meg, hogy a numpy a sajátvektorokat 1 hosszúre normálja!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Egyenletek megoldása "
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:06.991548Z",
"start_time": "2020-02-24T14:31:06.984867Z"
}
},
"outputs": [],
"source": [
"x, y, z = sp.symbols(\"x y z\")"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:07.006546Z",
"start_time": "2020-02-24T14:31:06.993406Z"
}
},
"outputs": [],
"source": [
"egy1=77*x+6 - 160"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:07.391310Z",
"start_time": "2020-02-24T14:31:07.008628Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ 2\\right]$"
],
"text/plain": [
"[2]"
]
},
"execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.solve(egy1,x)"
]
},
{
"cell_type": "code",
"execution_count": 108,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:07.885783Z",
"start_time": "2020-02-24T14:31:07.392803Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[ - \\sqrt{5}, \\ \\sqrt{5}\\right]$"
],
"text/plain": [
"[-√5, √5]"
]
},
"execution_count": 108,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.solve(x**2 - 5, x)"
]
},
{
"cell_type": "code",
"execution_count": 109,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:08.420286Z",
"start_time": "2020-02-24T14:31:07.890394Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJCAYAAADkZNYtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyklEQVQYGU2QvQ3CMBCFDWQAoKRkBBB9CkZAMAFQUkapktZsgJgAwQgU6SPYAErKKBuY7xlb4aTn9+Pc6Zyec86oyrIcQjtvjFnAWzAFm5DV/SBEloajgK7BGSzxGaxBNuEwBJpopUO18ApoumoMrNEaRVHMxBH4E3hEH9mvweSn7++ONfLS2Z/ya/yHNC7x2vEWczJ5kwRxReuBd3gPWvQbjpXjM62hSUITGpv4hZhMd/o7ZpCm6QfWa0dgzmVeVdULHMCEzJH5lb58Ulq0QJNfvQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$\\displaystyle x$"
],
"text/plain": [
"x"
]
},
"execution_count": 109,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:08.428102Z",
"start_time": "2020-02-24T14:31:08.422615Z"
}
},
"outputs": [],
"source": [
"e1 = 3*z + 2*y + 1*x - 7\n",
"e2 = 4*z + 0*y + 2*x - 8\n",
"e3 = 1*z + 4*y + 3*x - 9"
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:08.890159Z",
"start_time": "2020-02-24T14:31:08.429994Z"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAMkAAAAyCAYAAAD2k//fAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJ20lEQVR4Ae2d6ZHVOBDHH9QEwBEBsxlwRLCQASwRABlA8Y1vU0sGQAQsmwEQAUcGQAQLkwH7/3nUGtmW9XzOs2V3lca63d3qVrcOv7n0+/fvXQpevHhxrPJX1FH8XqruVrZxYCkckCzfFK4fFU4Uf5nC+3KqUI2fqvybwneFB6m6W9nGgSVxQLL9Vfgi088V/6aAMYjCpSZLokZ/qwVK8kTx19HWM8kUfswK7xRuKX4aQ8vRQ9F/Cn8o/K08lD9bCGg2Gt8qD+HIDkRX4e04wq7p+Uh5UVkIiVedK0pjUVAS5KcmE1ElUcW7avBe4bXiT/ScHTji3gixnwq3FVCUq8qvMUZ5X1SGWf1Xz52eMIa8e4rXmEKdJYNoYsCZNJ4p/gFa9CR9U08miGxA9NhYvlK8cJv0NFcqKvRV4l0fP5T/WfHakqLJ3XpGR2owSwVxuJ0KvwcOx7fkxUDlj5V/Rc9CQaijOIpEOpx9KMoFUAisRqEgjiiEKbsJQTQxUV4TrX5doTjW8rNCq/F18oC3dFdxFKwETUqCJQkZXGq0sAR+Z8zF+KR8mILwZAOi576IYaBLLrLysZq1WTIDwqE3pvyMeZfxtYkW2S9BTUkCoWG2zQEgGpesCsbYGlOqFReWxvpjZXMZv0b2B7IaG1/WngCueBsweai5o0eJ1rEXJ6rPryhgYgq5a6nCBZYhFN9FO9bkoYJtVLxTXi7eQTEsoofJgHhsDK8Xlc4W5C7a75FSkn49zquVMS81q2blbon9Rs9tCVCxtmRIFP+lwI6PX5vNa6h6YwM9MW+ASQIwfpylevytuVs9+lh6E5txlk4HimACwS5WaU0i4v5ReBPUWTy9joBHPEWXVxTFUZBTV25ulEt2f+SuJCmX0ayM+a7duTffFjHBYMsbJWrro8+XugAzKQTKcEOBnc6nCpzvHSuwMQPEeHFW0vJv1u4WDFSAFTbDhmyxvMFMDDs9ZDygF8FpAgQoK4BuEVQ6rlAeygIMHt/cLQlMYrEaEwyzJFktZh29NgFAfxUGC021w5mmcbk+OAUahGLWlsRxhoM1m1VCZt1S4usYTAw7nUGcAzRorgL0YlmzmhRED+ckHCjesLHUk0mCNQo0D4ZcLIktvs06eMaIYSxgfzpmFvmOiX8pUSz6fOUMIqKN3R5mUD8x5EyvaMVLqK49mSS4c/hVz8FQu7vlGPpLPc/23pZRLVxtxmTWYPaAKbgT71Xmd3ccTQgNvisL9TsK3OUahYnqa3Yg2qDX3C4mj2zpdbQyBkZv5zMhJyNRuV+0uyXCWl3fVz2Uo7Swg6M5g2j2ZyQ50wltU9Oai7uVuxxs9B2QA5uSHJD526tnyYHaujamJCyEAFyUDTYOrIIDziWHVlvXeLpjSsIiGOCjqw02DqyJA2zkmPx7umML9+cq5RZpq/101UPz+LAJYNeIbVWsETdQgU+qk7xUp3K+o2fbMpvFdcAXrl5/Ubq628Y9quTGg8oXxRfhywGe7ThyTYRzKGSB7zo8/UpHQXUOTe+JEGNn7HGIb8mSqMC+5OrycQ7fir8k6AWfFDjYgSnsrqBAfr9e8RqoHnVgZFZ3ikQPPzAAT7DIVR5wRsMhWCMsjS8OX8YeGWJSRC6gm2cbBTm4HAhP8AZXPgVGJgvwSqJMiGPw/lS81dUF1cOChAJwqjSDb0xhERSWK1kG9UEbvk0f5XS03PthUqKFGZUJA2DC+VnEzv+Ql7TUC+QLY43sMGHybb1NtI/OyW6OzYVe4YE3g/xi/RnH3ZEiaDA3RPHH/NG+4m2AD+dDhULQ/VUPlSXdCXuB6qEorUH10z8W1rqn84rq89J5anAMd9UOKpl4MOMh4PdW88LyIj4lX0amd6f+vBwojkzhtnOA2Xpsu9SFQarfWg5Ut/X4qi6n9XgAuF6vwjVJdbYDjySoAxMEqxcTCCsb7an3tiZ4tJd26Ej4FYKhJ8qAwJhl3SmP2Ym8pCVReWdQ3wfni3DATbmv5+SHmRdAL9Zxd1kv4tKbfdf7Q/HCxHQdIbUzgfCLdOVdIXTtK6P6WFJvWR1d8AmeVyeYxZPtZID1aPjLJYsbf+HP5gPrq8J9DNck+GJ8vfZRlfyiRekoqA4KwB0pBh2gPYPvza7SLF5PKVwaCO8xBhc+hvyADXvXI4fg1VB61Z61KDLhraajA2FbDAh/1tDQgoIUE9lRiL0y8cUQejRp30KaegQWaQhUyV1z/djiVcVxcG25WMaMu++d8U5GzjWc9ByKEwpSmGxQVH/GMyaUJBgOqjQUh+R7KLR36dnrXWqH9wFNtOeJP39dAWFruy5Fhg4qB8Kd8XmqgB54S+8tiQoM2OXiG2kapACfmlmDeo8dc/ihAbbPaj8I19SR6mJpEKbPTXXa5quvY96tMMgKBDgxMw7pq/DL1QfboDDfBGbveiTAoRNf1K4zD4J39aX3ofrgd72gF/lh0Y5clIRN6UYIcJic3kYkNOFTJlxK1jB2VZ5Zgd2uYiuPRksBEcfMhRVsAtzBq02F1XzX3yhft9G3+sOU39VzMovpcO7FA9d2NHqr/JwiPYTeKj7qi92ymjU9qlZU2nzoITNopNsLybqjtzBLGw3hS5nZuu643BHj/EZE2Nm+uNqVFEJp+ImFbXVusK//RPkQHvSmN4HP1EVD6PW4ufEhXZOdmJL4hkuMiFg7xPLoK6/YiNCztcA7pvGBVl/AquF6GLCA5UO21jhYw65PvaMzD0agtyuao9XvQ2/i5aW1NfVyU5KmjQLWBLYeSPCnVMTaxm9llkraJbBa/JAzaxG22Fmr7V2LtOs6WasvD4bSm0RqwsK+9LZGKSslkRDWZmnl4factOaIq6h2QxRkF8OlKw596sfeq7y9PFCdQfT2wXWMNn3p7fLu2O5Wl/azrisG4maxU+e382aN8ATIrY0HU9CbtZJI5phBCWuGtfFgdHqzVRI3o7DdehHrgFkq4dp4MBW92SqJpJaT39p23iyleTqk1saDSejNWUnYgq1t500nj7PseW08mITeLJVEZpeDu2OF01mK7gUgtTYeTElvlkoiGbzt5HDNlmRtPJiM3lyVxK7UrNaSaJJYGw8mozelJP6K9wV4B2O/gnMRFKTpNHbs982xv7XxYCi9jfJeuwXMaMu/4zYkt0Frd4DmKA0bThsHhnJAst54+73JknC2wBmDmbChOGztNw7MnQPcEgfenj3O/zYpiV0p5+OZDTYOZM0BZwyQdb4lwW0rQVRJXEUuvPGPGtl73mDjQJYccAry0REXvSkeVRIaqDHWhFD89hB5G2wcyIkDknFcLL7CZVlxS+noDY3owj1khBoeK118PKT4tpAPmbPFF8sByTILdSzIieLJzwT+B/42r8jCkwQFAAAAAElFTkSuQmCC\n",
"text/latex": [
"$\\displaystyle \\left\\{ x : \\frac{10}{7}, \\ y : \\frac{6}{7}, \\ z : \\frac{9}{7}\\right\\}$"
],
"text/plain": [
"{x: 10/7, y: 6/7, z: 9/7}"
]
},
"execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.solve([e1,e2,e3],[x,y,z])"
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:08.895355Z",
"start_time": "2020-02-24T14:31:08.892718Z"
}
},
"outputs": [],
"source": [
"def f(x):\n",
" return x**2 - 4"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:09.004684Z",
"start_time": "2020-02-24T14:31:08.897042Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2.0\n"
]
}
],
"source": [
"# Numerikus gyökkereséssel scipy segítségével\n",
"from scipy import optimize\n",
"\n",
"gyok = optimize.brentq(f,0,10)\n",
"print(gyok)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Szimbolikus függvényekből numerikus függvények \n",
"\n",
"Ezt a legegyszerűbben [*lambda function*](#Sajatfuggvenyek)-ök segítségével tehetjük meg, de van más mód is rá: [lásd ezt az összefoglalót](https://docs.sympy.org/latest/modules/numeric-computation.html)."
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:09.381881Z",
"start_time": "2020-02-24T14:31:09.006558Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle a x^{2} + b x + c$"
],
"text/plain": [
" 2 \n",
"a⋅x + b⋅x + c"
]
},
"execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a,b,c,x = sp.symbols('a,b,c,x')\n",
"adat = [(a, 3.0), (b,-2.), (c,1.0)]\n",
"\n",
"masodfoku_x = a*x**2 + b*x + c\n",
"\n",
"masodfoku_x"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:09.782575Z",
"start_time": "2020-02-24T14:31:09.383607Z"
}
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 2.0$"
],
"text/plain": [
"2.0"
]
},
"execution_count": 115,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"masodfoku_num = sp.lambdify(x,masodfoku_x.subs(adat))\n",
"masodfoku_num(1.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Egyszerű ábrák készítése "
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:10.018942Z",
"start_time": "2020-02-24T14:31:09.785732Z"
}
},
"outputs": [],
"source": [
"import numpy as np # Ez egyszerűsíti a sok függvénnyel való munkát\n",
"import matplotlib.pyplot as plt # matplotlib csomag python rajzolójának behívása"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Függvények ábrázolása egyszerű diagramokon"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:10.025434Z",
"start_time": "2020-02-24T14:31:10.021305Z"
}
},
"outputs": [],
"source": [
"def f(x):\n",
" return np.sin(x)*(1-4*np.exp(0.7*x)/(x**3))\n",
"\n",
"x1 = np.linspace(2,10,30)\n",
"y1 = f(x1)"
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {
"ExecuteTime": {
"end_time": "2020-02-24T14:31:10.345881Z",
"start_time": "2020-02-24T14:31:10.027431Z"
},
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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\n",
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