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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Vr&G9$! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%J============================= ============G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f or i from 1 to 15 do\nx:=Pi;\nx:=evalf(x,i):\ny:=Zaokr(x,i^2);\n(x=con vert(x,rational,i))=(y=evalf(y,i)), `+`/`-`,1/i^2;\n#print(B[1],B[2],B [3]);\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%//$\"\"$\"\"!F&/F &F%*&%\"+G\"\"\"%\"-G!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG %#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"#J!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% //$\"#J!\"\"\"\"$/F($F(\"\"!*&%\"+G\"\"\"%\"-GF'#F.\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"xG$\"$9$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG#\"#A\"\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%//$\"$9$!\"##\"#A\"\"(/F(F%*&%\"+G\" \"\"%\"-G!\"\"#F.\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"%UJ!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG#\"#A\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% //$\"%UJ!\"$#\"#A\"\"(/F($\"%VJF'*&%\"+G\"\"\"%\"-G!\"\"#F0\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"&;9$!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"yG#\"#A\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%//$\"&;9$!\"%#\"$b$ \"$8\"/#\"#A\"\"($\"&H9$F'*&%\"+G\"\"\"%\"-G!\"\"#F3\"#D" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"xG$\"'fTJ!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG#\"#A\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%//$\"'fTJ!\"&#\"$b$\"$8\"/#\"#A\"\"( $\"''G9$F'*&%\"+G\"\"\"%\"-G!\"\"#F3\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"($fTJ!\" '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG#\"#A\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%//$\"($fTJ!\"'#\"$b$\"$8\"/#\"#A\"\"($\"(dG9$F'*&% \"+G\"\"\"%\"-G!\"\"#F3\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG %#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\")FfTJ!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG#\"$,#\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%//$\")FfTJ!\"(#\"$b$\"$8\"/#\"$,#\"#k$\")]iSJF'*&%\"+G \"\"\"%\"-G!\"\"#F3F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"*l#fTJ!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG#\"$X#\"#y" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%//$\"*l#fTJ!\")#\"'tD5\"&]E$/#\"$X#\"#y$\"*kD59$F'*&%\"+G\"\"\"%\"-G !\"\"#F3\"#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+aEfTJ!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"yG#\"$6$\"#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% //$\"+aEfTJ!\"*#\"'[V5\"&:K$/#\"$6$\"#**$\"+TTTTJF'*&%\"+G\"\"\"%\"-G! \"\"#F3\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\",Ol#fTJ!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(5419351/1725033,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0#)*e`EfTJ!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "\332kol: napsat rychlej\232\355 proceduru!" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%)binomialG6$\"#5%\"rG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Zaokr(.4827586207,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Zaokr(0.1,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(5/7-0.71);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")Vr&G%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "abs(evalf(7/10-0.71));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*++++\"!#5" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Interpol ace polynomy" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "polynom 1. stupn \354: p\370\355mka" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unassi gn('x','y','p','q');" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "p\370\355k lad:" }}}{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 proc h\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\355mk y, kter\341 nen\355 svisl\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 do n\355 oba body a vy\370e\232\355me ro vnice pro nezn\341m\351 parametry:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "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\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%r ceAG/\"\"\",&*&$\"\"&!\"\"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 "Nzn\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 28 "Tot\351\236 m\371\236eme ud \354lat obecn\354. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "read lib(unassign):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "unassign( 'A','B','p','q'); # Zru\232\355me p\370i\370azen\355 n\354kter\375ch h odnot prom\354n\375" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "prim ka;" }}{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 vynikne dosazenim prvniho bodu\nrceB: =subs(x=B[1],y=B[2],primka); #druha dosazenim druheho\nparam:=solve(\{ rceA,rceB\},\{p,q\}); #vyresime obe rovnice vzhledem k promenym p a q \nassign(param); #timto prikazem promenym p a q priradime hodnoty" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rceAG/&%\"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<$/%\"pG*&,&&%\"AG6#\"\"#!\"\"&%\"BGF,\"\"\"F1,&&F+6#F1 F.&F0F4F1F./%\"qG,$*&,&*&F/F1F3F1F1*&F5F1F*F1F.F1F2F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*(,&&%\"AG6#\"\"#!\"\"&%\"BGF*\"\"\"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 -1 140 "To \236e jsme museli cel\375 v\375po\350 et opakovat n\341s vede k opr\341vn\354n\351 domn\354nce, \236e by byl o v\375hodn\351 jej zachovat pro dal\232\355 pou\236it\355 v podob\354 procedury." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 518 "line:=proc( A,B) # jm\351nu line je nyni prirazena procedura se dvema parametry. p rocedura\nlocal x; # deklarace lokalnich promenych. lokalni promene si po provedeni procedury uchovaji tytez hodnoty jake mely pred jejim sp ustenim \n if A[1]=B[1] then # vyloucime pripad svisle primky\n p rint(`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 unappl y zajisti, ze vysledek bude funkce (nezavisla na oznaceni promenych)\n fi;\nend; # konec procedury" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%line Gj+6$%\"AG%\"BG6#%\"xG6\"F+@%/&9$6#\"\"\"&9%F0-%&printG6#%Aprimka~je~s visla,~nejde~o~funkciG-%(unapplyG6$,&*(,&&F36#\"\"#F1&F/F?!\"\"F1,&F2F 1F.FBFB8$F1F1*&,&*&FAF1F2F1FB*&F.F1F>F1F1F1FCFBFBFDF+F+F+6$\"\"!FJ" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 259 " Pomoc\355 t\351to procedury 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 %%PlinGj+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$F 7&F?6#F1-%(unapplyG6$-%*piecewiseG6#F7F:F.F.F.6$\"\"!F[o" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Polynomy vy\232\232\355ch stup\361\371" } }{EXCHG {PARA 5 "" 0 "" {TEXT -1 18 "Lagrange\371v polynom" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "P\370\355klad. Uva\236ujme body:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "unassign('X','Y');\nn:=5;\n for i from 1 to n do\nX[i]:=i;\nY[i]:=evalf(ln(i));\nod:\ni:='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 78 "Hled\341me polynom, kter\375 proch \341z\355 v\232emi t\354mito body. Najdeme jej ve tvaru sou\350tu" }} {PARA 0 "" 0 "" {TEXT -1 9 "polynom\371 " }{XPPEDIT 18 0 "P[i];" "6#&% \"PG6#%\"iG" }{TEXT -1 21 " s touto vlastnostmi:" }}{PARA 15 "> " 0 " " {MPLTEXT 1 0 38 "P[i](x[j]) = piecewise(i <> j,0,y[i]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"PG6#%\"iG6#&%\"xG6#%\"jG-%*PIECEWISEG6$7$ \"\"!0F(F-7$&%\"yGF'%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "for j from 1 to n do\ncitatel[j]:=simplify(product(x- X[i],i=1..n)/(x-X[j])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "jmenovat el[j]:=subs(x=X[j],citatel[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " P[j]:=citatel[j]/jmenovatel[j]*Y[j];\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(citatelG6#\"\"\"**,&%\"xGF'\"\"#!\"\"F',&F*F'\"\"$F ,F',&F*F'\"\"%F,F',&F*F'\"\"&F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%+jmenovatelG6#\"\"\"\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6 #\"\"\"$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(citatelG6#\"\"# **,&%\"xG\"\"\"F+!\"\"F+,&F*F+\"\"$F,F+,&F*F+\"\"%F,F+,&F*F+\"\"&F,F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%+jmenovatelG6#\"\"#!\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6#\"\"#,$*,$\"+,`Cb6!#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#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6#\"\"$,$*,$\"+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+\"\"$F,F+,&F*F+\"\"&F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%+jmenovatelG6#\"\"%!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6# \"\"%,$*,$\"+-1\\5B!#5\"\"\",&%\"xGF-F-!\"\"F-,&F/F-\"\"#F0F-,&F/F-\" \"$F0F-,&F/F-\"\"&F0F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(citate lG6#\"\"&**,&%\"xG\"\"\"F+!\"\"F+,&F*F+\"\"#F,F+,&F*F+\"\"$F,F+,&F*F+ \"\"%F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%+jmenovatelG6#\"\"&\"# C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6#\"\"&,$*,$\"++8*fq'!#6\" \"\",&%\"xGF-F-!\"\"F-,&F/F-\"\"#F0F-,&F/F-\"\"$F0F-,&F/F-\"\"%F0F-F- " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A tedy hledan\375 polynom je: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Lagrange:=(sum(P[i],i=1..n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)LagrangeG,**,$\"+,`Cb6!#5\"\"\",& %\"xGF*F*!\"\"F*,&F,F*\"\"$F-F*,&F,F*\"\"%F-F*,&F,F*\"\"&F-F*F-*,$\"+A 2`YFF)F*F+F*,&F,F*\"\"#F-F*F0F*F2F*F**,$\"+-1\\5BF)F*F+F*F7F*F.F*F2F*F -*,$\"++8*fq'!#6F*F+F*F7F*F.F*F0F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "neboli" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Lagrange:=simplify( Lagrange);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)LagrangeG,,*&$\"++^gg [!#7\"\"\")%\"xG\"\"%F*!\"\"*&$\"+5dD#p(!#6F*)F,\"\"$F*F**&$\"+8DhQ[!# 5F*)F,\"\"#F*F.*&$\"+6@=z;!\"*F*F,F*F*$\"+7GQn7F>F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Op\354t m\371\236eme napsat proced\371ru, kter \341 polynom vytvo\370\355. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "Lagrange:=proc(X,Y) # Vstup: pole bodu a pole hodnot. Vystup: L agrangeuv polynom (funkce)\nlocal C,J,N,x,i,j,substitut,xxx;\ni:='i'; \nXpom:=convert(X,list);\nN:=nops(Xpom);\nfor j from 1 to N do\nC[j]:= product((x-X[i]),i=1..N)/(x-X[j]);\nJ[j]:=subs(x=X[j],C[j]);\n#print(C [j],J[j]);\nod;\ni:='i';\nxxx:=sum(C[i]/J[i]*Y[i],i=1..N);\nunapply(xx x,x)\nend;" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, `Xpom` is implicit ly declared local to procedure `Lagrange`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)LagrangeGj+6$%\"XG%\"YG6+%\"CG%\"JG%\"NG%\"xG%\"iG% \"jG%*substitutG%$xxxG%%XpomG6\"F3C)>8(.F6>8,-%(convertG6$9$%%listG>8& -%%nopsG6#F9?(8)\"\"\"FFF@%%trueGC$>&8$6#FE*&-%(productG6$,&8'FF&F=6#F 6!\"\"/F6;FFF@FF,&FRFF&F=FLFUFU>&8%FL-%%subsG6$/FRFYFJ>F6F7>8+-%$sumG6 $*(&FKFTFF&FfnFTFU&9%FTFFFV-%(unapplyG6$F]oFRF3F3F36$\"\"!Fjo" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "nebo ekvivaletn\354 s vzu\236it \355m ji\236 hotov\351 procedury " }{HYPERLNK 17 "interp" 2 "interp" " " }{TEXT -1 32 " v maple (interpolce polynomem)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Lagrange2:=proc(X,Y)\nunapply(interp(convert( X,list),convert(Y,list), x),x)\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*Lagrange2Gj+6$%\"XG%\"YG6\"F)F)-%(unapplyG6$-%'interpG6%-%(conver tG6$9$%%listG-F16$9%F4%\"xGF8F)F)F)6$\"\"!F:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Splainy" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "Nev \375hoou Lagrangeova polynomu je to, 6e m8me-li mnoho bod\371, dostabn eme polznom p\370\355li\232 vysok\351ho stzupn\354. Nev\375hodou po \+ \350\341stech afinn\355\355 aproximace je v tom, \236e dostaneme funkc i, kter\341 nem\341 v bodech, jimi\236 ji prokl\341d\341me derivaci. N ajdeme aproximaci po \350\341stech polynomialn\355 funkc\355, kter\341 bude m\355t v ka\236d\351m bod\354 derivaci. Jako p\370\355klad uvede me 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 dru h\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 "Polynom druh\351ho stupn\354 zavis\355 na t\370ech parametrech. Hled\341me takov\351 hod noty parmetr\371, aby m\354l polynom p\370edepsan\351 hodnoty v bodec h " }{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#&%\"xG 6#%\"jG" }{TEXT -1 49 ". Polynom mus\355 spl\362ovat n\341sleduj\355c \355 t\370i rovnice:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "dd d:=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*%\"cGF*&%\"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:F 0F0&FF0F7*(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],\nsubs(x=x[2],p)=y[2],\nsu bs(x=x[3],p)=y[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4PrvniSoustava RovnicG6%/,(*&%\"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 "kt er\351 maj\355 \370e\232en\355:" }{MPLTEXT 1 0 38 "\nsolve(\{PrvniSous tavaRovnic\},\{a,b,c\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/%\"aG*& ,.*&&%\"xG6#\"\"\"F,&%\"yG6#\"\"$F,!\"\"*&&F*F/F,&F.F+F,F,*&&F*6#\"\"# F,F-F,F,*&F6F,F4F,F1*&F)F,&F.F7F,F,*&F3F,F;F,F1F,,.*&F6F,)F)F8F,F1*&F6 F,)F3F8F,F,*&F3F,F?F,F,*&F)F,FAF,F1*&F)F,)F6F8F,F,*&F3F,FEF,F1F1/%\"bG ,$*&,.*&FAF,F4F,F,*&FAF,F;F,F1*&FEF,F4F,F1*&F-F,FEF,F,*&F-F,F?F,F1*&F; F,F?F,F,F,F=F1F1/%\"cG*&,.*(FAF,F6F,F4F,F,*(FAF,F)F,F;F,F1*(FEF,F3F,F4 F,F1*(F;F,F3F,F?F,F,*(F-F,F6F,F?F,F1*(FEF,F)F,F-F,F,F,F=F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Nyn\355 ji\236 hledan\341 procedura:" }}} {EXCHG {PARA 0 "> " 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 procha zet prvnimi trema body\n\nQ:=z%%Spl2Gj+6$%\"xG%\"yG6-%\"NG%\"KG%\"j G%\"aG%\"bG%\"cG%\"QG%\"zG%\"pG%\"iG%$derG6\"F5C,>8$-%%nopsG6#9$>8%,&F 8\"\"\"\"\"#!\"\">8',$*&,.*&&F<6#\"\"$F@&9%6#F@F@F@*&&F8(*&,.*&FgnF@FQF@FB*&FgnF@FWF@F@*&FjnF@FQF@F@*&FLF@FenF@F@*&FL F@FjnF@FB*&FWF@FenF@FBF@FYFB>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@F YFB>&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@FeqF@F@F@FirFB>Fho*&,,*(FerF@FhrF@FeqF@F@*&Fe rF@FarF@F@*(FhqF@FeqF@FgrF@FB*&F_rF@FgrF@F@**FAF@F_rF@FhrF@FhqF@FBF@Fi rFB>&Fcp6#F^qFdp@%2F^qF>>Fjp6%Fjp2FgpFhrF[t>Fjp6$FjpF[t>Fjp-%*piecewis eG6#Fjp-%(unapplyG6$FjpFgpF5F5F56$\"\"!F\\u" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 8 "Priklad:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 " with(plots):\nN:=7;f:=x->ln(x)*sin(x);\nx:=seq(round(2*i+sin(i)^2),i=1 ..N);\ny:=seq(evalf(f(x[i])),i=1..N);x[N];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGj+ 6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%#lnG6#9$\"\"\"-%$sinGF/F1F(F(F(6# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG6)\"\"$\"\"&\"\"'\"\"* \"#6\"#7\"#9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG6)$\"+^ " 0 "" {MPLTEXT 1 0 169 "Body:=pointplot([seq([x[i],y[i]],i=1..N)], color=NAV Y,symbolsize=15):\nAproximace:=plot(Spl2([x],[y])(z),z=x[1]..x[N],colo r=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$$\"3um;a3!GU:%F_o$!3>:o 4RhtS9F_o7$$\"33+]iI%)=jUF_o$!3?)RgCc\"e/:F_o7$$\"3UL$3F&)[@P%F_o$!3m^ >9BIX`:F_o7$$\"3Wm;ae5(\\[%F_o$!3cKF-grF)e\"F_o7$$\"3Y**\\PkKz(f%F_o$! 3f8g63r/2;F_o7$$\"3Mm;/^TZ9ZF_o$!3.;xcA=d4;F_o7$$\"3@L$3x.b6$[F_o$!37! 3&f;a#\\f\"F_o7$$\"3cm;HF&=&F[o7$$\"3#*****\\UE&)=iF_o$!3+B]ck!)pO:F[o7$$\"3'H L3x[JtU'F_o$\"3_#Ropxo3T\"F[o7$$\"3+nm;**HBvmF_o$\"3!=x2S& f=zFB\"F_o7$$\"3unmTg(GY/)F_o$\"3L#=d\"o<0U7F_o7$$\"3!p;zp&)eG:)F_o$\" 3&eVgipT2C\"F_o7$$\"31m;a`*)3h#)F_o$\"3FMNb:[-I7F_o7$$\"3')*\\iStHyP)F _o$\"3J\\5uh(>z?\"F_o7$$\"3oLLe90d%\\)F_o$\"3YKkq@.([<\"F_o7$$\"3xJ$3x B#4P()F_o$\"3wq!)o7uAr5F_o7$$\"3)***\\i5\"3#[*)F_o$\"3()Q9t_a^D%*F[o7$ $\"3ULL3P!>i<*F_o$\"3#R2]WLQih(F[o7$$\"3&*)****\\jw)**\\()yBAk*F_o$\"33&o4Pt&fSCF[o7$$\"3O**\\PfK>l)*F_o$!3=!y 8fXfCR(!#>7$$\"3&***\\7%Gw7,\"!#;$!3Ca&4YW[L![F[o7$$\"3fmm;7:_L5Fjz$!3 :iF2i%4R$*)F[o7$$\"3/++DTIFd5Fjz$!3Ss@\"fSVWQ\"F_o7$$\"3m;/,\\U.o5Fjz$ !3&H)*3qcLRi\"F_o7$$\"3GL3xcazy5Fjz$!3%=^y\"o$HS(=F_o7$$\"3g;/E*Hg04\" Fjz$!3uc^j\"f&ef@F_o7$$\"34++vT^K-6Fjz$!3Au5'=!pgbCF_o7$$\"3C$3x;%\\R8 6Fjz$!3QAql3AiwEF_o7$$\"3dmTgTZYC6Fjz$!3hk,Z'eA'3GF_o7$$\"3C$eRU_dt7\" Fjz$!3+jw;LqWGGF_o7$$\"3#***\\(oI]-8\"Fjz$!3=@23@F>UGF_o7$$\"3g;/^*3VJ 8\"Fjz$!3cm&4-lf)\\GF_o7$$\"3GLe9se.O6Fjz$!35*>a0#yW^GF_o7$$\"3'*\\7ya 'G*Q6Fjz$!390X6Ks&p%GF_o7$$\"3kmmTP9#=9\"Fjz$!3l%[!*[)yQOGF_o7$$\"3K$3 _+A9Z9\"Fjz$!3?3B))y(R(>GF_o7$$\"3++vo-qgZ6Fjz$!3a\"p*39H,(z#F_o7$$\"3 ?$3-y*)>*e6Fjz$!3I6Z^XVw\\EF_o7$$\"3emm\"HzK-<\"Fjz$!3jB\"RkM[&4CF_o7$ $\"3A$3xcMw?=\"Fjz$!39e4zoMXe?F_o7$$\"3')*\\P%)*)>R>\"Fjz$!38<:v(*[Y0; 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beta:=7; F:=1/1 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "N:=ceil(-(alpha-beta)/F);\nx :=seq(alpha+j*F,j=0..N);\ny:=seq(evalf(f(x[i])),i=1..N+1);\nPuvB:=poin tplot([seq([x[i],y[i]],i=1..N+1)]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&alphaG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG#\"\"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG67 \"\"&#\"#^\"#5#\"#EF&#\"#`F)#\"#FF&#\"#6\"\"##\"#GF&#\"#dF)#\"#HF&#\"# fF)\"\"'#\"#hF)#\"#JF&#\"#jF)#\"#KF&#\"#8F2#\"#LF&#\"#nF)#\"#MF&#\"#pF )\"\"(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG67$!+.Dbq9!#5$\"+i]FP8 !\"*$\"+_`2eDF+$\"+HH(*=KF+$\"+&yF2;$F+$\"+&4?-S#F+$\"+ltJE6F+$!+1#pOY $F($!+`;]a;F+$!+DU*\\Z#F+$!+s[4/EF+$!+= " 0 "" {MPLTEXT 1 0 61 "F:=1/7;\nN:=floor(-(alpha-beta)/F);\nz:=seq(alph a+j*F,j=0..N);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG#\"\"\"\"\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG61\"\"&#\"#O\"\"(#\"#PF)#\"#QF)#\"#RF)#\"#SF)#\"# TF)\"\"'#\"#VF)#\"#WF)#\"#XF)#\"#YF)#\"#ZF)#\"#[F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "LinA:=seq(Plin([x],[y])(z[i]),i=1..N+1); \nBilA:=seq(Spl2([x],[y])(z[i]),i=1..N+1);\n#PolA:=seq(Lagrange([x],[y ])(z[i]),i=1..N+1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%LinAG61$!)Eb q9!\")$\"*8v/'=F($\"*/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=BLU E):\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,symbol=cross,symbolsize=20):\nLB:=poi ntplot(\n [seq(\n [z[i],LinA[i]]\n ,i=1..N+1\n \+ )],\ncolor=RED,symbol=box,symbolsize=20):\nBB:=pointplot(\n [seq( \n [z[i],BilA[i]]\n ,i=1..N+1\n )],\ncolor=GREEN,s ymbol=diamond,symbolsize=20):\nPB:=pointplot(\n [seq(\n [z[i] ,PolA[i]]\n ,i=1..N+1\n )],\ncolor=BLUE,symbol=circle,s ymbolsize=20):\ndisplay([k1,k2,k3,PuvB],title=`funkce (cerne) a jeji a proximace`);\n\n\ndisplay(\{k1,RB\},title=`Skutecne hodnoty`);\ndispla y(\{k2,LB\},title=`hodnoty ziskane z linerani aproximace`);\ndisplay( \{k3,BB\},title=`hodnoty ziskane z aproximace kubickymi splainy`);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(\{\nRB\n,\nLB\n,BB\n#,PB \n\},title=`ruzne hodnoty pro prevzorkovani`);" }}{PARA 8 "" 1 "" {TEXT -1 53 "Error, (in plots/pointplot) incorrect first argument\n" } }{PARA 13 "" 1 "" {GLPLOT2D 921 449 449 {PLOTDATA 2 "6)-%'CURVESG6$7cp 7$$\"\"&\"\"!$!3=\">LH]_0Z\"!#=7$$\"3Rmm;arz@]!#<$\"3^w#Hf&zC4=F-7$$\" 3mLLL3VfV]F1$\"31]aZ9#*f(4&F-7$$\"3?+]i&*)fD1&F1$\"364:o7xbNzF-7$$\"3s mm\"H[D:3&F1$\"3'*HR:07`s5F17$$\"3U**\\PpU&G5&F1$\"3E5)[2A9uP\"F17$$\" 3+LL$e0$=C^F1$\"3a6\"pva9'p;F17$$\"3ULLLBKlX^F1$\"3!)>tCevgZ>F17$$\"3% HLL3RBr;&F1$\"3_,m3J^B1AF17$$\"3%om;zjf)4_F1$\"3wo*=A^f>l#F17$$\"3WLLe 4;[\\_F1$\"3!\\J&\\4G')oHF17$$\"3'pm;zt%**p_F1$\"3c;6gpD;#4$F17$$\"3e* **\\i'y]!H&F1$\"3Qjr'zl*)e=$F17$$\"34LLe>_6,`F1$\"3O7XtOoUAKF17$$\"3hm m\"HdAiQm$Q@G$F17$$\"3Y+++:1^V`F1$\"3sZ(p&4kB&G$F17$$\"3InmT ]R3a`F1$\"3mZC+3[&*zKF17$$\"3CLL$eGdYO&F1$\"35'[c-[6jE$F17$$\"31++D@1B v`F1$\"3^jnx:xMWKF17$$\"3[L$e9d#)pR&F1$\"3C[R7!e(HtJF17$$\"3\"pmm;_M(= aF1$\"3%*\\!ye&**>oIF17$$\"37LL$3y_qX&F1$\"3an$*=@-M/GF17$$\"3'****** \\1!>+bF1$\"3U-OmI>?)R#F17$$\"3w*****\\Z/Na&F1$\"3#R16b_]F*=F17$$\"3m* ****\\$fC&e&F1$\"3L`@wI()HN8F17$$\"3Qm;Hd&)>/cF1$\"3Y=5e>)*)f1\"F17$$ \"36LLez6:BcF1$\"3')4`&3s3%**yF-7$$\"3[+]i!o(oXcF1$\"3_fW2&**fPc%F-7$$ \"3'pmm;=C#ocF1$\"3.EcPW\\v57F-7$$\"3Amm;abJ(o&F1$!31i=*z7N9h\"F-7$$\" 3QmmmEpS1dF1$!3S$*)=K4H%*Q%F-7$$\"3LL$e9t9'GdF1$!31$p6AT.:`(F-7$$\"3G+ +DOD#3v&F1$!3grhl\"GHS0\"F17$$\"3/M$ek?![qdF1$!3$z%Qn`SB18F17$$\"3!pmm m(y8!z&F1$!3h\"Qdy+3Ea\"F17$$\"3[++DOIFLeF1$!3OX#R#f_x%*>F17$$\"3!**** \\(3zMueF1$!3NseO2J+DBF17$$\"3YL$e*olx&*eF1$!3[Iww_:V`CF17$$\"3-nm;H_? 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Ykuste tot\351\236 pro funkci:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:=x->ln(x)^2+sin(3*x)-x*sin(5*x)+cos(20*x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGj+6#%\"xG6\"6$%)operatorG%&arr owGF(,**$)-%#lnG6#9$\"\"#\"\"\"F4-%$sinG6#,$*&\"\"$F4F2F4F4F4*&F2F4-F6 6#,$*&\"\"&F4F2F4F4F4!\"\"-%$cosG6#,$*&\"#?F4F2F4F4F4F(F(F(6#\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 0 {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 0 {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(6#\"\"\"F0&%\"fGF/F0!\"\"*&&F(6#\"\"#F0&F2F6F0F3*&&F(6# \"\"$F0&F2F;F0F3&%\"gGF/F0F0&F2F)F3/F5F5/F.F./F:F:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$xxxG<&/&%\"aG6#\"\"%,$*&,**&&F(6#\"\"\"F1&%\"fGF0F 1F1*&&F(6#\"\"#F1&F3F6F1F1*&&F(6#\"\"$F1&F3F;F1F1&%\"gGF6!\"\"F1&F3F)F @F@/F5F5/F/F//F:F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG<&/&%\"aG 6#\"\"#F'/&F(6#\"\"%,$*&,**&&F(6#\"\"\"F5&%\"fGF4F5F5*&F'F5&F7F)F5F5*& &F(6#\"\"$F5&F7F%$xxxG<&/&%\"aG6#\"\"$,$*&,**&&F(6#\"\"\"F1&%\" fGF0F1F1*&&F(6#\"\"#F1&F3F6F1F1*&&F(6#\"\"%F1&F3F;F1F1&%\"gGF;!\"\"F1& F3F)F@F@/F:F:/F5F5/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Aproximace line\341rn\355 \+ kombinac\355 funkc\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "\332loh a aproximovat funkci line\341rn\355 kombinac\355 zadan\375ch funkc\355 :\nPokud jsme o fourierovu \370adu, nap\370\355klad, vol\355me p\370 \355slu\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. Vystup: prvnich n scitancu fourierovy rady (pocitaji se i nul ove). -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*eva lf(int(f(x)*cos(2*i*Pi*x/beta),x=0..beta)));\n b[i]:=evalf(2/beta*eva lf(int(f(x)*sin(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*P i*x/beta),i=1..N-1),x)\nelse\n unapply(a[0]/2+sum(a[i]*cos(2*i*Pi*x/b eta)+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 de clared 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 procedure `Fourier`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Fourie rGj+6%%\"FG%\"nG%%betaG6(%\"iG%\"NG%\"xG%\"fG%\"aG%\"bG6\"F1C(>8'-9$6# 8&@%/-%$modG6$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#-%$in tG6$*&-F4F7F@-%$cosG6#,$*,F?F@FJF@%#PiGF@F8F@FYFZF@F@/F8;FMFYF@F@>&8)F S-FU6#,$*(F?F@FYFZ-FU6#-Fhn6$*&F[oF@-%$sinGF^oF@FboF@F@>FJFK@%F:-%(una pplyG6$,&*&FDF@&FR6#FMF@F@-%$sumG6$,&*&FQF@F\\oF@F@*&FeoF@F`pF@F@/FJ;F @,&FBF@F@FZF@F8-Fep6$,(*&FDF@FipF@F@F[qF@*&&FR6#FBF@-F]o6#,$*,F?F@FBF@ FaoF@F8F@FYFZF@F@F@F8F1F1F16%FMFMFM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "\332loha aproximovat hodnoty hodnotami line\341rn\355 kombinac e funkc\355:\nPokud nechceme nejprve syst\351m funkc\355 respektive ve ktory jejich hodnot nahradit ortonorm\341ln\355m syst\351mem, mus\355m e poka\236d\351 \370e\232it soustavu line\341rn\355ch rovnic pro koefi cienty:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 601 "Aproximace:=pro c(X,Y,F) #vstup: pole bodu, pole hodnot, pole funkci. Vystup: linearni kombinace zadanych funkci, ktera nejlepe aproximuje zadane hodnoty.\n local i,j,k,PX,PF,v,Koeficienty,MaticeSoustavy,PraveStrany\n;\nPF:=nop s(F); # Pocet 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);\nMaticeSous tavy:=\nmatrix(PF,PF,\n[[sum((v[i][k]*v[j][k]),k=1..PX) $j=1..PF] $i=1 ..PF]\n);\nPraveStrany:=vector([sum(Y[j]*v[i][j],j=1..PX) $i=1..PF]); \ni:='i';\nKoeficienty:=linsolve(MaticeSoustavy,PraveStrany):\nunapply (sum(Koeficienty[i]*F[i],i=1..PF),x);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+AproximaceGj+6%%\"XG%\"YG%\"FG6+%\"iG%\"jG%\"kG%#PXG %#PFG%\"vG%,KoeficientyG%/MaticeSoustavyG%,PraveStranyG6\"F4C->8(-%%no psG6#9&>8'-F96#9$>8$.FB>8%.FE>8&.FH>8)-%&evalfG6#-%\"$G6$7#-FP6$--%(un applyG6$&F;6#FB%\"xG6#&F@6#FE/FE;\"\"\"F=/FB;F[oF7>8+-_%'linalgG%'matr ixG6%F7F77#-FP6$7#-FP6$-%$sumG6$*&&&FKFZ6#FHF[o&&FKFhnFapF[o/FHFjn/FEF ]oF\\o>8,-_Fbo%'vectorG6#7#-FP6$-F\\p6$*&&9%FhnF[o&F`pFhnF[oFinF\\o>FB FC>8*-_Fbo%)linsolveG6$F_oFgp-FW6$-F\\p6$*&&FgqFZF[oFYF[oF\\oFenF4F4F4 6%\"\"!Fcr\"*)yg`9" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "Metoda ne jmen\232\355ch \350tverc\371" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "H led\341me p\370\355mku, kter\341 m\341 nejmen\232\355 sou\350et \350tv erc\371 vzd\341lenost\355 od zadan\375ch bod\371. Vzd\341lenosti m\354 \370\355me ve svisl\351m sm\354ru. Jde o extrem\341ln\355 \372lohu, k teru vy\370e\232\355me standatn\355 metodou 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#>%'p rimkaG/%\"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:=rhs(primka)-lhs(prim ka)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'PrimkaG/,(*&%\"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 v bodu na primce [x[0],?]. Hled\341m glob\341mn\355 minimum 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%\"qGF4F5F/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:=unapp ly(rho,a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoGj+6$%\"aG%\"bG6 \"6$%)operatorG%&arrowGF)*$*&,(*&9$\"\"\"%\"pGF2F29%!\"\"%\"qGF2\"\"#, &F2F2*$)F3F7F2F2F5#F2F7F)F)F)6$\"\"!F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Soucet ctvercu vzdalenosti:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "i:='i';\nDelta:=Sum(rho(x[i],y[i])^2\n,i=1..N)=sum(rh o(x[i],y[i])^2\n,i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG/-%$SumG6$*&,(*&&%\"xG6#% \"iG\"\"\"%\"pGF0F0&%\"yGF.!\"\"%\"qGF0\"\"#,&F0F0*$)F1F6F0F0F4/F/;F0% \"NG,&*(F " 0 "" {MPLTEXT 1 0 20 "#Delta:=rhs(Delta);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Dp:=Diff(lhs(Delta),p)=simp lify(diff(rhs(Delta),p));\nDq:=Diff(lhs(Delta),q)=simplify(diff(rhs(De lta),q));\n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#DpG/-%%DiffG6$-%$S umG6$*&,(*&&%\"xG6#%\"iG\"\"\"%\"pGF3F3&%\"yGF1!\"\"%\"qGF3\"\"#,&F3F3 *$)F4F9F3F3F7/F2;F3%\"NGF4,$*(F9F3,&*(F?F3)F8F9F3F4F3F3-%$sumG6$,0*&)F /F9F3F4F3F7*(F/F3F%#DqG/-%%DiffG6$-%$SumG6$*&,(*&&%\"xG6#%\"iG\"\"\"%\"p GF3F3&%\"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:F6F4 F6F7F6F6F6)%#_ZG\"\"#F6F6*&,**$)F8FCF6F6*&-F.6$*$)F:FCF6F4F6F7F6F<*$)F -FCF6F<*&-F.6$*$)F0FCF6F4F6F7F6F6F6FBF6F6F,F6F=F " 0 "" {MPLTEXT 1 0 18 "xxx:=all values(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$xxxG6$<$/%\"pG,$*&#\" \"\"\"\"#F,*&,&*&-%$sumG6$&%\"xG6#%\"iG/F7;F,%\"NGF,-F26$&%\"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,FOF,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?F GF,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#>%$pr1G/%\"YG,(*&,&#\"#;\"#6 !\"\"*&\"#))F-\">##\"\"\"\"\"#F2F2%\"XGF2F2#\"$@#\"#AF2*(\"\"*F2\"$w \"F-F0F1F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pr2G/%\"YG,(*&,&#\"#; \"#6!\"\"*&\"#))F-\">##\"\"\"\"\"#F-F2%\"XGF2F2#\"$@#\"#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(pl ots):" }}}{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$$\"\")F)$\"\"%F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq([x[i],y[i]],i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7$\"\"\"\"\"$7$\"\"#F'7$\"\"(\"\"&7$\"\")\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "Reseni:=fsolve(\{rhs(Dp)=0,rhs(Dq)=0\},\{p,q\}#,\{p =-1..-0.10,q=0..10\}\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ReseniG <$/%\"pG$!+82o>K!\"*/%\"qG$\"+@j&))z\"!\")" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 " res_a:=subs(Reseni,primka_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&res_aG/%#yyG,&*&$\"+82o>K!\"*\"\"\"%#xxGF,!\"\"$\"+@ j&))z\"!\")F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "A:=pointpl ot([seq([x[i],y[i]],i=1..N)]):\nB:=plot(rhs(pr1),X=0..10):\nC:=plot(rh s(pr2),X=0..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display (\{A,B,C\});" }}{PARA 13 "" 1 "" {GLPLOT2D 460 460 460 {PLOTDATA 2 "6' -%'CURVESG6$7S7$$\"\"!F)$\"3\\!*o1%)eM-@!#<7$$\"3emmm;arz@!#=$\"3G,1PA c/q@F,7$$\"3[LL$e9ui2%F0$\"3\"z7!o-3&*GAF,7$$\"3nmmm\"z_\"4iF0$\"3;y'Q 'Qe>&H#F,7$$\"39ommT&phN)F0$\"3!oy'p*)*z=O#F,7$$\"3KLLe*=)H\\5F,$\"3M \"=DE?Z#GCF,7$$\"3smm\"z/3uC\"F,$\"3_/Wg/\"y(*[#F,7$$\"3!****\\7LRDX\" F,$\"3w`\\f*y*[`DF,7$$\"3%om;zR'ok;F,$\"32*o_`]!Q>EF,7$$\"33++D1J:w=F, $\"3IM$\\(3*f]o#F,7$$\"3oLLL3En$4#F,$\"3-nw;[#>Ev#F,7$$\"3#pmmT!RE&G#F ,$\"3xZ15^b77GF,7$$\"3D+++D.&4]#F,$\"3[nO$F,7$$\"33++D\"o7Tv$F,$\"3[Wf?/]LoKF,7$$\"3?LLL$Q*o]RF,$\"3%Ho'[ B(*QHLF,7$$\"3m++D\"=lj;%F,$\"3y.,>FkP'R$F,7$$\"3S++vV&RY2aF,$\"3'*>_1>%[=y$F,7$$\"3Znm;zXu9cF,$\"3cC9%)*HGi %QF,7$$\"34+++]y))GeF,$\"3!)4\"fi,RF\"RF,7$$\"3H++]i_QQgF,$\"3yn(\\kx1 y(RF,7$$\"3b++D\"y%3TiF,$\"3,Bh:!=j2/%F,7$$\"3+++]P![hY'F,$\"3cR!*pKbm 5TF,7$$\"3iKLL$Qx$omF,$\"3kw:\\0fZtTF,7$$\"3Y+++v.I%)oF,$\"3CYK0&QR0C% F,7$$\"3?mm\"zpe*zqF,$\"3)Q.H^')38I%F,7$$\"3;,++D\\'QH(F,$\"3!p(\\#[F,7$$\"31,++]Qk\\*)F,$\"3!Ra(>4S ,#)[F,7$$\"3![LL3dg6<*F,$\"3yX@+W[\"3&\\F,7$$\"3%ymmmw(Gp$*F,$\"3%*)3K 38^B,&F,7$$\"3C++D\"oK0e*F,$\"3-(\\^tkhz2&F,7$$\"35,+v=5s#y*F,$\"3#4%Q P$Gf29&F,7$$\"#5F)$\"3s*zt%>RC3_F,-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F( -F$6$7S7$F($\"37?Uo?j&))z\"!#;7$F.$\"3%eaBQWw'G$\"3x\"QO;M9)H:Fe[l7$FC$\"3_[TS1e,h9Fe[ l7$FH$\"3IZ$=uvIsR\"Fe[l7$FM$\"3\"Fe[l7$Ffn$\"3;FE7*egZ7\"Fe[l7$F[o$\"37fB84V2j5Fe[ l7$F`o$\"3,n**=#o,j$**F,7$Feo$\"3gz)4*H)3!R#*F,7$Fjo$\"3ijW^hz.n&)F,7$ F_p$\"3oa<0PS#o&zF,7$Fdp$\"3x#>6(f1AJsF,7$Fip$\"3+R?b@%Rlh'F,7$F^q$\"3 `re(R,>:!fF,7$Fcq$\"3yNas(z/'o_F,7$Fhq$\"3_5<*4f(>uXF,7$F]r$\"3p]9=F ,7$Fas$\"3U\"R!3@[>'>\"F,7$Ffs$\"3gq&HviBEy&F07$F[t$!3p9_N0@;7*)!#>7$F `t$!3AuZ&*)zXfy(F07$Fet$!3_`zEe$4JX\"F,7$Fjt$!3tPqC*3Pd5#F,7$F_u$!3hCs ^9*o.$GF,7$Fdu$!3]6\\#)QG[\"[$F,7$Fiu$!37'z20$fowTF,7$F^v$!3EG%o!fKk1[ F,7$Fcv$!3uP/s7IN&\\&F,7$Fhv$!3\"G\"3]=_PVhF,7$F]w$!3quDlU^u?oF,7$Fbw$ !3=<2Yb!)*H[(F,7$Fgw$!3sK_w![:j<)F,7$F\\x$!3<5T1kD1W))F,7$Fax$!3fCae,/ $p_*F,7$Ffx$!3R_R'[P9/-\"Fe[l7$F[y$!3g#[#>[Kk#3\"Fe[l7$F`y$!3oIbx@\"Fe[l7$Fjy$!359:P3$pdG\"Fe[l7$F_z$!3)RHGu\\n3N \"Fe[l7$Fdz$!3:!QZ>RC3U\"Fe[lFhz-%'POINTSG6&7$$\"\"\"F)$\"\"$F)7$$\"\" #F)F`el7$$\"\"(F)$\"\"&F)7$$\"\")F)$\"\"%F)-%+AXESLABELSG6%%\"XGQ!6\"- %%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FdzFefl" 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 "evalf(\nsubs(xxx[1],rhs(Delt a))\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%(backsubG%%bandG%&basisG%' bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%) colspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG%*crossprodG %%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprod G%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG% &equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG% *frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hada mardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*indexfunc G%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jaco bianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'m atrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normaliz eG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+ randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rref G%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*submatrixG%*subvec torG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*tr ansposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(subs(xxx[1],hessian(rh s(Delta),[p,q])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$ $\"+\\b[B@!\"($\"+oaF$G$!\")7$F+$\"+/x;'H(!\"*Q(pprint06\"" }}}{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:=n ops(x);\nxxx := \{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*su m(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)*s um(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))*sum(x[i],i = 1 .. N)-sum(y[i],i = 1 .. N))/N, p = 1/2/(s um(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*su m(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/(su m(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-su m(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))*sum(x[i],i = 1 .. N) -sum(y[i],i = 1 .. N))/N\};\n\npr1:=subs(xxx[1],Y=p*X+q);\npr2:=subs(x xx[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(Delta))\n)\n<\nevalf (\nsubs(xxx[2],rhs(Delta))\n)\n\nthen pr_:=pr1 else pr_:=pr2\nfi;\nuna pply(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 local to procedure \+ `RP1`\n" }}{PARA 7 "" 1 "" {TEXT -1 63 "Warning, `pr2` is implicitly d eclared local to procedure `RP1`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%$RP1Gj+6$%\"xG%\"yG6-%\"pG%\"qG%\"NG%&DeltaG%\"XG%\"YG%#prG%$pr_G%$x xxG%$pr1G%$pr2G6\"F5C)>8&-%%nopsG6#9$>8,6$<$/8%,$*&,&*&#\"\"\"\"\"#FH* (,&*&-%$sumG6$&F<6#%\"iG/FR;FHF8FH-FN6$&9%FQFSFHFH*&-FN6$*&FPFHFWFHFSF HF8FH!\"\"Fgn,,*$)FUFIFHFH*$)FMFIFHFgn*&-FN6$*$)FPFIFHFSFHF8FHFH*&-FN6 $*$)FWFIFHFSFHF8FHFgn-%%sqrtG6#,:*$)FU\"\"%FHFH*(FIFHFjnFHF\\oFHFH**FI FHFjnFHF^oFHF8FHFH**FIFHFjnFHFcoFHF8FHFgn*$)FMF]pFHFH**FIFHF\\oFHF^oFH F8FHFgn**FIFHF\\oFHFcoFHF8FHFH*&)F^oFIFH)F8FIFHFH**FIFHF^oFHFgpFHFcoFH Fgn*&)FcoFIFHFgpFHFH*,\"\")FHFMFHFUFHFZFHF8FHFgn*(F]pFH)FZFIFHFgpFHFHF HFHFMFHFHFHFUFgnFHF8FgnFgn/8$,$*&FGFH*&FKFgnFhnFHFHFH<$/F`q,$*&FGFH*&F KFgn,,FinFHF[oFgnF]oFHFboFgnFgoFgnFHFHFH/FB,$*&,&*&FGFH*(FKFgnFiqFHFMF HFHFHFUFgnFHF8FgnFgn>8--%%subsG6$&F>6#FH/8),&*&F`qFH8(FHFHFBFH>8.-Fcr6 $&F>6#FIFgr>8'/-%$SumG6$*$)-%$rhoG6$FPFWFIFHFS-FNFgs@%2-%&evalfG6#-Fcr 6$Fer-%$rhsG6#Fcs-Fat6#-Fcr6$F`sFet>8+Far>F]uF]s-%(unapplyG6$-Fft6#F]u F[sF5F5F56$\"\"!Feu" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Priklad:" } {MPLTEXT 1 0 190 "\nwith(plots):\nN:=30;\nx:=seq(evalf(i/3+(sin(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*si n(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=B F($\"+\"e!*e8'!#5$\"+)[E%y=F($\"+thSgmF($\"+uuaQgF($\"+w9:_!*F($\"+Dg/ 4>F($\"+y44^dF($\"*G*)zH%F($\"+FKCQ5!\")$\"+(\\]ef&F($\"+C`NX8FN$\"+\" e*=3LF($\"+tpc,\"*F($\"+hDE99F($\"+;$GB;\"FN$\"+Z'HJ_(F($\"+rre(\\\"FN $\"+=n&fU(F($\"+^jwa**F($\"+k1hV_F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%\"yG6@$!+6GX\\U!\"*$\"+Q/cg=!\")$!+H\\4(*[F($\"+n55Z5F+$!+#)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 71 "primka:=RP1([x],[y])('w');\npr:=plot(primka,w=-4..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$$\"+Dg/4>F)$\"+.k-HlF)7$$\"+y4 4^dF)$\"+\\>5#\\\"F17$$\"*G*)zH%F)$\"+W1j?xF)7$$\"+FKCQ5F1$\"+#=Qg*>F1 7$$\"+(\\]ef&F)$\"+\")\\x18F17$$\"+C`NX8F1$\"+CHVUJF17$$\"+\"e*=3LF)$ \"+Q2*)*G\"F17$$\"+tpc,\"*F)$\"+**[*[`#F17$$\"+hDE99F)$\"+CEF')\\F)7$$ \"+;$GB;\"F1$\"+8NxdAF17$$\"+Z'HJ_(F)$\"+mrxs$\"3_\"oV;#*>?U#Fcu7$$\"3#Q++]xCr'RFbw$\"3:%pZ)f @uEMFcu7$$\"3'Q++v)>a!*zFbw$\"3..&RT0-D\"Fcu$\"37 &=JHHdzI&Fcu7$$\"3v****\\()pz1KssFcu7$$\"3D+]i&Q(3VDFcu$\"3G)Q?yEVx9)Fcu7$$\"35++]2RN;IFcu$ \"3o5Po'*Qq)=*Fcu7$$\"3`+++IFF*Q\\\"F[[l7$$\"3X+]7$4(*Q/'Fcu$\"3CIR;&\\& y%e\"F[[l7$$\"3O***\\(=Yj*['Fcu$\"3Sc/%=\\EGo\"F[[l7$$\"3%*)*\\iN1j_pF cu$\"39FsYLNm%y\"F[[l7$$\"39+]PH+nbtFcu$\"3y&orv(HJt=F[[l7$$\"3+++D;O' 4z(Fcu$\"3#HQNe!o0p>F[[l7$$\"3$******\\[k1C)Fcu$\"3q=@(*[&pz1#F[[l7$$ \"3[)**\\7041o)Fcu$\"37q\\(3POZ;#F[[l7$$\"3!***\\iS!yi5*Fcu$\"3'zDBnCj $eAF[[l7$$\"3Y***\\(y3\"*y&*Fcu$\"3mRuSH*>BO#F[[l7$$\"3&)****\\]#f.+\" F[[l$\"3YD%fCjHdX#F[[l7$$\"33++vyIqX5F[[l$\"3%RKp$GWYbDF[[l7$$\"3#**\\ ilK\"z'3\"F[[l$\"35+!pp(*Qek#F[[l7$$\"3.++DM;rJ6F[[l$\"3_s[jHAkWFF[[l7 $$\"3y*\\iI9yR<\"F[[l$\"3>DfZ!R3w$GF[[l7$$\"3;+DJB)e\"=7F[[l$\"3Qok`:[ yMHF[[l7$$\"3y***\\P^`8E\"F[[l$\"3'[]mK`#zHIF[[l7$$\"3\")*\\i:NulI\"F[ [l$\"3YRG()3oDHJF[[l7$$\"3++++$RF,N\"F[[l$\"3q48*ys_]A$F[[l7$$\"3%*** \\Pomm%R\"F[[l$\"3s^))>\"*y,BLF[[l7$$\"37+D1Or$)Q9F[[l$\"3UjE\\`=P%f_\"F[[l$\"3h& =`Lzkh \"F[[l$\"3IUa!oq`3!QF[[l7$$\"3=+v$RTrVl\"F[[l$\"3-3Z6oWC%*QF[[l7$$\"#< F`u$\"3[****R&f0Y*RF[[l-%'COLOURG6&%$RGBG$\"*++++\"F1$F`uF`uF]el-%+AXE SLABELSG6$%\"wGQ!6\"-%%VIEWG6$;F=Fcdl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Regresni prim ka v ine petrice:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "RP2:= proc(X,Y)\nlocal Primka,x,p,q;\nPrimka:=x->p*x+q; #obecna rovnice prim ky\nSoucetCtvercu:=sum(((Y[i]-Primka(X[i]))^2),i=1..nops(X)); #soucet \+ druhych mocnin odchylek\nRce:=diff(SoucetCtvercu,p)=0,diff(SoucetCtver cu,q)=0; # hledame minimum, tedy hledame parametry p a q tak, aby deri vace `SoucetCtvercu` podle p i podle q byla 0\nParam:=solve(\{Rce\},\{ p,q\}); # parametry najdeme resenim vynikle rovnice\nunapply(subs(Para m,Primka(x)),x); #vystup 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 im plicitly declared local to procedure `RP2`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `Param` is implicitly declared local to procedure `RP2 `\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RP2Gj+6$%\"XG%\"YG6)%'Primka G%\"xG%\"pG%\"qG%.SoucetCtvercuG%$RceG%&ParamG6\"F1C'>8$j+6#F+F16$%)op eratorG%&arrowGF1,&*&T#\"\"\"9$F=F=T%F=F1F16&F,8&F-8'6#\"\"!>8(-%$sumG 6$*$),&&9%6#%\"iGF=-F46#&F>FO!\"\"\"\"#F=/FP;F=-%%nopsG6#F>>8)6$/-%%di ffG6$FFFAFD/-Fjn6$FFFBFD>8*-%&solveG6$<#Ffn<$FAFB-%(unapplyG6$-%%subsG 6$F`o-F46#8%F^pF1F1F16$FDFD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "V \+ maple je vestavena funkce " }{HYPERLNK 17 "leastsquare" 2 "leastsquare " "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Porovnani obou primek:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "primka:=RP2([x],[y])('w');\npr2:=p lot(primka,w=-4..17,color=NAVY):\n#display(\{bb,pr\});\n\n#pr2:=plot(R P2,w=1..19):\ndisplay(\{bb,pr,pr2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'primkaG,&*&$\"+oiJj?!\"*\"\"\"%\"wGF*F*$\"+4+9\\KF)F*" }}{PARA 13 "" 1 "" {GLPLOT2D 413 413 413 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"% \"\"!$!3/+++j]7/]!#<7$$!32++]i(fAa$F-$!3#p1[W=i'fSF-7$$!3$)**\\PHC)R9$ F-$!3t(fNQ3!*yB$F-7$$!3%****\\P\"z2'p#F-$!3>veB59s8BF-7$$!37++D'R/_C#F -$!3CVxwYnU$Q\"F-7$$!3%***\\(=!QZ'z\"F-$!3Wi\\W9h`vX!#=7$$!3K+]P*4V/Q \"F-$\"3!=@7,0$\\3SFG7$$!3o***\\P/um\\*FG$\"3?GPO/en*G\"F-7$$!3^***\\P k&eT]FG$\"3HPNZT9!*3AF-7$$!32B+](oZy+'!#>$\"3B]rI5\"z^7$F-7$$\"3#Q++]x Cr'RFG$\"3aMVaa!*zFG$\"3%3e%HO:%y*[F-7$$\"3[++]#o& *>D\"F-$\"3o8!o\"oISKeF-7$$\"3v****\\()pz1!3x'F-7$$\"3++ +]d\"F_q7$ $\"3O***\\(=Yj*['F-$\"3I&GC\"p3$Rm\"F_q7$$\"3%*)*\\iN1j_pF-$\"3%R-)[!f h%fO C$>F_q7$$\"3$******\\[k1C)F-$\"3BMI+5PAD?F_q7$$\"3[)**\\7041o)F-$\"3l; #f!)=)*f6#F_q7$$\"3!***\\iS!yi5*F-$\"3$GF:J;FQ?#F_q7$$\"3Y***\\(y3\"*y &*F-$\"3+0Z[liM,BF_q7$$\"3&)****\\]#f.+\"F_q$\"3/P4I;:(*)Q#F_q7$$\"33+ +vyIqX5F_q$\"3?gDyu,`#[#F_q7$$\"3#**\\ilK\"z'3\"F_q$\"3u1x\"\\A3tc#F_q 7$$\"3.++DM;rJ6F_q$\"39\"*G`EI**fEF_q7$$\"3y*\\iI9yR<\"F_q$\"3?(Gd%[!=4OQGF_q7$$\"3y***\\P^`8E\"F_q$\"3EO $>bA&[FHF_q7$$\"3\")*\\i:NulI\"F_q$\"3\"pXIe6!z?IF_q7$$\"3++++$RF,N\"F _q$\"3FH$\\Z\"Ql5JF_q7$$\"3%***\\Pomm%R\"F_q$\"3Z)\\&>ECb-KF_q7$$\"37+ D1Or$)Q9F_q$\"3g&GS22!p$H$F_q7$$\"3-++]3_Uz9F_q$\"3-Ut)35OuP$F_q7$$\"3 (***\\()>P%f_\"F_q$\"3EA0<-&=MZ$F_q7$$\"35+++J/bn:F_q$\"3e6F41jEfNF_q7 $$\"3'**\\iI'=\">h\"F_q$\"3sW!R0(zz]OF_q7$$\"3=+v$RTrVl\"F_q$\"3)fS\"f _aSQPF_q7$$\"#h^D=Fa[l7$$\"+@-!y1$Fj[l$\"+J$H8L \"Fa[l7$$\"++eF=BFj[l$\"+mmqL&*Fj[l7$$\"+\"e!*e8'!#5$\"+]'y@I(Fj[l7$$ \"+)[E%y=Fj[l$\"+%e@&\\jFj[l7$$\"+thSgmFj[l$\"+$GWS.#Fa[l7$$\"+uuaQgFj [l$\"+:u6)y\"Fa[l7$$\"+w9:_!*Fj[l$\"+Adh8DFa[l7$$\"+Dg/4>Fj[l$\"+.k-Hl Fj[l7$$\"+y44^dFj[l$\"+\\>5#\\\"Fa[l7$$\"*G*)zH%Fj[l$\"+W1j?xFj[l7$$\" +FKCQ5Fa[l$\"+#=Qg*>Fa[l7$$\"+(\\]ef&Fj[l$\"+\")\\x18Fa[l7$$\"+C`NX8Fa [l$\"+CHVUJFa[l7$$\"+\"e*=3LFj[l$\"+Q2*)*G\"Fa[l7$$\"+tpc,\"*Fj[l$\"+* *[*[`#Fa[l7$$\"+hDE99Fj[l$\"+CEF')\\Fj[l7$$\"+;$GB;\"Fa[l$\"+8NxdAFa[l 7$$\"+Z'HJ_(Fj[l$\"+mrxs$!3kc/@H\\@%Q#F-7$FC$!3O2kqQ;A(R\"F-7 $FI$!3OyitW^_@[FG7$FN$\"3ga<(zg&[`YFG7$FS$\"3[MlDGbDX9F-7$FX$\"3_\"oV; #*>?U#F-7$Fhn$\"3:%pZ)f@uEMF-7$F]o$\"3..&RT0-KssF-7$Fap$\"3G)Q?yEVx9)F-7$Ffp$ \"3o5Po'*Qq)=*F-7$F[q$\"3tUhB4M025F_q7$Faq$\"3s-/mC7j46F_q7$Ffq$\"37-G Xs*H/?\"F_q7$F[r$\"3=K%3'*e]+I\"F_q7$F`r$\"3I'Qk`(H\"\\R\"F_q7$Fer$\"3 /nH=B>*Q\\\"F_q7$Fjr$\"3CIR;&\\&y%e\"F_q7$F_s$\"3Sc/%=\\EGo\"F_q7$Fds$ \"39FsYLNm%y\"F_q7$Fis$\"3y&orv(HJt=F_q7$F^t$\"3#HQNe!o0p>F_q7$Fct$\"3 q=@(*[&pz1#F_q7$Fht$\"37q\\(3POZ;#F_q7$F]u$\"3'zDBnCj$eAF_q7$Fbu$\"3mR uSH*>BO#F_q7$Fgu$\"3YD%fCjHdX#F_q7$F\\v$\"3%RKp$GWYbDF_q7$Fav$\"35+!pp (*Qek#F_q7$Ffv$\"3_s[jHAkWFF_q7$F[w$\"3>DfZ!R3w$GF_q7$F`w$\"3Qok`:[yMH F_q7$Few$\"3'[]mK`#zHIF_q7$Fjw$\"3YRG()3oDHJF_q7$F_x$\"3q48*ys_]A$F_q7 $Fdx$\"3s^))>\"*y,BLF_q7$Fix$\"3UjE\\`= " 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 0 {PARA 3 "" 0 "" {TEXT -1 20 "Komparace aproximaci" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Aproximujeme funkci f na intervalu " } {XPPEDIT 18 0 "`<,>`(alpha,beta);" "6#-%$<,>G6$%&alphaG%%betaG" } {TEXT -1 201 " . Stupen aproximace je N: to znamena v pripade aproxim ace polynomeme bude mit tennto stupen n, u Fourierovy rqady vezmeme n- ty castecny soucet. Porovcname kvalitu aproximace ve dvou metrik\341ch . rvn\355: " }{XPPEDIT 18 0 "rho(f,g)=Int(abs(f-g),w=alpha..beta);" "6 #/-%$rhoG6$%\"fG%\"gG-%$IntG6$-%$absG6#,&F'\"\"\"F(!\"\"/%\"wG;%&alpha G%%betaG" }{TEXT -1 10 " a druha: " }{XPPEDIT 18 0 "rho(f,g) = Int(abs (f-g)+abs(diff(f-g,w)),w = alpha .. beta);" "6#/-%$rhoG6$%\"fG%\"gG-%$ IntG6$,&-%$absG6#,&F'\"\"\"F(!\"\"F1-F.6#-%%diffG6$,&F'F1F(F2%\"wGF1/F 9;%&alphaG%%betaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "readlib(mtaylor);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#j+6\"6*%\"fG%\"kG%\"vG%\"mG%\"nG%\"sG%\"tG%\"wG6#%aoCopyright~(c)~19 91~by~the~University~of~Waterloo.~All~rights~reserved.GF$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)-%$mapG6$j+6#%\"xGF$F$F$@%-F>6$9$%\"=G-%$rhsG6#F]o \"\"!F$F$F$6#\"*;4(o`F8>F8-Fen6$j+FhnF$F$F$@%F[o-%$lhsGFaoF]oF$F$F$6#F doF8>8'-%%nopsGFE@$0F^p-F`p6#<#FCYQHvariables~(2nd~argument)~must~be~u niqueF$@%/9#F;>8(%&OrderG>F\\q&F46#\"\"$@%/Fjp\"\"%>8+&F46#Fdq>Ffq7#-% \"$G6$F6F^p@$4-F>6$F8<$-F@6#FS-FIFdrYQQ2nd~argument~(the~variable(s))~ must~be~a~name(s)F$@$34-F>6$F\\q%*nonnegintG0F\\q%)infinityGYQX3rd~arg ument~(the~order)~must~be~a~non-negative~integerF$@$54-F>6$Ffq-FI6#%'p osintG0-F`p6#FfqF^pYQen4th~argument~(weights)~must~be~a~list~of~positi ve~integersF$>F2-%%subsG6$7#-%$seqG6$/&F86#8%,&*&FhtF6)8*&FfqFitF6F6&F YFitF6/Fjt;F6F^pF2>F2-Fen6&%(collectG-Fat6$/-%\"OGF5Fbo-%'taylorG6%F2F ^uF\\qF8.%,distributedG>F2-Fat6$7#-Fet6$/Fht,&FhtF6F`u!\"\"Fau-Fat6$/F ^uF6F2F$F$F$6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "unassign ('x');\nf:=x->exp(x)*log(x);\nN:=5;\nalpha:=1;\nbeta:=2;" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGj+6#%\"xG 6\"6$%)operatorG%&arrowGF(*&-%$expG6#9$\"\"\"-%$logGF/F1F(F(F(6#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%b etaG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "Porovnani:=pr oc(f,N,alpha,beta)\nlocal x,y,w,n,i,A1,A2,A3;\nA1:=unapply(evalf(mtayl or(f(w),w=(alpha+beta)/2,N+1)),w); # Tayloruv polynom ve stredu ionter valu 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,beta-alpha))('x'-alpha),'x');\nprint(`aproximujeme f unkci`,f,`temito funkcemi:`);\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=alpha..beta)\n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "r1:=simplify(evalf(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(r 3),w=alpha..beta))],\n[evalf(Int(abs(r1)+abs(diff(r1,w)),w=alpha..beta )),\nevalf(Int(abs(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 `Porov nani`\n" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, `psi` is implicitly d eclared local 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 declar ed local to procedure `Porovnani`\n" }}{PARA 7 "" 1 "" {TEXT -1 68 "Wa rning, `r3` is implicitly declared local to procedure `Porovnani`\n" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*PorovnaniGj+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*(,&FPFNFRFNF NFT!\"\"8(FNFN/Fhn;\"\"!FT>8%-FX6$-FA6#-FG6#&FV6#Fhn/Fhn;FNFS>8*-%)Lag rangeG6$7#FV7#F]o>8,j+6#F,F96$%)operatorG%&arrowGF9,&FGFNT$FNF9F96$F)F P6#F[o>8-j+FbpF9FcpF9FfpF9F9Fhp6#F[o>8+-F>6$--%(FourierG6%-%\"@G6$FGF` pFT,&FRFNFPFgn6#,&.FVFNFPFgnF\\r-%&printG6%%4aproximujeme~funkciGFG%1t emito~funkcemi:G-F^r6'-%%sortG6$-%'normalG6#-F<6#F\\rF\\r%\"|+G-Fer6$- Fhr6#-FioF[sF\\rF\\s-Fer6$-Fhr6#-F_qF[sF\\r-F^r6#-%%plotG6$<&-F8.-%)simplifyG6#-FA6#,&FFFNF]tFgn>8/-Fet6#-FA6#,&FF FNF^tFgn>80-Fet6#-FA6#,&FFFNF_tFgn-F^r6#%_oVzdalenost~aproximaci~od~pu vodni~funkcer~v~prvni~a~ve~druhe~metrice:G6$7%-FA6#-%$IntG6$-%$absG6#F ctF`t-FA6#-F`v6$-Fcv6#F[uF`t-FA6#-F`v6$-Fcv6#FbuF`t7%-FA6#-F`v6$,&FbvF N-Fcv6#-%%diffG6$FctFIFNF`t-FA6#-F`v6$,&FivFN-Fcv6#-Fjw6$FbuFIFNF`t-FA 6#-F`v6$,&F_wFN-Fcv6#-Fjw6$F[uFIFNF`tF9F9F96&F[oF[o\"&>.$\"'L)*e" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Porovnani(x->exp(x)*log(x),5 ,1,2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%4aproximujeme~funkciGj+6# %\"xG6\"6$%)operatorG%&arrowGF'*&-%$expG6#9$\"\"\"-%$logGF.F0F'F'F'6# \"\"!%1temito~funkcemi:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6',.*&$\"+^0 Ao\"*!#6\"\"\")%\"xG\"\"&F(F(*&$\"+l?$e!R!#5F()F*\"\"%F(!\"\"*&$\"+\\C 6C:!\"*F()F*\"\"$F(F(*&$\"+*3h%zLL3x&)*3,\"F-$\"3!)\\iXa_1?AF-7$$\"3hmm;arz@5F- $\"372/#ejY3+#F-7$$\"35+D\"y%*z7.\"F-$\"3od1NSD!>\"=F-7$$\"3OL$e9ui2/ \"F-$\"36Zr\"*)3$QE;F-7$$\"3smm\"z_\"4i5F-$\"3JH&Qr6+*H7F-7$$\"3qmmT&p hN3\"F-$\"3)>K!>*f+jt)!#=7$$\"3UL$e*=)H\\5\"F-$\"3aKj`%GG5w&FL7$$\"3sm ;z/3uC6F-$\"3q#Ru4'>_0OFL7$$\"3-+]7LRDX6F-$\"3Y@6f=)eU/#FL7$$\"3IL$ekG he:\"F-$\"3a\\mXp&HL^\"FL7$$\"3em;zR'ok;\"F-$\"3s(p*3%G=0<\"FL7$$\"3y* 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P\370ed pokl\341d\341e, \236e " }{XPPEDIT 18 0 "x(t);" "6#-%\"xG6#%\"tG" } {TEXT -1 16 " pl\362uje rovnici:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r[1]:=diff(x(t),t)=a*x(t)*g(x(t)/k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"/-%%diffG6$-%\"xG6#%\"tGF/*(%\"aGF' F,F'-%\"gG6#*&F,F'%\"kG!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "pro n\354jkou funkci " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT -1 13 " a konstanty " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 3 " a " } {XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 11 ". \nRovnice " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6#\"\"\"" }{TEXT -1 63 " vyjad\370uje, jak rychlost r\371stu populce z\341vis\355 na jej\355 velikosti" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Malthus\371v p\370edpoklad " }{XPPEDIT 18 0 "g( y) = 1;" "6#/-%\"gG6#%\"yG\"\"\"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Za p\370edpokladu, \236e funkce je konstantn\355" }{MPLTEXT 1 0 10 "\ng:=y->1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGj+6#%\"yG6\" 6$%)operatorG%&arrowGF(\"\"\"F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "m\341 rovnice tvar:" }{MPLTEXT 1 0 13 "\nMalt:=r[1];\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MaltG/-%%diffG6$-%\"xG6#%\"tGF,*&% \"aG\"\"\"F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "a jej\355 \370e \232en\355 je:" }{MPLTEXT 1 0 25 "\nM:=dsolve(Malt);\ng:='g':" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG/-%\"xG6#%\"tG*&%$_C1G\"\"\"-%$e xpG6#*&%\"aGF,F)F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "C1: =solve(subs(t=0,a=.4,k=1,rhs(M))=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G,$*&\"#5\"\"\"-%$expG6#\"\"!!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(subs(_C1=C1,a=.4,rhs(M)),t=0..10);" }} {PARA 13 "" 1 "" {GLPLOT2D 315 315 315 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"\"!F)$\"#5F)7$$\"3emmm;arz@!#=$\"3%)3*>/Y-64\"!#;7$$\"3[LL$e9ui2%F/ $\"3wKH()zm4x6F27$$\"3nmmm\"z_\"4iF/$\"3e\\!=bG\"F27$$\"39ommT&phN) F/$\"3Q<\"[t#y)oR\"F27$$\"3KLLe*=)H\\5!#<$\"3ykV`@:F27$$\"3smm\"z/3 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"" 1 "" {XPPMATH 20 "6#7#%(declareG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"xG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%%%GompG/%#x'G,$*(%\"aG\"\"\"% \"xGF*-%#lnG6#*&F+F*%\"kG!\"\"F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%*infolevelG6#%'dsolveG\"\"$" }}{PARA 6 "" 1 "" {TEXT -1 29 "Methods for first order ODEs:" }}{PARA 6 "" 1 "" {TEXT -1 37 "--- Trying clas sification methods ---" }}{PARA 6 "" 1 "" {TEXT -1 19 "trying a quadra ture" }}{PARA 6 "" 1 "" {TEXT -1 23 "trying 1st order linear" }}{PARA 6 "" 1 "" {TEXT -1 16 "trying Bernoulli" }}{PARA 6 "" 1 "" {TEXT -1 16 "trying separable" }}{PARA 6 "" 1 "" {TEXT -1 23 "<- separable succ essful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG/%\"xG*&-%$expG6#-F)6# ,$*&%\"aG\"\"\",&%\"tGF0%$_C1GF0F0!\"\"F0%\"kGF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "C1:=solve(subs(t=0,a=.4,k=1,rhs(G))=.2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G^$$!+)[7(*=\"!\"*$!+M;)R&yF(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(RealDomain);\nG2:=dsol ve(\{Gomp,x(0)=.1\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7H%#ImG%#ReG% \"^G%'arccosG%(arccoshG%'arccotG%(arccothG%'arccscG%(arccschG%'arcsecG %(arcsechG%'arcsinG%(arcsinhG%'arctanG%(arctanhG%$cosG%%coshG%$cotG%%c othG%$cscG%%cschG%%evalG%$expG%'expandG%&limitG%#lnG%$logG%$secG%%sech G%'signumG%)simplifyG%$sinG%%sinhG%&solveG%%sqrtG%%surdG%$tanG%%tanhG " }}{PARA 6 "" 1 "" {TEXT -1 29 "Methods for first order ODEs:" }} {PARA 6 "" 1 "" {TEXT -1 37 "--- Trying classification methods ---" }} {PARA 6 "" 1 "" {TEXT -1 19 "trying a quadrature" }}{PARA 6 "" 1 "" {TEXT -1 23 "trying 1st order linear" }}{PARA 6 "" 1 "" {TEXT -1 16 "t rying Bernoulli" }}{PARA 6 "" 1 "" {TEXT -1 16 "trying separable" }} {PARA 6 "" 1 "" {TEXT -1 23 "<- separable successful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G2G/%\"xG*&-%$expG6#*&,&-%#lnG6#,$*&\"\"\"F2*&\"# 5F2%\"kGF2!\"\"F2F2*(^#\"\"#F2%#PiGF2%&_Z18|irGF2F2F2-F)6#*&%\"aGF2%\" tGF2F6F2F5F2" }}}{EXCHG {PARA 0 "> " 0 "" 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