{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 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 Input" 2 19 "" 0 1 255 0 0 1 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 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 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 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -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 Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -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 "Bulle t Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 41 "Koment\341\370e k p\355se mk\341m z numerick\375ch metod" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "readlib(isolate):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "V n\341sleduj\355c\355m textu je n\354kolik aproxinma\350 n\355ch mtod t\370i cbi\350en\355:" }}{PARA 15 "" 0 "" {TEXT -1 40 "P \370evzorkov\341n\355 pomoc\355 interpolace hodnot " }}{PARA 15 "" 0 " " {TEXT -1 34 "Regresn\355 p\370\355mk v r\371zn\375ch metrik\341ch" } }{PARA 15 "" 0 "" {TEXT -1 27 "Porovn\341n\355 aproximace funkce" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Interpolace polynomy" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "polynom 1. stupn\354: p\370\355mka" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unassign('x','y','p','q');" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "p\370\355klad:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Uva\236ujme body" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=[0.5,1];\nB:=[1,7.5];\n" }{TEXT -1 49 "chceme naj \355t rovnici p\370\355mky, kter\341 jimi proch\341z\355." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7$$\"\"&!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7$\"\"\"$\"#v!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "obecn\375 tvar rovnice p\370\355mky, kter\341 nen\355 svi sl\341 je:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "primka:=y=p*x +q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'primkaG/%\"yG,&*&%\"pG\"\"\" %\"xGF*F*%\"qGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Dosad\355me d o n\355 oba body a vy\370e\232\355me rovnice pro nezn\341m\351 paramet ry:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "rceA:=subs(x=A[1],y =A[2],primka); #prvni rovnice vynikne dosazenim prvniho bodu\nrceB:=su bs(x=B[1],y=B[2],primka); #druha dosazenim druheho\nparam:=solve(\{rce A,rceB\},\{p,q\}); #vyresime obe rovnice vyhledem k promenym p a q\nas sign(param); #timto prikazem promenym p a q priradime hodnoty\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rceAG/\"\"\",&*&$\"\"&!\"\"F&%\"pGF &F&%\"qGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rceBG/$\"#v!\"\",&%\" pG\"\"\"%\"qGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%¶mG<$/%\"pG$ \"#8\"\"!/%\"qG$!+++++b!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Nz n\355 m\341 p\370\355mka tvar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&$\"#8\"\"! \"\"\"%\"xGF*F*$\"+++++b!\"*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Tot\351\236 m\371\236eme ud\354lat obecn\354. Zru\232\355me p\370 i\370azen\355 n\354kter\375ch hodnot prom\354n\375m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "readlib(unassign):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "unassign('A','B','p','q');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"yG,&*&%\"pG\"\"\"%\"xGF(F(%\"qGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "rceA:=subs(x=A[1],y=A[2],primka); #prvni rovnice vyn ikne dosazenim prvniho bodu\nrceB:=subs(x=B[1],y=B[2],primka); #druha \+ dosazenim druheho\nparam:=solve(\{rceA,rceB\},\{p,q\}); #vyresime obe \+ rovnice vyhledem k promenym p a q\nassign(param); #timto prikazem prom enym p a q priradime hodnoty" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rce AG/&%\"AG6#\"\"#,&*&%\"pG\"\"\"&F'6#F-F-F-%\"qGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rceBG/&%\"BG6#\"\"#,&*&%\"pG\"\"\"&F'6#F-F-F-%\"qGF- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%¶mG<$/%\"qG,$*&,&*&&%\"AG6# \"\"#\"\"\"&%\"BG6#F0F0!\"\"*&&F-F3F0&F2F.F0F0F0,&F1F0F6F4F4F4/%\"pG*& ,&F7F0F,F4F0F8F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*(,&&%\"BG6#\"\"#\"\"\"&% \"AGF*!\"\"F,,&&F)6#F,F,&F.F2F/F/%\"xGF,F,*&,&*&F-F,F1F,F/*&F3F,F(F,F, F,F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "To \236e jsme museli cel\375 v\375po\350et opakovat n\341s vede k opr\341vn\354n\351 domn \354nce, \236e by bylo v\375hodn\351 jej zachovat pro dal\232\355 pou \236it\355v podob\354 procedury." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "line:=proc(A,B) # jm0nu line je nyni prirazena proce dura se dvema parametry. procedura\nlocal x; # deklarace lokalnich pro menych. lokalni promene si po provedeni procedury uchopvaji tytez hiod noty jake mely pred jejim spustenim \n if A[1]=B[1] then # vyloucime \+ pripad svisle primky\n print(`primka je svisla, nejde o funkci`)\n else\nunapply( (B[2]-A[2])/(B[1]-A[1])*x-(-A[2]*B[1]+A[1]*B[2])/(B[1 ]-A[1]),x ); # prikaz unapply zajisti, ze vysledek bude funkce (nezavi sla na oznaceni promenych)\nfi;\nend; # konec procedury" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%%lineGf*6$%\"AG%\"BG6#%\"xG6\"F+@%/&9$6#\"\"\" &9%F0-%&printG6#%Aprimka~je~svisla,~nejde~o~funkciG-%(unapplyG6$,&*(,& &F36#\"\"#F1&F/F?!\"\"F1,&F2F1F.FBFB8$F1F1*&,&*&FAF1F2F1FB*&F.F1F>F1F1 F1FCFBFBFDF+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 259 " Pomoc\355 t \351to procedurz m\371\236eme vytvo\370it proceduru, jej\355m\236 v \375stupem bude lomen\341 \350\341ra, kter\341 proch\341z\355 zadan \375mi body. Jejich prvn\355 sou\370adnice jsou v prvn\355 prom\354nn \351 argumentu, druh\351 sou\370adnice jsou ve druh\351 prom\354nn\351 . P\370\355kazem nops se sjist\355, kolik jich vlastn\354 je." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "Plin:=proc(X,Y)\nlocal N,i, x;\nN:=nops(X);\nparam:=x%%PlinGf*6$%\"XG%\"YG6& %\"NG%\"iG%\"xG%¶mG6\"F.C'>8$-%%nopsG6#9$>8'6$28&&F56#\"\"\"&9%FF76%F72F:&F56#,&FAF=F=F=--%%lineG6$7$&F56# FA&F?FQ7$FH&F?FI6#F:>F76$F7&F?6#F1-%(unapplyG6$-%*piecewiseG6#F7F:F.F. F." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Polynomy vz\232\232\355ch stup\361\371" }}{EXCHG {PARA 5 "" 0 "" {TEXT -1 18 "Lagrange\371v pol ynom" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "P\370\355klad. Uva\236ujm e body:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "unassign('X','Y ');\nn:=5;\nfor i from 1 to n do\nX[i]:=i;\nY[i]:=evalf(ln(i));\nod:\n i:='i':\n[X[i],Y[i]] $i=1..n;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'7$\"\"\"$\"\"!F&7$\"\"#$ \"+1=ZJp!#57$\"\"$$\"+*G7')4\"!\"*7$\"\"%$\"+hVH'Q\"F07$\"\"&$\"+7zV4; F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Hled\341me polynom, kter \375 proch\341z\355 v\232emi t\354mito body. Najdeme jej ve tvaru sou \350tu:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "for j from 1 to \+ n do\ncitatel[j]:=simplify(product(x-X[i],i=1..n)/(x-X[j])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "jmenovatel[j]:=subs(x=X[j],citatel[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "clen[j]:=citatel[j]/jmenovatel[j] *Y[j];\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(citatelG6#\"\"\"**, &%\"xGF'\"\"#!\"\"F',&F*F'\"\"$F,F',&F*F'\"\"%F,F',&F*F'\"\"&F,F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%+jmenovatelG6#\"\"\"\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%clenG6#\"\"\"$\"\"!F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%(citatelG6#\"\"#**,&%\"xG\"\"\"F+!\"\"F+,&F*F+\"\" $F,F+,&F*F+\"\"%F,F+,&F*F+\"\"&F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%+jmenovatelG6#\"\"#!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%cle nG6#\"\"#,$*,$\"+,`Cb6!#5\"\"\",&%\"xGF-F-!\"\"F-,&F/F-\"\"$F0F-,&F/F- \"\"%F0F-,&F/F-\"\"&F0F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(cita telG6#\"\"$**,&%\"xG\"\"\"F+!\"\"F+,&F*F+\"\"#F,F+,&F*F+\"\"%F,F+,&F*F +\"\"&F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%+jmenovatelG6#\"\"$\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%clenG6#\"\"$,$*,$\"+A2`YF!# 5\"\"\",&%\"xGF-F-!\"\"F-,&F/F-\"\"#F0F-,&F/F-\"\"%F0F-,&F/F-\"\"&F0F- F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(citatelG6#\"\"%**,&%\"xG\"\" \"F+!\"\"F+,&F*F+\"\"#F,F+,&F*F+\"\"$F,F+,&F*F+\"\"&F,F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%+jmenovatelG6#\"\"%!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%clenG6#\"\"%,$*,$\"+-1\\5B!#5\"\"\",&%\"xGF-F-!\"\" F-,&F/F-\"\"#F0F-,&F/F-\"\"$F0F-,&F/F-\"\"&F0F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(citatelG6#\"\"&**,&%\"xG\"\"\"F+!\"\"F+,&F*F+\"\"#F ,F+,&F*F+\"\"$F,F+,&F*F+\"\"%F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%+jmenovatelG6#\"\"&\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%clenG 6#\"\"&,$*,$\"++8*fq'!#6\"\"\",&%\"xGF-F-!\"\"F-,&F/F-\"\"#F0F-,&F/F- \"\"$F0F-,&F/F-\"\"%F0F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A t edy hledan\375 polynom je:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Lagra nge:=(sum(clen[i],i=1..n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Lagr angeG,**,$\"+,`Cb6!#5\"\"\",&%\"xGF*F*!\"\"F*,&F,F*\"\"$F-F*,&F,F*\"\" %F-F*,&F,F*\"\"&F-F*F-*,$\"+A2`YFF)F*F+F*,&F,F*\"\"#F-F*F0F*F2F*F**,$ \"+-1\\5BF)F*F+F*F7F*F.F*F2F*F-*,$\"++8*fq'!#6F*F+F*F7F*F.F*F0F*F*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "neboli" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Lagrange:=simplify(sum(clen[i],i=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)LagrangeG,,*&$\"++^gg[!#7\"\"\")%\"xG\"\"%F* !\"\"*&$\"+5dD#p(!#6F*)F,\"\"$F*F**&$\"+8DhQ[!#5F*)F,\"\"#F*F.*&$\"+6@ =z;!\"*F*F,F*F*$\"+7GQn7F>F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "O p\354t m\371\236eme napsat proced\371ru, kter\341 polunom vztvo\370 \355. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "Lagrange:=proc(X ,Y) # Vstup: pole bodu a pole hodnot. Vystup: Lagrangeuv polynom (funk ce)\nlocal C,J,N,x,i,j,substitut,xxx;\ni:='i';\nXpom:=convert(X,list); \nN:=nops(Xpom);\nfor j from 1 to N do\nC[j]:=product((x-X[i]),i=1..N) /(x-X[j]);\nJ[j]:=subs(x=X[j],C[j]);\n#print(C[j],J[j]);\nod;\ni:='i'; \nxxx:=sum(C[i]/J[i]*Y[i],i=1..N);\nunapply(xxx,x)\nend;" }}{PARA 7 " " 1 "" {TEXT -1 69 "Warning, `Xpom` is implicitly declared local to pr ocedure `Lagrange`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)LagrangeGf* 6$%\"XG%\"YG6+%\"CG%\"JG%\"NG%\"xG%\"iG%\"jG%*substitutG%$xxxG%%XpomG6 \"F3C)>8(.F6>8,-%(convertG6$9$%%listG>8&-%%nopsG6#F9?(8)\"\"\"FFF@%%tr ueGC$>&8$6#FE*&-%(productG6$,&8'FF&F=6#F6!\"\"/F6;FFF@FF,&FRFF&F=FLFUF U>&8%FL-%%subsG6$/FRFYFJ>F6F7>8+-%$sumG6$*(&FKFTFF&FfnFTFU&9%FTFFFV-%( unapplyG6$F]oFRF3F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "nebo ekv ivaletn\354 s vzu\236it\355m ji\236 hotov\351 procedury " }{HYPERLNK 17 "interp" 2 "interp" "" }{TEXT -1 32 " v maple (interpolce polynomem )." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Lagrange2:=proc(X,Y) \nunapply(interp(convert(X,list),convert(Y,list), x),x)\nend;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Lagrange2Gf*6$%\"XG%\"YG6\"F)F)-%(u napplyG6$-%'interpG6%-%(convertG6$9$%%listG-F16$9%F4%\"xGF8F)F)F)" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Splainy" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 368 "Nev\375hoou Lagrangeova polynomu je to, 6e m8me-li mno ho bod\371, dostabneme polznom p\370\355li\232 vysok\351ho stzupn\354. Nev\375hodou po \350\341stech afinn\355\355 aproximace je v tom, \236 e dostaneme funkci, kter\341 nem\341 v bodech, jimi\236 ji prokl\341d \341me derivaci. Najdeme aproximaci po \350\341stech polynomialn\355 f unkc\355, kter\341 bude m\355t v ka\236d\351m bod\354 derivaci. Jako p \370\355klad uvedeme po \350\341stech kvadratick\375 pol\375nom:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "unssign('a','b','c','x');" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(unssignG6&%\"aG%\"bG%\"cG%\"xG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:=a*x^2+b*x+c; # obecn\375 tvar polznomu druh\351ho stupn\354" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*&%\"aG\"\"\")%\"xG\"\" #F(F(*&%\"bGF(F*F(F(%\"cGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "P olynom druh\351ho stupn\354 zavis\355 na t\370ech parametrech. Hled \341me takov\351 hodnotz parmetr\371, aby m\354l polynom p\370edepsan \351 hodnoty v bodech " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "x[j+1];" "6#&%\"xG6#,&%\"jG\"\"\"F(F( " }{TEXT -1 32 ", a p\370edepsanou derivaci v bod\354 " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" }{TEXT -1 49 ". Polynom mus\355 spl\362ov at ns\371eduj\355c\355 t\370i rovnioce:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "ddd:=diff(p,x);\nSoustavaRovnic:=\nsubs(x=x[j],ddd)= der,\nsubs(x=x[j],p)=y[j],\nsubs(x=x[j+1],p)=y[j+1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dddG,&*(\"\"#\"\"\"%\"aGF(%\"xGF(F(%\"bGF(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/SoustavaRovnicG6%/,&*(\"\"#\"\"\"% \"aGF*&%\"xG6#\"\"'F*F*%\"bGF*%$derG/,(*&F+F*)F,F)F*F**&F0F*F,F*F*%\"c GF*&%\"yGF./,(*&F+F*)&F-6#\"\"(F)F*F**&F0F*F>F*F*F7F*&F9F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Jejich \370e\232en\355 je" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(\{SoustavaRovnic\},\{a,b,c\}) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/%\"aG,$*&,(*$)&%\"xG6#\"\"'\" \"#\"\"\"F0*$)&F,6#\"\"(F/F0F0*(F/F0F3F0F+F0!\"\"F7,**&F+F0%$derGF0F7& %\"yGF-F0*&F3F0F:F0F0&FF0 F7*(F+F0F:F0F2F0F0*&F;F0F2F0F7**F/F0F;F0F3F0F+F0F0F0F(F7F7/%\"bG*&,**& F*F0F:F0F7*(F/F0F+F0F;F0F0*(F/F0F+F0F>F0F7*&F:F0F2F0F0F0F(F7" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Polznom, kter\375proch\341z\355 t \370emi zadan\375mi body zase spl\362uje rovnice" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "PrvniSoustavaRovnic:=\nsubs(x=x[1],p)=y[1],\n subs(x=x[2],p)=y[2],\nsubs(x=x[3],p)=y[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4PrvniSoustavaRovnicG6%/,(*&%\"aG\"\"\")&%\"xG6#F*\" \"#F*F**&%\"bGF*F,F*F*%\"cGF*&%\"yGF./,(*&F)F*)&F-6#F/F/F*F**&F1F*F9F* F*F2F*&F4F:/,(*&F)F*)&F-6#\"\"$F/F*F**&F1F*FAF*F*F2F*&F4FB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "kter\351 maj\355 \370e\232en\355:" } {MPLTEXT 1 0 38 "\nsolve(\{PrvniSoustavaRovnic\},\{a,b,c\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/%\"cG*&,.*()&%\"xG6#\"\"$\"\"#\"\"\"&F+6# F.F/&%\"yG6#F/F/F/*(F)F/&F+F4F/&F3F1F/!\"\"*()F0F.F/F*F/F2F/F8*(F7F/F* F/)F6F.F/F/*(&F3F,F/F0F/FF/F/F/,.*&F0F/FF/F8*&F*F/F2F/F/*&F0F/F>F/F/*&F0F/F2F/F8*&F6F/F7F/F/*&F*F/F7F/F8F/F@ F8/%\"bG,$*&,.*&F)F/F2F/F/*&F)F/F7F/F8*&F:F/F2F/F8*&F>F/F:F/F/*&F>F/F< F/F8*&F7F/F " 0 "" {MPLTEXT 1 0 1372 "Spl2:=proc(x,y)\nlocal N,K,j,a,b,c,Q,z;\nN:=nops(x) ; # pocet bodu\nK:=N-2; # pocet polynomu\na := -(x[3]*y[1]-x[1]*y[3]-y [1]*x[2]+x[2]*y[3]-x[3]*y[2]+x[1]*y[2])/(-x[2]*x[3]^2+x[1]^2*x[2]+x[1] *x[3]^2+x[3]*x[2]^2-x[3]*x[1]^2-x[1]*x[2]^2); \nb := (-x[1]^2*y[3]+x[1 ]^2*y[2]+x[2]^2*y[3]+y[1]*x[3]^2-y[1]*x[2]^2-y[2]*x[3]^2)/(-x[2]*x[3]^ 2+x[1]^2*x[2]+x[1]*x[3]^2+x[3]*x[2]^2-x[3]*x[1]^2-x[1]*x[2]^2); \nc := (x[1]^2*x[2]*y[3]-x[1]^2*x[3]*y[2]-x[2]^2*x[1]*y[3]+y[2]*x[1]*x[3]^2- y[1]*x[2]*x[3]^2+x[2]^2*x[3]*y[1])/(-x[2]*x[3]^2+x[1]^2*x[2]+x[1]*x[3] ^2+x[3]*x[2]^2-x[3]*x[1]^2-x[1]*x[2]^2);\np[1]:=a*z^2+b*z+c;\n# Pvni ( kubicky) polynom vytvorim tak, ze bude prochazet prvnimi trema body\n \nQ:=z%%Spl2Gf*6$%\"xG%\"yG6-%\"NG%\"KG%\"jG%\"aG%\"bG%\"cG%\"QG%\"zG %\"pG%\"iG%$derG6\"F5C,>8$-%%nopsG6#9$>8%,&F8\"\"\"\"\"#!\"\">8',$*&,. *&&F<6#\"\"$F@&9%6#F@F@F@*&&F8(*&,.*&FgnF@FQF@F B*&FgnF@FWF@F@*&FjnF@FQF@F@*&FLF@FenF@F@*&FLF@FjnF@FB*&FWF@FenF@FBF@FY FB>8)*&,.*(FgnF@FSF@FQF@F@*(FgnF@FIF@FWF@FB*(FjnF@FPF@FQF@FB*(FWF@FPF@ FenF@F@*(FLF@FSF@FenF@FB*(FjnF@FIF@FLF@F@F@FYFB>&8,FN,(*&FDF@)8+FAF@F@ *&F^oF@FgpF@F@FhoF@>8*6$2FgpFIFbp?(8-FAF@F>%%trueGC)>8&,&F^qF@F@F@>8., &*(FAF@FDF@&F<6#FbqF@F@F^oF@>F^o,$*&,**(FAF@FhqF@&FMFiqF@FB*(FAF@FhqF@ &FM6#,&FbqF@F@F@F@F@*&)FhqFAF@FeqF@F@*&FeqF@)&FFD*&,*F_rFB*&FhrF@FeqF@FBFarF@*&FhqF@F eqF@F@F@FirFB>Fho*&,,*(FerF@FhrF@FeqF@F@*&FerF@FarF@F@*(FhqF@FeqF@FgrF @FB*&F_rF@FgrF@F@**FAF@F_rF@FhrF@FhqF@FBF@FirFB>&Fcp6#F^qFdp@%2F^qF>>F jp6%Fjp2FgpFhrF[t>Fjp6$FjpF[t>Fjp-%*piecewiseG6#Fjp-%(unapplyG6$FjpFgp F5F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Priklad:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "with(plots):\nN:=7;f:=x->ln(x)*sin (x);\nx:=seq(round(2*i+sin(i)^2),i=1..N);\ny:=seq(evalf(f(x[i])),i=1.. N);x[N];" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoord s has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(*&-%#lnG6#9$\"\"\"-%$sinGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG6)\"\"$\"\"&\"\"'\"\"*\"#6\"#7\"#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"yG6)$\"+^ " 0 "" {MPLTEXT 1 0 169 "Bod y:=pointplot([seq([x[i],y[i]],i=1..N)], color=NAVY,symbolsize=15):\nAp roximace:=plot(Spl2([x],[y])(z),z=x[1]..x[N],color=RED):\nFce:=plot(f( z),z=x[1]..x[N],color=BLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display(\{Body,Aproximace,Fce\});" }}{PARA 13 "" 1 "" {GLPLOT2D 450 450 450 {PLOTDATA 2 "6'-%'POINTSG6+7$$\"\"$\"\"!$\"+^O]:!#=7$$\"3QLL$epo(RK!#<$!3Oq5L\"\\Us9$F[o7$$\"3am;/c ,R[MF_o$!3U#H6)Q]pWmF[o7$$\"3GLL32o+$o$F_o$!3+mRF_o$!3I&o/)zr,_7F_o7$$\"3um;a3!GU:%F_o$!3>:o4RhtS9F_o7$$\"33+]iI %)=jUF_o$!3?)RgCc\"e/:F_o7$$\"3UL$3F&)[@P%F_o$!3m^>9BIX`:F_o7$$\"3Wm;a 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\"3M!p$*4g=\\u\"F_o7$F^el$\"3gc5w'fLD:#F_o7$Fbfl$\"3&yOcwL6&QCF_o7$FH$ \"3nOt\"GgpUh#F_o-FM6&FOF\\hlF\\hlFjgl-%+AXESLABELSG6%Q\"z6\"Q!Fgjm-%% FONTG6#FX-%%VIEWG6$;F'FHFX" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Priklad: prevzorkovani" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "unassign('x','y','z','N','i','j','a lpha','beta');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Uva\236ujme n \354jakou funkci, nap\370\355klad:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Asuume some function, for instance:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "f:=x->#ln(x)^2+sin(3*x)-x*sin(5*x)+cos(20*x);\nx^2/ 100+3*sin(5*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(,&*&#\"\"\"\"$+\"F/*$)9$\"\"#F/F/F/*&\"\"$F/-%$s inG6#,$*&\"\"&F/F3F/F/F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "jej\355 hodnoty na intervalu " }{XPPEDIT 18 0 "`<,>`(alpha,beta); " "6#-%$<,>G6$%&alphaG%%betaG" }{TEXT -1 45 " zaznamen\341me p\370i v zorkovacv\355 frekvenci 10Hz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alpha:=5; beta:=7; F:=1/10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "N:=ceil(-(alpha-beta)/F);\nx:=seq(alpha+j*F,j=0..N);\ny:=seq(eval f(f(x[i])),i=1..N+1);\nPuvB:=pointplot([seq([x[i],y[i]],i=1..N+1)]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG #\"\"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG67\"\"&#\"#^\"#5#\"#EF&#\"#`F)#\"#FF&#\"# 6\"\"##\"#GF&#\"#dF)#\"#HF&#\"#fF)\"\"'#\"#hF)#\"#JF&#\"#jF)#\"#KF&#\" #8F2#\"#LF&#\"#nF)#\"#MF&#\"#pF)\"\"(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG67$!+.Dbq9!#5$\"+i]FP8!\"*$\"+_`2eDF+$\"+HH(*=KF+$\"+&yF2;$F+ $\"+&4?-S#F+$\"+ltJE6F+$!+1#pOY$F($!+`;]a;F+$!+DU*\\Z#F+$!+s[4/EF+$!+= " 0 "" {MPLTEXT 1 0 61 "F:=1/7;\nN:=floor(-(alpha -beta)/F);\nz:=seq(alpha+j*F,j=0..N);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG#\"\"\"\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG61\"\"&#\"#O\"\"(#\"#PF)#\"# QF)#\"#RF)#\"#SF)#\"#TF)\"\"'#\"#VF)#\"#WF)#\"#XF)#\"#YF)#\"#ZF)#\"#[F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "LinA:=seq(Plin([x], [y])(z[i]),i=1..N+1);\nBilA:=seq(Spl2([x],[y])(z[i]),i=1..N+1);\n#PolA :=seq(Lagrange([x],[y])(z[i]),i=1..N+1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%LinAG61$!)Ebq9!\")$\"*8v/'=F($\"*/fX7$F($\"*$*RM%HF($\"*b*G! \\\"F($!)GVK`F($!*gaL7#F($!*'[4/EF($!*A_<]\"F($\")g*)yVF($\"*c%%BilAG61$!(+%p9!\"($\")p'G*=F($\"(Lq<$!\"'$\"(Xu-$F-$ \")O/6:F($!(yFl&F($!)`zy@F($!)+3/EF($!(C/e\"F-$\"(Q(4WF($\"(awU#F-$\"( H%*Q$F-$\")\"H/%HF($\")P[)H\"F($!(!=VzF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "k1:=plot(f(w),w=z[1]..z[N+1],color=BLACK):\nk2:=plot (Plin([x],[y])(w),w=z[1]..z[N+1],color=RED):\nk3:=plot(Spl2([x],[y])(w ),w=z[1]..z[N+1],color=BLUE):\n#plot(Lagrange([x],[y])(w),w=z[1]..z[N+ 1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 730 "RB:=pointplot(\n [seq(\n \+ [z[i],evalf(f(z[i]))]\n ,i=1..N+1\n )]\n,color=BLACK,sy mbol=cross,symbolsize=20):\nLB:=pointplot(\n [seq(\n [z[i],Li nA[i]]\n ,i=1..N+1\n )],\ncolor=RED,symbol=box,symbolsi ze=20):\nBB:=pointplot(\n [seq(\n [z[i],BilA[i]]\n ,i=1 ..N+1\n )],\ncolor=GREEN,symbol=diamond,symbolsize=20):\nPB:= pointplot(\n [seq(\n [z[i],PolA[i]]\n ,i=1..N+1\n \+ )],\ncolor=BLUE,symbol=circle,symbolsize=20):\ndisplay([k1,k2,k3,P uvB],title=`funkce (cerne) a jeji aproximace`);\n\n\ndisplay(\{k1,RB\} ,title=`Skutecne hodnoty`);\ndisplay(\{k2,LB\},title=`hodnoty ziskane \+ z linerani 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Ykuste tot\351\236 pro funkci:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:=x->ln(x)^2+s in(3*x)-x*sin(5*x)+cos(20*x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)-%#lnG6#9$\"\"#\"\"\"F4-%$ sinG6#,$*&\"\"$F4F2F4F4F4*&F2F4-F66#,$*&\"\"&F4F2F4F4F4!\"\"-%$cosG6#, $*&\"#?F4F2F4F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Ortonorm\341ln\355 vektory " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Vektory " }{XPPEDIT 18 0 "v[1]. ..v[n];" "6#;&%\"vG6#\"\"\"&F%6#%\"nG" }{TEXT -1 81 " jsou ortonorm \341ln\355, pokud jsou normovan\351 a ka\236d\351 dva jsou ortogon \341ln\355, tj, je-li" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "phi" "6#%$phiG " }{TEXT -1 30 " skal\341rn\355 sou\350in, mus\355 platit " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }{XPPEDIT 19 1 "phi(v[i],v[j]):=p iecewise(i=j,0,1);" "6#>-%$phiG6$&%\"vG6#%\"iG&F(6#%\"jG-%*piecewiseG6 %/F*F-\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%$phiG6$&%\"vG6 #%\"iG&F(6#%\"jG-%*PIECEWISEG6$7$\"\"!/F*F-7$\"\"\"%*otherwiseG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "M\341me-li naj\355t vektory, kter \351 generuj\355 stejn\375 prostor, jako nez\341visl\351 vektory " } {XPPEDIT 18 0 "v[1]...v[n];" "6#;&%\"vG6#\"\"\"&F%6#%\"nG" }{TEXT -1 75 " , m\371\236eme vz\355t vektory takov\351, \236e prvn\355 bude ten , kter\375 z\355sk\341me normov\341n\355m " }{XPPEDIT 18 0 "v[1]" "6#& %\"vG6#\"\"\"" }{TEXT -1 6 ", tj. " }{XPPEDIT 18 0 "u[1]=v[1]/phi(v[1] ,v[1])" "6#/&%\"uG6#\"\"\"*&&%\"vG6#F'F'-%$phiG6$&F*6#F'&F*6#F'!\"\"" }{TEXT -1 32 ", druh\375 bude line\341rn\355 kombinac\355 " }{XPPEDIT 18 0 "u[1]" "6#&%\"uG6#\"\"\"" }{TEXT -1 3 " a " }{XPPEDIT 18 0 "v[2] " "6#&%\"vG6#\"\"#" }{TEXT -1 86 ". Abychom zajistili, \236e vektory b udou nad\341le line\341rn\354 nez\341visl\351, zvol\355me koeficient u " }{XPPEDIT 18 0 "v[2]" "6#&%\"vG6#\"\"#" }{TEXT -1 8 " 1. Tj.:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rce[1]:=u[2]=v[2]+c[1]*u[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"\"/&%\"uG6#\"\"#,&&% \"vGF+F'*&&%\"cGF&F'&F*F&F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 " a koeficient " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\"\"" }{TEXT -1 20 " vypo\350\355t\341me tak, \236e " }{XPPEDIT 18 0 "rce[1]" "6#&%$rceG6# \"\"\"" }{TEXT -1 21 " vyn\341sob\355me skal\341rn\354 " }{XPPEDIT 18 0 "u[1]" "6#&%\"uG6#\"\"\"" }{TEXT -1 6 ", tj.:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "rce[1]:=0=phi(v[2],u[1])+c[1]*phi(u[1],u[1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"\"/\"\"!,&-%$phiG6$& %\"vG6#\"\"#&%\"uGF&F'*&&%\"cGF&F'-F,6$F2F2F'F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "xxx:=isolate(rce[1],c[1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$xxxG/&%\"cG6#\"\"\",$*&-%$phiG6$&%\"vG6#\"\"#&%\"u GF(F)-F-6$F3F3!\"\"F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "d\341le \+ rekursivn\354. Z rovnice:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "rce[n]:=u[n]=v[n]+sum(c[i]*u[i],i=1..n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"&/&%\"uGF&,,&%\"vGF&\"\"\"*&&%\"cG6#F.F.& F*F2F.F.*&&F16#\"\"#F.&F*F6F.F.*&&F16#\"\"$F.&F*F;F.F.*&&F16#\"\"%F.&F *F@F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Z\355sk\341me " }{XPPEDIT 18 0 " n-1 " "6# ,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 12 " rovnic pro " }{XPPEDIT 18 0 "n-1 " "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]" "6#&% \"cG6#%\"iG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "P ak v\232echny vektory vyd\354l\355me jejich velikost\355." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Pozn\341mky:" }}{PARA 15 "" 0 "" {TEXT -1 109 "Pokud jsou vektory n-tice \350\355sel, je n\341soben\355 vektoru \+ skal\341rem (zna\350eno dosud hv\354zdi\350kou) prov\341d\354no p\370 \355kazem " }{HYPERLNK 17 "scalarmul " 2 "scalarmul" "" }{TEXT -1 35 "a skal\341rn\355 sou\350in vektor\371 p\370\355kazem " }{HYPERLNK 17 "multiply" 2 "linalg,multiply" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 101 "V takov\351m p\372\370\355pad\354 lze pou\236\355t k n alezen\355 ortogon\341ln\355ho syst\351mu )vektory nejsou ale normovan \351) p\370\355kaz " }{HYPERLNK 17 "GramSchmidt" 2 "GramSchmidt" "" } {TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Kontrola" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "Prov\341d\355 se kontrola toho, z da jsou v\232echny vektory normovan\351, pak toho, zda jsou ka\236d \351 dva r\371zn\351 vektory ortogon\341ln\355 a nakonec toho, zda vek tory generuj\355 tent\375\236 prostor, jako zadan\351 vektory" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Pro \350ty\370i vektory z " } {XPPEDIT 18 0 "Re^n" "6#)%#ReG%\"nG" }{TEXT -1 91 " se zkoum\341 hodno st matice, kter\341 vznikne p\370id\341n\355m zadan\375ch vektor\371 \+ k vypo\350\355tan\375m vektor\371m:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "if add(rank(matrix( 5,7,[w[kk],v[i] $i=1..4])),kk=1..4)<>\n4*rank(matrix(4,7,[v[i] $i=1..4 ])) then\nkomentC:=`vektory negeneruji zadany prostor`;\nelse \nkoment C:=``;\nfi;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(komentCG%Bvektory~negeneruji~zadany~prostorG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Pokud jsou vektory 4 funkce, zkoum\341 se, zda je mo\236no line\341rn\355 kombinac\355 vyj\341d \370it hodnoty ve \350ty\370ech bodech. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "komentC:=``;\nfor k from 1 to 4 do\n\nxxx:=solve(\n \{subs(x=i,\nsum(a[j]*f[j],j=1..4)\n=g[k]) $i=1..5\}\n,\n\{a[j] $j=1.. 4\});\n\nif xxx= NULL then komentC:=`funkce negeneruji tentyz prostor \+ jako zadane funkce` else fi;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%(komentCG%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG<&/&%\"aG6# \"\"\"F'/&F(6#\"\"$F,/&F(6#\"\"%F0/&F(6#\"\"#*&,**&F'F*&%\"fGF)F*!\"\" *&F,F*&F;F-F*F<*&F0F*&F;F1F*F<&%\"gGF)F*F*&F;F5F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG<&/&%\"aG6#\"\"\"F'/&F(6#\"\"%,$*&,**&F'F*&%\"fG F)F*F**&&F(6#\"\"#F*&F4F7F*F**&&F(6#\"\"$F*&F4F%$xxxG<&/&%\"aG6# \"\"$,$*&,**&&F(6#\"\"\"F1&%\"fGF0F1F1*&&F(6#\"\"#F1&F3F6F1F1*&&F(6#\" \"%F1&F3F;F1F1&%\"gGF)!\"\"F1&F3F)F@F@/F/F//F:F:/F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG<&/&%\"aG6#\"\"#,$*&,**&&F(6#\"\"\"F1&%\"fG F0F1F1*&&F(6#\"\"$F1&F3F6F1F1*&&F(6#\"\"%F1&F3F;F1F1&%\"gGF;!\"\"F1&F3 F)F@F@/F/F//F5F5/F:F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Aproximace line\341rn\355 kom binac\355 funkc\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "\332loha a proximovat funkci line\341rn\355 kombinac\355 zadan\375ch funkc\355:\n Pokud jsme o fourierovu \370adu, nap\370\355klad, vol\355me p\370\355s lu\232n\375 ortonorm\341ln\355 syst\351m. Koeficienty pak m\371\236eme po\350\355tat p\370\355mo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 671 "Fourier:=proc(F,n,beta) #vstup: funkce a prirozene cislo n. Vyst up: prvnich n scitancu fourierovy rady (pocitaji se i nulove). -a[0]+. ..+a[(n-1)/2]*sin/cos (x*((n-1)/2))- n intervalu 0..beta.\nlocal i,N,x ;\nf:=F(x);\nif n mod 2 = 1 then N:=(n+1)/2\n else\n N:=n/2; \nfi; i:='i';\n\nfor i from 0 to N do\n a[i]:=evalf(2/beta*evalf(int(f(x)*c os(2*i*Pi*x/beta),x=0..beta)));\n b[i]:=evalf(2/beta*evalf(int(f(x)*s in(2*i*Pi*x/beta),x=0..beta)));\nod;\ni:='i';\nif n mod 2 = 1 then\n \+ unapply(a[0]/2+sum(a[i]*cos(2*i*Pi*x/beta)+b[i]*sin(2*i*Pi*x/beta),i=1 ..N-1),x)\nelse\n unapply(a[0]/2+sum(a[i]*cos(2*i*Pi*x/beta)+b[i]*sin (2*i*Pi*x/beta),i=1..N-1)+a[N]*cos(N*2*Pi*x/beta),x);\nfi\nend;\n" }} {PARA 7 "" 1 "" {TEXT -1 65 "Warning, `f` is implicitly declared local to procedure `Fourier`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `a` is implicitly declared local to procedure `Fourier`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `b` is implicitly declared local to procedu re `Fourier`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(FourierGf*6%%\"FG %\"nG%%betaG6(%\"iG%\"NG%\"xG%\"fG%\"aG%\"bG6\"F1C(>8'-9$6#8&@%/-%$mod G6$9%\"\"#\"\"\">8%,&#F@F?F@*&FDF@F>F@F@>FB,$*&FDF@F>F@F@>8$.FJ?(FJ\" \"!F@FB%%trueGC$>&8(6#FJ-%&evalfG6#,$*(F?F@9&!\"\"-FU6#-%$intG6$*&-F4F 7F@-%$cosG6#,$*,F?F@FJF@%#PiGF@F8F@FYFZF@F@/F8;FMFYF@F@>&8)FS-FU6#,$*( F?F@FYFZ-FU6#-Fhn6$*&F[oF@-%$sinGF^oF@FboF@F@>FJFK@%F:-%(unapplyG6$,&* &FDF@&FR6#FMF@F@-%$sumG6$,&*&FQF@F\\oF@F@*&FeoF@F`pF@F@/FJ;F@,&FBF@F@F ZF@F8-Fep6$,(*&FDF@FipF@F@F[qF@*&&FR6#FBF@-F]o6#,$*,F?F@FBF@FaoF@F8F@F YFZF@F@F@F8F1F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "\332loha ap roximovat hodnoty hodnotami line\341rn\355 kombinace funkc\355:\nPokud nechceme nejprve syst\351m funkc\355 respektive vektory jejich hodnot nahradit ortonorm\341ln\355m syst\351mem, mus\355me poka\236d\351 \+ \370e\232it soustavu line\341rn\355ch rovnic pro koeficienty:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 601 "Aproximace:=proc(X,Y,F) #vs tup: pole bodu, pole hodnot, pole funkci. Vystup: linearni kombinace z adanych funkci, ktera nejlepe aproximuje zadane hodnoty.\nlocal i,j,k, PX,PF,v,Koeficienty,MaticeSoustavy,PraveStrany\n;\nPF:=nops(F); # Poce t funkci\nPX:=nops(X); # Pocet bodu\ni:='i';j:='j';k:='k';\nv:=evalf\n ([ unapply(F[i],x)(X[j]) $j=1..PX] $i=1..PF);\nMaticeSoustavy:=\nmatr ix(PF,PF,\n[[sum((v[i][k]*v[j][k]),k=1..PX) $j=1..PF] $i=1..PF]\n);\nP raveStrany:=vector([sum(Y[j]*v[i][j],j=1..PX) $i=1..PF]);\ni:='i';\nKo eficienty:=linsolve(MaticeSoustavy,PraveStrany):\nunapply(sum(Koeficie nty[i]*F[i],i=1..PF),x);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+A proximaceGf*6%%\"XG%\"YG%\"FG6+%\"iG%\"jG%\"kG%#PXG%#PFG%\"vG%,Koefici entyG%/MaticeSoustavyG%,PraveStranyG6\"F4C->8(-%%nopsG6#9&>8'-F96#9$>8 $.FB>8%.FE>8&.FH>8)-%&evalfG6#-%\"$G6$7#-FP6$--%(unapplyG6$&F;6#FB%\"x G6#&F@6#FE/FE;\"\"\"F=/FB;F[oF7>8+-%'matrixG6%F7F77#-FP6$7#-FP6$-%$sum G6$*&&&FKFZ6#FHF[o&&FKFhnF_pF[o/FHFjn/FEF]oF\\o>8,-%'vectorG6#7#-FP6$- Fjo6$*&&9%FhnF[o&F^pFhnF[oFinF\\o>FBFC>8*-_%'linalgG%)linsolveG6$F_oFe p-FW6$-Fjo6$*&&FdqFZF[oFYF[oF\\oFenF4F4F4" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 25 "Metoda nejmen\232\355ch \350tverc\371" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Hled\341me p\370\355mku, kter\341 m\341 nejmen \232\355\351 sou\350et \350tverc\371 vzd\341kleost\355 od zadan\375ch \+ bvod\371. Vzd\341lenosti m\354\370\355me ve svisl\351m sm\354ru. Jde \+ o extrem\341ln\355 \372lohu, kteru vy\370e\232\355me standatn\355 meto dou infinitezim\341ln\355ho po\350tu." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "unassign('y','p','q','x','yy','xx','X','Y','N');" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Vydaleno\351st bodu od primky:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "primka:=y=p*x+q;\nprimka_:=yy=p*xx+q;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'primkaG/%\"yG,&*&%\"pG\"\"\"%\"xGF*F*%\"qGF* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(primka_G/%#yyG,&*&%\"pG\"\"\"%# xxGF*F*%\"qGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Primka:=r hs(primka)-lhs(primka)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Primka G/,(*&%\"pG\"\"\"%\"xGF)F)%\"qGF)%\"yG!\"\"\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 132 "mam bod [a,b] a na primce zvolim x[0]. Vzdalenost bodu [a,b] od vbodu na primce [x[0],?]. Hled\341m glob\341mn\355 mini mum v lok\341ln\355m minimu." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "rho:=(((a-x[0])^2+(b-subs(x=x[0],rhs(primka)))^2))^(1/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG*$,&*$),&%\"aG\"\"\"&%\"xG6#\" \"!!\"\"\"\"#F+F+*$),(%\"bGF+*&%\"pGF+F,F+F0%\"qGF0F1F+F+#F+F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xxx:=Diff(rho,x[0])=diff(rho ,x[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG/-%%DiffG6$*$,&*$),& %\"aG\"\"\"&%\"xG6#\"\"!!\"\"\"\"#F/F/*$),(%\"bGF/*&%\"pGF/F0F/F4%\"qG F4F5F/F/#F/F5F0,$*&F=F/*&F*#F4F5,(*&F5F/F.F/F4*&F5F/F0F/F/*(F5F/F8F/F; F/F4F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z:=solve(rhs(xxx)=0,x[0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&,(%\"aG\"\"\"*&%\"pGF(%\"bGF(F (*&F*F(%\"qGF(!\"\"F(,&F(F(*$)F*\"\"#F(F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rho:=simplify(subs(x[0]=z,rho));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG*$*&,(*&%\"aG\"\"\"%\"pGF*F*%\"bG!\"\"%\"qGF* \"\"#,&F*F**$)F+F/F*F*F-#F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rho:=unapply(rho,a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rho Gf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)*$*&,(*&9$\"\"\"%\"pGF2F29%! \"\"%\"qGF2\"\"#,&F2F2*$)F3F7F2F2F5#F2F7F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Soucet ctvercu vzdalenosti:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Delta:=Sum(rho(x[i],y[i])^2\n,i=1..N)=sum(rho(x[ i],y[i])^2\n,i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG/-%$ SumG6$*&,(*&&%\"xG6#%\"iG\"\"\"%\"pGF0F0&%\"yGF.!\"\"%\"qGF0\"\"#,&F0F 0*$)F1F6F0F0F4/F/;F0%\"NG,&*(F " 0 "" {MPLTEXT 1 0 20 "#Delta:=r hs(Delta);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Dp:=Diff(l hs(Delta),p)=simplify(diff(rhs(Delta),p));\nDq:=Diff(lhs(Delta),q)=sim plify(diff(rhs(Delta),q));\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#D pG/-%%DiffG6$-%$SumG6$*&,(*&&%\"xG6#%\"iG\"\"\"%\"pGF3F3&%\"yGF1!\"\"% \"qGF3\"\"#,&F3F3*$)F4F9F3F3F7/F2;F3%\"NGF4,$*(F9F3,&*(F?F3)F8F9F3F4F3 F3-%$sumG6$,0*&)F/F9F3F4F3F7*(F/F3F%#DqG/-%%DiffG6$-%$SumG6$*&,(*&&%\"xG 6#%\"iG\"\"\"%\"pGF3F3&%\"yGF1!\"\"%\"qGF3\"\"#,&F3F3*$)F4F9F3F3F7/F2; F3%\"NGF8,$*(F9F3,&*&F?F3F8F3F3-%$sumG6$,&F.F3F5F7F=F3F3F:F7F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{rhs(Dp)=0,rhs(Dq)=0 \},\{p,q\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/%\"pG-%'RootOfG6#,* *&,&*&-%$sumG6$&%\"xG6#%\"iG/F3;\"\"\"%\"NGF6-F.6$&%\"yGF2F4F6!\"\"*&- F.6$*&F0F6F:F6F4F6F7F6F6F6)%#_ZG\"\"#F6F6*&,**$)F8FCF6F6*&-F.6$*$)F:FC F6F4F6F7F6F<*$)F-FCF6F<*&-F.6$*$)F0FCF6F4F6F7F6F6F6FBF6F6F,F6F=F " 0 "" {MPLTEXT 1 0 18 "xxx:=allvalues(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$xxxG6 $<$/%\"pG,$*&#\"\"\"\"\"#F,*&,&*&-%$sumG6$&%\"xG6#%\"iG/F7;F,%\"NGF,-F 26$&%\"yGF6F8F,!\"\"*&-F26$*&F4F,F=F,F8F,F:F,F,F?,,*$)F;F-F,F?*&-F26$* $)F=F-F,F8F,F:F,F,*$)F1F-F,F,*&-F26$*$)F4F-F,F8F,F:F,F?*$,:*$)F;\"\"%F ,F,**F-F,FFF,FHF,F:F,F?*(F-F,FFF,FMF,F,**F-F,FFF,FOF,F:F,F,*&)FHF-F,)F :F-F,F,**F-F,FHF,F:F,FMF,F,**F-F,FHF,FgnF,FOF,F?*$)F1FWF,F,**F-F,FMF,F OF,F:F,F?*&)FOF-F,FgnF,F,*,\"\")F,F1F,F;F,FAF,F:F,F?*(FWF,)FAF-F,FgnF, F,F+F,F,F,F,/%\"qG,$*&,&*&F+F,*(F/F?FDF,F1F,F,F,F;F?F,F:F?F?<$/F(,$*&F +F,*&F/F?,,FEF?FGF,FLF,FNF?FSF?F,F,F,/Fdo,$*&,&*&F+F,*(F/F?F_pF,F1F,F, F,F;F?F,F:F?F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "N:=4; x:=[1,2,7,8];y:=[3,2,5 ,4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"xG7&\"\"\"\"\"#\"\"(\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7&\"\"$\"\"#\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "xxx;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"qG,&# \"$@#\"#A\"\"\"*(\"\"*F*\"$w\"!\"\"\">##F*\"\"#F./%\"pG,&#\"#;\"#6F. *&\"#))F.F/F0F*<$/F%,&F'F**(F,F*F-F.F/F0F*/F3,&#F6F7F.*&F9F.F/F0F." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "pr1:=subs(xxx[1],Y=p*X+q); \npr2:=subs(xxx[2],Y=p*X+q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pr1 G/%\"YG,(*&,&#\"#;\"#6!\"\"*&\"#))F-\">##\"\"\"\"\"#F2F2%\"XGF2F2#\" $@#\"#AF2*(\"\"*F2\"$w\"F-F0F1F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $pr2G/%\"YG,(*&,&#\"#;\"#6!\"\"*&\"#))F-\">##\"\"\"\"\"#F-F2%\"XGF2F 2#\"$@#\"#AF2*(\"\"*F2\"$w\"F-F0F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"YG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "pointplot([seq([x[i],y[i]],i=1..N)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 460 460 460 {PLOTDATA 2 "6#-%'POINTSG6&7$$\"\"\"\"\"!$\"\"$F )7$$\"\"#F)F-7$$\"\"(F)$\"\"&F)7$$\"\")F)$\"\"%F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "seq([x[i],y[i]],i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7$\"\"\"\"\"$7$\"\"#F'7$\"\"(\"\"&7$\"\")\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "Reseni:=fsolve(\{rhs(Dp)=0,rhs(Dq)=0\},\{p,q\}#,\{p =-1..-0.10,q=0..10\}\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ReseniG <$/%\"qG$\"+@j&))z\"!\")/%\"pG$!+82o>K!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 " res_a:=subs(Reseni,primka_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&res_aG/%#yyG,&*&$\"+82o>K!\"*\"\"\"%#xxGF,!\"\"$\"+@ j&))z\"!\")F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "A:=pointpl ot([seq([x[i],y[i]],i=1..N)]):\nB:=plot(rhs(pr1),X=0..10):\nC:=plot(rh s(pr2),X=0..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display (\{A,B,C\});" }}{PARA 13 "" 1 "" {GLPLOT2D 460 460 460 {PLOTDATA 2 "6' -%'CURVESG6$7S7$$\"\"!F)$\"37?Uo?j&))z\"!#;7$$\"3emmm;arz@!#=$\"3%eaBQ Ww'G\"F,7$$\"3oLLL3En$4#FE$\"3;FE7*egZ7\"F,7$$\"3#pmmT!RE&G#FE$ \"37fB84V2j5F,7$$\"3D+++D.&4]#FE$\"3,n**=#o,j$**FE7$$\"3;+++vB_6(f1AJsFE7$$\"3nLLLLY.KNFE$\"3+R? b@%Rlh'FE7$$\"33++D\"o7Tv$FE$\"3`re(R,>:!fFE7$$\"3?LLL$Q*o]RFE$\"3yNas (z/'o_FE7$$\"3m++D\"=lj;%FE$\"3_5<*4f(>uXFE7$$\"3S++vV&RFE7$$\"3=nm\"zRQb@&FE$\"3U\"R !3@[>'>\"FE7$$\"3:++v=>Y2aFE$\"3gq&HviBEy&F07$$\"3Znm;zXu9cFE$!3p9_N0@ ;7*)!#>7$$\"34+++]y))GeFE$!3AuZ&*)zXfy(F07$$\"3H++]i_QQgFE$!3_`zEe$4JX \"FE7$$\"3b++D\"y%3TiFE$!3tPqC*3Pd5#FE7$$\"3+++]P![hY'FE$!3hCs^9*o.$GF E7$$\"3iKLL$Qx$omFE$!3]6\\#)QG[\"[$FE7$$\"3Y+++v.I%)oFE$!37'z20$fowTFE 7$$\"3?mm\"zpe*zqFE$!3EG%o!fKk1[FE7$$\"3;,++D\\'QH(FE$!3uP/s7IN&\\&FE7 $$\"3%HL$e9S8&\\(FE$!3\"G\"3]=_PVhFE7$$\"3s++D1#=bq(FE$!3quDlU^u?oFE7$ $\"3\"HLL$3s?6zFE$!3=<2Yb!)*H[(FE7$$\"3a***\\7`Wl7)FE$!3sK_w![:j<)FE7$ $\"3enmmm*RRL)FE$!3<5T1kD1W))FE7$$\"3%zmmTvJga)FE$!3fCae,/$p_*FE7$$\"3 ]MLe9tOc()FE$!3R_R'[P9/-\"F,7$$\"31,++]Qk\\*)FE$!3g#[#>[Kk#3\"F,7$$\"3 ![LL3dg6<*FE$!3oIbx@\"F,7$$\"3C ++D\"oK0e*FE$!359:P3$pdG\"F,7$$\"35,+v=5s#y*FE$!3)RHGu\\n3N\"F,7$$\"#5 F)$!3:!QZ>RC3U\"F,-%'COLOURG6&%$RGBG$Fgz!\"\"F(F(-F$6$7S7$F($\"3\\!*o1 %)eM-@FE7$F.$\"3G,1PAc/q@FE7$F4$\"3\"z7!o-3&*GAFE7$F9$\"3;y'Q'Qe>&H#FE 7$F>$\"3!oy'p*)*z=O#FE7$FC$\"3M\"=DE?Z#GCFE7$FI$\"3_/Wg/\"y(*[#FE7$FN$ \"3w`\\f*y*[`DFE7$FS$\"32*o_`]!Q>EFE7$FX$\"3IM$\\(3*f]o#FE7$Fgn$\"3-nw ;[#>Ev#FE7$F\\o$\"3xZ15^b77GFE7$Fao$\"3[nO$FE7$F_q$\"3[Wf?/]LoKFE7$Fdq$\"3%Ho'[B(*QHLFE7$Fiq$\"3y .,>FkP'R$FE7$F^r$\"3;t745O;gMFE7$Fcr$\"3zPPtV)=n_$FE7$Fhr$\"3;&*G^Ct$y e$FE7$F]s$\"3=tJ&G'=w`OFE7$Fbs$\"3idZVI*QAs$FE7$Fgs$\"3'*>_1>%[=y$FE7$ F\\t$\"3cC9%)*HGi%QFE7$Fbt$\"3!)4\"fi,RF\"RFE7$Fgt$\"3yn(\\kx1y(RFE7$F \\u$\"3,Bh:!=j2/%FE7$Fau$\"3cR!*pKbm5TFE7$Ffu$\"3kw:\\0fZtTFE7$F[v$\"3 CYK0&QR0C%FE7$F`v$\"3)Q.H^')38I%FE7$Fev$\"3!p(\\#[FE7$F] y$\"3!Ra(>4S,#)[FE7$Fby$\"3yX@+W[\"3&\\FE7$Fgy$\"3%*)3K38^B,&FE7$F\\z$ \"3-(\\^tkhz2&FE7$Faz$\"3#4%QP$Gf29&FE7$Ffz$\"3s*zt%>RC3_FEFjz-%'POINT SG6&7$$\"\"\"F)$\"\"$F)7$$\"\"#F)F_el7$$\"\"(F)$\"\"&F)7$$\"\")F)$\"\" %F)-%+AXESLABELSG6%Q\"X6\"Q!F_fl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FfzFd fl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eval f(\nsubs(xxx[1],rhs(Delta))\n);\nevalf(\nsubs(xxx[2],rhs(Delta))\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*<7Ne\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(y[;/%!\")" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"YG,(*&,&#!#;\"#6\"\"\"*&#F+\"#))F+-%%sqrtG6#\">#F+F+F+%\"XGF+ F+#\"$@#\"#AF+*&#\"\"*\"$w\"F+*$F/F+F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 ^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%* WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsu bG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)cholesk yG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)co pyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG %(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenve ctsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibon acciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genm atrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)i hermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimi larG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)li nsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multipl yG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potent ialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspa ceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG% *submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)to eplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vecto rG%*wronskianG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(sub s(xxx[1],hessian(rhs(Delta),[p,q])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$$\"+\\b[B@!\"($\"+oaF$G$!\")7$F+$\"+/x;'H(!\"*Q(ppr int06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3195 "RP1:=proc(x,y)\nlocal p,q,N,Delta ,X,Y,pr,pr_;\nN:=nops(x);\nxxx := \{q = -(1/2/(sum(x[i],i = 1 .. N)*su m(y[i],i = 1 .. N)-sum(x[i]*y[i],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^ 2-sum(x[i],i = 1 .. N)^2+sum(x[i]^2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. \+ N)*N+sqrt(sum(y[i],i = 1 .. N)^4+2*sum(y[i],i = 1 .. N)^2*sum(x[i],i = 1 .. N)^2+2*sum(y[i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N-2*sum(y[i ],i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[i],i = 1 .. N)^4-2*sum( x[i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N+2*sum(x[i],i = 1 .. N)^2*s um(y[i]^2,i = 1 .. N)*N+sum(x[i]^2,i = 1 .. N)^2*N^2-2*sum(x[i]^2,i = \+ 1 .. N)*N^2*sum(y[i]^2,i = 1 .. N)+sum(y[i]^2,i = 1 .. N)^2*N^2-8*sum( x[i],i = 1 .. N)*sum(y[i],i = 1 .. N)*sum(x[i]*y[i],i = 1 .. N)*N+4*su m(x[i]*y[i],i = 1 .. N)^2*N^2))*sum(x[i],i = 1 .. N)-sum(y[i],i = 1 .. N))/N, p = 1/2/(sum(x[i],i = 1 .. N)*sum(y[i],i = 1 .. N)-sum(x[i]*y[ i],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^2-sum(x[i],i = 1 .. N)^2+sum(x [i]^2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. N)*N+sqrt(sum(y[i],i = 1 .. N) ^4+2*sum(y[i],i = 1 .. N)^2*sum(x[i],i = 1 .. N)^2+2*sum(y[i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N-2*sum(y[i],i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[i],i = 1 .. N)^4-2*sum(x[i],i = 1 .. N)^2*sum(x[i]^2, i = 1 .. N)*N+2*sum(x[i],i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[ i]^2,i = 1 .. N)^2*N^2-2*sum(x[i]^2,i = 1 .. N)*N^2*sum(y[i]^2,i = 1 . . N)+sum(y[i]^2,i = 1 .. N)^2*N^2-8*sum(x[i],i = 1 .. N)*sum(y[i],i = \+ 1 .. N)*sum(x[i]*y[i],i = 1 .. N)*N+4*sum(x[i]*y[i],i = 1 .. N)^2*N^2) )\}, \{p = 1/2/(sum(x[i],i = 1 .. N)*sum(y[i],i = 1 .. N)-sum(x[i]*y[i ],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^2-sum(x[i],i = 1 .. N)^2+sum(x[ i]^2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. N)*N-sqrt(sum(y[i],i = 1 .. N)^ 4+2*sum(y[i],i = 1 .. N)^2*sum(x[i],i = 1 .. N)^2+2*sum(y[i],i = 1 .. \+ N)^2*sum(x[i]^2,i = 1 .. N)*N-2*sum(y[i],i = 1 .. N)^2*sum(y[i]^2,i = \+ 1 .. N)*N+sum(x[i],i = 1 .. N)^4-2*sum(x[i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N+2*sum(x[i],i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[i ]^2,i = 1 .. N)^2*N^2-2*sum(x[i]^2,i = 1 .. N)*N^2*sum(y[i]^2,i = 1 .. N)+sum(y[i]^2,i = 1 .. N)^2*N^2-8*sum(x[i],i = 1 .. N)*sum(y[i],i = 1 .. N)*sum(x[i]*y[i],i = 1 .. N)*N+4*sum(x[i]*y[i],i = 1 .. N)^2*N^2)) , q = -(1/2/(sum(x[i],i = 1 .. N)*sum(y[i],i = 1 .. N)-sum(x[i]*y[i],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^2-sum(x[i],i = 1 .. N)^2+sum(x[i]^ 2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. N)*N-sqrt(sum(y[i],i = 1 .. N)^4+2 *sum(y[i],i = 1 .. N)^2*sum(x[i],i = 1 .. N)^2+2*sum(y[i],i = 1 .. N)^ 2*sum(x[i]^2,i = 1 .. N)*N-2*sum(y[i],i = 1 .. N)^2*sum(y[i]^2,i = 1 . . N)*N+sum(x[i],i = 1 .. N)^4-2*sum(x[i],i = 1 .. N)^2*sum(x[i]^2,i = \+ 1 .. N)*N+2*sum(x[i],i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[i]^2 ,i = 1 .. N)^2*N^2-2*sum(x[i]^2,i = 1 .. N)*N^2*sum(y[i]^2,i = 1 .. N) +sum(y[i]^2,i = 1 .. N)^2*N^2-8*sum(x[i],i = 1 .. N)*sum(y[i],i = 1 .. N)*sum(x[i]*y[i],i = 1 .. N)*N+4*sum(x[i]*y[i],i = 1 .. N)^2*N^2))*su m(x[i],i = 1 .. N)-sum(y[i],i = 1 .. N))/N\};\n\npr1:=subs(xxx[1],Y=p* X+q);\npr2:=subs(xxx[2],Y=p*X+q);\n\nDelta:=Sum(rho(x[i],y[i])^2\n,i=1 ..N)=sum(rho(x[i],y[i])^2\n,i=1..N);\n\nif\nevalf(\nsubs(xxx[1],rhs(De lta))\n)\n<\nevalf(\nsubs(xxx[2],rhs(Delta))\n)\n\nthen pr_:=pr1 else \+ pr_:=pr2\nfi;\nunapply(rhs(pr_),X)\nend;" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `xxx` is implicitly declared local to procedure `RP1`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `pr1` is implicitly declared l ocal to procedure `RP1`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `pr 2` is implicitly declared local to procedure `RP1`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RP1Gf*6$%\"xG%\"yG6-%\"pG%\"qG%\"NG%&DeltaG%\"XG% \"YG%#prG%$pr_G%$xxxG%$pr1G%$pr2G6\"F5C)>8&-%%nopsG6#9$>8,6$<$/8%,$*&, &*&#\"\"\"\"\"#FH*(,&*&-%$sumG6$&F<6#%\"iG/FR;FHF8FH-FN6$&9%FQFSFHFH*& -FN6$*&FPFHFWFHFSFHF8FH!\"\"Fgn,,*$)FUFIFHFH*$)FMFIFHFgn*&-FN6$*$)FPFI FHFSFHF8FHFH*&-FN6$*$)FWFIFHFSFHF8FHFgn-%%sqrtG6#,:*$)FU\"\"%FHFH*(FIF HFjnFHF\\oFHFH**FIFHFjnFHF^oFHF8FHFH**FIFHFjnFHFcoFHF8FHFgn*$)FMF]pFHF H**FIFHF\\oFHF^oFHF8FHFgn**FIFHF\\oFHFcoFHF8FHFH*&)F^oFIFH)F8FIFHFH**F IFHF^oFHFgpFHFcoFHFgn*&)FcoFIFHFgpFHFH*,\"\")FHFMFHFUFHFZFHF8FHFgn*(F] pFH)FZFIFHFgpFHFHFHFHFMFHFHFHFUFgnFHF8FgnFgn/8$,$*&FGFH*&FKFgnFhnFHFHF H<$/FB,$*&,&*&FGFH*(FKFgn,,FinFHF[oFgnF]oFHFboFgnFgoFgnFHFMFHFHFHFUFgn FHF8FgnFgn/F`q,$*&FGFH*&FKFgnF[rFHFHFH>8--%%subsG6$&F>6#FH/8),&*&F`qFH 8(FHFHFBFH>8.-Fcr6$&F>6#FIFgr>8'/-%$SumG6$*$)-%$rhoG6$FPFWFIFHFS-FNFgs @%2-%&evalfG6#-Fcr6$Fer-%$rhsG6#Fcs-Fat6#-Fcr6$F`sFet>8+Far>F]uF]s-%(u napplyG6$-Fft6#F]uF[sF5F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Pri klad:" }{MPLTEXT 1 0 190 "\nwith(plots):\nN:=30;\nx:=seq(evalf(i/3+(si n(i))*3+4*cos(i*3)),i=1..N);\n\n#x:=seq(i,i=1..N);\ny:=seq(evalf(2*x[i ]+3+3*sin(x[i])+4*cos(2*x[i])),i=1..N);\nbb:=pointplot([seq([x[i],y[i] ],i=1..N)]):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"xG6@$!+*pBA5\"!\"*$\"+%4S_B(F($!+C5;@AF( $\"+#oT$QCF($!+4y&)[UF($\"+Q.-.QF($\"+)3wL6#F($\"+OuXJtF($\"+@-!y1$F($ \"++eF=BF($\"+\"e!*e8'!#5$\"+)[E%y=F($\"+thSgmF($\"+uuaQgF($\"+w9:_!*F ($\"+Dg/4>F($\"+y44^dF($\"*G*)zH%F($\"+FKCQ5!\")$\"+(\\]ef&F($\"+C`NX8 FN$\"+\"e*=3LF($\"+tpc,\"*F($\"+hDE99F($\"+;$GB;\"FN$\"+Z'HJ_(F($\"+rr e(\\\"FN$\"+=n&fU(F($\"+^jwa**F($\"+k1hV_F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG6@$!+6GX\\U!\"*$\"+Q/cg=!\")$!+H\\4(*[F($\"+n55Z5 F+$!+#)H-:_F($\"+S?%\\u*F($\"+*4/)GzF($\"+>h^D=F+$\"+J$H8L\"F+$\"+mmqL &*F($\"+]'y@I(F($\"+%e@&\\jF($\"+$GWS.#F+$\"+:u6)y\"F+$\"+Adh8DF+$\"+. k-HlF($\"+\\>5#\\\"F+$\"+W1j?xF($\"+#=Qg*>F+$\"+\")\\x18F+$\"+CHVUJF+$ \"+Q2*)*G\"F+$\"+**[*[`#F+$\"+CEF')\\F($\"+8NxdAF+$\"+mrxs " 0 "" {MPLTEXT 1 0 70 "primka:=RP1([x],[y])('w');\npr:=plot(primka,w=1..17): \ndisplay(\{bb,pr\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'primkaG,&*&$\"+hI_*>#!\"*\"\"\"%\"wGF*F*$ \"+h ^D=F17$$\"+@-!y1$F)$\"+J$H8L\"F17$$\"++eF=BF)$\"+mmqL&*F)7$$\"+\"e!*e8 '!#5$\"+]'y@I(F)7$$\"+)[E%y=F)$\"+%e@&\\jF)7$$\"+thSgmF)$\"+$GWS.#F17$ $\"+uuaQgF)$\"+:u6)y\"F17$$\"+w9:_!*F)$\"+Adh8DF17$$\"+Dg/4>F)$\"+.k-H lF)7$$\"+y44^dF)$\"+\\>5#\\\"F17$$\"*G*)zH%F)$\"+W1j?xF)7$$\"+FKCQ5F1$ \"+#=Qg*>F17$$\"+(\\]ef&F)$\"+\")\\x18F17$$\"+C`NX8F1$\"+CHVUJF17$$\"+ \"e*=3LF)$\"+Q2*)*G\"F17$$\"+tpc,\"*F)$\"+**[*[`#F17$$\"+hDE99F)$\"+CE F')\\F)7$$\"+;$GB;\"F1$\"+8NxdAF17$$\"+Z'HJ_(F)$\"+mrxs0_,]Fcu$\"33r/3kB^b8F\\x7$$\"3!********zN![`Fcu$\"3Je`,*>H o&Fcu$\"3[076U#y^]\"F\\x7$$\"3gmmmO%4_)fFcu$ \"3;:jy4q(=d\"F\\x7$$\"37LLL`MzXjFcu$\"3A\\\\z&H)=^;F\\x7$$\"3#GLLLTb7 l'Fcu$\"3F\\x7 $$\"35+++qKy%*zFcu$\"3\"*)>1(3u)Q,#F\\x7$$\"3kKLLL=kP$)Fcu$\"3I)\\x.%* *H*3#F\\x7$$\"3ELLLBI\\_')Fcu$\"3aFjp&=_&e@F\\x7$$\"3_mmmmD5#**)Fcu$\" 38ig#z3]KB#F\\x7$$\"3NlmmO9'[M*Fcu$\"39#zz[@S3J#F\\x7$$\"3=******p!R>l *Fcu$\"3;*=$[[EQyBF\\x7$$\"3#emmmK\"f$)**Fcu$\"3]\"HgrKI8X#F\\x7$$\"3& ******f0AE.\"F\\x$\"35L#prT#pEDF\\x7$$\"3%)*****>kTh1\"F\\x$\"3w09d>&> /g#F\\x7$$\"3))*****\\ct&)4\"F\\x$\"39#[A@Gah)Q (p'HF\\x7$$\"3)******zQ=qE\"F\\x$\"3^Xq\"[b_A/$F\\x7$$\"3;LLLU9A*H\"F \\x$\"3!y%yX:R38JF\\x7$$\"3\"******H\"H)GL\"F\\x$\"3AcjX'4Br=$F\\x7$$ \"3GLLL`Jzl8F\\x$\"3AvD,H*4&fKF\\x7$$\"3$******\\7Z-S\"F\\x$\"3O\\kBMA HNLF\\x7$$\"3rmmm%RIMV\"F\\x$\"3bkw5,'z#3MF\\x7$$\"3immm!3ltY\"F\\x$\" 3+eC`D(>H[$F\\x7$$\"3JLLLq(=5]\"F\\x$\"3aG?/#zTpb$F\\x7$$\"3'******f,V >`\"F\\x$\"3K\"*)HNQg\\i$F\\x7$$\"3KLLL\"p&Qn:F\\x$\"3h$4p_OIksPF\\x7$$\"3/+++H_)Gj\"F\\x$\"3wl<&)3^) p%QF\\x7$$\"3/+++j`Bl;F\\x$\"3l07hu*R\"=RF\\x7$$\"# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Regresni primka v ine petrice:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "RP2:=proc(X,Y)\nlocal Prim ka,x,p,q;\nPrimka:=x->p*x+q; #obecna rovnice primky\nSoucetCtvercu:=su m(((Y[i]-Primka(X[i]))^2),i=1..nops(X)); #soucet druhych mocnin odchyl ek\nRce:=diff(SoucetCtvercu,p)=0,diff(SoucetCtvercu,q)=0; # hledame mi nimum, tedy hledame parametry p a q tak, aby derivace `SoucetCtvercu` \+ podle p i podle q byla 0\nParam:=solve(\{Rce\},\{p,q\}); # parametry n ajdeme resenim vynikle rovnice\nunapply(subs(Param,Primka(x)),x); #vys tup je ve tvaru funkce\nend;" }}{PARA 7 "" 1 "" {TEXT -1 73 "Warning, \+ `SoucetCtvercu` is implicitly declared local to procedure `RP2`\n" }} {PARA 7 "" 1 "" {TEXT -1 63 "Warning, `Rce` is implicitly declared loc al to procedure `RP2`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `Para m` is implicitly declared local to procedure `RP2`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RP2Gf*6$%\"XG%\"YG6)%'PrimkaG%\"xG%\"pG%\"qG%.Sou cetCtvercuG%$RceG%&ParamG6\"F1C'>8$f*6#F+F16$%)operatorG%&arrowGF1,&*& T#\"\"\"9$F=F=T%F=F1F16&F,8&F-8'>8(-%$sumG6$*$),&&9%6#%\"iGF=-F46#&F>F M!\"\"\"\"#F=/FN;F=-%%nopsG6#F>>8)6$/-%%diffG6$FDFA\"\"!/-Fhn6$FDFBFjn >8*-%&solveG6$<#FZ<$FAFB-%(unapplyG6$-%%subsG6$F_o-F46#8%F]pF1F1F1" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "V maple je vestavena funkce " } {HYPERLNK 17 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E\"F\\x$\"3^Xq\"[b_A/$F\\x7$$\"3;LLLU9A*H\"F\\x$\"3!y%yX:R38JF\\x7$$\" 3\"******H\"H)GL\"F\\x$\"3AcjX'4Br=$F\\x7$$\"3GLLL`Jzl8F\\x$\"3AvD,H*4 &fKF\\x7$$\"3$******\\7Z-S\"F\\x$\"3O\\kBMAHNLF\\x7$$\"3rmmm%RIMV\"F\\ x$\"3bkw5,'z#3MF\\x7$$\"3immm!3ltY\"F\\x$\"3+eC`D(>H[$F\\x7$$\"3JLLLq( =5]\"F\\x$\"3aG?/#zTpb$F\\x7$$\"3'******f,V>`\"F\\x$\"3K\"*)HNQg\\i$F \\x7$$\"3KLLL\"p&Qn:F\\x$\"3h$4p_OIksPF\\x7$$\"3/+++H_)Gj\"F\\x$\"3wl<&)3^)p%QF\\x7$$\"3/+++j`Bl;F\\x $\"3l07hu*R\"=RF\\x7$$\"# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Komparace aproximaci" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Aproximujeme funkci f na interva lu " }{XPPEDIT 18 0 "`<,>`(alpha,beta);" "6#-%$<,>G6$%&alphaG%%betaG" }{TEXT -1 201 " . Stupen aproximace je N: to znamena v pripade aproxi mace polynomeme bude mit tennto stupen n, u Fourierovy rqady vezmeme n -ty castecny soucet. Porovcname kvalitu aproximace ve dvou metrik\341c h. rvn\355: " }{XPPEDIT 18 0 "rho(f,g)=Int(abs(f-g),w=alpha..beta);" " 6#/-%$rhoG6$%\"fG%\"gG-%$IntG6$-%$absG6#,&F'\"\"\"F(!\"\"/%\"wG;%&alph aG%%betaG" }{TEXT -1 10 " a druha: " }{XPPEDIT 18 0 "rho(f,g) = Int(ab s(f-g)+abs(diff(f-g,w)),w = alpha .. beta);" "6#/-%$rhoG6$%\"fG%\"gG-% $IntG6$,&-%$absG6#,&F'\"\"\"F(!\"\"F1-F.6#-%%diffG6$,&F'F1F(F2%\"wGF1/ F9;%&alphaG%%betaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "readlib(mtaylor);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#f*6\"6*%\"fG%\"kG%\"vG%\"mG%\"nG%\"sG%\"tG%\"wG6#%aoCopyright~(c)~19 91~by~the~University~of~Waterloo.~All~rights~reserved.GF$C2>8$&9\"6#\" \"\">8&&F46#\"\"#@&-%%typeG6$F8%$setG>F87#-%#opG6#F84-F>6$F8%%listG>F8 7#F8@$4-F>6$F8-FI6#<$%%nameG/FS%*algebraicGYQGinvalid~2nd~argument~(ex pansion~point)F$>8)-%$mapG6$f*6#%\"xGF$F$F$@%-F>6$9$%\"=G-%$rhsG6#F]o \"\"!F$F$F$F8>F8-Fen6$f*FhnF$F$F$@%F[o-%$lhsGFaoF]oF$F$F$F8>8'-%%nopsG FE@$0F[p-F]p6#<#FCYQHvariables~(2nd~argument)~must~be~uniqueF$@%/9#F;> 8(%&OrderG>Fip&F46#\"\"$@%/Fgp\"\"%>8+&F46#Faq>Fcq7#-%\"$G6$F6F[p@$4-F >6$F8<$-F@6#FS-FIFarYQQ2nd~argument~(the~variable(s))~must~be~a~name(s )F$@$34-F>6$Fip%*nonnegintG0Fip%)infinityGYQX3rd~argument~(the~order)~ must~be~a~non-negative~integerF$@$54-F>6$Fcq-FI6#%'posintG0-F]p6#FcqF[ pYQen4th~argument~(weights)~must~be~a~list~of~positive~integersF$>F2-% %subsG6$7#-%$seqG6$/&F86#8%,&*&FetF6)8*&FcqFftF6F6&FYFftF6/Fgt;F6F[pF2 >F2-Fen6&%(collectG-F^t6$/-%\"OGF5Fbo-%'taylorG6%F2F[uFipF8.%,distribu tedG>F2-F^t6$7#-Fbt6$/Fet,&FetF6F]u!\"\"F^u-F^t6$/F[uF6F2F$F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "unassign('x');\nf:=x->exp(x) *log(x);\nN:=5;\nalpha:=1;\nbeta:=2;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arr owGF(*&-%$expG6#9$\"\"\"-%$logGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alpha G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "Porovnani:=proc(f,N,alpha,beta)\nl ocal x,y,w,n,i,A1,A2,A3;\nA1:=unapply(evalf(mtaylor(f(w),w=(alpha+beta )/2,N+1)),w); # Tayloruv polynom ve stredu iontervalu alpha.. beta" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "x:=seq(alpha+(alpha+beta)/N*i,i=0.. N);\ny:=seq(evalf(f(x[i])),i=1..N+1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A2:=(Lagrange([x],[y]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 245 " phi:=x->x+alpha:\npsi:=x->x+alpha:\nA3:=unapply((Fourier((f@phi),N,bet a-alpha))('x'-alpha),'x');\nprint(`aproximujeme funkci`,f,`temito funk cemi:`);\nprint(\nsort(normal(A1('x')),'x'),`\\n`,\nsort(normal(A2('x' )),'x'),`\\n`,\nsort(normal(A3('x')),'x')\n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "print(\nplot(\{\nf(w)\n,A1(w)\n,A2(w)\n,A3(w)\n\},w=a lpha..beta)\n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "r1:=simplify(eva lf(f(w)-A1(w)));\nr2:=simplify(evalf(f(w)-A2(w)));\nr3:=simplify(evalf (f(w)-A3(w)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "print(`Vzdalenost aproximaci od puvodni funkcer v prvni a ve druhe metrice:`);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "[evalf(Int(abs(r1),w=alpha..beta)) ,\nevalf(Int(abs(r2),w=alpha..beta)),\nevalf(Int(abs(r3),w=alpha..beta ))],\n[evalf(Int(abs(r1)+abs(diff(r1,w)),w=alpha..beta)),\nevalf(Int(a bs(r2)+abs(diff(r3,w)),w=alpha..beta)),\nevalf(Int(abs(r3)+abs(diff(r2 ,w)),w=alpha..beta))];\nend;" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, \+ `phi` is implicitly declared local to procedure `Porovnani`\n" }} {PARA 7 "" 1 "" {TEXT -1 69 "Warning, `psi` is implicitly declared loc al to procedure `Porovnani`\n" }}{PARA 7 "" 1 "" {TEXT -1 68 "Warning, `r1` is implicitly declared local to procedure `Porovnani`\n" }} {PARA 7 "" 1 "" {TEXT -1 68 "Warning, `r2` is implicitly declared loca l to procedure `Porovnani`\n" }}{PARA 7 "" 1 "" {TEXT -1 68 "Warning, \+ `r3` is implicitly declared local to procedure `Porovnani`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*PorovnaniGf*6&%\"fG%\"NG%&alphaG%%betaG6/ %\"xG%\"yG%\"wG%\"nG%\"iG%#A1G%#A2G%#A3G%$phiG%$psiG%#r1G%#r2G%#r3G6\" F9C1>8)-%(unapplyG6$-%&evalfG6#-%(mtaylorG6%-9$6#8&/FI,&*&#\"\"\"\"\"# FN9&FNFN*&FMFN9'FNFN,&FNFN9%FNFI>8$-%$seqG6$,&FPFN*(,&FPFNFRFNFNFT!\" \"8(FNFN/Fhn;\"\"!FT>8%-FX6$-FA6#-FG6#&FV6#Fhn/Fhn;FNFS>8*-%)LagrangeG 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