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\vskip 1.5em} +%opening +\newcommand{\snovdn}{Potence z racionalnimi eksponenti } +\newcommand{\predmdn}{mat} +\newcommand{\predmkaj}{domace\_naloge} +\newcommand{\stevilkadn}{30} +\newcommand{\cm}[1]{\SI{#1}{\centi\meter}} +\newcommand{\kmh}[1]{\SI{#1}{\kilo\meter\per\hour}} +\makeatletter +\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname} +\newcommand{\@xslalph}[1]{% + \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i% + \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}% + \or t\or u\or v\or z\or \v{z} + \else\@ctrerr\fi% +} +\AddEnumerateCounter{\xslalph}{\@xslalph}{m} +\makeatother +\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma +\newcommand*\textfrac[2]{ + \frac{\text{#1}}{\text{#2}} +} +\NewTasks[counter-format=\xslalph*),label-format=\bfseries]{primerTasks}(3) +\newcommand\vektor{\overrightarrow} +\newcommand{\iic}{I\textsuperscript{2}C } +\title{% + \snovdn --- \stevilkadn. domača naloga + \\ + \large Matematika, Gimnazija Bežigrad} +\author{\begin{tabular}{rl} + \textbf{Profesor:} & prof. Vilko Domajnko \\ + \textbf{Avtor:} & Anton Luka Šijanec, 2. a +% \textbf{Avtor:} & Anton Luka Šijanec \\ & Member 2 \\ & Member 3 +\end{tabular}} +\newcommand\hcancel[2][black]{\setbox0=\hbox{$0#2$}% +\rlap{\raisebox{.45\ht0}{\textcolor{#1}{\rule{\wd0}{1pt}}}}#2} +\everymath{\displaystyle} % https://tex.stackexchange.com/a/32847/212260 +\begin{document} +\maketitle +\begin{abstract} +Ta dokument vsebuje navodila in rešitve domače naloge snovi \textit{\snovdn}pri matematiki, ki sem jih spisal sam. +Kjer je bilo potrebno izbrati neke poljubne naloge, sem jih vedno izbiral naključno. +\end{abstract} +\paragraph{Navodilo naloge} \textbf{listi}: \textit{Potence z racionalnimi eksponenti} / 1, ... 6 (povsod po en primer) +%\tableofcontents +\begin{enumerate}[label=\textbf{\arabic*.}] + \item Zapiši v obliki potence z racionalnim eksponentom: + \begin{enumerate}[label=\textbf{\xslalph*)}] + \setcounter{enumii}{3} + \item $\sqrt{a^3}\cdot\sqrt{a^5}=a^4$ + \end{enumerate} + \item Zapiši v obliki potence z racionalnim eksponentom: + \begin{enumerate}[label=\textbf{\xslalph*)}] + \item $\sqrt{a\sqrt{a}}=\sqrt{a\cdot a^\frac{1}{2}}=\sqrt{a^\frac{3}{2}}=\left(a^\frac{3}{2}\right)^\frac{1}{2}=a^\frac{3}{4}$ + \end{enumerate} + \item Izračunaj: + \begin{enumerate}[label=\textbf{\xslalph*)}] + \setcounter{enumii}{5} + \item + $$ + \left(\sqrt[6]{9-4\sqrt{5}+\sqrt[3]{2-\sqrt{5}}}\right)\sqrt[3]{2+\sqrt{5}} + =\left(\sqrt[6]{\left(2-\sqrt{5}\right)^2}+\sqrt[3]{2+\sqrt{5}}\right)\sqrt[3]{2+\sqrt{5}}= + $$ + $$ + =\left(\sqrt[3]{\sqrt{5}-2}+\sqrt[3]{2-\sqrt{5}}\right)\sqrt[3]{2+\sqrt{5}} + =\sqrt[3]{\left(\sqrt{5}-2\right)\cdot\left(2+\sqrt{5}\right)}+\sqrt[3]{\left(2-\sqrt{5}\right)\cdot\left(2+\sqrt{5}\right)}= + $$ + $$ + =\sqrt[3]{\left(\sqrt{5}-2\right)\cdot\left(\sqrt{5}+2\right)}+\sqrt[3]{4-5}= + \sqrt[3]{5-4}+\sqrt[3]{-1}=1-1=0 + $$ + \end{enumerate} + \item Poenostavi in zapiši v obliki ulomka (brez negativnih eksponentov): + \begin{enumerate}[label=\textbf{\xslalph*)}] + \setcounter{enumii}{1} + \item $\left(2x^\frac{1}{5}y^\frac{4}{5}\right)^{-5}=\frac{1}{\left(2x^\frac{1}{5}y^\frac{4}{5}\right)^5}=\frac{1}{32xy^4}$ + \end{enumerate} + \item Poenostavi: + \begin{enumerate}[label=\textbf{\xslalph*)}] + \setcounter{enumii}{1} + \item $\sqrt{a^3\cdot\sqrt[3]{\frac{1}{a^2}\cdot\sqrt{a}}}=\sqrt{a^3\cdot a^\frac{-1}{2}}=\sqrt{a^\frac{5}{2}} + =\left(a^\frac{5}{2}\right)^\frac{1}{2}=a^\frac{5}{4}=\sqrt[4]{a^5}=a\sqrt[4]{a}$ + \end{enumerate} + \item Poenostavi: + \begin{enumerate}[label=\textbf{\xslalph*)}] + \setcounter{enumii}{1} + \item Vau, štiri korene smo spremenili v enega! Spodaj je praktična zanimivost. + $$ + \sqrt[3]{\frac{x^2}{x-1}}\cdot\sqrt{\frac{\left(x-1\right)^2}{x^5}}\cdot\sqrt[6]{\frac{x^{20}}{\left(x+ 1\right)^3}}\cdot\sqrt{\frac{x+1}{x^3}}= + \sqrt[6]{\left(\frac{x^2}{x-1}\right)^2\cdot\left(\frac{\left(x-1\right)^2}{x^5}\right)^3\cdot\frac{x^{20}}{\left(x+1}\right)^3\cdot\left(\frac{x+1}{x^3}\right)^3}= + $$ + $$ + =\sqrt[6]{\left(\frac{x^2}{x-1}\right)^2\cdot\left(\frac{\left(x-1\right)^2}{x^5}\right)^3 + \cdot\frac{x^{20}}{\left(x+1\right)^3}\cdot\left(\frac{x+1}{x^3}\right)^3}= + \sqrt[6]{\frac{x^4}{\left(x-1\right)^2}\cdot\frac{\left(x-1\right)^6}{x^{15}} + \cdot\frac{x^{20}}{\left(x+1\right)^3}\cdot\frac{\left(x+1\right)^3}{x^9}}= + $$ + $$ + =\sqrt[6]{\left(x-1\right)^4\cdot x^5\cdot\frac{1}{x^5}}= + \sqrt[6]{\left(x-1\right)^4}=\sqrt[3]{\left(x-1\right)^2} + $$ + \paragraph{Zanimivost} Želim preveriti, kako taka optimizacija vpliva v resničnem življenju. Recimo, da se ta račun v računalniku izvede desetmilijonkrat v sklopu nekega procesa. Koliko procesorskega časa smo pridobili s tem, da smo štiri korene spremenili v enega? Kljub vsemu smo na koncu optimizacije dobili šesti koren, kar nikakor ni mačji kašelj za X86 arhitekturo. Najlažje je računalniku računati korene kvadratnih eksponentov (2, 4, 16, ...). + \lstinputlisting[language=Octave]{kompleksnost.c} + +\texttt{a@upor:[0]\$ ./kompleksnost \\ +neoptimizirana enacba je za 10e6 iteracij porabila 15.330899 sekund, optimizirana pa 4.157521 \\ +rezultat = 278495353224.555294, rezultat2 = 278495353224.555814 \\ +a@upor:[0]\$ \\} + + Ojej! Mislim, da je razlika očitna. Neoptimizirana enačba je porabila 10 sekund več. Optimizacija je torej zelo pomembna. Ampak zakaj tega namesto nas ne delajo računalniki? V GNU prevajalniku C jezika sem avtomatilno optimizacijo nastavil na nivo \texttt{0}, kar pomeni, da prevajalnik ne dela optimizacij, v navodilih za uporabo pa sem, poleg drugih, zasledil še eno opcijo, \texttt{fast}. Opcija \texttt{fast} je sicer eksperimentalna in nepriporočena, vendar dela čudeže! Če program prevedem z optimizacijskim nivojem \texttt{fast}, bo računanje neoptimizirane enačbe in optimizirane trajalo približno enako dolgo. + +\texttt{a@upor:[0]\$ gcc kompleksnost.c -pedantic -Wall -Wextra -g -okompleksnost -lm -Ofast \\ +a@upor:[0]\$ ./kompleksnost \\ +neoptimizirana enacba je za 10e6 iteracij porabila 5.326709 sekund, optimizirana pa 4.017244 \\ +rezultat = 278495353224.555294, rezultat2 = 278495353224.555814 \\ +a@upor:[0]\$ \\} + \end{enumerate} +\end{enumerate} +\section{Zaključek} +Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in +\hologo{LaTeX}\footnote{Za izdelavo dokumenta potrebujete \texttt{TeXLive 2020}.} +izvorna koda, zgodovina sprememb in prejšnje različice, je na voljo v mojem šolskem Git repozitoriju na +\url{https://git.sijanec.eu/sijanec/sola-gimb-2} v mapi +\href{https://git.sijanec/sola-gimb-2/src/branch/master/\predmdn/\predmkaj/\stevilkadn/}{/\predmdn/\predmkaj/\stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/\predmdn/\predmkaj/\stevilkadn/dokument.pdf} in/ali \url{https://git.sijanec.eu/sijanec/sola-gimb-2/raw/branch/master/\predmdn/\predmkaj/\stevilkadn/dokument.pdf}. +\if\razhroscevanje1 +\vfill +\section*{Razhroščevalne informacije} +Te informacije so generirane, ker je omogočeno razhroščevanje. Pred objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroscevanje} na 0 v začetku dokumenta. + +Grafi imajo natančnost \functionSamples\space točk na graf. + +Konec generiranja dokumenta: \today\ ob \currenttime\footnote{To ne nakazuje dejanskega časa, ko je bil dokument napisan, temveč čas, ko je bi dokument generiran v PDF/DVI obliko. Isto velja za datum v glavi dokumenta. Če berete direktno iz LaTeX datoteke, bo to vedno današnji datum.}%\input|"date -Ins" + +Dokument se je generiral R0qK1KR2 \SI{}{\second}. +\fi +% \item $$$$ aaasecgeninsaaa R0qK1KR2 +\end{document} diff --git a/mat/domace_naloge/30/kompleksnost b/mat/domace_naloge/30/kompleksnost new file mode 100755 index 0000000..31f2eff Binary files /dev/null and b/mat/domace_naloge/30/kompleksnost differ diff --git a/mat/domace_naloge/30/kompleksnost.c b/mat/domace_naloge/30/kompleksnost.c new file mode 100644 index 0000000..ae36c99 --- /dev/null +++ b/mat/domace_naloge/30/kompleksnost.c @@ -0,0 +1,27 @@ +/* gcc kompleksnost.c -pedantic -E -Wall -Wextra -g -okompleksnost -lm -O0 */ +#include +#include +#include +#include +#define KOLIKOKRAT 10000001 +#define nrx(x, y) powl((x), 1.0 / (y)) /* nti koren izpeljemo iz potence na 1/eks */ +int main (void) { /* za kvadratne in kubicne korene uporabimo sqrt in cbrt */ + long double rezultat = 0; /* da prevajalnik ne bi optimiziral */ + long double rezultat2 = 0; + clock_t cas_zacetek, cas_konec, cas2_konec; + cas_zacetek = clock(); + for (unsigned long long int x = 2; x <= KOLIKOKRAT; x++) + rezultat += cbrtl(powl(x, 2)/(x-1))*sqrtl(powl(x-1, 2)/powl(x, 5))*nrx(powl(x, 20)/powl(x+1, 3), 6)*sqrtl((x+1)/powl(x, 3)); + cas_konec = clock(); + for (unsigned long long int x = 2; x <= KOLIKOKRAT; x++) + rezultat2 += nrx(pow(x-1, 2), 3); + cas2_konec = clock(); + fprintf(stdout, "neoptimizirana enacba je za 10e6 iteracij porabila %lf sekund, optimizirana pa %lf\n" + "rezultat = %Lf, rezultat2 = %Lf\n", + ((double) (cas_konec-cas_zacetek))/CLOCKS_PER_SEC, + ((double) (cas2_konec-cas_konec))/CLOCKS_PER_SEC, + rezultat, + rezultat2 + ); + return 0; +} diff --git a/mat/domace_naloge/30/stderr.log b/mat/domace_naloge/30/stderr.log new file mode 100644 index 0000000..e69de29 diff --git a/mat/domace_naloge/30/stdout.log b/mat/domace_naloge/30/stdout.log new file mode 100644 index 0000000..4b3f4f9 --- /dev/null +++ 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