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(*** Wolfram CDF File ***)
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Cell["\<\
Jak ji\[ZHacek] bylo nazna\[CHacek]eno, interval je rozd\[EHacek]len \
na d\[IAcute]lky a sestaven lichob\[EHacek]\[ZHacek]n\[IAcute]k se \[SHacek]\
\[IAcute]\[RHacek]kou d\[IAcute]lku a d\[EAcute]lkami stran podle funk\
\[CHacek]n\[IAcute]ch hodnot na prav\[EAcute]m a lev\[EAcute]m bodu \
d\[IAcute]lku. Vypo\[CHacek]\[IAcute]tat obsah lichob\[EHacek]\[ZHacek]n\
\[IAcute]ku je snadn\[EAcute]: \[SHacek]\[IAcute]\[RHacek]ka je d\[AAcute]na \
a v\[YAcute]\[SHacek]ka je pr\[URing]m\[EHacek]rem lev\[EAcute] a prav\
\[EAcute] funk\[CHacek]n\[IAcute] hodnoty. To je z\[AAcute]sadn\[IAcute] rozd\
\[IAcute]l proti obd\[EAcute]ln\[IAcute]k\[URing]m, kde se bere jako v\
\[YAcute]\[SHacek]ka jen hodnota jedin\[AAcute] (kdekoliv z \
\[SHacek]\[IAcute]\[RHacek]ky d\[IAcute]lku).
P\[RHacek]i bli\[ZHacek]\[SHacek]\[IAcute]m pohledu na obr\[AAcute]zek lze \
vysledovat, \[ZHacek]e p\[URing]vodn\[IAcute] k\[RHacek]ivka je nahrazena \
\[OpenCurlyDoubleQuote]jakousi\[CloseCurlyDoubleQuote] lomenou \[CHacek]\
\[AAcute]rou sestavenou z \[UAcute]se\[CHacek]ek. Lomen\[AAcute] \[CHacek]\
\[AAcute]ra l\[EAcute]pe charakterizuje pr\[URing]b\[EHacek]h funkce proti \
obd\[EAcute]ln\[IAcute]kov\[EAcute] metod\[EHacek].\
\>", "Text",
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Cell["P\[RHacek]esnost v\[YAcute]po\[CHacek]tu", "Section",
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Podobn\[EHacek] jako u obd\[EAcute]lnikov\[EAcute] metody p\[RHacek]esnost v\
\[YAcute]po\[CHacek]tu ovliv\[NHacek]uj\[IAcute] zejm\[EAcute]na dva faktory: \
sama funkce, ale tu t\[EHacek]\[ZHacek]ko ovlivn\[IAcute]me a tak\[EAcute] po\
\[CHacek]et d\[IAcute]lk\[URing] na kter\[EAcute] interval rozd\[EHacek]l\
\[IAcute]me. Teoreticky \[CHacek]\[IAcute]m v\[EHacek]t\[SHacek]\[IAcute] po\
\[CHacek]et d\[IAcute]lk\[URing], t\[IAcute]m v\[EHacek]t\[SHacek]\[IAcute] p\
\[RHacek]esnost.
Ze vzorce pro odhad chyby je mo\[ZHacek]no stanovit p\[RHacek]ibli\[ZHacek]n\
\[YAcute] po\[CHacek]et d\[IAcute]lk\[URing] (segment\[URing]) stejn\[EAcute] \
d\[EAcute]lky, p\[RHacek]i zadan\[EAcute] maxim\[AAcute]ln\[IAcute] povolen\
\[EAcute] chyb\[EHacek]. Odhad chyby:\
\>", "Text",
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p\[RHacek]edpokl\[AAcute]d\[AAcute]me, \[ZHacek]e \[CHacek]ten\[AAcute]\
\[RHacek] neum\[IAcute] derivovat, tak druhou derivaci na\[SHacek]\[IAcute] \
funkce nalezne system ",
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\[URing]\[ZHacek]eme vylou\[CHacek]it. V\[YAcute]sledek je tedy 134, tedy ",
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jedn\[EAcute] tis\[IAcute]ciny. Proto\[ZHacek]e se jedn\[AAcute] o nejhor\
\[SHacek]\[IAcute] mo\[ZHacek]n\[YAcute] odhad, je mo\[ZHacek]n\[EAcute], \
\[ZHacek]e po\[ZHacek]adovan\[EAcute] p\[RHacek]esnosti dos\[AAcute]hneme i p\
\[RHacek]i men\[SHacek]\[IAcute]m po\[CHacek]tu d\[IAcute]lk\[URing], ale 135 \
je jistota."
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V literatu\[RHacek]e je \[CHacek]asto uv\[AAcute]d\[EHacek]na je\[SHacek]t\
\[EHacek] jedna numerick\[AAcute] metoda pro integraci, tzv Simpsonovo \
pravidlo. Tato metoda nahrazuje p\[URing]vodni funkci (k\[RHacek]ivku) na dan\
\[EAcute]m \[OpenCurlyDoubleQuote]dvoud\[IAcute]lku\[CloseCurlyDoubleQuote] \
nikoliv \[UAcute]se\[CHacek]kou, ale \[CHacek]\[AAcute]st\[IAcute] paraboly. \
Tyto paraboly je\[SHacek]t\[EHacek] l\[EAcute]pe vystihuj\[IAcute] \
pr\[URing]b\[EHacek]h p\[URing]vodn\[IAcute] funkce a doch\[AAcute]z\[IAcute] \
k men\[SHacek]\[IAcute]m chyb\[AAcute]m. ALE v\[YAcute]po\[CHacek]et je o n\
\[EHacek]co slo\[ZHacek]it\[EHacek]j\[SHacek]\[IAcute] a trv\[AAcute] d\
\[EAcute]le, ALE zase je mo\[ZHacek]n\[EAcute] zmen\[SHacek]it po\[CHacek]et \
d\[IAcute]lk\[URing] pro zadanou p\[RHacek]esnost. Ot\[AAcute]zkou \
z\[URing]st\[AAcute]v\[AAcute] zda-li se vyplat\[IAcute] tuto metodu pou\
\[ZHacek]\[IAcute]t m\[IAcute]sto metody \
lichob\[EHacek]\[ZHacek]n\[IAcute]kov\[EAcute]...\
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Dal\[SHacek]\[IAcute] metody jsou zalo\[ZHacek]eny na my\[SHacek]lence \
automatick\[EAcute] volby kroku. Tedy metota (a\[THacek] u\[ZHacek] pou\
\[ZHacek]\[IAcute]v\[AAcute] obd\[EAcute]ln\[IAcute]ky nebo lichob\[EHacek]\
\[ZHacek]n\[IAcute]ky) si automaticky vol\[IAcute] \
\[SHacek]\[IAcute]\[RHacek]ku d\[IAcute]lku. Je-li \[OpenCurlyDoubleQuote]zm\
\[EHacek]na\[CloseCurlyDoubleQuote] na k\[RHacek]ivce mal\[AAcute] vol\
\[IAcute] se krok \[SHacek]ir\[SHacek]\[IAcute] (a t\[IAcute]m je \
v\[YAcute]po\[CHacek]et rychlej\[SHacek]\[IAcute] bez ztr\[AAcute]ty p\
\[RHacek]esnosti), nebo m\[EHacek]n\[IAcute]-li k\[RHacek]ivka sv\[URing]j pr\
\[URing]b\[EHacek]h hodn\[EHacek]-prudce, vol\[IAcute] se krok men\[SHacek]\
\[IAcute], jemn\[EHacek]j\[SHacek]\[IAcute], aby byla p\[RHacek]esnost zachov\
\[AAcute]na. Ot\[AAcute]zkou z\[URing]st\[AAcute]v\[AAcute] zda-li se vyplat\
\[IAcute] tuto metodu pou\[ZHacek]\[IAcute]t m\[IAcute]sto metody lichob\
\[EHacek]\[ZHacek]n\[IAcute]kov\[EAcute]...\
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/* *
* Auto Generated Java Class.
*/
public class Integral {
public static double f (double x) {
return x*x*x - 7*x*x + 14*x - 2;
}
public static double lichobeznik (double a, double b, long dilek) {
double suma = 0;
final double DELTAX = (b - a)/dilek;
for (int i = 0; i < dilek; i++) {
suma = suma + (DELTAX*( (f (a + i*DELTAX) + f (a + (i + \
1)*DELTAX))/2 ));
}
return suma;
}
public static void main (String[] args) {
System.out.print ( lichobeznik (1, 4, 3000) );
}
}\
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Lichob\[EHacek]\[ZHacek]n\[IAcute]kov\[AAcute] metoda je p\[RHacek]esn\
\[EHacek]j\[SHacek]\[IAcute] a efektivn\[EHacek]j\[SHacek]\[IAcute] ne\
\[ZHacek] metoda obd\[EAcute]ln\[IAcute]kov\[AAcute], zn\[AAcute]me i odhad \
chyby. Animace n\[AAcute]m uka\[AAcute]\[ZHacek]e jak p\[RHacek]esnost v\
\[YAcute]po\[CHacek]tu z\[AAcute]vys\[IAcute] na po\[CHacek]tu d\[IAcute]lk\
\[URing] nad intervalem. Je zde uveden i k\[OAcute]d v jazyce Java.\
\>", "Text",
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"1/ Sestavte tabulku velikost\[IAcute] chyb pro 10, 15, 20, 30, 50, 100 a \
150 d\[IAcute]lk\[URing].\n2/ Kolik d\[IAcute]lk\[URing] je pot\[RHacek]eba \
zvolit pro p\[RHacek]esnost pod hranic\[IAcute] 13/55555?\n3/ Integrujte \
funkci sinus na intervalu <0, 1> porovnejte v\[YAcute]sledek s obd\[EAcute]ln\
\[IAcute]kovou metodou. Experimentujte se stejn\[YAcute]mi po\[CHacek]ty d\
\[IAcute]lk\[URing], nebo pokuste se naj\[IAcute]t po\[CHacek]et d\[IAcute]lk\
\[URing] aby byla obd\[EAcute]ln\[IAcute]kov\[AAcute] metoda stejn\[AAcute] \
jako lichob\[EHacek]\[ZHacek]n\[IAcute]kov\[AAcute].\n4/ Integruj funkce nad \
intervalem <0, 1> : ",
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Cell[BoxData[
FormBox[
SuperscriptBox["x", "2"], TraditionalForm]]],
", ",
Cell[BoxData[
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", sin(x), cos(x), log(x), ",
Cell[BoxData[
FormBox[
SuperscriptBox["10", "x"], TraditionalForm]]],
". Experimentuj s po\[CHacek]tem d\[IAcute]lk\[URing] p\[RHacek]i integraci. \
Porovnej v\[YAcute]sledky z Java programu s v\[YAcute]sledky Wolfram \
Mathamatici.\n5/ Experimentuj s vysok\[YAcute]m po\[CHacek]tem d\[IAcute]lk\
\[URing] v jazyce Java na intervalem . Prvn\[EHacek] nech v\[YAcute]po\
\[CHacek]ty b\[EHacek]z\[EHacek]t v datov\[EAcute]m typu float a pak v \
double. Sleduj p\[RHacek]esnost v\[YAcute]po\[CHacek]tu a jeho d\[EAcute]lku/\
\[CHacek]as.\n6/ Optimalizujte k\[OAcute]d v jazyce Java, tak aby se funk\
\[CHacek]n\[IAcute] hodnoty nepo\[CHacek]\[IAcute]taly 2x."
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Cell["\<\
http://www.kvd.zcu.cz/cz/materialy/numet/_numet.html#_Toc501178915
http://reference.wolfram.com/mathematica/ref/Integrate.html
http://reference.wolfram.com/mathematica/ref/NIntegrate.html
http://www.matematika.cz/urcity-integral\
\>", "Text",
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