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V pripade ze \[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] a \
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Ted pridame do Kernelu dalsi pravidlo:\
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Cell["Rozdil mnoziny1 a mnoziny2", "Text",
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Cell[TextData[{
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Cell["Vsechno je vyraz! ", "Subsection",
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Cell[TextData[{
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Tady vidime, ze Equal je hlavicka, pak leva strana, prava strana a uvnitr je \
vyraz na leve strane.\
\>", "Text",
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Cell[TextData[{
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vyraz jde ji aplikovat na jakykoliv vyraz.\nPomoci funkce ",
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Cell["Prvni prvek vyrazu je v nasem pripade leva strana.", "Text",
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Cell[TextData[{
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Kdyz jsme deklarovali promenou ",
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prvni pozici leve strany. "
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Cell[TextData[{
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Cell["\<\
Muzeme vybrat predposledni prvek (zaporny cislo znamena ze pocitame s konce)\
\>", "Text",
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Cell["Lze vybrat tzv span tj. kousek od 5. prvku az po 9.", "Text",
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Cell["\<\
Vygenerujeme pole s pseudonahodnych cisel s poctem radku 10 a sloupcu 2\
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 "podporuje programovani pomoci ruznych paradigmat. Avsak funkcionalni \
programovani je jedna ze zakladnich v jazyce. Funkcionalni programovani \
muzeme definovat jako zpusob vypoctu, ktery se ve sve podstate chova jako \
matematicke funkce, tim padem se data nemeni a vysledky vypoctu jsou kopie \
puvodnich dat. Tento koncept byl prevzat z matematiky, kde funkce je \
zobrazeni z jedne mnoziny do druhe, tim padem pri vypoctu se prvni mnozina \
nemeni. Samozrejme to neni cely popis funkcionalniho programovani ale shrnuje \
jeho podstatu. Dalsi problem s definici je, ze prace s pameti pocitace \
limituje realizci funkce jako matiamticke, protoze pokud mame dve velke \
mnoziny, data pro program budou potrebovat obrovskou pamet. Navic vzdycky \
pracujeme s vice daty a funkcemi. "
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vstupu a pro stejne vstupy vzdycky budou stejne vystupy. Takze ve sve \
podstate to je matimaticka funkce, ktera je napsana v programovacim jazyce. \
Krome toho lambda funkce lze predavat jako argument, lambda funkce muze vrace \
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Cell[TextData[{
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imperativniho programovani, kde uzivatel presne definuje jake kroky ma \
provest pocitac. V pripade programovani pomoci pravidel programator pridava \
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prikladu 1+1, kde tyto dve cisla muzeme secist a dostaneme nove cislo 2. \
Podobne vypada ",
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Cell["\<\
nejprve si ujasneme Set a SetDelayed
Mam rad tenhle priklad... Random[] byvsi zavolano, vrati (psuedo) nahodne \
cislo z intervalu (0,1).
\[OpenCurlyDoubleQuote]=\[CloseCurlyDoubleQuote], tedy \
\[OpenCurlyDoubleQuote]Set\[CloseCurlyDoubleQuote], vzdycky nejprve VYHODNOTI \
pravou stranu a pak vyslednou hodnotu oznaci jako hodnotu leve strany. \
Jelikoz probehlo vyhodnoceni, Set (\[OpenCurlyDoubleQuote]=\
\[CloseCurlyDoubleQuote]) ma vystup, protoze to  vyhodnotil Kernel a dal \
vystup.
Takze kdyz napisu rn1=Randon[], 
tak to da vystup a ten bude roven vygenerovaneho Random[] :\
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Cell["\<\
ekvivalentni syntaxe, jasne, vygenerovalo se nove nahodne cislo\
\>", "Text",
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Cell["\<\
stejne tak, kdyz se zeptam co je rn1 zac, tak dostanu cislo, ktere vzniklo \
vyhodnocenim prave strany v okamziku poslani rn1=Random[] (v ekvivalentni \
alternativni suntaxi Set[rn1,Random[]]) do Kernelu\
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Cell["\<\
Podivejme se na SetDelayed, ergo \[OpenCurlyDoubleQuote]:=\
\[CloseCurlyDoubleQuote]
to, co je na prave strane \[OpenCurlyDoubleQuote]:=\[CloseCurlyDoubleQuote] \
se nevyhodnoti, ani kdyz dame \[OpenCurlyDoubleQuote]shift + enter\
\[CloseCurlyDoubleQuote], tedy to posleme do Kernelu\
\>", "Text",
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Cell["\<\
zadny vystup... Kernel ma jen informaci, ze \[OpenCurlyDoubleQuote]az prijde \
na rn2, ma udelat Random[]\[CloseCurlyDoubleQuote]
kdyz se zeptame, co je rn2 zac, dostaneme informaci, ze az na nej prijde, \
nahradi se nahodnym cislem\
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Cell["\<\
kdyz ale vygenerujeme 5 krat rn2, dostaneme ruzna cisla,  pokazde se \
vygeneruje nove nahodne cislo\
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Cell["\<\
tak pojdme na ty funkce:-) 
... ale vlastne jeste ne. Potrebujeme patterny.
Pattern je vlastne cokoli: Kyncl se svym rodnym cislem a genetickou informaci \
je zcela konkretni pattern, unikatni, tak jako napriklad v Mathematice cislo \
1 nebo \[Pi].
Na uplnem protipolu jsou tri podtrzitka ___ , coz znamena, ze tam, kde jsou, \
muze byt jeden, zadny, nebo vice vyrazu pripustnych v Mathematice.
Nejobvyklejsi je jedno podtrzitko, _, coz znamena \
\[OpenCurlyDoubleQuote]jeden jakykoli v Mathematice pripustny vyraz\
\[CloseCurlyDoubleQuote].
Kdyz napiseme: 
x:_
, tak to znamena, ze si jmenem x pojmenovame cokoli, protoze podtrzitko \
znamena \[OpenCurlyDoubleQuote]jedno cokoli\[CloseCurlyDoubleQuote], tedy \
jeden jakykoli v Mathematice pripustny vyraz.
Obecne si v Mathematice dvojteckou pojmenovame. 
Jenze tohle se pouziva tak casto, ze se misto toho temer vyhradne pouziva \
zkracena syntaxe, tedy:
x_
s uplne stejnym vyznamem.  
tak uz konecne na ty funkce...
Funkci muzeme chapat jako trvale nahrazovaci pravidlo.
Kdyz napisu (v plne syntaxi by to bylo f[x:_]:=x+1+x)
f[x_]:=x+1+x;
, tak Mathematice rikam, ze kdyz se nekde vyskytne f[cokoli], nahradi se to \
cokoli+1+cokoli.
Jak to funguje (i to SEt Delayed, cili \[OpenCurlyDoubleQuote]:=\
\[CloseCurlyDoubleQuote]) si ukazeme:
\
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Cell["\<\
vidime, ze Mathematica 
1/ v okamziku poslani f[x_]:=x+1+x do Kernelu nedala zadny vysledek a pravou \
stranu nevyhodnotila, nesecetla x+x=2x. 
2/ vidime, ze x ma na obou stranach vyblitou zelenou barvu: je to jen lokalni \
pojmenovani podtrzitka a s zadnou globalni promennou nema nic \
spolecneho..(tedy pojmenovani toho jednoho objektu, parametru (argumentu) \
funkce f) 
zkusme to pomoci Set, tedy \[OpenCurlyDoubleQuote]=\[CloseCurlyDoubleQuote]\
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Cell["\<\
vidime, ze 
1/ po poslani f[x_]=x+1+x do Kernelu Mathematica vratila 1+2x. Pravou stranu \
vyrazu f[x_]=x+1+x tedy vyhodnotila a vratila vysledek.
2/ x ma na obou stranach definice ruznou barvu. Pri vyhodnoceni prave starny \
se pouzije globalni x. Jelikoz x zadnou globalni hodnotu nemelo (je modre a \
pojistili jsme si to v ClearAll[f,x]), tak se vyhodnocenim nezmenilo. Funkce \
v tomto pripade funguje ale nakonec dobre: \
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Cell["\<\
Kdyby nejakou hodnotu melo, tak by funkce nefungovala, jak asi chceme:\
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Cell["\<\
a skutecne i podle helpu vrati f 1+2opice nezavisle na hodnote sveho \
parametru:\
\>", "Text",
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Cell["\<\
Podobne nebezpeci pri pouziti Set Delayed, tedy \[OpenCurlyDoubleQuote]:=\
\[CloseCurlyDoubleQuote], nehrozi:\
\>", "Text",
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Cell["\<\
Vypada to, ze pouziti Set, tedy \[OpenCurlyDoubleQuote]=\
\[CloseCurlyDoubleQuote] je k nicemu. Neni... obcas potrebujeme pravou stranu \
nejprve vyhodnotit, napriklad pri integrovani a derivovani. Pokud ale \
pouzijeme Set, doporucuju vzdycky vycistit jmeno promenne.
Ukazme si, jak SetDelayed nefunguje:\
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Cell["\<\
v okamziku poslani vyrazu f[x_]:=Integrate[2x,x] do kernelu si nic \
nevyhodnotilo, nic neintegrovalo. Pri zavolani funkce na parametr jedna se \
ale pri integraci dosadilo za x i za dx... coz je nesmysl. Jeste se \
podivejme, jak vidi funkci help:\
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Kdyz bude v listu vic pravidel zameny, pouzije se prvni relevantni, pokud \
jsou pravidle v listu pravidel jen jako prvky listu:\
\>", "Text",
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Cell["\<\
Pokud jsou v listu dosazeni pravidla v samostatnych listech, dostaneme list \
vysledku:\
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Cell["\<\
Chceme-li do vyrazu dosadit konkretni vyraz, je uplne jedno, pouzijeme-li \
Rule (tedy \[OpenCurlyDoubleQuote]\[Rule]\[CloseCurlyDoubleQuote]) nebo \
RuleDelayed (tedy \[OpenCurlyDoubleQuote]\[RuleDelayed]\
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Cell["\<\
Pred dalsim su uvedomme, ze podstatne je to, co \[OpenCurlyDoubleQuote]vidi \
Kernel\[CloseCurlyDoubleQuote], ne to, co vidime my. Napriklad my vidime ve \
FrontEndu (to je to, v cem tohle ctete:-) 
a\[Equal]b a Kernel \[OpenCurlyDoubleQuote]vidi\[CloseCurlyDoubleQuote] vyraz \
jako jeho FullForm (zjednodusene vyjadreno), tedy jako Equal[a,b]) a podle \
toho provede (/.) pozadovane zameny (\[Rule] nebo \[RuleDelayed]), tedy je \
provede ve vnitrni reprezentaci vyrazu, ne v tom, co vidime... ostatne, \
zkusme si to, zamenime \[OpenCurlyDoubleQuote]rovna \
se\[CloseCurlyDoubleQuote] za \[OpenCurlyDoubleQuote]nasobeni\
\[CloseCurlyDoubleQuote]:\
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Cell["\<\
pouziti \[OpenCurlyDoubleQuote]{FullForm[vyrazZmeneny],FullForm[vyraz]}/.\
FullForm\[Rule]TreeForm\[CloseCurlyDoubleQuote] v predchozim je tak trosku \
vtip:-)... ale vlastne to zapada: proste jsme opet pouzili pravidlo zameny. \
Ze muzeme vyuzit i \[OpenCurlyDoubleQuote]zachovani listovatosti\
\[CloseCurlyDoubleQuote] vidime na prikladu (srovnej s x/.{{x\[Rule]Pi},{x\
\[Rule]E}}}, jen se zamenuje \[OpenCurlyDoubleQuote]form\
\[CloseCurlyDoubleQuote] (promenna bez hodnoty, samotne \
\[OpenCurlyDoubleQuote]form\[CloseCurlyDoubleQuote] se vlastne nepouzije, \
protoze se pri pouziti zameni za FullForm nebo TreeForm a teprve tyto \
existujici funkce se vyhodnoti.))\
\>", "Text",
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oby cviceni je hezke zkusit si predchozi naprogramovat pomoci @@ a/@ a # a &, \
ale tady nam jde o pravidla, tak to ukazu na pozadani:-)
Pro dalsi se odpichneme od predchoziho prikladu a zkusme jak bychom \
\[OpenCurlyDoubleQuote]to udelali, kdybychom nevedeli, ze \
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reprezentace \[OpenCurlyDoubleQuote]a\[Equal]b \[OpenCurlyDoubleQuote], ale \
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\[OpenCurlyDoubleQuote], zavorky na leve strane ve vyrazu (lhs_\[Equal]rhs_)\
\[RuleDelayed]lhs*rhs jsou zbytecne, jen pro pochopeni*)
(*lhs a rhs pouzivam jako se pouziva v helpu, zkratka left hand side, rihgt \
hand side\
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podobne s funkcemi, schvalne napisu stejny pripad a uvidite, ze chapat funkce \
jako (globalni) pravidla zamen docela sedi...:\
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podivejme se, co se ale stane, kdyz treba \[OpenCurlyDoubleQuote]rhs\
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podivejme se, co se ale stane, kdyz treba \[OpenCurlyDoubleQuote]rhs\
\[CloseCurlyDoubleQuote] nebude panensky modre ciste... muze to byt \
ok...pokud chceme tenhle vysledek:-), ale vetsinou asi ne\
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ukazme si, jak to dopadne pri RuleDelayed (tedy \[RuleDelayed]) a SetDelayed \
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podle dvojtecky navic na zacatku:-)\
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Cell["\<\
Budeme se zabyvat pouzitim patternu, ne patterny obecne... kdo chce, hledejte \
Pattern v helpu a SeeAlso atd...
Samosebou stale plati, ze VSECHNO je pattern. Ja, ty, on... ale i elektron a \
to jako pojem i elektron itself atd. V dalsim jde o to jak zapsat paterny \
vykazujici jisty stupen obecnosti, takze se v nich bude vyskytovat podtrzitko \
(tedy \[OpenCurlyDoubleQuote]_\[CloseCurlyDoubleQuote])
zakladni syntaxe je:
\[OpenCurlyDoubleQuote]jmenoPatternu\[CloseCurlyDoubleQuote]:\
\[CloseCurlyDoubleQuote]pattern\[CloseCurlyDoubleQuote]
(puvodni syntaxe, chvili ji budu pouzivat)
Jako opakovani ukazeme:
(\[OpenCurlyDoubleQuote]pattern\[CloseCurlyDoubleQuote] je {cokoli1,cokoli2}, \
ve vnitrni syntaxi List[cokoli1,cokoli2]).
(kdyz to nedam ve spravnem tvaru, je to jiny pattern a funkce \
\[OpenCurlyDoubleQuote]nefunguje\[CloseCurlyDoubleQuote], ma mit jeden \
parametr, dvojici, tedy f[{cokoli1,cokoli2}], pro f[cokoli1,cokoli2] Kernel \
zadne pravidlo nema, vyhodnoceni tedy nic nezmeni)\
\>", "Text",
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Cell["\<\
Obvykle zapisujeme patterny tak, ze se odpichneme od \
\[OpenCurlyDoubleQuote]obecneho\[CloseCurlyDoubleQuote] a pak \
\[OpenCurlyDoubleQuote]obecne\[CloseCurlyDoubleQuote] omezime nejakymi \
podminkami.
Zakladni pouziti je
jmenoPatternu:_typTohoNaCoSeMaVztahovat, napriklad
n:_Integer
Ukazeme si to na prikladu, kdy hledame delitele parametru funkce... protoze \
to ma dobry smysl jen pro celociselne parametry a nechceme chybove hlasky, \
kdyz nekdo zada necelociselny parametr, omezime platnost patternu na cela \
cisla:\
\>", "Text",
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Cell["\<\
Jenze typu (Real,Integer,Complex,Subscript,List..) mame relativne malo. \
Pomuzeme si tim, ze omezeni patternu muzeme udelat pomoci otazniku a boolske \
funkce (co vraci True nebo False)
Ukazeme si to na prikladu (funkce \[OpenCurlyDoubleQuote]test\
\[CloseCurlyDoubleQuote] ma schvalne parametr \[OpenCurlyDoubleQuote]y\
\[CloseCurlyDoubleQuote] aby bylo jasnejsi, ze u SetDelayed (tedy \
\[OpenCurlyDoubleQuote]:=\[CloseCurlyDoubleQuote]) na pojmenovani promenne \
nezalezi... nikdy se nepouzije, pri volani ma paremetr uz obecne jinou \
hodnotu):\
\>", "Text",
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Cell["a v \[OpenCurlyDoubleQuote]normalni\[CloseCurlyDoubleQuote] syntaxi:", \
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Cell["znak \[OpenCurlyDoubleQuote]&\[CloseCurlyDoubleQuote] ma velikou \
prioritu a tak to bez zavorky nefunguje:", "Text",
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Cell["\<\
v predchozi ukazce to funguje dobre, co kdyz ale listDvojic bude obasahovat \
jen DVE dvojice?\
\>", "Text",
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Cell["\<\
..to asi nechceme. Jde o to, ze pattern {u_,i_} splnuje jak {2,1}, tak {4,2}, \
ale i {{2,1},{4,2}}, temi \[OpenCurlyDoubleQuote]jakymikoli dvema prvky\
\[CloseCurlyDoubleQuote] mohou totiz byt i listy {2,1} a {4,2}. Pokud si \
nejsme jisti, zda neco splnuje pattern, mame prikaz MatchQ. Pokud vnitrni \
funkce konci Q (asi od \
\[OpenCurlyDoubleQuote]question\[CloseCurlyDoubleQuote]), tak vraci True nebo \
False, jejich tam docela dost, muzes je zkoumat napriklad kdyz do Kernelu \
posles ?*Q\
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Cell["\<\
Vyresit to muzeme mnoha zpusoby... ze dvou ukazanych je ten druhy lepsi, \
proudy a napeti nebudou obecne cela cisla (Integer), ale mela by mit ciselnou \
hodnotu\
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Cell["\<\
patterny se nam hodi uz proto, ze mame v Mathematice mocne prikazy, ktere je \
vyuzivaji, napriklad Cases*)
(*Cases pouzivam nejcasteji v syntaxi \
Cases[vCemHledam,pattern,{0,\[Infinity]}], kde {0,\[Infinity]} rika, ze se ma \
prohledat vyraz vCemHledam cely. V podstate je vetsinou hlavni problem \
vymyslet spravny pattern.
Take obvas chceme ztransformovat vyrazy... ukazme si to treba na seriovem RLC \
obvodu, vstupni napeti je uin[t] \
\>", "Text",
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Matematicka optimalizce je hledanim optimalniho parametru dle kriteria. Tim \
se mysly v inzenirske praxi napriklad navrh zarizeni s optimalnimy parametry. \
Jeste castejsi ulohou inzenyra byva nalezeni vhodneho modelu zarizeni z \
namerenych dat tzv regrese. V obou uvedenych pripadech pracujeme s modelem a \
matematickou optimalizaci. \
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Neni to nejlepsi zapis ucelove funkce protoze predchozi defenice muzou \
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Lepsi predelat to na funkci s lokalnimy promennymi\
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 "Mnozina diferencialnich rovnic ktere muzeme vyresit analitciky je velmi \
mala v porovnani se vsemi typy diferencialnich rovnic pouzivanych v \
inzenyrske praxi.\nProto pro reseni rovnic ktere nemaji analyticke reseni \
pouzivame numericke metody.\nNejznamejsi a nejjdnodussi metoda pro numerecke \
reseni dif. rovnic je Eulerova metoda.\npro obcnou diferencialni rovnici:\n",
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Cell["\<\
Podivame se na reseni na kratsim useku a vidime ze Euler aproximuje reseni s \
nejakou chybou 
to je treba se uvedomit ze numericke reseni vzdycky ma nejakou konecnou \
presnost a neni to reseni spravnejsi definici je aproximace.\
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Uzivatel muze definovat metodu pro reseni.\
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