--- /dev/null
+\def\scharfs{\char"19}
+\input lecnotes.tex
+\def\imply{\Rightarrow}
+\prednaska{9}{Fourierova transformace}{}
+
+Násobení polynomù mù¾e mnohým pøipadat jako pomìrnì (algoritmicky) snadný
+problém. Asi ka¾dého hned napadne \uv{hloupý} algoritmus -- vezmeme
+koeficienty prvního polynomu a~vynásobíme ka¾dý se v¹emi koeficienty druhého
+polynomu a~pøíslu¹nì u~toho seèteme i~exponenty (stejnì jako to dìláme, kdy¾
+násobíme polynomy na~papíøe). Pokud stupeò prvního polynomu je $n$ a~druhého
+$m$, strávíme tím èas $\Omega(mn)$. Pro $m=n$ je to kvadraticky pomalé.
+Na~první pohled se mù¾e zdát, ¾e rychleji to prostì nejde (pøeci musíme
+v¾dy vynásobit \uv{ka¾dý s~ka¾dým}). Ve skuteènosti to ale rychleji fungovat
+mù¾e, ale k~tomu je potøeba znát trochu tajemný algoritmus FFT neboli {\I Fast
+Fourier Transform}.
+
+
+\ss{Trochu algebry na~zaèátek}
+Celé polynomy oznaèujeme velkými písmeny, jednotlivé èleny polynomù pøíslu¹nými
+malými písmeny (pø.: polynom $W$ stupnì $d$ má koeficienty $w_{0}, w_{1},
+w_{2},\ldots, w_{d}$).
+
+Libovolný polynom $P$ stupnì (nejvý¹e) $d$ lze reprezentovat
+jednak jeho koeficienty, tedy èísly $p_{0}, p_{1}, \ldots ,p_{d}$, druhak
+i~pomocí hodnot:
+
+\>{\bf Lemma:} Polynom stupnì nejvý¹e $d$ je jednoznaènì urèeò svými
+hodnotami v~$d+1$ rùzných bodech.
+
+\>{\it Dùkaz:}
+Polynom stupnì $d$ má maximálnì $d$ koøenù (indukcí -- je-li
+$k$ koøenem $P$, pak lze $P$ napsat jako $(x-k)Q$ kde $Q$ je polynom stupnì
+o~jedna men¹í, pøitom polynom stupnì 1 má jediný koøen); uvá¾íme-li
+dva rùzné polynomy $P$ a~$Q$ stupnì $d$ nabývající v~daných bodech stejných
+hodnot, tak $P-Q$ je polynom stupnì maximálnì $d$, ka¾dé
+z $x_{0}\ldots x_{d}$ je koøenem tohoto polynomu $\imply$ spor, polynom stupnì
+$d$ má $d+1$ koøenù $\imply$ $P-Q$ musí být nulový polynom $\imply$ $P=Q$.
+\qed
+\medskip
+
+Pov¹imnìme si jedné skuteènosti -- máme-li dva polynomy $A$ a~$B$ stupnì $d$
+a~body $x_{0}, \ldots, x_{n}$, dále polynom $C=A \cdot B$ (stupnì $2d$), pak
+platí $C(x_{j}) = A(x_{j}) \cdot B(x_{j})$ pro $j = 0,1,2, \ldots, n$. Toto
+èiní tento druhý zpùsob reprezentace polynomu velice atraktivním pro násobení
+-- máme-li $A$ i $B$ reprezentované hodnotami v $n \geq 2d+1$ bodech, pak
+snadno (v $\Theta(n)$) spoèteme takovou reprezentaci $C$.
+Problémem je, ¾e typicky máme polynom zadaný koeficienty, a~ne hodnotami
+v~bodech. Tím pádem potøebujeme nìjaký hodnì rychlý algoritmus (tj.
+rychlej¹í ne¾ kvadratický, jinak bychom si nepomohli oproti hloupému
+algoritmu) na~pøevod polynomu z jedné reprezentace do druhé a~zase zpìt.
+
+\s{Idea, jak by mìl algoritmus pracovat:}
+\algo
+\:Vybereme $n\geq 2d+1$ bodù $x_{0}, x_{1}, \ldots , x_{n-1}$.
+\:V tìchto bodech vyhodnotíme polynomy $A$ a~$B$.
+\:Nyní ji¾ v~lineárním èase získáme hodnoty polynomu $C$ v~tìchto bodech:
+ $C(x_i) = A(x_i)\cdot B(x_i)$
+\:Pøevedeme hodnoty polynomu $C$ na~jeho koeficienty.
+\endalgo
+
+\>Je vidìt, ¾e klíèové jsou kroky 2 a~4.
+Celý trik spoèívá v~chytrém vybrání onìch bodù, ve kterých budeme polynomy
+vyhodnocovat -- zvolí-li se obecná $x_j$, tak se to rychle neumí, pro speciální
+$x_j$ ale uká¾eme, ¾e to rychle jde.
+
+\ss{Vyhodnocení polynomu metodou Rozdìl a~panuj (algoritmus FFT):}
+Mìjme polynom $P$ stupnì $\leq d$ a~chtìjme jej vyhodnotit v~$n$ bodech.
+Vybereme si body tak, aby byly spárované, èili $\pm x_{0}, \pm x_{1},
+\ldots , \pm x_{n/2-1} $. To nám výpoèet urychlí, proto¾e pak se druhé
+mocniny $x_{j}$ shodují s~druhými mocninami $-x_{j}$.
+
+Polynom $P$ rozlo¾íme na~dvì èásti, první obsahuje èleny se sudými exponenty,
+druhá s~lichými:
+$$P(x) = (p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{d-2}x^{d-2}) + (p_{1}x^{1} +
+ p_{3}x^{3} + \ldots + p_{d-1}x^{d-1})$$
+\>se zavedením znaèení:
+$$P_s(t) = p_0t^0 + p_{2}t^{1} + \ldots + p_{d - 2}t^{d - 2\over 2}$$
+$$P_l(t) = p_1t^0 + p_3t^1 + \ldots + p_{d - 1}t^{d - 2\over 2}$$
+
+\>bude $P(x) = P_s(x^{2}) + xP_l(x^{2})$ a~$P(-x) = P_s(x^{2}) -
+xP_l(x^{2})$. Jinak øeèeno, vyhodnocování polynomu $P$ v~$n$ bodech se nám
+smrskne na~vyhodnocení $P_s$ a~$P_l$ v~$n/2$ bodech -- oba jsou polynomy
+stupnì nejvý¹e $d/2$ a~vyhodnocujeme je v~$x^{2}$ (vyu¾íváme
+rovnosti $(x_{i})^{2} = (-x_{i})^{2}$).
+
+\s{Pøíklad:}
+$3 + 4x + 6x^{2} + 2x^{3} + x^{4} + 10x^{5} = (3 + 6x^{2} + x^{4}) + x(4 +
+2x^{2} + 10x^{4})$.
+
+Teï nám ov¹em vyvstane problém s~oním párováním -- druhá mocnina pøece nemù¾e
+být záporná a~tím pádem u¾ v~druhé úrovni rekurze body spárované nebudou.
+Z~tohoto dùvodu musíme pou¾ít komplexní èísla -- tam druhé mocniny záporné býti
+mohou.
+
+% komplex
+\h{Komplexní intermezzo}
+\def\i{{\rm i}}
+\def\\{\hfil\break}
+
+\s{Základní operace}
+
+\itemize\ibull
+\:Definice: ${\bb C} = \{a + b\i \mid a,b \in {\bb R}\}$
+
+\:Sèítání: $(a+b\i)\pm(p+q\i) = (a\pm p) + (b\pm q)\i$. \\
+Pro $\alpha\in{\bb R}$ je $\alpha(a+b\i) = \alpha a + \alpha b\i$.
+
+\:Komplexní sdru¾ení: $\overline{a+b\i} = a-b\i$. \\
+$\overline{\overline x} = x$, $\overline{x\pm y} = \overline{x} \pm
+\overline{y}$, $\overline{x\cdot y} = \overline x \cdot \overline y$, $x
+\cdot \overline x \in {\bb R}$.
+
+\:Absolutní hodnota: $\vert x \vert = \sqrt{x\cdot\overline{x}}$, tak¾e $\vert
+a+b\i \vert = \sqrt{a^2+b^2}$. \\
+Také $\vert \alpha x \vert = \vert \alpha\vert \cdot \vert x \vert$.
+
+\:Dìlení: $x/y = (x\cdot \overline{y}) / (y \cdot \overline{y})$.
+\endlist
+
+\s{Gau{\scharfs}ova rovina a goniometrický tvar}
+
+\itemize\ibull
+\:Komplexním èíslùm pøiøadíme body v~${\bb R}^2$: $a+b\i \leftrightarrow (a,b)$.
+
+\:$\vert x\vert$ je vzdálenost od~bodu $(0,0)$.
+
+\:$\vert x\vert = 1$ pro èísla le¾ící na~jednotkové kru¾nici ({\I komplexní jednotky}). \\
+Pak platí $x=\cos\varphi + \i\sin\varphi$ pro nìjaké $\varphi\in\left[
+0,2\pi \right)$.
+
+\:Pro libovolné $x\in{\bb C}$: $x=\vert x \vert \cdot (\cos\varphi(x) +
+\i\sin\varphi(x))$. \\
+Èíslu $\varphi(x)\in\left[ 0,2\pi \right)$ øíkáme {\I argument}
+èísla~$x$, nìkdy znaèíme $\mathop{\rm arg} x$.
+
+\:Navíc $\varphi({\overline{x}}) = -\varphi(x)$.
+\endlist
+
+\s{Exponenciální tvar}
+
+\itemize\ibull
+\:Eulerova formule: $e^{i\varphi} = \cos\varphi + \i\sin\varphi$.
+
+\:Ka¾dé $x\in{\bb C}$ lze tedy zapsat jako $\vert x\vert \cdot e^{\i\cdot
+\varphi(x)}$.
+
+\:Násobení: $xy = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right) \cdot
+ \left(\vert y\vert\cdot e^{\i\cdot\varphi(y)}\right) = \vert x\vert
+ \cdot \vert y\vert \cdot e^{\i\cdot(\varphi(x) + \varphi(y))}$. \\
+(absolutní hodnoty se násobí, argumenty sèítají)
+
+\:Umocòování: $x^\alpha = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right)^
+ \alpha = {\vert x\vert}^\alpha\cdot e^{\i \alpha \varphi(x)}$.
+
+\:Odmocòování: $\root n\of x = {\vert x\vert}^{1/n} \cdot e^{\i\cdot
+\varphi(x)/n}$. \\
+Pozor -- odmocnina není jednoznaèná: $1^4=(-1)^4=\i^4=(-\i)^4=1$.
+\endlist
+
+\s{Odmocniny z~jednièky}
+
+\itemize\ibull
+\:Je-li nìjaké $x\in{\bb C}$ $n$-tou odmocninou z~jednièky, musí platit:
+$\vert x \vert = 1$, tak¾e $x=e^{\i\varphi}$ pro nìjaké~$\varphi$.
+Proto $x^n = e^{\i\varphi n} = \cos{\varphi n} + \i\sin\varphi n = 1$.
+Platí tedy $\varphi n = 2k\pi$ pro nìjaké $k\in{\bb Z}$.
+
+\:Z~toho plyne: $\varphi = 2k\pi/n$ \\
+(pro $k=0,\ldots,n-1$ dostáváme rùzné $n$-té odmocniny).
+
+\:Obecné odmocòování: $\root n \of x = {\vert x\vert}^{1/n} \cdot e^{\i\varphi
+ (x)/n} \cdot u$, kde $u=\root n\of 1$.
+
+\:Je-li $x$ odmocninou z 1, pak $\overline{x} = x^{-1}$ -- je toti¾ $1 = \vert
+x\cdot \overline{x}\vert = x\cdot \overline{x}$.
+\endlist
+
+\s{Primitivní odmocniny}
+
+\s{Definice:} $x$ je {\I primitivní} $k$-tá odmocnina z 1 $\equiv x^k=1 \land \forall j: 0<j<k \Rightarrow x^j \neq 1$.
+
+\>Tuto definici splòují napøíklad èísla $\omega = e^{2\pi \i / k}$ a $\overline\omega = e^{-2\pi\i/k}$.
+Platí toti¾, ¾e $\omega^j = e^{2\pi\i j/k}$, co¾ je rovno~1 právì tehdy,
+je-li $j$ násobkem~$k$ (jednotlivé mocniny èísla~$\omega$ postupnì obíhají
+jednotkovou kru¾nici). Analogicky pro~$\overline\omega$.
+
+\>Uka¾me si nìkolik pozorování fungujících pro libovolné èíslo~$\omega$,
+které je primitivní $k$-tou odmocninou z~jednièky (nìkdy budeme potøebovat,
+aby navíc $k$ bylo sudé):
+
+\itemize\ibull
+\:Pro $0\leq j<l<k$ je $\omega^j \neq \omega^l$, nebo» $\omega^l /
+\omega^j = \omega^{l-j} \neq 1$, proto¾e $l-j < k$ a $\omega$ je primitivní.
+\:$\omega^{k/2} = -1$, proto¾e $(\omega^{k/2})^2 = 1$, a tedy
+$\omega^{k/2}$ je druhá odmocnina z~1. Takové odmocniny jsou dvì:
+1 a $-1$, ov¹em 1 to být nemù¾e, proto¾e $\omega$ je primitivní.
+\:$\omega^j = - \omega^{k/2 + j}$ -- pøímý dùsledek pøedchozího bodu, pro
+nás ale velice zajímavý: $\omega^0,\omega^1,\ldots,\omega^{k-1}$ jsou po dvou
+spárované.
+\:$\omega^2$ je $k/2$-tá primitivní odmocnina z 1 -- dosazením.
+\endlist
+
+\h{Konec intermezza}
+% end komplex
+
+\bigskip
+
+Vra»me se nyní k algoritmu. Z poslední èásti komplexního intermezza se zdá,
+¾e by nemusel být ¹patný nápad zkusit vyhodnocovat polynom v mocninách
+$n$-té primitivní odmocniny z~jedné (tedy za $x_0,x_1,\ldots,x_{n-1}$
+z~pùvodního algoritmu zvolíme $\omega^0,\omega^1,\ldots,\omega^{n-1}$).
+Aby nám v¹e vycházelo pìknì, zvolíme $n$ jako mocninu dvojky.
+
+\ss{Celý algoritmus bude vypadat takto:}
+\>FFT($P$, $ \omega$)
+
+\>{\sl Vstup:} $p_{0}, \ldots , p_{n-1}$, koeficienty polynomu $P$ stupnì
+nejvý¹e $n-1$, a~$\omega$,
+$n$-tá primitivní odmocina z jedné.
+
+\>{\sl Výstup:} Hodnoty polynomu v~bodech $1, \omega, \omega^{2}, \ldots ,
+\omega^{n - 1}$, èili èísla $P(1), P(\omega), P(\omega^{2}),$ $\ldots ,
+P(\omega^{n - 1})$.
+
+\algo
+\:Pokud $n = 1$, vrátíme $p_{0}$ a~skonèíme.
+\:Jinak rozdìlíme $P$ na èleny se sudými a lichými exponenty (jako v pùvodní
+my¹lence) a~rekurzivnì zavoláme FFT($P_s$, $\omega^{2}$) a~FFT($P_l$,
+$\omega^{2}$) -- $P_l$ i~$P_s$ jsou stupnì max. $n/2-1$, $\omega^2$ je
+$n/2$-tá primitivní odmocnina, a mocniny $\omega^2$ jsou stále po dvou
+spárované ($n$ je mocnina dvojky, a tedy i $n/2$ je sudé; popø. $n=2$ a je to
+zøejmé).
+\:Pro $j = 0, \ldots , n/2 - 1$ spoèítáme:
+
+\:\qquad $P(\omega^{j}) = P_s(\omega^{2j}) + \omega^{j}\cdot P_l(\omega^{2j})$.
+
+\:\qquad $P(\omega^{j+n/2})=P_s(\omega^{2j})-\omega^{j}\cdot P_l(\omega^{2j})$.
+
+\endalgo
+
+
+\s{Èasová slo¾itost:}
+\>$T(n)=2T(n/2) + \Theta(n) \Rightarrow$ slo¾itost $\Theta(n \log n)$, jako
+MergeSort.
+
+
+Máme tedy algoritmus, který pøevede koeficienty polynomu na~hodnoty tohoto
+polynomu v~rùzných bodech. Potøebujeme ale také algoritmus, který doká¾e
+reprezentaci polynomu pomocí hodnot pøevést zpìt na~koeficienty polynomu.
+K~tomu nám pomù¾e podívat se na~ná¹ algoritmus trochu obecnìji.
+
+
+\s{Definice:}
+\>{\I Diskrétní Fourierova transformace} $(DFT)$
+je zobrazení $f: { {\bb C} ^n} \rightarrow { {\bb C} ^n}$, kde $$y=f(x) \equiv
+\forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} \cdot \omega ^{jk}$$
+(DFT si lze mimo jiné pøedstavit jako funkci vyhodnocující polynom
+s~koeficienty $x_k$ v~bodech $\omega^j$). Takovéto zobrazení je lineární
+a~tedy popsatelné maticí $\Omega$ s~prvky $\Omega_{jk} = \omega^{jk}$.
+Chceme-li umìt pøevádìt z~hodnot polynomu na koeficienty, zajímá nás inverze
+této matice.
+
+
+\ss{Jak najít inverzní matici?}
+Znaème $\overline{\Omega}$ matici, její¾ prvky
+jsou komplexnì sdru¾ené odpovídajícím prvkùm $\Omega$, a vyu¾ijme následující
+lemma:
+
+\ss{Lemma:}$\quad \Omega\cdot \overline{\Omega} = n\cdot E$.
+
+\proof $$ (\Omega\cdot \overline{\Omega})_{jk} = \sum_{l=0}^{n-1} \omega^{jl}
+\cdot \overline{\omega^{lk}} = \sum \omega^{jl} \cdot \overline{\omega}^{lk} =
+\sum \omega^{jl} \cdot \omega^{-lk} = \sum \omega^{l(j-k)}\hbox{.}$$
+\itemize\ibull
+\:Pokud $j=k$, pak $ \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
+
+\:Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické øady
+s kvocientem $\omega ^{(j-k) }$ a~dostaneme ${{\omega^{(j-k)n} -1} \over
+{\omega^{(j-k)} -1}} ={1-1 \over \neq 0} = 0$ (
+$\omega^{j-k} - 1$ je jistì $\neq 0$, nebo» $\omega$ je $n$-tá primitivní
+odmoncina a $j-k<n$).
+\endlist
+\qed
+
+
+\>Na¹li jsme inverzi:
+$\Omega({1 \over n} \overline{\Omega}) = {1 \over n}\Omega \cdot
+\overline{\Omega} = E$, \quad $\Omega^{-1}_{jk} = {1 \over n}\overline{\omega^
+{jk}} = {1 \over n}\omega^{-jk} = {1 \over n} {(\omega^{-1})}^{jk}$, \quad
+(pøipomínáme, $\omega^{-1}$ je $\overline{\omega}$).
+
+Vyhodnocení polynomu lze provést vynásobením $\Omega$, pøevod do pùvodní
+reprezentace vynásobením $\Omega^{-1}$. My jsme si ale v¹imli chytrého
+spárování, a vyhodnocujeme polynom rychleji ne¾ kvadraticky (proto FFT,
+jako¾e {\it fast}, ne jako {\it fuj}). Uvìdomíme-li si, ¾e $\overline \omega =
+\omega^{-1}$ je také $n$-tá primitivní odmocnina z 1 (má akorát
+úhel s opaèným znaménkem), tak mù¾eme stejným trikem vyhodnotit i~zpìtný
+pøevod -- nejprve vyhodnotíme $A$ a $B$ v $\omega^j$, poté pronásobíme
+hodnoty a~dostaneme tak hodnoty polynomu $C=A\cdot B$, a pustíme na nì
+stejný algoritmus s~$\omega^{-1}$ (hodnoty $C$
+vlastnì budou v~algoritmu \uv{koeficienty polynomu}). Nakonec jen získané
+hodnoty vydìlíme $n$ a~máme chtìné koeficienty.
+
+\s{Výsledek:} Pro $n= 2^k$ lze DFT na~${\bb C}^n$ spoèítat v~èase $\Theta(n
+\log n)$ a~DFT$^{-1}$ takté¾.
+
+\s{Dùsledek:}
+Polynomy stupnì $n$ lze násobit v~èase $\Theta(n \log n)$:
+$\Theta(n \log n)$ pro vyhodnocení, $\Theta(n)$ pro vynásobení a~$\Theta(n
+\log n)$ pro pøevedení zpìt.
+
+\s{Pou¾ití FFT:}
+
+\itemize\ibull
+
+\:Zpracování signálu -- rozklad na~siny a~cosiny o~rùzných frekvencích
+$\Rightarrow$ spektrální rozklad.
+\:Komprese dat -- napøíklad formát JPEG.
+\:Násobení dlouhých èísel v~èase $\Theta(n \log n)$.
+\endlist
+
+\s{Paralelní implementace FFT}
+
+\figure{img.eps}{Pøíklad prùbìhu algoritmu na~vstupu velikosti 8}{3in}
+
+Zkusme si prùbìh algoritmu FFT znázornit graficky (podobnì, jako jsme kreslili
+hradlové sítì). Na~levé stranì obrázku se nachází vstupní vektor $x_0,\ldots,x_{n-1}$
+(v~nìjakém poøadí), na~pravé stranì pak výstupní vektor $y_0,\ldots,y_{n-1}$.
+Sledujme chod algoritmu pozpátku: Výstup spoèítáme z~výsledkù \uv{polovièních}
+transformací vektorù $x_0,x_2,\ldots,x_{n-2}$ a $x_1,x_3,\ldots,x_{n-1}$.
+Èerné krou¾ky pøitom odpovídají výpoètu lineární kombinace $a+\omega^kb$,
+kde $a,b$ jsou vstupy krou¾ku a $k$~nìjaké pøirozené èíslo závislé na poloze
+krou¾ku. Ka¾dá z~polovièních transformací se poèítá analogicky z~výsledkù
+transformace velikosti $n/4$ atd. Celkovì výpoèet probíhá v~$\log_2 n$ vrstvách
+po~$\Theta(n)$ operacích.
+
+Jeliko¾ operace na~ka¾dé vrstvì probíhají na sobì nezávisle, mù¾eme je poèítat
+paralelnì. Ná¹ diagram tedy popisuje hradlovou sí» pro paralelní výpoèet FFT
+v~èase $\Theta(\log n\cdot T)$ a prostoru $\O(n\cdot S)$, kde $T$ a~$S$ znaèí
+èasovou a prostorovou slo¾itost výpoètu lineární kombinace dvou komplexních èísel.
+
+\s{Cvièení:} Doka¾te, ¾e permutace vektoru $x_0,\ldots,x_{n-1}$ odpovídá bitovému
+zrcadlení, tedy ¾e na pozici~$b$ shora se vyskytuje prvek $x_d$, kde~$d$ je
+èíslo~$b$ zapsané ve~dvojkové soustavì pozpátku.
+
+\s{Nerekurzivní FFT}
+
+Obvod z~pøedchozího obrázku také mù¾eme vyhodnocovat po~hladinách zleva doprava,
+èím¾ získáme elegantní nerekurzivní algoritmus pro výpoèet FFT v~èase $\Theta(n\log n)$
+a prostoru $\Theta(n)$:
+
+\algo
+\algin $x_0,\ldots,x_{n-1}$
+\:Pro $j=0,\ldots,n-1$ polo¾íme $y_k\= x_{r(k)}$, kde $r$ je funkce bitového zrcadlení.
+\:Pøedpoèítáme tabulku $\omega^0,\omega^1,\ldots,\omega^{n-1}$.
+\:$b\= 1$ \cmt{velikost bloku}
+\:Dokud $b<n$, opakujeme:
+\::Pro $j=0,\ldots,n-1$ s~krokem~$2b$ opakujeme: \cmt{zaèátek bloku}
+\:::Pro $k=0,\ldots,b-1$ opakujeme: \cmt{pozice v~bloku}
+\::::$\alpha\=\omega^{nk/2b}$
+\::::$(y_{j+k},y_{j+k+b}) \= (y_{j+k}+\alpha\cdot y_{j+k+b}, y_{j+k}-\alpha\cdot y_{j+k+b})$.
+\::$b\= 2b$
+\algout $y_0,\ldots,y_{n-1}$
+\endalgo
+
+\s{FFT v~koneèných tìlesech}
+
+Nakonec dodejme, ¾e Fourierovu transformaci lze zavést nejen nad tìlesem
+komplexních èísel, ale i v~nìkterých koneèných tìlesech, pokud zaruèíme
+existenci primitivní $n$-té odmocniny z~jednièky. Napøíklad v~tìlese ${\bb
+Z}_p$ pro prvoèíslo $p=2^k+1$ platí $2^k=-1$, tak¾e $2^{2k}=1$
+a $2^0,2^1,\ldots,2^{2k-1}$ jsou navzájem rùzná, tak¾e èíslo~2 je~primitivní
+$2k$-tá odmocnina z~jedné. To se nám ov¹em nehodí pro algoritmus FFT, jeliko¾
+$2k$ bude málokdy mocnina dvojky.
+
+Zachrání nás ov¹em algebraická vìta, která øíká, ¾e multiplikativní grupa\foot{To je
+mno¾ina v¹ech nenulových prvkù tìlesa s~operací násobení.}
+libovolného koneèného tìlesa je cyklická, tedy ¾e v¹echny nenulové prvky tìlesa lze
+zapsat jako mocniny nìjakého èísla~$g$ (generátoru). Napøíklad pro $p=2^{16}+1=65\,537$
+je jedním takovým generátorem èíslo~$3$. Jeliko¾ mezi èísly $g^1,g^2,\ldots,g^{p-1}$
+se musí ka¾dý nenulový prvek tìlesa vyskytnout právì jednou, je~$g$ primitivní
+$2^k$-tou odmocninou z~jednièky, tak¾e mù¾eme poèítat FFT pro libovolný vektor,
+jeho¾ velikost je mocnina dvojky men¹í nebo rovná $2^k$.\foot{Bli¾¹í prùzkum
+na¹ich úvah o~FFT dokonce odhalí, ¾e není ani potøeba tìleso. Postaèí libovolný
+komutativní okruh, ve~kterém existuje pøíslu¹ná primitivní odmocnina
+z~jednièky, její multiplikativní inverze (ta ov¹em existuje v¾dy, proto¾e
+$\omega^{-1} = \omega^{n-1}$) a multiplikativní inverze èísla~$n$. To nám
+poskytuje je¹tì daleko více volnosti ne¾ tìlesa, ale není snadné takové okruhy
+hledat.}
+
+\bye
--- /dev/null
+P=7-fft
+
+include ../Makerules
--- /dev/null
+%!PS-Adobe-2.0 EPSF-1.2\r
+%%Creator: Xara X\r
+%%For: (Unregistered user) (Unregistered company)\r
+%%Title: (velikost8.xar)\r
+%%CreationDate: (11/02/08) (09:37 PM)\r
+%%BoundingBox: 7 12 447 369\r
+%%HiResBoundingBox: 7.148 12.362 446.455 368.162\r
+%%AWColourTable\r
+%%+h (Red) 0.0 100.0 100.0\r
+%%+h (Orange-Red) 15.0 100.0 100.0\r
+%%+h (Orange) 30.0 100.0 100.0\r
+%%+h (Orange-Yellow) 45.0 100.0 100.0\r
+%%+h (Yellow) 60.0 100.0 100.0\r
+%%+h (Yellow-Chartreuse) 75.0 100.0 100.0\r
+%%+h (Chartreuse) 90.0 100.0 100.0\r
+%%+h (Chartreuse-Green) 105.0 100.0 100.0\r
+%%+h (Green) 120.0 100.0 100.0\r
+%%+h (Green-SpringGreen) 135.0 100.0 100.0\r
+%%+h (Spring Green) 150.0 100.0 100.0\r
+%%+h (SpringGreen-Cyan) 165.0 100.0 100.0\r
+%%+h (Cyan) 180.0 100.0 100.0\r
+%%+h (Sky Blue) 195.0 100.0 100.0\r
+%%+h (Mid Blue) 210.0 100.0 100.0\r
+%%+h (MidBlue-Blue) 225.0 100.0 100.0\r
+%%+h (Blue) 240.0 100.0 100.0\r
+%%+h (Blue-Indigo) 255.0 100.0 100.0\r
+%%+h (Indigo) 270.0 100.0 100.0\r
+%%+h (Violet) 285.0 100.0 100.0\r
+%%+h (Magenta) 300.0 100.0 100.0\r
+%%+h (Magenta-Crimson) 315.0 100.0 100.0\r
+%%+h (Crimson) 330.0 100.0 100.0\r
+%%+h (Crimson-Red) 345.0 100.0 100.0\r
+%%+h (Black) 0.0 0.0 0.0\r
+%%+t (90% Black) 90\r
+%%+t (80% Black) 80\r
+%%+t (70% Black) 70\r
+%%+t (60% Black) 60\r
+%%+t (50% Black) 50\r
+%%+t (40% Black) 40\r
+%%+t (30% Black) 30\r
+%%+t (20% Black) 20\r
+%%+t (10% Black) 10\r
+%%+h (White) 0.0 0.0 100.0\r
+%%EndComments\r
+%%BeginProlog\r
+\r
+%%BeginResource: procset XaraStudio1Dict\r
+% Copyright (c) 1995,1996 Xara Ltd\r
+/XaraStudio1Dict 300 dict def XaraStudio1Dict begin\r
+/bd{bind def}bind def/ld{load def}bind def/xd{exch def}bind def/sv{save}bd\r
+/rs{restore}bd/gs{gsave}bd/gr{grestore}bd/bg{begin}bd/en{end}bd/level2\r
+/languagelevel where{pop languagelevel 2 ge}{false}ifelse def/setseps{\r
+/v_gseps xd}bd/setplate{/v_plate xd}bd/setkgray{/v_keyg xd}bd/setmono{\r
+/v_mono xd}bd/rgb2gray{0.109 mul exch 0.586 mul add exch 0.305 mul\r
+add}bd/cmyk2rgb{3{dup 5 -1 roll add dup 1 gt{pop 1}if 1 exch sub exch}repeat\r
+pop}bd/rgb2cmyk{3{1.0 exch sub 3 1 roll}repeat 3 copy 2 copy gt{exch}if\r
+pop 2 copy gt{exch}if pop dup 0.5 gt{0.5 sub dup 3{5 1 roll dup 3 1\r
+roll sub}repeat 5 1 roll pop}{pop 0}ifelse}bd/cmyk2hsb{3{dup 5 -1 roll\r
+add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop rgb2hsb}bd/rgb2hsb{setrgbcolor\r
+currenthsbcolor}bd/readcurve{exch 255.0 mul 0.5 add cvi get 255.0 div}bd\r
+/rgb2devcmyk{3 copy dup 3 1 roll eq 3 1 roll eq v_keyg 1 eq and and{pop\r
+pop 1 exch sub 0 0 0 4 -1 roll}{/ucurve where{pop 3{1.0 exch sub 3\r
+1 roll}repeat 3 copy 2 copy gt{exch}if pop 2 copy gt{exch}if pop dup\r
+ucurve readcurve exch bcurve readcurve clamp01 3{5 1 roll dup 3 1 roll\r
+sub clamp01}repeat 5 1 roll pop 4 1 roll ycurve readcurve 4 1 roll\r
+mcurve readcurve 4 1 roll ccurve readcurve 4 1 roll}{rgb2cmyk}ifelse}ifelse}def\r
+/rgb2keyG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop}{Max3}ifelse\r
+1 exch sub bcurve readcurve clamp01}bd/rgb2key{Max3 1 exch sub bcurve\r
+readcurve clamp01}bd/rgb2cyanG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop\r
+pop pop 0}{rgb2cyan}ifelse}bd/rgb2cyan{3 copy Max3 1 exch sub ucurve\r
+readcurve 4 1 roll pop pop 1 exch sub exch sub ccurve readcurve clamp01}bd\r
+/rgb2magentaG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2magenta}ifelse}bd\r
+/rgb2magenta{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll pop\r
+1 exch sub 3 1 roll pop sub mcurve readcurve clamp01}bd/rgb2yellowG{3\r
+copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2yellow}ifelse}bd\r
+/rgb2yellow{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll 1 exch\r
+sub 4 1 roll pop pop sub ycurve readcurve clamp01}bd/rgb2plategray{v_keyg\r
+0 eq v_plate v_cpky eq{{rgb2key}{rgb2keyG}ifelse}{v_plate v_cpyl eq{{rgb2yellow}{rgb2yellowG}ifelse}{v_plate\r
+v_cpmg eq{{rgb2magenta}{rgb2magentaG}ifelse}{v_plate v_cpcy eq{{rgb2cyan}{rgb2cyanG}ifelse}{{rgb2key}{rgb2keyG}ifelse}ifelse}ifelse}ifelse}ifelse\r
+1 exch sub setgray}bd/dc{0 def}bd/aca{/v_cpnone 0 def/v_cpcy 1 def\r
+/v_cpyl 2 def/v_cpmg 3 def/v_cpky 4 def/v_gseps 0 def/v_keyg 0 def\r
+/v_plate v_cpnone def/v_mono 0 def/v_wr dc/v_fc dc/v_fm dc/v_fy dc\r
+/v_fk dc/v_fg dc/v_fr dc/v_fg dc/v_fb dc/v_sc dc/v_sm dc/v_sy dc/v_sk\r
+dc/v_sg dc/v_sr dc/v_sg dc/v_sb dc/v_sct 0 def/v_fct 0 def/v_ft 0 def\r
+/v_cxe 0 def/v_cxm 0 def/v_sa -1 def/v_ea -1 def/sR dc/sG dc/sB dc\r
+/mR dc/mG dc/mB dc/eR dc/eG dc/eB dc/sC dc/sM dc/sY dc/sK dc/eC dc\r
+/eM dc/eY dc/eK dc/sH dc/sS dc/sV dc/eH dc/eS dc/eV dc/sGy dc/eGy\r
+dc/mGy dc/ci_datasrc dc/ci_matrix dc/ci_dataleft dc/ci_buf dc/ci_dataofs\r
+dc/ci_y dc/rciBuf dc/cbslw dc/cmiBuf dc/cPalette dc/cpci_datasrc dc\r
+/cpci_matrix dc/cpci_bpp dc/cpci_y dc/cpci_sampsleft dc/cpci_nextcol\r
+dc/cpci_buf dc/startX dc/startY dc/endX dc/endY dc/endX2 dc/endY2 dc\r
+/fillX dc/urx dc/ury dc/llx dc/lly dc/incD dc/distance dc/slice dc\r
+/startangle dc/Steps dc/incH dc/incS dc/incV dc/incR dc/incG dc/incB\r
+dc/incGy dc 0.25 setlinewidth [] 0 setdash 0 setlinejoin 0 setlinecap}bd\r
+aca/setplategray{v_plate v_cpky eq{1 exch sub setgray pop pop pop}{v_plate\r
+v_cpyl eq{pop 1 exch sub setgray pop pop}{v_plate v_cpmg eq{pop pop\r
+1 exch sub setgray pop}{v_plate v_cpcy eq{pop pop pop 1 exch sub setgray}{1\r
+exch sub setgray pop pop pop}ifelse}ifelse}ifelse}ifelse}bd/setplatecolor{v_plate\r
+v_cpky eq{1 exch sub 0 0 0 4 -1 roll setcmykcolor pop pop pop}{v_plate\r
+v_cpyl eq{pop 1 exch sub 0 0 0 4 2 roll setcmykcolor pop pop}{v_plate\r
+v_cpmg eq{pop pop 1 exch sub 0 0 0 4 1 roll setcmykcolor pop}{v_plate\r
+v_cpcy eq{pop pop pop 1 exch sub 0 0 0 setcmykcolor}{1 exch sub 0 0\r
+0 4 -1 roll setcmykcolor pop pop pop}ifelse}ifelse}ifelse}ifelse}bd\r
+/setcmykcolor where{pop}{/setcmykcolor{cmyk2rgb setrgbcolor}bd}ifelse\r
+/setlogcmykcolor{v_gseps 1 eq{v_mono 1 eq{1 exch sub setgray pop pop\r
+pop}{setcmykcolor}ifelse}{v_mono 1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd\r
+/setlogrgbcolor{v_gseps 1 eq{v_mono 1 eq{rgbtoplategray}{rgb2devcmyk\r
+setplatecolor}ifelse}{v_mono 1 eq{rgb2gray setgray}{systemdict begin\r
+setrgbcolor end}ifelse}ifelse}bd/setfillcolor{v_fct 0 eq{v_fc v_fm\r
+v_fy v_fk setlogcmykcolor}{v_fr v_fg v_fb setlogrgbcolor}ifelse}bd\r
+/setstrokecolor{v_sct 0 eq{v_sc v_sm v_sy v_sk setlogcmykcolor}{v_sr\r
+v_sg v_sb setlogrgbcolor}ifelse}bd/setgfillcmyk{v_gseps 1 eq{v_mono\r
+1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono\r
+1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd/setgfillrgb{v_gseps\r
+1 eq{v_mono 1 eq{rgb2plategray}{rgb2devcmyk setplatecolor}ifelse}{v_mono\r
+1 eq{rgb2gray setgray}{systemdict begin setrgbcolor end}ifelse}ifelse}bd\r
+/setgfillhsb{v_gseps 1 eq{v_mono 1 eq{systemdict begin sethsbcolor\r
+currentrgbcolor end rgb2plategray}{systemdict begin sethsbcolor currentrgbcolor\r
+end rgb2devcmyk setplatecolor}ifelse}{v_mono 1 eq{systemdict begin\r
+sethsbcolor currentgray end setgray}{systemdict begin sethsbcolor end}ifelse}ifelse}bd\r
+/Max{2 copy lt{exch}if pop}bd/Max3{2 copy lt{exch}if pop 2 copy lt{exch}if\r
+pop}bd/Min{2 copy gt{exch}if pop}bd/Min3{2 copy gt{exch}if pop 2 copy\r
+gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max\r
+1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace\r
+setcolor}bd/ssg{setgray}bd/p_render{}def/p_count 0 def/vis_flag true\r
+def/DataString 3 string def/DataSrc{currentfile DataString readhexstring\r
+pop}bd/DataStr1 1 string def/DataStr2 1 string def/DataStr3 1 string\r
+def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd\r
+/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch\r
+def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def\r
+/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1\r
+roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft\r
+exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get\r
+255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub\r
+def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1\r
+roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for\r
+m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3\r
+index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring\r
+pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate\r
+scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def\r
+currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring\r
+pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter\r
+/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw\r
+string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
+DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind\r
+true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index\r
+/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1\r
+gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
+DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def\r
+currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd\r
+/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile\r
+cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch\r
+def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch\r
+def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init\r
+0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft\r
+1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup\r
+cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll\r
+1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for\r
+pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp\r
+1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop\r
+/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift\r
+1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft\r
+0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if\r
+cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put\r
+/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring\r
+pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup\r
+1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd\r
+/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx\r
+] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx\r
+/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih\r
+0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag\r
+0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop\r
+/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print\r
+flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd\r
+/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save\r
+restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto\r
+0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc\r
+imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto\r
+0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc\r
+imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0\r
+0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2\r
+not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode\r
+filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode\r
+filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource\r
+false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def\r
+/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{\r
+/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0\r
+eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc\r
+exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring\r
+exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1\r
+ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill\r
+0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity\r
+def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc\r
+0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc\r
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc\r
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2\r
+restore}if\r
+level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd\r
+/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd\r
+/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0\r
+statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo\r
+where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse\r
+end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict\r
+/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict\r
+and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3\r
+ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{\r
+/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut\r
+colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse\r
+/expandfactor xd [/expandbw load/exec load dup currenttransfer exch\r
+] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not\r
+endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def\r
+/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval\r
+def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def\r
+/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil\r
+] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil\r
+] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{\r
+/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut\r
+def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if\r
+not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x\r
+mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts\r
+/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [\r
+setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true\r
+3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{\r
+/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round\r
+cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts\r
+/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add\r
+string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut\r
+cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0\r
+def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup\r
+rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch\r
+get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint\r
+yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd\r
+/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if\r
+/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat\r
+[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc\r
+nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup\r
+not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch\r
+3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index\r
+def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{\r
+/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and\r
+lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd\r
+/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load\r
+forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring\r
+length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8\r
+eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12\r
+createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc\r
+/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if\r
+not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw\r
+ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload\r
+not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr\r
+str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval\r
+def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr\r
+2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put\r
+dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc\r
+/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload\r
+/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1\r
+restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB\r
+def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4\r
+eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]\r
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
+[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace\r
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
+[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def\r
+/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2\r
+not{dontloadlevel2 restore}if\r
+/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen\r
+pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs\r
+llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag\r
+2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag\r
+endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX\r
+sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX\r
+startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop\r
+/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup\r
+currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print\r
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
+/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print\r
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
+/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20\r
+gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush\r
+exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
+/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd\r
+/lineargfill{initgfill{false initgfx/distance endX startX sub endY\r
+startY sub Pythag def/incD distance Steps div def endY startY sub endX\r
+startX sub atan newpath llx lly urx ury Bx startX startY translate\r
+rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath\r
+llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath\r
+fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd\r
+/radialgfill{initgfill{false initgfx/distance endX startX sub endY\r
+startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx\r
+lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath\r
+startX startY distance 0 360 arc closepath distance incD add dup/distance\r
+exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill\r
+pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath\r
+llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2\r
+startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx\r
+/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx\r
+startX sub abs llx startX sub abs Max ury startY sub abs lly startY\r
+sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius\r
+exch def true initgfx [ endX startX sub endY startY sub startY endY\r
+sub endX startX sub startX startY ] concat/slice 180 Steps div def\r
+/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle\r
+neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice\r
+add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse\r
+startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath\r
+llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy\r
+eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps\r
+Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy\r
+exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch\r
+def}bd/linearfill{initfill{false initfx endX startX sub endY startY\r
+sub Pythag/distance exch def/incD distance Steps div def endY startY\r
+sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate\r
+rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
+newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH\r
+sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0\r
+def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX\r
+exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx\r
+endX startX sub endY startY sub Pythag/distance exch def/incD distance\r
+Steps div def v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
+newpath llx lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepfx\r
+newpath startX startY distance 0 360 arc closepath distance incD add\r
+dup/distance exch def 0 rmoveto startX startY distance 0 360 arc closepath\r
+S_eofill pop}for}if}bind def/ellipticalfill{initfill{true initfx v_cxe\r
+0 eq{sR sG sB setgfillrgb}{sH sS sV setgfillhsb}ifelse newpath llx\r
+lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2 startX\r
+sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepfx\r
+/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for v_cxe 0\r
+eq{sR sG sB eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def\r
+/eG exch def/eR exch def}if}if}bd/conicalfill{initfill{urx startX\r
+sub abs llx startX sub abs Max ury startY sub abs lly startY sub abs\r
+Max Pythag startY endY sub startX endX sub Pythag div/radius exch def\r
+true initfx [ endX startX sub endY startY sub startY endY sub endX\r
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+0 def Steps -1 0{pop stepfx 0 0 moveto 0 0 radius startangle neg dup\r
+slice add arc fill 0 0 moveto 0 0 radius startangle dup slice add dup\r
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+startX endX eq startY endY eq and v_cxe 2 ne sR eR eq sG eG eq and\r
+sB eB eq and and or dup{sR sG sB setgfillrgb newpath llx lly urx ury\r
+Bx S_eofill}if not}bd/initfx{/flag exch def v_cxe 0 eq{flag{sR sG sB\r
+eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def/eG exch def\r
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+sB sub abs 256 mul Max FindMinSteps Min 256 Min ceiling 1 Max/Steps\r
+exch def/incR eR sR sub Steps div def/incG eG sG sub Steps div def\r
+/incB eB sB sub Steps div def sR/mR exch def sG/mG exch def sB/mB\r
+exch def}{sR sG sB rgb2hsb/sV exch def/sS exch def/sH exch def eR eG\r
+eB rgb2hsb/eV exch def/eS exch def/eH exch def eH sH sub abs v_cxe\r
+1 eq{dup 0.5 gt{1 exch sub}if}{dup 0.5 lt{1 exch sub}if}ifelse 256\r
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+sS sub Steps div def/incV eV sV sub Steps div def}{/incH eH sH sub\r
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+eq{mR mG mB setgfillrgb mR incR add/mR exch def mG incG add/mG exch\r
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+def sV incV add/sV exch def}ifelse}bd\r
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+pop Xa}ifelse}bd/XX{0 eq{X}{pop pop XA}ifelse}bd/awr{/v_wr exch def}bd\r
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+grestore}bd/*u{/p_count p_count 1 add def}bd/*U{/p_count p_count 1\r
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+setstrokecolor stroke}bd p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd\r
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+/endY exch def/endX exch def/startY exch def/startX exch def/eGy exch\r
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+exch def/startY exch def/startX exch def/eB exch def/eG exch def/eR\r
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+/endY exch def/endX exch def/startY exch def/startX exch def pop pop\r
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+exch def/eR exch def}bd/axm{/endY exch def/endX exch def/startY exch\r
+def/startX exch def}bd/alyr{pop pop 1 eq{true}{false}ifelse/vis_flag\r
+exch def pop pop}bd\r
+/t{moveto show}bd/ts{moveto false charpath S}bd/tf{moveto true charpath\r
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+/u{}bd/U{}bd/anu{pop}bd\r
+end\r
+%%EndResource\r
+%%EndProlog\r
+%%BeginSetup\r
+save XaraStudio1Dict begin\r
+%%EndSetup\r
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-\def\scharfs{\char"19}
-\input lecnotes.tex
-\def\imply{\Rightarrow}
-\prednaska{9}{Fourierova transformace}{\vbox{\hbox{(K.Jakubec, M.Polák
- a~G.Ocsovszky,}\hbox{\ V.Tùma, M.Kozák)}}}
-
-Násobení polynomù mù¾e mnohým pøipadat jako pomìrnì (algoritmicky) snadný
-problém. Asi ka¾dého hned napadne \uv{hloupý} algoritmus -- vezmeme
-koeficienty prvního polynomu a~vynásobíme ka¾dý se v¹emi koeficienty druhého
-polynomu a~pøíslu¹nì u~toho seèteme i~exponenty (stejnì jako to dìláme, kdy¾
-násobíme polynomy na~papíøe). Pokud stupeò prvního polynomu je $n$ a~druhého
-$m$, strávíme tím èas $\Omega(mn)$. Pro $m=n$ je to kvadraticky pomalé.
-Na~první pohled se mù¾e zdát, ¾e rychleji to prostì nejde (pøeci musíme
-v¾dy vynásobit \uv{ka¾dý s~ka¾dým}). Ve skuteènosti to ale rychleji fungovat
-mù¾e, ale k~tomu je potøeba znát trochu tajemný algoritmus FFT neboli {\I Fast
-Fourier Transform}.
-
-
-\ss{Trochu algebry na~zaèátek}
-Celé polynomy oznaèujeme velkými písmeny, jednotlivé èleny polynomù pøíslu¹nými
-malými písmeny (pø.: polynom $W$ stupnì $d$ má koeficienty $w_{0}, w_{1},
-w_{2},\ldots, w_{d}$).
-
-Libovolný polynom $P$ stupnì (nejvý¹e) $d$ lze reprezentovat
-jednak jeho koeficienty, tedy èísly $p_{0}, p_{1}, \ldots ,p_{d}$, druhak
-i~pomocí hodnot:
-
-\>{\bf Lemma:} Polynom stupnì nejvý¹e $d$ je jednoznaènì urèeò svými
-hodnotami v~$d+1$ rùzných bodech.
-
-\>{\it Dùkaz:}
-Polynom stupnì $d$ má maximálnì $d$ koøenù (indukcí -- je-li
-$k$ koøenem $P$, pak lze $P$ napsat jako $(x-k)Q$ kde $Q$ je polynom stupnì
-o~jedna men¹í, pøitom polynom stupnì 1 má jediný koøen); uvá¾íme-li
-dva rùzné polynomy $P$ a~$Q$ stupnì $d$ nabývající v~daných bodech stejných
-hodnot, tak $P-Q$ je polynom stupnì maximálnì $d$, ka¾dé
-z $x_{0}\ldots x_{d}$ je koøenem tohoto polynomu $\imply$ spor, polynom stupnì
-$d$ má $d+1$ koøenù $\imply$ $P-Q$ musí být nulový polynom $\imply$ $P=Q$.
-\qed
-\medskip
-
-Pov¹imnìme si jedné skuteènosti -- máme-li dva polynomy $A$ a~$B$ stupnì $d$
-a~body $x_{0}, \ldots, x_{n}$, dále polynom $C=A \cdot B$ (stupnì $2d$), pak
-platí $C(x_{j}) = A(x_{j}) \cdot B(x_{j})$ pro $j = 0,1,2, \ldots, n$. Toto
-èiní tento druhý zpùsob reprezentace polynomu velice atraktivním pro násobení
--- máme-li $A$ i $B$ reprezentované hodnotami v $n \geq 2d+1$ bodech, pak
-snadno (v $\Theta(n)$) spoèteme takovou reprezentaci $C$.
-Problémem je, ¾e typicky máme polynom zadaný koeficienty, a~ne hodnotami
-v~bodech. Tím pádem potøebujeme nìjaký hodnì rychlý algoritmus (tj.
-rychlej¹í ne¾ kvadratický, jinak bychom si nepomohli oproti hloupému
-algoritmu) na~pøevod polynomu z jedné reprezentace do druhé a~zase zpìt.
-
-\s{Idea, jak by mìl algoritmus pracovat:}
-\algo
-\:Vybereme $n\geq 2d+1$ bodù $x_{0}, x_{1}, \ldots , x_{n-1}$.
-\:V tìchto bodech vyhodnotíme polynomy $A$ a~$B$.
-\:Nyní ji¾ v~lineárním èase získáme hodnoty polynomu $C$ v~tìchto bodech:
- $C(x_i) = A(x_i)\cdot B(x_i)$
-\:Pøevedeme hodnoty polynomu $C$ na~jeho koeficienty.
-\endalgo
-
-\>Je vidìt, ¾e klíèové jsou kroky 2 a~4.
-Celý trik spoèívá v~chytrém vybrání onìch bodù, ve kterých budeme polynomy
-vyhodnocovat -- zvolí-li se obecná $x_j$, tak se to rychle neumí, pro speciální
-$x_j$ ale uká¾eme, ¾e to rychle jde.
-
-\ss{Vyhodnocení polynomu metodou Rozdìl a~panuj (algoritmus FFT):}
-Mìjme polynom $P$ stupnì $\leq d$ a~chtìjme jej vyhodnotit v~$n$ bodech.
-Vybereme si body tak, aby byly spárované, èili $\pm x_{0}, \pm x_{1},
-\ldots , \pm x_{n/2-1} $. To nám výpoèet urychlí, proto¾e pak se druhé
-mocniny $x_{j}$ shodují s~druhými mocninami $-x_{j}$.
-
-Polynom $P$ rozlo¾íme na~dvì èásti, první obsahuje èleny se sudými exponenty,
-druhá s~lichými:
-$$P(x) = (p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{d-2}x^{d-2}) + (p_{1}x^{1} +
- p_{3}x^{3} + \ldots + p_{d-1}x^{d-1})$$
-\>se zavedením znaèení:
-$$P_s(t) = p_0t^0 + p_{2}t^{1} + \ldots + p_{d - 2}t^{d - 2\over 2}$$
-$$P_l(t) = p_1t^0 + p_3t^1 + \ldots + p_{d - 1}t^{d - 2\over 2}$$
-
-\>bude $P(x) = P_s(x^{2}) + xP_l(x^{2})$ a~$P(-x) = P_s(x^{2}) -
-xP_l(x^{2})$. Jinak øeèeno, vyhodnocování polynomu $P$ v~$n$ bodech se nám
-smrskne na~vyhodnocení $P_s$ a~$P_l$ v~$n/2$ bodech -- oba jsou polynomy
-stupnì nejvý¹e $d/2$ a~vyhodnocujeme je v~$x^{2}$ (vyu¾íváme
-rovnosti $(x_{i})^{2} = (-x_{i})^{2}$).
-
-\s{Pøíklad:}
-$3 + 4x + 6x^{2} + 2x^{3} + x^{4} + 10x^{5} = (3 + 6x^{2} + x^{4}) + x(4 +
-2x^{2} + 10x^{4})$.
-
-Teï nám ov¹em vyvstane problém s~oním párováním -- druhá mocnina pøece nemù¾e
-být záporná a~tím pádem u¾ v~druhé úrovni rekurze body spárované nebudou.
-Z~tohoto dùvodu musíme pou¾ít komplexní èísla -- tam druhé mocniny záporné býti
-mohou.
-
-% komplex
-\h{Komplexní intermezzo}
-\def\i{{\rm i}}
-\def\\{\hfil\break}
-
-\s{Základní operace}
-
-\itemize\ibull
-\:Definice: ${\bb C} = \{a + b\i \mid a,b \in {\bb R}\}$
-
-\:Sèítání: $(a+b\i)\pm(p+q\i) = (a\pm p) + (b\pm q)\i$. \\
-Pro $\alpha\in{\bb R}$ je $\alpha(a+b\i) = \alpha a + \alpha b\i$.
-
-\:Komplexní sdru¾ení: $\overline{a+b\i} = a-b\i$. \\
-$\overline{\overline x} = x$, $\overline{x\pm y} = \overline{x} \pm
-\overline{y}$, $\overline{x\cdot y} = \overline x \cdot \overline y$, $x
-\cdot \overline x \in {\bb R}$.
-
-\:Absolutní hodnota: $\vert x \vert = \sqrt{x\cdot\overline{x}}$, tak¾e $\vert
-a+b\i \vert = \sqrt{a^2+b^2}$. \\
-Také $\vert \alpha x \vert = \vert \alpha\vert \cdot \vert x \vert$.
-
-\:Dìlení: $x/y = (x\cdot \overline{y}) / (y \cdot \overline{y})$.
-\endlist
-
-\s{Gau{\scharfs}ova rovina a goniometrický tvar}
-
-\itemize\ibull
-\:Komplexním èíslùm pøiøadíme body v~${\bb R}^2$: $a+b\i \leftrightarrow (a,b)$.
-
-\:$\vert x\vert$ je vzdálenost od~bodu $(0,0)$.
-
-\:$\vert x\vert = 1$ pro èísla le¾ící na~jednotkové kru¾nici ({\I komplexní jednotky}). \\
-Pak platí $x=\cos\varphi + \i\sin\varphi$ pro nìjaké $\varphi\in\left[
-0,2\pi \right)$.
-
-\:Pro libovolné $x\in{\bb C}$: $x=\vert x \vert \cdot (\cos\varphi(x) +
-\i\sin\varphi(x))$. \\
-Èíslu $\varphi(x)\in\left[ 0,2\pi \right)$ øíkáme {\I argument}
-èísla~$x$, nìkdy znaèíme $\mathop{\rm arg} x$.
-
-\:Navíc $\varphi({\overline{x}}) = -\varphi(x)$.
-\endlist
-
-\s{Exponenciální tvar}
-
-\itemize\ibull
-\:Eulerova formule: $e^{i\varphi} = \cos\varphi + \i\sin\varphi$.
-
-\:Ka¾dé $x\in{\bb C}$ lze tedy zapsat jako $\vert x\vert \cdot e^{\i\cdot
-\varphi(x)}$.
-
-\:Násobení: $xy = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right) \cdot
- \left(\vert y\vert\cdot e^{\i\cdot\varphi(y)}\right) = \vert x\vert
- \cdot \vert y\vert \cdot e^{\i\cdot(\varphi(x) + \varphi(y))}$. \\
-(absolutní hodnoty se násobí, argumenty sèítají)
-
-\:Umocòování: $x^\alpha = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right)^
- \alpha = {\vert x\vert}^\alpha\cdot e^{\i \alpha \varphi(x)}$.
-
-\:Odmocòování: $\root n\of x = {\vert x\vert}^{1/n} \cdot e^{\i\cdot
-\varphi(x)/n}$. \\
-Pozor -- odmocnina není jednoznaèná: $1^4=(-1)^4=\i^4=(-\i)^4=1$.
-\endlist
-
-\s{Odmocniny z~jednièky}
-
-\itemize\ibull
-\:Je-li nìjaké $x\in{\bb C}$ $n$-tou odmocninou z~jednièky, musí platit:
-$\vert x \vert = 1$, tak¾e $x=e^{\i\varphi}$ pro nìjaké~$\varphi$.
-Proto $x^n = e^{\i\varphi n} = \cos{\varphi n} + \i\sin\varphi n = 1$.
-Platí tedy $\varphi n = 2k\pi$ pro nìjaké $k\in{\bb Z}$.
-
-\:Z~toho plyne: $\varphi = 2k\pi/n$ \\
-(pro $k=0,\ldots,n-1$ dostáváme rùzné $n$-té odmocniny).
-
-\:Obecné odmocòování: $\root n \of x = {\vert x\vert}^{1/n} \cdot e^{\i\varphi
- (x)/n} \cdot u$, kde $u=\root n\of 1$.
-
-\:Je-li $x$ odmocninou z 1, pak $\overline{x} = x^{-1}$ -- je toti¾ $1 = \vert
-x\cdot \overline{x}\vert = x\cdot \overline{x}$.
-\endlist
-
-\s{Primitivní odmocniny}
-
-\s{Definice:} $x$ je {\I primitivní} $k$-tá odmocnina z 1 $\equiv x^k=1 \land \forall j: 0<j<k \Rightarrow x^j \neq 1$.
-
-\>Tuto definici splòují napøíklad èísla $\omega = e^{2\pi \i / k}$ a $\overline\omega = e^{-2\pi\i/k}$.
-Platí toti¾, ¾e $\omega^j = e^{2\pi\i j/k}$, co¾ je rovno~1 právì tehdy,
-je-li $j$ násobkem~$k$ (jednotlivé mocniny èísla~$\omega$ postupnì obíhají
-jednotkovou kru¾nici). Analogicky pro~$\overline\omega$.
-
-\>Uka¾me si nìkolik pozorování fungujících pro libovolné èíslo~$\omega$,
-které je primitivní $k$-tou odmocninou z~jednièky (nìkdy budeme potøebovat,
-aby navíc $k$ bylo sudé):
-
-\itemize\ibull
-\:Pro $0\leq j<l<k$ je $\omega^j \neq \omega^l$, nebo» $\omega^l /
-\omega^j = \omega^{l-j} \neq 1$, proto¾e $l-j < k$ a $\omega$ je primitivní.
-\:$\omega^{k/2} = -1$, proto¾e $(\omega^{k/2})^2 = 1$, a tedy
-$\omega^{k/2}$ je druhá odmocnina z~1. Takové odmocniny jsou dvì:
-1 a $-1$, ov¹em 1 to být nemù¾e, proto¾e $\omega$ je primitivní.
-\:$\omega^j = - \omega^{k/2 + j}$ -- pøímý dùsledek pøedchozího bodu, pro
-nás ale velice zajímavý: $\omega^0,\omega^1,\ldots,\omega^{k-1}$ jsou po dvou
-spárované.
-\:$\omega^2$ je $k/2$-tá primitivní odmocnina z 1 -- dosazením.
-\endlist
-
-\h{Konec intermezza}
-% end komplex
-
-\bigskip
-
-Vra»me se nyní k algoritmu. Z poslední èásti komplexního intermezza se zdá,
-¾e by nemusel být ¹patný nápad zkusit vyhodnocovat polynom v mocninách
-$n$-té primitivní odmocniny z~jedné (tedy za $x_0,x_1,\ldots,x_{n-1}$
-z~pùvodního algoritmu zvolíme $\omega^0,\omega^1,\ldots,\omega^{n-1}$).
-Aby nám v¹e vycházelo pìknì, zvolíme $n$ jako mocninu dvojky.
-
-\ss{Celý algoritmus bude vypadat takto:}
-\>FFT($P$, $ \omega$)
-
-\>{\sl Vstup:} $p_{0}, \ldots , p_{n-1}$, koeficienty polynomu $P$ stupnì
-nejvý¹e $n-1$, a~$\omega$,
-$n$-tá primitivní odmocina z jedné.
-
-\>{\sl Výstup:} Hodnoty polynomu v~bodech $1, \omega, \omega^{2}, \ldots ,
-\omega^{n - 1}$, èili èísla $P(1), P(\omega), P(\omega^{2}),$ $\ldots ,
-P(\omega^{n - 1})$.
-
-\algo
-\:Pokud $n = 1$, vrátíme $p_{0}$ a~skonèíme.
-\:Jinak rozdìlíme $P$ na èleny se sudými a lichými exponenty (jako v pùvodní
-my¹lence) a~rekurzivnì zavoláme FFT($P_s$, $\omega^{2}$) a~FFT($P_l$,
-$\omega^{2}$) -- $P_l$ i~$P_s$ jsou stupnì max. $n/2-1$, $\omega^2$ je
-$n/2$-tá primitivní odmocnina, a mocniny $\omega^2$ jsou stále po dvou
-spárované ($n$ je mocnina dvojky, a tedy i $n/2$ je sudé; popø. $n=2$ a je to
-zøejmé).
-\:Pro $j = 0, \ldots , n/2 - 1$ spoèítáme:
-
-\:\qquad $P(\omega^{j}) = P_s(\omega^{2j}) + \omega^{j}\cdot P_l(\omega^{2j})$.
-
-\:\qquad $P(\omega^{j+n/2})=P_s(\omega^{2j})-\omega^{j}\cdot P_l(\omega^{2j})$.
-
-\endalgo
-
-
-\s{Èasová slo¾itost:}
-\>$T(n)=2T(n/2) + \Theta(n) \Rightarrow$ slo¾itost $\Theta(n \log n)$, jako
-MergeSort.
-
-
-Máme tedy algoritmus, který pøevede koeficienty polynomu na~hodnoty tohoto
-polynomu v~rùzných bodech. Potøebujeme ale také algoritmus, který doká¾e
-reprezentaci polynomu pomocí hodnot pøevést zpìt na~koeficienty polynomu.
-K~tomu nám pomù¾e podívat se na~ná¹ algoritmus trochu obecnìji.
-
-
-\s{Definice:}
-\>{\I Diskrétní Fourierova transformace} $(DFT)$
-je zobrazení $f: { {\bb C} ^n} \rightarrow { {\bb C} ^n}$, kde $$y=f(x) \equiv
-\forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} \cdot \omega ^{jk}$$
-(DFT si lze mimo jiné pøedstavit jako funkci vyhodnocující polynom
-s~koeficienty $x_k$ v~bodech $\omega^j$). Takovéto zobrazení je lineární
-a~tedy popsatelné maticí $\Omega$ s~prvky $\Omega_{jk} = \omega^{jk}$.
-Chceme-li umìt pøevádìt z~hodnot polynomu na koeficienty, zajímá nás inverze
-této matice.
-
-
-\ss{Jak najít inverzní matici?}
-Znaème $\overline{\Omega}$ matici, její¾ prvky
-jsou komplexnì sdru¾ené odpovídajícím prvkùm $\Omega$, a vyu¾ijme následující
-lemma:
-
-\ss{Lemma:}$\quad \Omega\cdot \overline{\Omega} = n\cdot E$.
-
-\proof $$ (\Omega\cdot \overline{\Omega})_{jk} = \sum_{l=0}^{n-1} \omega^{jl}
-\cdot \overline{\omega^{lk}} = \sum \omega^{jl} \cdot \overline{\omega}^{lk} =
-\sum \omega^{jl} \cdot \omega^{-lk} = \sum \omega^{l(j-k)}\hbox{.}$$
-\itemize\ibull
-\:Pokud $j=k$, pak $ \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
-
-\:Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické øady
-s kvocientem $\omega ^{(j-k) }$ a~dostaneme ${{\omega^{(j-k)n} -1} \over
-{\omega^{(j-k)} -1}} ={1-1 \over \neq 0} = 0$ (
-$\omega^{j-k} - 1$ je jistì $\neq 0$, nebo» $\omega$ je $n$-tá primitivní
-odmoncina a $j-k<n$).
-\endlist
-\qed
-
-
-\>Na¹li jsme inverzi:
-$\Omega({1 \over n} \overline{\Omega}) = {1 \over n}\Omega \cdot
-\overline{\Omega} = E$, \quad $\Omega^{-1}_{jk} = {1 \over n}\overline{\omega^
-{jk}} = {1 \over n}\omega^{-jk} = {1 \over n} {(\omega^{-1})}^{jk}$, \quad
-(pøipomínáme, $\omega^{-1}$ je $\overline{\omega}$).
-
-Vyhodnocení polynomu lze provést vynásobením $\Omega$, pøevod do pùvodní
-reprezentace vynásobením $\Omega^{-1}$. My jsme si ale v¹imli chytrého
-spárování, a vyhodnocujeme polynom rychleji ne¾ kvadraticky (proto FFT,
-jako¾e {\it fast}, ne jako {\it fuj}). Uvìdomíme-li si, ¾e $\overline \omega =
-\omega^{-1}$ je také $n$-tá primitivní odmocnina z 1 (má akorát
-úhel s opaèným znaménkem), tak mù¾eme stejným trikem vyhodnotit i~zpìtný
-pøevod -- nejprve vyhodnotíme $A$ a $B$ v $\omega^j$, poté pronásobíme
-hodnoty a~dostaneme tak hodnoty polynomu $C=A\cdot B$, a pustíme na nì
-stejný algoritmus s~$\omega^{-1}$ (hodnoty $C$
-vlastnì budou v~algoritmu \uv{koeficienty polynomu}). Nakonec jen získané
-hodnoty vydìlíme $n$ a~máme chtìné koeficienty.
-
-\s{Výsledek:} Pro $n= 2^k$ lze DFT na~${\bb C}^n$ spoèítat v~èase $\Theta(n
-\log n)$ a~DFT$^{-1}$ takté¾.
-
-\s{Dùsledek:}
-Polynomy stupnì $n$ lze násobit v~èase $\Theta(n \log n)$:
-$\Theta(n \log n)$ pro vyhodnocení, $\Theta(n)$ pro vynásobení a~$\Theta(n
-\log n)$ pro pøevedení zpìt.
-
-\s{Pou¾ití FFT:}
-
-\itemize\ibull
-
-\:Zpracování signálu -- rozklad na~siny a~cosiny o~rùzných frekvencích
-$\Rightarrow$ spektrální rozklad.
-\:Komprese dat -- napøíklad formát JPEG.
-\:Násobení dlouhých èísel v~èase $\Theta(n \log n)$.
-\endlist
-
-\s{Paralelní implementace FFT}
-
-\figure{img.eps}{Pøíklad prùbìhu algoritmu na~vstupu velikosti 8}{3in}
-
-Zkusme si prùbìh algoritmu FFT znázornit graficky (podobnì, jako jsme kreslili
-hradlové sítì). Na~levé stranì obrázku se nachází vstupní vektor $x_0,\ldots,x_{n-1}$
-(v~nìjakém poøadí), na~pravé stranì pak výstupní vektor $y_0,\ldots,y_{n-1}$.
-Sledujme chod algoritmu pozpátku: Výstup spoèítáme z~výsledkù \uv{polovièních}
-transformací vektorù $x_0,x_2,\ldots,x_{n-2}$ a $x_1,x_3,\ldots,x_{n-1}$.
-Èerné krou¾ky pøitom odpovídají výpoètu lineární kombinace $a+\omega^kb$,
-kde $a,b$ jsou vstupy krou¾ku a $k$~nìjaké pøirozené èíslo závislé na poloze
-krou¾ku. Ka¾dá z~polovièních transformací se poèítá analogicky z~výsledkù
-transformace velikosti $n/4$ atd. Celkovì výpoèet probíhá v~$\log_2 n$ vrstvách
-po~$\Theta(n)$ operacích.
-
-Jeliko¾ operace na~ka¾dé vrstvì probíhají na sobì nezávisle, mù¾eme je poèítat
-paralelnì. Ná¹ diagram tedy popisuje hradlovou sí» pro paralelní výpoèet FFT
-v~èase $\Theta(\log n\cdot T)$ a prostoru $\O(n\cdot S)$, kde $T$ a~$S$ znaèí
-èasovou a prostorovou slo¾itost výpoètu lineární kombinace dvou komplexních èísel.
-
-\s{Cvièení:} Doka¾te, ¾e permutace vektoru $x_0,\ldots,x_{n-1}$ odpovídá bitovému
-zrcadlení, tedy ¾e na pozici~$b$ shora se vyskytuje prvek $x_d$, kde~$d$ je
-èíslo~$b$ zapsané ve~dvojkové soustavì pozpátku.
-
-\s{Nerekurzivní FFT}
-
-Obvod z~pøedchozího obrázku také mù¾eme vyhodnocovat po~hladinách zleva doprava,
-èím¾ získáme elegantní nerekurzivní algoritmus pro výpoèet FFT v~èase $\Theta(n\log n)$
-a prostoru $\Theta(n)$:
-
-\algo
-\algin $x_0,\ldots,x_{n-1}$
-\:Pro $j=0,\ldots,n-1$ polo¾íme $y_k\= x_{r(k)}$, kde $r$ je funkce bitového zrcadlení.
-\:Pøedpoèítáme tabulku $\omega^0,\omega^1,\ldots,\omega^{n-1}$.
-\:$b\= 1$ \cmt{velikost bloku}
-\:Dokud $b<n$, opakujeme:
-\::Pro $j=0,\ldots,n-1$ s~krokem~$2b$ opakujeme: \cmt{zaèátek bloku}
-\:::Pro $k=0,\ldots,b-1$ opakujeme: \cmt{pozice v~bloku}
-\::::$\alpha\=\omega^{nk/2b}$
-\::::$(y_{j+k},y_{j+k+b}) \= (y_{j+k}+\alpha\cdot y_{j+k+b}, y_{j+k}-\alpha\cdot y_{j+k+b})$.
-\::$b\= 2b$
-\algout $y_0,\ldots,y_{n-1}$
-\endalgo
-
-\s{FFT v~koneèných tìlesech}
-
-Nakonec dodejme, ¾e Fourierovu transformaci lze zavést nejen nad tìlesem
-komplexních èísel, ale i v~nìkterých koneèných tìlesech, pokud zaruèíme
-existenci primitivní $n$-té odmocniny z~jednièky. Napøíklad v~tìlese ${\bb
-Z}_p$ pro prvoèíslo $p=2^k+1$ platí $2^k=-1$, tak¾e $2^{2k}=1$
-a $2^0,2^1,\ldots,2^{2k-1}$ jsou navzájem rùzná, tak¾e èíslo~2 je~primitivní
-$2k$-tá odmocnina z~jedné. To se nám ov¹em nehodí pro algoritmus FFT, jeliko¾
-$2k$ bude málokdy mocnina dvojky.
-
-Zachrání nás ov¹em algebraická vìta, která øíká, ¾e multiplikativní grupa\foot{To je
-mno¾ina v¹ech nenulových prvkù tìlesa s~operací násobení.}
-libovolného koneèného tìlesa je cyklická, tedy ¾e v¹echny nenulové prvky tìlesa lze
-zapsat jako mocniny nìjakého èísla~$g$ (generátoru). Napøíklad pro $p=2^{16}+1=65\,537$
-je jedním takovým generátorem èíslo~$3$. Jeliko¾ mezi èísly $g^1,g^2,\ldots,g^{p-1}$
-se musí ka¾dý nenulový prvek tìlesa vyskytnout právì jednou, je~$g$ primitivní
-$2^k$-tou odmocninou z~jednièky, tak¾e mù¾eme poèítat FFT pro libovolný vektor,
-jeho¾ velikost je mocnina dvojky men¹í nebo rovná $2^k$.\foot{Bli¾¹í prùzkum
-na¹ich úvah o~FFT dokonce odhalí, ¾e není ani potøeba tìleso. Postaèí libovolný
-komutativní okruh, ve~kterém existuje pøíslu¹ná primitivní odmocnina
-z~jednièky, její multiplikativní inverze (ta ov¹em existuje v¾dy, proto¾e
-$\omega^{-1} = \omega^{n-1}$) a multiplikativní inverze èísla~$n$. To nám
-poskytuje je¹tì daleko více volnosti ne¾ tìlesa, ale není snadné takové okruhy
-hledat.}
-
-\bye
+++ /dev/null
-P=9-fft
-
-include ../Makerules
+++ /dev/null
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-%%Creator: Xara X\r
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-%%Title: (velikost8.xar)\r
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-%%BeginProlog\r
-\r
-%%BeginResource: procset XaraStudio1Dict\r
-% Copyright (c) 1995,1996 Xara Ltd\r
-/XaraStudio1Dict 300 dict def XaraStudio1Dict begin\r
-/bd{bind def}bind def/ld{load def}bind def/xd{exch def}bind def/sv{save}bd\r
-/rs{restore}bd/gs{gsave}bd/gr{grestore}bd/bg{begin}bd/en{end}bd/level2\r
-/languagelevel where{pop languagelevel 2 ge}{false}ifelse def/setseps{\r
-/v_gseps xd}bd/setplate{/v_plate xd}bd/setkgray{/v_keyg xd}bd/setmono{\r
-/v_mono xd}bd/rgb2gray{0.109 mul exch 0.586 mul add exch 0.305 mul\r
-add}bd/cmyk2rgb{3{dup 5 -1 roll add dup 1 gt{pop 1}if 1 exch sub exch}repeat\r
-pop}bd/rgb2cmyk{3{1.0 exch sub 3 1 roll}repeat 3 copy 2 copy gt{exch}if\r
-pop 2 copy gt{exch}if pop dup 0.5 gt{0.5 sub dup 3{5 1 roll dup 3 1\r
-roll sub}repeat 5 1 roll pop}{pop 0}ifelse}bd/cmyk2hsb{3{dup 5 -1 roll\r
-add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop rgb2hsb}bd/rgb2hsb{setrgbcolor\r
-currenthsbcolor}bd/readcurve{exch 255.0 mul 0.5 add cvi get 255.0 div}bd\r
-/rgb2devcmyk{3 copy dup 3 1 roll eq 3 1 roll eq v_keyg 1 eq and and{pop\r
-pop 1 exch sub 0 0 0 4 -1 roll}{/ucurve where{pop 3{1.0 exch sub 3\r
-1 roll}repeat 3 copy 2 copy gt{exch}if pop 2 copy gt{exch}if pop dup\r
-ucurve readcurve exch bcurve readcurve clamp01 3{5 1 roll dup 3 1 roll\r
-sub clamp01}repeat 5 1 roll pop 4 1 roll ycurve readcurve 4 1 roll\r
-mcurve readcurve 4 1 roll ccurve readcurve 4 1 roll}{rgb2cmyk}ifelse}ifelse}def\r
-/rgb2keyG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop}{Max3}ifelse\r
-1 exch sub bcurve readcurve clamp01}bd/rgb2key{Max3 1 exch sub bcurve\r
-readcurve clamp01}bd/rgb2cyanG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop\r
-pop pop 0}{rgb2cyan}ifelse}bd/rgb2cyan{3 copy Max3 1 exch sub ucurve\r
-readcurve 4 1 roll pop pop 1 exch sub exch sub ccurve readcurve clamp01}bd\r
-/rgb2magentaG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2magenta}ifelse}bd\r
-/rgb2magenta{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll pop\r
-1 exch sub 3 1 roll pop sub mcurve readcurve clamp01}bd/rgb2yellowG{3\r
-copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2yellow}ifelse}bd\r
-/rgb2yellow{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll 1 exch\r
-sub 4 1 roll pop pop sub ycurve readcurve clamp01}bd/rgb2plategray{v_keyg\r
-0 eq v_plate v_cpky eq{{rgb2key}{rgb2keyG}ifelse}{v_plate v_cpyl eq{{rgb2yellow}{rgb2yellowG}ifelse}{v_plate\r
-v_cpmg eq{{rgb2magenta}{rgb2magentaG}ifelse}{v_plate v_cpcy eq{{rgb2cyan}{rgb2cyanG}ifelse}{{rgb2key}{rgb2keyG}ifelse}ifelse}ifelse}ifelse}ifelse\r
-1 exch sub setgray}bd/dc{0 def}bd/aca{/v_cpnone 0 def/v_cpcy 1 def\r
-/v_cpyl 2 def/v_cpmg 3 def/v_cpky 4 def/v_gseps 0 def/v_keyg 0 def\r
-/v_plate v_cpnone def/v_mono 0 def/v_wr dc/v_fc dc/v_fm dc/v_fy dc\r
-/v_fk dc/v_fg dc/v_fr dc/v_fg dc/v_fb dc/v_sc dc/v_sm dc/v_sy dc/v_sk\r
-dc/v_sg dc/v_sr dc/v_sg dc/v_sb dc/v_sct 0 def/v_fct 0 def/v_ft 0 def\r
-/v_cxe 0 def/v_cxm 0 def/v_sa -1 def/v_ea -1 def/sR dc/sG dc/sB dc\r
-/mR dc/mG dc/mB dc/eR dc/eG dc/eB dc/sC dc/sM dc/sY dc/sK dc/eC dc\r
-/eM dc/eY dc/eK dc/sH dc/sS dc/sV dc/eH dc/eS dc/eV dc/sGy dc/eGy\r
-dc/mGy dc/ci_datasrc dc/ci_matrix dc/ci_dataleft dc/ci_buf dc/ci_dataofs\r
-dc/ci_y dc/rciBuf dc/cbslw dc/cmiBuf dc/cPalette dc/cpci_datasrc dc\r
-/cpci_matrix dc/cpci_bpp dc/cpci_y dc/cpci_sampsleft dc/cpci_nextcol\r
-dc/cpci_buf dc/startX dc/startY dc/endX dc/endY dc/endX2 dc/endY2 dc\r
-/fillX dc/urx dc/ury dc/llx dc/lly dc/incD dc/distance dc/slice dc\r
-/startangle dc/Steps dc/incH dc/incS dc/incV dc/incR dc/incG dc/incB\r
-dc/incGy dc 0.25 setlinewidth [] 0 setdash 0 setlinejoin 0 setlinecap}bd\r
-aca/setplategray{v_plate v_cpky eq{1 exch sub setgray pop pop pop}{v_plate\r
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-1 exch sub setgray pop}{v_plate v_cpcy eq{pop pop pop 1 exch sub setgray}{1\r
-exch sub setgray pop pop pop}ifelse}ifelse}ifelse}ifelse}bd/setplatecolor{v_plate\r
-v_cpky eq{1 exch sub 0 0 0 4 -1 roll setcmykcolor pop pop pop}{v_plate\r
-v_cpyl eq{pop 1 exch sub 0 0 0 4 2 roll setcmykcolor pop pop}{v_plate\r
-v_cpmg eq{pop pop 1 exch sub 0 0 0 4 1 roll setcmykcolor pop}{v_plate\r
-v_cpcy eq{pop pop pop 1 exch sub 0 0 0 setcmykcolor}{1 exch sub 0 0\r
-0 4 -1 roll setcmykcolor pop pop pop}ifelse}ifelse}ifelse}ifelse}bd\r
-/setcmykcolor where{pop}{/setcmykcolor{cmyk2rgb setrgbcolor}bd}ifelse\r
-/setlogcmykcolor{v_gseps 1 eq{v_mono 1 eq{1 exch sub setgray pop pop\r
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-/setlogrgbcolor{v_gseps 1 eq{v_mono 1 eq{rgbtoplategray}{rgb2devcmyk\r
-setplatecolor}ifelse}{v_mono 1 eq{rgb2gray setgray}{systemdict begin\r
-setrgbcolor end}ifelse}ifelse}bd/setfillcolor{v_fct 0 eq{v_fc v_fm\r
-v_fy v_fk setlogcmykcolor}{v_fr v_fg v_fb setlogrgbcolor}ifelse}bd\r
-/setstrokecolor{v_sct 0 eq{v_sc v_sm v_sy v_sk setlogcmykcolor}{v_sr\r
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-1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono\r
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-gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max\r
-1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace\r
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-def/DataString 3 string def/DataSrc{currentfile DataString readhexstring\r
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-/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch\r
-def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def\r
-/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1\r
-roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft\r
-exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get\r
-255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub\r
-def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1\r
-roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for\r
-m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3\r
-index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring\r
-pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate\r
-scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def\r
-currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring\r
-pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter\r
-/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw\r
-string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
-DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind\r
-true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index\r
-/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1\r
-gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
-DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def\r
-currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd\r
-/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile\r
-cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch\r
-def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch\r
-def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init\r
-0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft\r
-1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup\r
-cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll\r
-1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for\r
-pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp\r
-1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop\r
-/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift\r
-1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft\r
-0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if\r
-cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put\r
-/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring\r
-pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup\r
-1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd\r
-/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx\r
-] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx\r
-/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih\r
-0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag\r
-0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop\r
-/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print\r
-flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd\r
-/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save\r
-restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto\r
-0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc\r
-imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto\r
-0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc\r
-imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0\r
-0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2\r
-not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode\r
-filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode\r
-filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource\r
-false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def\r
-/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{\r
-/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0\r
-eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc\r
-exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring\r
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-[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc\r
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projects / ads2.git / commitdiff