{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plo t" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 32 "Relaxa\350n\355 Gauss-S eidelova metoda" }{TEXT 256 54 " pro dv\354 line\341rn\355 algebraick \351 rovnice o dvou nezn\341m\375ch" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 37 "\332LOHA. Pro danou \350tvercovou matic i " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 23 " \370\341du 2 a pro \+ vektor " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT 258 1 " " }{TEXT -1 17 " najd\354te vektor " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 27 " s pl\362uj\355c\355 syst\351m rovnic " }{XPPEDIT 18 0 "Ax = b;" "6#/%#A xG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "(P\370\355kazov\375) blok" }{TEXT -1 210 " je skupi na \350erven\354 zbarven\375ch \370\341dk\371, spojen\375ch zleva svis lou svorkou. Ka\236d\375 p\370\355kazov\375 blok je po otev\370en\355 \+ worksheeetu nutno aktivovat (spustit) um\355st\354n\355m kurzoru kamko liv do bloku a stisknut\355m tla\350\355tka ENTER. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "N\341sleduj\355c\355 v elk\375 blok tvo\370\355 popis procedury \"G_S_relax\" pro numerick \351 \370e\232en\355 \332LOHY relaxa\350n\355 Gauss-Seidelovou itera \350n\355 metodou a pro grafick\351 zn\341zorn\354n\355 vypo\350ten \375ch aproximac\355 \370e\232en\355. (" }{TEXT 259 64 "Blok je nutno \+ aktivovat, i kdy\236 aktivace \236\341dn\375 v\375po\350et nezah\341j \355!" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots):with(linalg):with(plottools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "G_S_relax:=proc(A::matrix,b::matrix,x0::al gebraic,y0::algebraic,epsilon::algebraic,omega::algebraic)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "local l,l1,l2,l3,l4,maxxx,i,ii,chyba,miny, maxy,minx,maxx,maxyy,AY,y1,AX,max_iter,x1,x2,xx,vx,vy,r,rov1,rov2,x,y, p,P1,P2,pr1,pr2,q,t;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "global indv ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xx:=vector([x,y]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vx:=array(0..44):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vy:=array(0..44):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "q:=true;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(xx):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rov1:=A[1,1]*x+A[1,2]*y:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rov2:=A[2,1]*x+A[2,2]*y:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x1:=solve(rov1=b[1,1],x):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AX:=unapply(x1,y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y1:=solve(rov2=b[2,1],y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AY :=unapply(y1,x):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "max_iter:=11: \+ # nejv\354t\232\355 p\370\355pustn\341 hodnota max_iter <= 4 0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vx[2]:=evalf(x0):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "vy[2]:=evalf(y0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "indv:=2:q:=true:r:=10:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "vx[1]:=vx[2]-1:vx[0]:=infinity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "vy[1]:=vy[2]-1:vy[0]:=infinity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "maxx:=vx[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "minx:=vx[2] ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "maxy:=vy[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "miny:=vy[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "chyba:=evalf((vx[2]-vx[1])^2+(vy[2]-vy[1])^2);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "for i from 2 to max_iter while sqrt(chyba)>epsilon \+ do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " indv:=indv+1;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 3 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " \+ vx[indv]:=vx[indv-1]+omega*(evalf(AX(vy[indv-1]))-vx[indv-1]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " vy[indv]:=vy[indv-1]+omega*(eval f(AY(vx[indv]))-vy[indv-1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " \+ chyba:=evalf((vx[indv]-vx[indv-1])^2+(vy[indv]-vy[indv-1])^2); " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 278 " if chyba>(vx[indv-1]-vx[indv-2])^2+(vy[indv-1]-vy[indv-2])^2 a nd (vx[indv- 1]-vx[indv-2])^2+(vy[indv-1]-vy[indv-2])^2 > \+ (vx[indv-2]-vx[indv-3])^2+(vy[indv-2]-vy[indv-3]) ^2 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " then " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 23 " q:=false; break;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " m axx:=max(maxx,vx[indv]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " minx :=min(minx,vx[indv]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " maxy:=m ax(maxy,vy[indv]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " miny:=min( miny,vy[indv]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "maxxx:=max(maxx-vx[indv],vx[indv]-minx);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "maxyy:=max(maxy-vy[indv],vy[indv]-m iny);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "maxxx:=max(maxxx,maxyy);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "maxx:=vx[indv]+maxxx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "minx:=vx[indv]-maxxx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "maxy:=vy[indv]+maxxx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "miny:=vy[indv]-maxxx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pr 1:=solve(rov1-b[1,1],y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pr2:=so lve(rov2-b[2,1],y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "P1:=plot(\{ pr1\},x=minx-(maxx-minx)/r..maxx+(maxx-minx)/r,y=miny-(maxy-miny)/r..m axy+(maxy-miny)/r,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 " P2:=plot(\{pr2\},x=minx-(maxx-minx)/r..maxx+(maxx-minx)/r,y=miny-(maxy -miny)/r..maxy+(maxy-miny)/r,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for ii f rom 3 to indv do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 " " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 79 " l1[ii]:=line([vx[ii-1],vy[ii-1]],[vx[ii],v y[ii-1]],color=brown, linestyle=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " l2[ii]:=line([vx[ii],vy[ii-1]],[vx[ii],vy[ii]],color=brown, line style=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " l[ii+1]:=point([vx[ ii],vy[ii]],color=red): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 139 " l[i i]:=point([vx[ii-1],vy[ii-1]],color=red): t[ii]:=TEXT([vx[ii-1]+ (maxx-minx)/30,vy[ii-1]+(maxy-miny)/30],sprintf(\"x(%d)\",ii-3)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "t[ii+1]:=TEXT([vx[ii]+(maxx-minx)/ 30,vy[ii]+(maxy-miny)/30],sprintf(\"x(%d)\",ii-2)): p[ii]: =display(\{P1,P2,l1[ii],l2[ii],seq(l[jj],jj=3..ii+1),seq(t[jj],jj=3..i i+1)\},insequence=false):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "if q then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "if sqrt(chyba)>epsilon then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " if chyba>=(vx[indv-2]-vx[indv-1])^2+(vy[indv-2]-vy [indv-1])^2 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " print(`Me toda pravd\354podobn\354 diverguje`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 " print(`Metod a konverguje pomalu. Zkuste zv\354t\232it hodnotu konstanty \"max_iter \", udavajici maximaln\355 po\350et iterac\355 v t\354le procedury`); \+ end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " else" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 " print(`Po\236adovan\351 p\370esnosti bylo d osa\236eno`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 131 " print(`V\375po\350et zastaven, itera\350n\355 posloupnost diverguje. \+ Zvolte jinak funkce F(x),G(x) nebo po\350\341te\350n\355 aprox imace x0, y0`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "printf(\"Nalezen\351 \370e\232en\355: x=% f y=%f\\n\",vx[indv],vy[indv]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display([seq(p[jj],jj=3..indv)] ,insequence=true);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected name s norm and trace have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefined\n" }}} {EXCHG {PARA 259 "" 0 "" {TEXT 262 21 "Prvn\355 \370e\232en\375 p\370 \355klad:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " V\375po\350et skon\350\355 ur\350en\355m aproximace " } {XPPEDIT 18 0 "x(k) = (x[k], y[k]);" "6#/-%\"xG6#%\"kG6$&F%6#F'&%\"yG6 #F'" }{TEXT -1 8 ", kde " }{TEXT 261 2 "k " }{TEXT -1 32 " je nejmen \232\355 index s vlastnost\355 " }}{PARA 257 "" 0 "" {TEXT -1 2 "||" } {XPPEDIT 18 0 "x(k)-x(k-1);" "6#,&-%\"xG6#%\"kG\"\"\"-F%6#,&F'F(F(!\" \"F," }{TEXT -1 3 "||<" }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }} {PARA 259 "" 0 "" {TEXT -1 129 " Po spu\232t\354n\355 procedury G -Srelax se na obr\341zku objev\355 modr\351 p\370\355mky v\232ech bod \371 spl\362uj\355c\355ch prvn\355 nebo druhou rovnici syst\351mu " } {XPPEDIT 18 0 "Ax = b;" "6#/%#AxG%\"bG" }{TEXT -1 16 " a aproximace \+ " }{XPPEDIT 18 0 "x(0);" "6#-%\"xG6#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "x(1);" "6#-%\"xG6#\"\"\"" }{TEXT -1 7 ". Bod " } {XPPEDIT 18 0 "x(1);" "6#-%\"xG6#\"\"\"" }{TEXT -1 18 " vznikne z bod u " }{XPPEDIT 18 0 "x(0);" "6#-%\"xG6#\"\"!" }{TEXT -1 44 " posunem \+ po dvou \372se\350k\341ch. Nejprve zm\354nou " }{XPPEDIT 18 0 "x;" "6 #%\"xG" }{TEXT -1 33 "-ov\351 sou\370adnice, tj. posunem od " } {XPPEDIT 18 0 "x(0);" "6#-%\"xG6#\"\"!" }{TEXT -1 37 " pod\351l \350 \341rkovan\351 rovnob\354\236ky s osou " }{XPPEDIT 18 0 "x;" "6#%\"xG " }{TEXT -1 9 " a\236 do " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" } {TEXT -1 22 "-n\341sobku vzd\341lenosti " }{XPPEDIT 18 0 "x(0);" "6#- %\"xG6#\"\"!" }{TEXT -1 52 " od bodu spl\362uj\355c\355ho prvn\355 ro vnici a potom zm\354nou " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 71 "-ov\351 sou\370adnice, tj. dal\232\355m posunem pod\351l \350\341rkov an\351 rovnob\354\236ky s osou " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 9 " a\236 do " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 56 "-n\341sobku vzd\341lenosti od bodu spl\362uj\355c\355ho druhou \+ rovnici." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A:=matrix(2,2,[[4,-1],[-2,-1]]):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "b:=matrix(2,1,[[1/3],[1]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x0:=2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "y0:=2:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "epsilon:=0.1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "omega:=0.8:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G_S _relax(A,b,x0,y0,epsilon,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C Po|itadovan|dy~p|czesnosti~bylo~dosa|itenoG" }}{PARA 6 "" 1 "" {TEXT -1 38 "Nalezen\351 \370e\232en\355: x=-.114031 y=-.768104" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%(ANIMATEG6&7,-%'CURVESG 6$7S7$$!3********fevNM!#<$!3RLLLxcj29!#;7$$!3'G%QLfv%4H$F.$!3aqo1dBr\\ 8F17$$!3_&**47j]\\;$F.$!3gJt\"ee8$*H\"F17$$!3\"RQFz!RDBIF.$!3'pG/l*[jU 7F17$$!3S`TyEzh!)GF.$!3w%*p//0e&=\"F17$$!3/95cp)f'QFF.$!3,Rx:\"G(zG6F1 7$$!3Rf*='yl/2EF.$!3;<4yk>:w5F17$$!3+$z(=R'o2Z#F.$!3f]%3!*yS;-\"F17$$! 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