{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Titl e" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 41 "Koment\341\370e k p\355se mk\341m z numerick\375ch metod" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "readlib(isolate):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "V n\341sleduj\355c\355m textu je n\354kolik aproxinma\350 n\355ch mtod t\370i cbi\350en\355:" }}{PARA 15 "" 0 "" {TEXT -1 40 "P \370evzorkov\341n\355 pomoc\355 interpolace hodnot " }}{PARA 15 "" 0 " " {TEXT -1 34 "Regresn\355 p\370\355mk v r\371zn\375ch metrik\341ch" } }{PARA 15 "" 0 "" {TEXT -1 27 "Porovn\341n\355 aproximace funkce" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Interpolace polynomy" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "polynom 1. stupn\354: p\370\355mka" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unassign('x','y','p','q');" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "p\370\355klad:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Uva\236ujme body" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=[0.5,1];\nB:=[1,7.5];\n" }{TEXT -1 49 "chceme naj \355t rovnici p\370\355mky, kter\341 jimi proch\341z\355." }}}{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;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Dosad\355 me do n\355 oba body a vy\370e\232\355me rovnice pro nezn\341m\351 par ametry:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "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 vyhledem k promenym p a \+ q\nassign(param); #timto prikazem promenym p a q priradime hodnoty\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Nzn\355 m\341 p\370\355mka tvar :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Tot\351\236 m\371\236eme ud\354lat obecn \354. Zru\232\355me p\370i\370azen\355 n\354kter\375ch hodnot prom \354n\375m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "readlib(unass ign):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unassign('A','B',' p','q');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "rceA:=subs(x=A[1],y=A[2],pr imka); #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(par am); #timto prikazem promenym p a q priradime hodnoty" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "To \236e jsme museli cel\375 v\375po\350et opakovat n \341s vede k opr\341vn\354n\351 domn\354nce, \236e by bylo v\375hodn \351 jej zachovat pro dal\232\355 pou\236it\355v podob\354 procedury. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "line:=proc(A,B) # jm0n u line je nyni prirazena procedura se dvema parametry. procedura\nloca l x; # deklarace lokalnich promenych. lokalni promene si po provedeni \+ procedury uchopvaji tytez hiodnoty jake mely pred jejim spustenim \n \+ if A[1]=B[1] then # vyloucime pripad svisle primky\n print(`primka je svisla, nejde o funkci`)\n else\nunapply( (B[2]-A[2])/(B[1]-A[1]) *x-(-A[2]*B[1]+A[1]*B[2])/(B[1]-A[1]),x ); # prikaz unapply zajisti, z e vysledek bude funkce (nezavisla na oznaceni promenych)\nfi;\nend; # \+ konec procedury" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 259 " Pomoc\355 t \351to procedurz m\371\236eme vytvo\370it proceduru, jej\355m\236 v \375stupem bude lomen\341 \350\341ra, kter\341 proch\341z\355 zadan \375mi body. Jejich prvn\355 sou\370adnice jsou v prvn\355 prom\354nn \351 argumentu, druh\351 sou\370adnice jsou ve druh\351 prom\354nn\351 . P\370\355kazem nops se sjist\355, kolik jich vlastn\354 je." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "Plin:=proc(X,Y)\nlocal N,i, x;\nN:=nops(X);\nparam:=x " 0 "" {MPLTEXT 1 0 108 "unassign('X','Y');\nn:=5;\nfor i f rom 1 to n do\nX[i]:=i;\nY[i]:=evalf(ln(i));\nod:\ni:='i':\n[X[i],Y[i] ] $i=1..n;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Hled\341me polyno m, kter\375 proch\341z\355 v\232emi t\354mito body. Najdeme jej ve tva ru sou\350tu:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "for j from 1 to n do\ncitatel[j]:=simplify(product(x-X[i],i=1..n)/(x-X[j])):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "jmenovatel[j]:=subs(x=X[j],citatel[ j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "clen[j]:=citatel[j]/jmenova tel[j]*Y[j];\nod;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A tedy hleda n\375 polynom je:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Lagrange:=(sum (clen[i],i=1..n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "neboli" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Lagrange:=simplify(sum(clen[i],i=1. .n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Op\354t m\371\236eme nap sat proced\371ru, kter\341 polunom vztvo\370\355. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "Lagrange:=proc(X,Y) # Vstup: pole bodu a p ole hodnot. Vystup: Lagrangeuv polynom (funkce)\nlocal C,J,N,x,i,j,sub stitut,xxx;\ni:='i';\nXpom:=convert(X,list);\nN:=nops(Xpom);\nfor j fr om 1 to N do\nC[j]:=product((x-X[i]),i=1..N)/(x-X[j]);\nJ[j]:=subs(x=X [j],C[j]);\n#print(C[j],J[j]);\nod;\ni:='i';\nxxx:=sum(C[i]/J[i]*Y[i], i=1..N);\nunapply(xxx,x)\nend;" }}}{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; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Splainy" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 368 "Nev\375hoou Lagrangeova polynomu je to, 6e m8me-l i 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 po \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\341stech kvadratick\375 pol\375nom: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "unssign('a','b','c','x' );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:=a*x^2+b*x+c; # obe cn\375 tvar polznomu druh\351ho stupn\354" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Polynom druh \351ho stupn\354 zavis\355 na t\370ech parametrech. Hled\341me takov \351 hodnotz parmetr\371, aby m\354l polynom p\370edepsan\351 hodnoty \+ v bodech " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[j+1];" "6#&%\"xG6#,&%\"jG\"\"\"F(F(" }{TEXT -1 32 ", a p\370edepsanou derivaci v bod\354 " }{XPPEDIT 18 0 "x[j];" "6#&%\"x G6#%\"jG" }{TEXT -1 49 ". Polynom mus\355 spl\362ovat ns\371eduj\355c \355 t\370i rovnioce:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "d dd:=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];" }}}{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\});" }}}{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];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "kter\351 ma j\355 \370e\232en\355:" }{MPLTEXT 1 0 38 "\nsolve(\{PrvniSoustavaRovni c\},\{a,b,c\});" }}}{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 v ytvorim tak, ze bude prochazet prvnimi trema body\n\nQ:=z " 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];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "Body:=pointplo t([seq([x[i],y[i]],i=1..N)], color=NAVY,symbolsize=15):\nAproximace:=p lot(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 "displa y(\{Body,Aproximace,Fce\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 22 "Priklad: prevzorkovani" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "unassign('x','y','z','N','i','j','alpha','beta');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Uva\236ujme n\354jakou funkci, \+ nap\370\355klad:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Asuume some f unction, for instance:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f :=x->#ln(x)^2+sin(3*x)-x*sin(5*x)+cos(20*x);\nx^2/100+3*sin(5*x);" }}} {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 vzorkovacv\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(evalf(f(x[i])),i=1..N+1);\nPuvB:=point plot([seq([x[i],y[i]],i=1..N+1)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Nyn\355 p\370edpokl\341dejme, \236e u\236 neznme p\371vodn\355 \+ funkci, ale jen jej\355 hodnoty " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 10 " v bodech " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 177 " . Chceme p\370evzorkovat hodnoty na n\354jkou jinou frkvenci. Nap\355k lad frekvenci 7 Hz. Pou\236ijeme Linearn\355 aproximaci, aproximaci k ubick\375mi splainy (tyto nab\355z\355 nap\370\355klad Photoshop). " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "F:=1/7;\nN:=floor(-(alpha- beta)/F);\nz:=seq(alpha+j*F,j=0..N);\n" }}}{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);" }}}{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#p lot(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`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Provnani presnosti:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " Sou\350et \350tverc\371; odchylek" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "sum((f(z[i])-Plin([x],[y])(z[i]))^2,i=1..N+1); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sum((f(z[i])-Spl2([x],[y])(z[i] ))^2,i=1..N+1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Nejv\354t\232 \355 odchzlka:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "max(seq(( evalf(f(z[i])-Plin([x],[y])(z[i]))^2),i=1..N+1));" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "max(seq((evalf(f(z[i])-Spl2([x],[y])(z[i]))^2),i=1. .N+1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Obecn\354 ov\232em nel ze \370\355ci, \236e by druh\341 metoda byla lep\232\355, ne\236 prvn \355. 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" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 20 "Ortonorm\341ln\355 vektory" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Vektory " }{XPPEDIT 18 0 "v[1]...v[n];" "6#;&%\"vG6#\"\"\" &F%6#%\"nG" }{TEXT -1 81 " jsou ortonorm\341ln\355, pokud jsou normova n\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 s ou\350in, mus\355 platit " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }{XPPEDIT 19 1 "phi(v[i],v[j]):=piecewise(i=j,0,1);" "6#>-%$phiG6$ &%\"vG6#%\"iG&F(6#%\"jG-%*piecewiseG6%/F*F-\"\"!\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "M\341me-li naj\355t vektory, kter\351 gen eruj\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]; " }}}{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 "rc e[1]:=0=phi(v[2],u[1])+c[1]*phi(u[1],u[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "xxx:=isolate(rce[1],c[1]);" }}}{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) ;" }}}{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\232ech ny 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 "multip ly" 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 "Prov\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 gener uj\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 hodnost matice, kter \341 vznikne p\370id\341n\355m zadan\375ch vektor\371 k vypo\350\355t an\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\nkome ntC:=`vektory negeneruji zadany prostor`;\nelse \nkomentC:=``;\nfi;\n " }}}{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;" }}}{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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "\332loha aproximovat hodnoty hodnotami line\341rn\355 kombinace funkc\355:\nPo kud nechceme nejprve syst\351m funkc\355 respektive vektory jejich hod not nahradit ortonorm\341ln\355m syst\351mem, mus\355me poka\236d\351 \+ \370e\232it soustavu line\341rn\355ch rovnic pro koeficienty:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 601 "Aproximace:=proc(X,Y,F) #vs tup: pole bodu, pole hodnot, pole funkci. Vystup: linearni kombinace z adanych funkci, ktera nejlepe aproximuje zadane hodnoty.\nlocal i,j,k, PX,PF,v,Koeficienty,MaticeSoustavy,PraveStrany\n;\nPF:=nops(F); # Poce t funkci\nPX:=nops(X); # Pocet bodu\ni:='i';j:='j';k:='k';\nv:=evalf\n ([ unapply(F[i],x)(X[j]) $j=1..PX] $i=1..PF);\nMaticeSoustavy:=\nmatr ix(PF,PF,\n[[sum((v[i][k]*v[j][k]),k=1..PX) $j=1..PF] $i=1..PF]\n);\nP raveStrany:=vector([sum(Y[j]*v[i][j],j=1..PX) $i=1..PF]);\ni:='i';\nKo eficienty:=linsolve(MaticeSoustavy,PraveStrany):\nunapply(sum(Koeficie nty[i]*F[i],i=1..PF),x);\nend;" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Metoda nejmen\232\355ch \350tverc\371" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Hled\341me p\370\355mku, kter\341 m\341 nejmen\232\355 \351 sou\350et \350tverc\371 vzd\341kleost\355 od zadan\375ch bvod\371 . Vzd\341lenosti m\354\370\355me ve svisl\351m sm\354ru. Jde o extrem \341ln\355 \372lohu, kteru vy\370e\232\355me standatn\355 metodou infi nitezim\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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Primka:=rhs(primka)-lhs(primka)=0; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "mam bod [a,b] a na primce zv olim x[0]. Vzdalenost bodu [a,b] od vbodu na primce [x[0],?]. Hled\341 m 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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xxx:=Diff( rho,x[0])=diff(rho,x[0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z:=solve(rhs(xxx)=0,x[ 0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rho:=simplify(subs( x[0]=z,rho));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rho:=unapp ly(rho,a,b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Soucet ctvercu vz dalenosti:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Delta:=Sum(rh o(x[i],y[i])^2\n,i=1..N)=sum(rho(x[i],y[i])^2\n,i=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#Delta:=rhs(Delta);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Dp:=Diff(lhs(Delta),p)=simplify(di ff(rhs(Delta),p));\nDq:=Diff(lhs(Delta),q)=simplify(diff(rhs(Delta),q) );\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{rhs(Dp)=0 ,rhs(Dq)=0\},\{p,q\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x xx:=allvalues(%);" }}}{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];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "xxx;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "pr1:=subs(xxx[1],Y=p*X+q);\npr2:=su bs(xxx[2],Y=p*X+q);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Y;" } }}{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)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq([x[i],y[i]],i=1..N);" }}} {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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " res_a:=subs(Reseni,primka_);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "A:=pointplot([seq([x[i],y[i]],i=1..N)]):\nB:=plot(rhs(pr1),X=0 ..10):\nC:=plot(rhs(pr2),X=0..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{A,B,C\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "evalf(\nsubs(xxx[1],rhs(Delta))\n);\nevalf(\nsubs(xxx [2],rhs(Delta))\n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (linalg);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(subs(xxx [1],hessian(rhs(Delta),[p,q])));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3195 "RP1:= proc(x,y)\nlocal p,q,N,Delta,X,Y,pr,pr_;\nN:=nops(x);\nxxx := \{q = -( 1/2/(sum(x[i],i = 1 .. N)*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+s um(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, p = 1/2/(sum(x[i],i = 1 .. N)*sum( y[i],i = 1 .. N)-sum(x[i]*y[i],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^2- sum(x[i],i = 1 .. N)^2+sum(x[i]^2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. N) *N+sqrt(sum(y[i],i = 1 .. N)^4+2*sum(y[i],i = 1 .. N)^2*sum(x[i],i = 1 .. N)^2+2*sum(y[i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N-2*sum(y[i], i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[i],i = 1 .. N)^4-2*sum(x[ i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N+2*sum(x[i],i = 1 .. N)^2*sum (y[i]^2,i = 1 .. N)*N+sum(x[i]^2,i = 1 .. N)^2*N^2-2*sum(x[i]^2,i = 1 \+ .. N)*N^2*sum(y[i]^2,i = 1 .. N)+sum(y[i]^2,i = 1 .. N)^2*N^2-8*sum(x[ i],i = 1 .. N)*sum(y[i],i = 1 .. N)*sum(x[i]*y[i],i = 1 .. N)*N+4*sum( x[i]*y[i],i = 1 .. N)^2*N^2))\}, \{p = 1/2/(sum(x[i],i = 1 .. N)*sum(y [i],i = 1 .. N)-sum(x[i]*y[i],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^2-s um(x[i],i = 1 .. N)^2+sum(x[i]^2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. N)* N-sqrt(sum(y[i],i = 1 .. N)^4+2*sum(y[i],i = 1 .. N)^2*sum(x[i],i = 1 \+ .. N)^2+2*sum(y[i],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N-2*sum(y[i],i = 1 .. N)^2*sum(y[i]^2,i = 1 .. N)*N+sum(x[i],i = 1 .. N)^4-2*sum(x[i ],i = 1 .. N)^2*sum(x[i]^2,i = 1 .. N)*N+2*sum(x[i],i = 1 .. N)^2*sum( y[i]^2,i = 1 .. N)*N+sum(x[i]^2,i = 1 .. N)^2*N^2-2*sum(x[i]^2,i = 1 . . N)*N^2*sum(y[i]^2,i = 1 .. N)+sum(y[i]^2,i = 1 .. N)^2*N^2-8*sum(x[i ],i = 1 .. N)*sum(y[i],i = 1 .. N)*sum(x[i]*y[i],i = 1 .. N)*N+4*sum(x [i]*y[i],i = 1 .. N)^2*N^2)), q = -(1/2/(sum(x[i],i = 1 .. N)*sum(y[i] ,i = 1 .. N)-sum(x[i]*y[i],i = 1 .. N)*N)*(sum(y[i],i = 1 .. N)^2-sum( x[i],i = 1 .. N)^2+sum(x[i]^2,i = 1 .. N)*N-sum(y[i]^2,i = 1 .. N)*N-s qrt(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(xxx[2],Y=p*X+q);\n\nDelta :=Sum(rho(x[i],y[i])^2\n,i=1..N)=sum(rho(x[i],y[i])^2\n,i=1..N);\n\nif \nevalf(\nsubs(xxx[1],rhs(Delta))\n)\n<\nevalf(\nsubs(xxx[2],rhs(Delta ))\n)\n\nthen pr_:=pr1 else pr_:=pr2\nfi;\nunapply(rhs(pr_),X)\nend;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Priklad:" }{MPLTEXT 1 0 190 "\nw ith(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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "primka:=RP1([x],[y])('w');\npr:=plo t(primka,w=1..17):\ndisplay(\{bb,pr\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Regresni primka 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 primky\nSoucetCtvercu:=sum(((Y[i]-Primka(X[i]))^2), i=1..nops(X)); #soucet druhych mocnin odchylek\nRce:=diff(SoucetCtverc u,p)=0,diff(SoucetCtvercu,q)=0; # hledame minimum, tedy hledame parame try p a q tak, aby derivace `SoucetCtvercu` podle p i podle q byla 0\n Param:=solve(\{Rce\},\{p,q\}); # parametry najdeme resenim vynikle rov nice\nunapply(subs(Param,Primka(x)),x); #vystup je ve tvaru funkce\nen d;" }}}{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 0 131 "primka:=RP2([x],[y])('w');\npr2:=plot(primka,w=1..15 ,color=NAVY):\n#display(\{bb,pr\});\n\n#pr2:=plot(RP2,w=1..19):\ndispl ay(\{bb,pr,pr2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 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);" }}}{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 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "Porovnani:=proc(f,N,alpha,beta)\nlocal 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(eva lf(f(x[i])),i=1..N+1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A2:=(Lagr ange([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'-a lpha),'x');\nprint(`aproximujeme funkci`,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 p uvodni funkcer v prvni a ve druhe metrice:`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "[evalf(Int(abs(r1),w=alpha..beta)),\nevalf(Int(abs(r 2),w=alpha..beta)),\nevalf(Int(abs(r3),w=alpha..beta))],\n[evalf(Int(a bs(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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Porovnani(x-> exp(x)*log(x),5,1,2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " Porovnani(x->sin(x)+cos(x),7,0,12);\n" }}}{EXCHG {PARA 13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }