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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(54193 51/1725033,15);" }}{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(.482758620 7,6);" }}{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 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 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 rovnice pro nezn\341m\351 parametry:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "rceA:=subs(x=A[1],y=A[2],p rimka); #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 vyhledem k promenym p a q\nassign(p aram); #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 "Nzn\355 m \341 p\370\355mka tvar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p rimka;" }}{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 "readlib(unassign):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "unassign('A','B','p','q'); # Zru\232\355me p\370i\370 azen\355 n\354kter\375ch hodnot prom\354n\375" }}}{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 vzhledem 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<$/%\"pG*&,&&%\"AG6#\"\"# !\"\"&%\"BGF,\"\"\"F1,&&F+6#F1F.&F0F4F1F./%\"qG,$*&,&*&F/F1F3F1F1*&F5F 1F*F1F.F1F2F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*(,&&%\"AG6#\"\"#!\"\"&%\"BG F*\"\"\"F/,&&F)6#F/F,&F.F2F/F,%\"xGF/F/*&,&*&F-F/F1F/F/*&F3F/F(F/F,F/F 0F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "To \236e jsme museli ce l\375 v\375po\350et opakovat n\341s vede k opr\341vn\354n\351 domn\354 nce, \236e by bylo v\375hodn\351 jej zachovat pro dal\232\355 pou\236i t\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. procedura\nlocal x; # deklarace lokalnich promenych. \+ lokalni promene si po provedeni procedury uchovaji tytez hodnoty jake \+ mely pred jejim spustenim \n if A[1]=B[1] then # vyloucime pripad svi sle primky\n print(`primka je svisla, nejde o funkci`)\n else\nun apply( (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 (nezavisla na ozn aceni 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>F1F1F1FCFBF BFDF+F+F+" }}}{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\355kaz em 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);\np aram:=x%%PlinGf*6$%\"XG%\"YG6&%\"NG%\"iG%\"xG%&par amG6\"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 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 -*,$\"++8*fq'!#6F*F+F*F7F*F.F*F0F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "neboli" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Lagrange:=simplify( Lagrange);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)LagrangeG,,*&$\"++^gg [!#7\"\"\")%\"xG\"\"%F*!\"\"*&$\"+5dD#p(!#6F*)F,\"\"$F*F**&$\"+8DhQ[!# 5F*)F,\"\"#F*F.*&$\"+6@=z;!\"*F*F,F*F*$\"+7GQn7F>F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Op\354t m\371\236eme napsat proced\371ru, kter \341 polynom vytvo\370\355. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "Lagrange:=proc(X,Y) # Vstup: pole bodu a pole hodnot. Vystup: L agrangeuv polynom (funkce)\nlocal C,J,N,x,i,j,substitut,xxx;\ni:='i'; \nXpom:=convert(X,list);\nN:=nops(Xpom);\nfor j from 1 to N do\nC[j]:= product((x-X[i]),i=1..N)/(x-X[j]);\nJ[j]:=subs(x=X[j],C[j]);\n#print(C [j],J[j]);\nod;\ni:='i';\nxxx:=sum(C[i]/J[i]*Y[i],i=1..N);\nunapply(xx x,x)\nend;" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, `Xpom` is implicit ly declared local to procedure `Lagrange`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)LagrangeGf*6$%\"XG%\"YG6+%\"CG%\"JG%\"NG%\"xG%\"iG% \"jG%*substitutG%$xxxG%%XpomG6\"F3C)>8(.F6>8,-%(convertG6$9$%%listG>8& -%%nopsG6#F9?(8)\"\"\"FFF@%%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]oFRF3F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "nebo ekvivaletn\354 s vzu\236it\355m ji\236 hot ov\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#>%*Lag range2Gf*6$%\"XG%\"YG6\"F)F)-%(unapplyG6$-%'interpG6%-%(convertG6$9$%% listG-F16$9%F4%\"xGF8F)F)F)" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "S plainy" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "Nev\375hoou Lagrangeova polynomu je to, 6e m8me-li mnoho bod\371, dostabneme polznom p\370 \355li\232 vysok\351ho stzupn\354. Nev\375hodou po \350\341stech afinn \355\355 aproximace je v tom, \236e dostaneme funkci, kter\341 nem\341 v bodech, jimi\236 ji prokl\341d\341me derivaci. Najdeme aproximaci p o \350\341stech polynomialn\355 funkc\355, kter\341 bude m\355t v ka \236d\351m bod\354 derivaci. Jako p\370\355klad uvedeme po \350\341ste ch kvadratick\375 pol\375nom:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "unssign('a','b','c','x');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%( unssignG6&%\"aG%\"bG%\"cG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:=a*x^2+b*x+c; # obecn\375 tvar polznomu druh\351ho stupn\354 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*&%\"aG\"\"\")%\"xG\"\"#F(F(*&%\"bGF(F*F(F(%\"cGF(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Polynom druh\351ho stupn\354 zavi s\355 na t\370ech parametrech. Hled\341me takov\351 hodnoty parmetr \371, aby m\354l polynom p\370edepsan\351 hodnoty v bodech " } {XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[j+1];" "6#&%\"xG6#,&%\"jG\"\"\"F(F(" }{TEXT -1 32 ", a p\370edeps anou derivaci v bod\354 " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" } {TEXT -1 49 ". Polynom mus\355 spl\362ovat n\341sleduj\355c\355 t\370 i rovnice:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "ddd:=diff(p, x);\nSoustavaRovnic:=\nsubs(x=x[j],ddd)=der,\nsubs(x=x[j],p)=y[j],\nsu bs(x=x[j+1],p)=y[j+1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dddG,&*( \"\"#\"\"\"%\"aGF(%\"xGF(F(%\"bGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%/SoustavaRovnicG6%/,&*(\"\"#\"\"\"%\"aGF*&%\"xG6#\"\"'F*F*%\"bGF*%$d erG/,(*&F+F*)F,F)F*F**&F0F*F,F*F*%\"cGF*&%\"yGF./,(*&F+F*)&F-6#\"\"(F) F*F**&F0F*F>F*F*F7F*&F9F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Jeji ch \370e\232en\355 je" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "so lve(\{SoustavaRovnic\},\{a,b,c\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# <%/%\"aG,$*&,(*$)&%\"xG6#\"\"'\"\"#\"\"\"F0*$)&F,6#\"\"(F/F0F0*(F/F0F3 F0F+F0!\"\"F7,**&F+F0%$derGF0F7&%\"yGF-F0*&F3F0F:F0F0&FF0F7*(F+F0F:F0F2F0F0*&F;F0F2F0F7**F/F0F; F0F3F0F+F0F0F0F(F7F7/%\"bG*&,**&F*F0F:F0F7*(F/F0F+F0F;F0F0*(F/F0F+F0F> F0F7*&F:F0F2F0F0F0F(F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Polznom , kter\375proch\341z\355 t\370emi zadan\375mi body zase spl\362uje rov nice" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "PrvniSoustavaRovnic :=\nsubs(x=x[1],p)=y[1],\nsubs(x=x[2],p)=y[2],\nsubs(x=x[3],p)=y[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4PrvniSoustavaRovnicG6%/,(*&%\"aG \"\"\")&%\"xG6#F*\"\"#F*F**&%\"bGF*F,F*F*%\"cGF*&%\"yGF./,(*&F)F*)&F-6 #F/F/F*F**&F1F*F9F*F*F2F*&F4F:/,(*&F)F*)&F-6#\"\"$F/F*F**&F1F*FAF*F*F2 F*&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#<%/%\"aG*&,.*&&%\"xG6#\"\"\"F, &%\"yG6#\"\"$F,!\"\"*&&F*F/F,&F.F+F,F,*&&F*6#\"\"#F,F-F,F,*&F6F,F4F,F1 *&F)F,&F.F7F,F,*&F3F,F;F,F1F,,.*&F6F,)F)F8F,F1*&F6F,)F3F8F,F,*&F3F,F?F ,F,*&F)F,FAF,F1*&F)F,)F6F8F,F,*&F3F,FEF,F1F1/%\"bG,$*&,.*&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,F4F,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 prochazet prvnimi tre ma body\n\nQ:=z%%Spl2Gf*6$%\"xG%\"yG6-%\"NG%\"KG%\"jG%\"aG%\"bG%\"c G%\"QG%\"zG%\"pG%\"iG%$derG6\"F5C,>8$-%%nopsG6#9$>8%,&F8\"\"\"\"\"#!\" \">8',$*&,.*&&F<6#\"\"$F@&9%6#F@F@F@*&&F8(*&,.* &FgnF@FQF@FB*&FgnF@FWF@F@*&FjnF@FQF@F@*&FLF@FenF@F@*&FLF@FjnF@FB*&FWF@ FenF@FBF@FYFB>8)*&,.*(FgnF@FSF@FQF@F@*(FgnF@FIF@FWF@FB*(FjnF@FPF@FQF@F B*(FWF@FPF@FenF@F@*(FLF@FSF@FenF@FB*(FjnF@FIF@FLF@F@F@FYFB>&8,FN,(*&FD F@)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@FBFa rF@*&FhqF@FeqF@F@F@FirFB>Fho*&,,*(FerF@FhrF@FeqF@F@*&FerF@FarF@F@*(Fhq F@FeqF@FgrF@FB*&F_rF@FgrF@F@**FAF@F_rF@FhrF@FhqF@FBF@FirFB>&Fcp6#F^qFd p@%2F^qF>>Fjp6%Fjp2FgpFhrF[t>Fjp6$FjpF[t>Fjp-%*piecewiseG6#Fjp-%(unapp lyG6$FjpFgpF5F5F5" }}}{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#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&-%#lnG6#9$\"\"\"-%$sinGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG6)\"\"$\"\"&\"\"'\"\"*\"#6\"#7\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG6)$\"+^ " 0 "" {MPLTEXT 1 0 169 "Body:=pointplot([s eq([x[i],y[i]],i=1..N)], color=NAVY,symbolsize=15):\nAproximace:=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$$\"3am;/c,R[MF_o$!3U#H 6)Q]pWmF[o7$$\"3GLL32o+$o$F_o$!3+mRF_o$!3I &o/)zr,_7F_o7$$\"3um;a3!GU:%F_o$!3>:o4RhtS9F_o7$$\"33+]iI%)=jUF_o$!3?) 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$%)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($!(!=VzF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "k1:=plot(f(w),w=z[1]..z[N+1],color=BLACK):\nk2:=plot (Plin([x],[y])(w),w=z[1]..z[N+1],color=RED):\nk3:=plot(Spl2([x],[y])(w ),w=z[1]..z[N+1],color=BLUE):\n#plot(Lagrange([x],[y])(w),w=z[1]..z[N+ 1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 730 "RB:=pointplot(\n [seq(\n \+ [z[i],evalf(f(z[i]))]\n ,i=1..N+1\n )]\n,color=BLACK,sy mbol=cross,symbolsize=20):\nLB:=pointplot(\n [seq(\n [z[i],Li nA[i]]\n ,i=1..N+1\n )],\ncolor=RED,symbol=box,symbolsi ze=20):\nBB:=pointplot(\n [seq(\n [z[i],BilA[i]]\n ,i=1 ..N+1\n )],\ncolor=GREEN,symbol=diamond,symbolsize=20):\nPB:= pointplot(\n [seq(\n [z[i],PolA[i]]\n ,i=1..N+1\n \+ )],\ncolor=BLUE,symbol=circle,symbolsize=20):\ndisplay([k1,k2,k3,P uvB],title=`funkce (cerne) a jeji aproximace`);\n\n\ndisplay(\{k1,RB\} ,title=`Skutecne hodnoty`);\ndisplay(\{k2,LB\},title=`hodnoty ziskane \+ z linerani 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" }{XPPEDIT 18 0 "u[1]= v[1]/phi(v[1],v[1])" "6#/&%\"uG6#\"\"\"*&&%\"vG6#F'F'-%$phiG6$&F*6#F'& F*6#F'!\"\"" }{TEXT -1 32 ", druh\375 bude line\341rn\355 kombinac\355 " }{XPPEDIT 18 0 "u[1]" "6#&%\"uG6#\"\"\"" }{TEXT -1 3 " a " } {XPPEDIT 18 0 "v[2]" "6#&%\"vG6#\"\"#" }{TEXT -1 86 ". Abychom zajisti li, \236e vektory budou nad\341le line\341rn\354 nez\341visl\351, zvol \355me koeficient u " }{XPPEDIT 18 0 "v[2]" "6#&%\"vG6#\"\"#" }{TEXT -1 8 " 1. Tj.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rce[1]:=u [2]=v[2]+c[1]*u[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$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 $&%\"vG6#\"\"#&%\"uGF(F)-F-6$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 "Pak v\232echny vektory vyd\354l\355me jej ich velikost\355." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Pozn\341mky: " }}{PARA 15 "" 0 "" {TEXT -1 109 "Pokud jsou vektory n-tice \350\355s el, 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 nalezen\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 "Pr ov\341d\355 se kontrola toho, zda 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 vektory generuj\355 tent\375\236 prostor, jak o 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 hodnost matice, kter\341 vznikne p\370id\341n\355m zada n\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(matr ix(4,7,[v[i] $i=1..4])) then\nkomentC:=`vektory negeneruji zadany pros tor`;\nelse \nkomentC:=``;\nfi;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(komentCG%Bvektory~negene ruji~zadany~prostorG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Pokud js ou vektory 4 funkce, zkoum\341 se, zda je mo\236no line\341rn\355 komb inac\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] $j=1..4\});\n\nif xxx= NULL then komentC:=`funkce negeneruji tentyz prostor jako zadane funkce` else fi;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(komentCG%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xx xG<&/&%\"aG6#\"\"%*&,**&&F(6#\"\"\"F0&%\"fGF/F0!\"\"*&&F(6#\"\"#F0&F2F 6F0F3*&&F(6#\"\"$F0&F2F;F0F3&%\"gGF/F0F0&F2F)F3/F5F5/F.F./F:F:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xxxG<&/&%\"aG6#\"\"%,$*&,**&&F(6#\" \"\"F1&%\"fGF0F1F1*&&F(6#\"\"#F1&F3F6F1F1*&&F(6#\"\"$F1&F3F;F1F1&%\"gG F6!\"\"F1&F3F)F@F@/F5F5/F/F//F:F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$xxxG<&/&%\"aG6#\"\"#F'/&F(6#\"\"%,$*&,**&&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 1 {PARA 3 "" 0 "" {TEXT -1 36 "Aproximace line\341rn\355 kombinac\355 funkc\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "\332loha aproximovat funkci line\341rn\355 kombinac\355 \+ zadan\375ch funkc\355:\nPokud jsme o fourierovu \370adu, nap\370\355kl ad, vol\355me p\370\355slu\232n\375 ortonorm\341ln\355 syst\351m. Koef icienty 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 (pocit aji se i nulove). -a[0]+...+a[(n-1)/2]*sin/cos (x*((n-1)/2))- n interv alu 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]:=eval f(2/beta*evalf(int(f(x)*cos(2*i*Pi*x/beta),x=0..beta)));\n b[i]:=eval f(2/beta*evalf(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*Pi*x/beta),i=1..N-1),x)\nelse\n unapply(a[0]/2+sum(a[i]*co s(2*i*Pi*x/beta)+b[i]*sin(2*i*Pi*x/beta),i=1..N-1)+a[N]*cos(N*2*Pi*x/b eta),x);\nfi\nend;\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `f` is i mplicitly declared local to procedure `Fourier`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `a` is implicitly declared local to procedure `F ourier`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `b` is implicitly d eclared local to procedure `Fourier`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(FourierGf*6%%\"FG%\"nG%%betaG6(%\"iG%\"NG%\"xG%\"fG%\"aG%\"bG 6\"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#-%$intG6$*&-F4F7F@-%$cosG6#,$*,F?F@FJF@%#PiGF@F8F@FYFZF@F@/F 8;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@*&Fe oF@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@F8F1F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "\332loha aproximovat hodnoty hodnotami line\341rn\355 ko mbinace funkc\355:\nPokud nechceme nejprve syst\351m funkc\355 respekt ive vektory jejich hodnot nahradit ortonorm\341ln\355m syst\351mem, mu s\355me poka\236d\351 \370e\232it soustavu line\341rn\355ch rovnic pro koeficienty:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 601 "Aproximac e:=proc(X,Y,F) #vstup: pole bodu, pole hodnot, pole funkci. Vystup: li nearni kombinace zadanych funkci, ktera nejlepe aproximuje zadane hodn oty.\nlocal i,j,k,PX,PF,v,Koeficienty,MaticeSoustavy,PraveStrany\n;\nP F:=nops(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);\nMati ceSoustavy:=\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) $i=1.. PF]);\ni:='i';\nKoeficienty:=linsolve(MaticeSoustavy,PraveStrany):\nun apply(sum(Koeficienty[i]*F[i],i=1..PF),x);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+AproximaceGf*6%%\"XG%\"YG%\"FG6+%\"iG%\"jG%\"kG%#PXG %#PFG%\"vG%,KoeficientyG%/MaticeSoustavyG%,PraveStranyG6\"F4C->8(-%%no psG6#9&>8'-F96#9$>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$*&&FgqFZF[oFYF[oF\\oFenF4F4F4 " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Metoda nejmen\232\355ch \+ \350tverc\371" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "Hled\341me p\370 \355mku, kter\341 m\341 nejmen\232\355 sou\350et \350tverc\371 vzd\341 lenost\355 od zadan\375ch bod\371. Vzd\341lenosti m\354\370\355me ve \+ svisl\351m sm\354ru. Jde o extrem\341ln\355 \372lohu, kteru vy\370e \232\355me standatn\355 metodou infinitezim\341ln\355ho po\350tu." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "unassign('y','p','q','x','yy ','xx','X','Y','N');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Vydaleno \351st bodu od primky:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "primka:=y=p*x+q;\nprimka_ :=yy=p*xx+q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'primkaG/%\"yG,&*&% \"pG\"\"\"%\"xGF*F*%\"qGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(primk a_G/%#yyG,&*&%\"pG\"\"\"%#xxGF*F*%\"qGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Primka:=rhs(primka)-lhs(primka)=0;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'PrimkaG/,(*&%\"pG\"\"\"%\"xGF)F)%\"qGF)%\"yG!\"\" \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "mam bod [a,b] a na prim ce zvolim x[0]. Vzdalenost bodu [a,b] od vbodu na primce [x[0],?]. Hle d\341m glob\341mn\355 minimum v lok\341ln\355m minimu." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "rho:=(((a-x[0])^2+(b-subs(x=x[0],rh s(primka)))^2))^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG*$,&* $),&%\"aG\"\"\"&%\"xG6#\"\"!!\"\"\"\"#F+F+*$),(%\"bGF+*&%\"pGF+F,F+F0% \"qGF0F1F+F+#F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xxx:=D iff(rho,x[0])=diff(rho,x[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xx xG/-%%DiffG6$*$,&*$),&%\"aG\"\"\"&%\"xG6#\"\"!!\"\"\"\"#F/F/*$),(%\"bG F/*&%\"pGF/F0F/F4%\"qGF4F5F/F/#F/F5F0,$*&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:=solv e(rhs(xxx)=0,x[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&,(%\"aG \"\"\"*&%\"pGF(%\"bGF(F(*&F*F(%\"qGF(!\"\"F(,&F(F(*$)F*\"\"#F(F(F." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rho:=simplify(subs(x[0]=z,r ho));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$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#>%$rhoGf*6$%\"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 75 "i:='i';\nDelta:=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#>%\"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 "#Delt a:=rhs(Delta);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Dp:=Di ff(lhs(Delta),p)=simplify(diff(rhs(Delta),p));\nDq:=Diff(lhs(Delta),q) =simplify(diff(rhs(Delta),q));\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#DpG/-%%DiffG6$-%$SumG6$*&,(*&&%\"xG6#%\"iG\"\"\"%\"pGF3F3&%\"yGF1! \"\"%\"qGF3\"\"#,&F3F3*$)F4F9F3F3F7/F2;F3%\"NGF4,$*(F9F3,&*(F?F3)F8F9F 3F4F3F3-%$sumG6$,0*&)F/F9F3F4F3F7*(F/F3F%#DqG/-%%DiffG6$-%$SumG6$*&,(*&&%\" xG6#%\"iG\"\"\"%\"pGF3F3&%\"yGF1!\"\"%\"qGF3\"\"#,&F3F3*$)F4F9F3F3F7/F 2;F3%\"NGF8,$*(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 <$/%\"pG$!+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# >%$RP1Gf*6$%\"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[sF5F5F5" }}}{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*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`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#>%$RP2Gf*6$%\"XG%\"YG6)%'Primka G%\"xG%\"pG%\"qG%.SoucetCtvercuG%$RceG%&ParamG6\"F1C'>8$f*6#F+F16$%)op eratorG%&arrowGF1,&*&T#\"\"\"9$F=F=T%F=F1F16&F,8&F-8'>8(-%$sumG6$*$),& &9%6#%\"iGF=-F46#&F>FM!\"\"\"\"#F=/FN;F=-%%nopsG6#F>>8)6$/-%%diffG6$FD FA\"\"!/-Fhn6$FDFBFjn>8*-%&solveG6$<#FZ<$FAFB-%(unapplyG6$-%%subsG6$F_ o-F46#8%F]pF1F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "V maple je v estavena 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 1 {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#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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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 1 {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#>%\"gGf*6#%\"yG6\" 6$%)operatorG%&arrowGF(\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "m\341 rovnice tvar:" }{MPLTEXT 1 0 13 "\nMalt:=r[1];\n" }}} {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\"\"\"-%$expG6#*&%\"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$$\"3smm\"z/3uC\"FE$\"3Y(>)*Qz 7qk\"F27$$\"3!****\\7LRDX\"FE$\"3gc:o*[`yy\"F27$$\"3%om;zR'ok;FE$\"3y6 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}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "C1 :=solve(subs(t=0,a=.4,k=1,rhs(G))=.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G^$$!+)[7(*=\"!\"*$!+M;)R&yF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(RealDomain);\nG2:=dsolve(\{Gomp,x(0)=.1\});" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7H%#ImG%#ReG%\"^G%'arccosG%(arccoshG%' arccotG%(arccothG%'arccscG%(arccschG%'arcsecG%(arcsechG%'arcsinG%(arcs inhG%'arctanG%(arctanhG%$cosG%%coshG%$cotG%%cothG%$cscG%%cschG%%evalG% $expG%'expandG%&limitG%#lnG%$logG%$secG%%sechG%'signumG%)simplifyG%$si nG%%sinhG%&solveG%%sqrtG%%surdG%$tanG%%tanhG" }}{PARA 6 "" 1 "" {TEXT -1 29 "Methods for first order ODEs:" }}{PARA 6 "" 1 "" {TEXT -1 37 "- -- Trying classification methods ---" }}{PARA 6 "" 1 "" {TEXT -1 19 "t rying a quadrature" }}{PARA 6 "" 1 "" {TEXT -1 23 "trying 1st order li near" }}{PARA 6 "" 1 "" {TEXT -1 16 "trying Bernoulli" }}{PARA 6 "" 1 "" {TEXT -1 16 "trying separable" }}{PARA 6 "" 1 "" {TEXT -1 23 "<- se parable successful" }}{PARA 11 "" 1 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animatecurves = true );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %*infolevelG6#%'dsolveG\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ivs G72/-%\"xG6#\"\"!$F*F*/F'$\"\"\"!\"\"/F'$\"\"#F//F'$\"\"$F//F'$\"\"%F/ /F'$\"\"&F//F'$\"\"'F//F'$\"\"(F//F'$\"\")F//F'$\"\"*F//F'$\"#5F//F'$ \"#6F//F'$\"#7F//F'$\"#8F//F'$\"#9F//F'$\"#:F/" }}{PARA 6 "" 1 "" {TEXT -1 69 "Warning, plot may be incomplete, the following errors(s) \+ were issued:" }}{PARA 6 "" 1 "" {TEXT -1 119 " cannot evaluate the s olution past the initial point, problem may be complex, initially sing ular or improperly set up" }}{PARA 13 "" 1 "" {GLPLOT2D 315 315 315 {PLOTDATA 2 "6'-%(ANIMATEG6;75-%'CURVESG6&7#7$$\"\"!F-F,-%&COLORG6&%$R GBG$\"\"*!\"\"F2$\"\"#F4-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"$-F(6&7#7 $F,$\"/++++++5!#9F.F7F;-F(6&7#7$F,$\"/++++++?FEF.F7F;-F(6&7#7$F,$\"/++ ++++IFEF.F7F;-F(6&7#7$F,$\"/++++++SFEF.F7F;-F(6&7#7$F,$\"/++++++]FEF.F 7F;-F(6&7#7$F,$\"/++++++gFEF.F7F;-F(6&7#7$F,$\"/++++++qFEF.F7F;-F(6&7# 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