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0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 41 "Koment\240\375e k p\241se
mk\240m z  numerick\354ch metod" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "readlib(isolate):with(linalg):\n" }{TEXT -1 182 " Tento text j
e ps\240n v jazyce Maple. V\347echny v\354po\237ty m\205\247ete ov\347
em stejn\330 dob\375e prov\240d\330t t\375eba v programu Matlab (co
\247 uje rovn\330\247 i s d\205vod\205, \247e jste s n\241m dob\375e o
bezn\240meni doporu\237eno)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Interpolace polynomy" }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 25 "polynom 1. stupn\330: p\375\241mka" }}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 24 "Lmena cara pouze obrazek" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "with(plots):\na:=rand(-9..9)
:\nA:=t->A(1)+a()/sqrt(abs(t+1)):\nA(1):=10:\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "B:=seq([n,evalf(A(n))],n=0..50);\npointplot([B],style
=POINT);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pointplot([
B],style=LINE);\n" }}}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "p\375\241kla
d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Uva\247ujme body" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=[0.5,1];\nB:=[1,7.5];\n" 
}{TEXT -1 49 "chceme naj\241t rovnici p\375\241mky, kter\240 jimi proc
h\240z\241." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7$$\"\"&!\"\"\"\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7$\"\"\"$\"#v!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "obecn\354 tvar rovnice p\375\241mk
y, kter\240 nen\241 svisl\240 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\241me do n\241 oba body a vy\375e\347\241me ro
vnice pro nezn\240m\202 parametry:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 284 "rceA:=subs(x=A[1],y=A[2],primka); #prvni rovnice vyn
ikne dosazenim prvniho bodu\nrceB:=subs(x=B[1],y=B[2],primka); #druha \+
dosazenim druheho\nparam:=solve(\{rceA,rceB\},\{p,q\}); #vyresime obe \+
rovnice vyhledem k promenym p a q\nassign(param); #timto prikazem prom
enym p a q priradime hodnoty\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%r
ceAG/\"\"\",&*&$\"\"&!\"\"F&%\"pGF&F&%\"qGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%rceBG/$\"#v!\"\",&%\"pG\"\"\"%\"qGF+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&paramG<$/%\"pG$\"#8\"\"!/%\"qG$!+++++b!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Nyn\241 m\240 p\375\241mka tvar:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%\"yG,&*&$\"#8\"\"!\"\"\"%\"xGF*F*$\"+++++b!\"*!\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Tot\202\247 m\205\247eme ud
\330lat obecn\330.  Zru\347\241me p\375i\375azen\241 n\330kter\354ch h
odnot  prom\330n\354m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "re
adlib(unassign):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unassig
n('A','B','p','q');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "primk
a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&%\"pG\"\"\"%\"xGF(F(%
\"qGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "rceA:=subs(x=A[1
],y=A[2],primka); #prvni rovnice vynikne dosazenim prvniho bodu\nrceB:
=subs(x=B[1],y=B[2],primka); #druha dosazenim druheho\nparam:=solve(\{
rceA,rceB\},\{p,q\}); #vyresime obe rovnice vyhledem k promenym p a q
\nassign(param); #timto prikazem promenym p a q priradime hodnoty" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rceAG/&%\"AG6#\"\"#,&*&%\"pG\"\"\"&
F'6#F-F-F-%\"qGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rceBG/&%\"BG6#
\"\"#,&*&%\"pG\"\"\"&F'6#F-F-F-%\"qGF-" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&paramG<$/%\"pG*&,&&%\"AG6#\"\"#\"\"\"&%\"BGF,!\"\"F.,&&F+6#F.
F.&F0F4F1F1/%\"qG*&,&*&F/F.F3F.F.*&F5F.F*F.F1F.F2F1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "primka;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%\"yG,&*(,&&%\"AG6#\"\"#\"\"\"&%\"BGF*!\"\"F,,&&F)6#F,F,&F.F2F/F/%\"
xGF,F,*&,&*&F-F,F1F,F,*&F3F,F(F,F/F,F0F/F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 139 "To \247e jsme museli cel\354 v\354po\237et opakovat n
\240s vede k opr\240vn\330n\202 domn\330nce, \247e by bylo v\354hodn
\202 jej zachovat pro dal\347\241 pou\247it\241v podob\330 procedury.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "line:=proc(A,B) # jmen
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" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%lineGf*6$%\"AG%
\"BG6#%\"xG6\"F+@%/&9$6#\"\"\"&9%F0-%&printG6#%Aprimka~je~svisla,~nejd
e~o~funkciG-%(unapplyG6$,&*(,&&F36#\"\"#F1&F/F?!\"\"F1,&F2F1F.FBFB8$F1
F1*&,&*&FAF1F2F1FB*&F.F1F>F1F1F1FCFBFBFDF+F+F+" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 259 " Pomoc\241 t\202to procedury m\205\247eme vytvo\375
it proceduru, jej\241m\247 v\354stupem bude lomen\240 \237\240ra, kter
\240 proch\240z\241 zadan\354mi body. Jejich prvn\241 sou\375adnice js
ou v prvn\241 prom\330nn\202 argumentu, druh\202 sou\375adnice jsou ve
 druh\202 prom\330nn\202. P\375\241kazem nops se sjist\241, kolik jich
 vlastn\330 je." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "Plin:=p
roc(X,Y)\nlocal N,i,x;\nN:=nops(X);\nparam:=x<X[1],Y[1];\nfor i from 1
 to N-1 do\n  param:=param, x<X[i+1],line([X[i],Y[i]],[X[i+1],Y[i+1]])
(x);\nod;\n  param:=param, Y[N];\nunapply(piecewise(param),x);\nend;" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Polynomy vz\347\347\241ch stu
p\345\205" }}{EXCHG {PARA 5 "" 0 "" {TEXT -1 18 "Lagrange\205v polynom
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "P\375\241klad. Uva\247ujme bo
dy:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "n:=5;\nfor 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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Hled\240me polynom, kte
r\354 proch\240z\241 v\347emi t\330mito body. Najdeme jej ve tvaru sou
\237tu:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "for j from 1 to \+
n do\ncitatel[j]:=simplify(product(x-X[i],i=1..n)/(x-X[j])):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "jmenovatel[j]:=subs(x=X[j],citatel[j]):" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "clen[j]:=citatel[j]/jmenovatel[j]
*Y[j];\nod;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A tedy hledan\354 \+
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\330t m\205\247eme napsat proc
ed\205ru, kter\240 polynom vytvo\375\241. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 339 "Lagrange:=proc(X,Y) # Vstup: pole bodu a pole hodn
ot. Vystup: Lagrangeuv polynom (funkce)\nlocal C,J,N,x,i,j,substitut,x
xx;\ni:='i';\nXpom:=convert(X,list);\nN:=nops(Xpom);\nfor j from 1 to \+
N do\nC[j]:=product((x-X[i]),i=1..N)/(x-X[j]);\nJ[j]:=subs(x=X[j],C[j]
);\n#print(C[j],J[j]);\nod;\ni:='i';\nxxx:=sum(C[i]/J[i]*Y[i],i=1..N);
\nunapply(xxx,x)\nend;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "nebo ek
vivaletn\330 s vzu\247it\241m ji\247 hotov\202 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 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "PoCastechP:=proc(X,Y,n)\nlocal N,i;\nN:=nops(x)\nLagr
ange2(X[i..i+n+1],Y[i..i+N+1]),\n\nLagrange" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "PoCastechP()" }}}{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 "Ortonorm
\240ln\241 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\240ln\241, pokud jsou normovan\202 a  ka\247d\202 d
va jsou ortogon\240ln\241, tj, je-li" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 
"phi" "6#%$phiG" }{TEXT -1 30 " skal\240rn\241 sou\237in, mus\241 plat
it " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }{XPPEDIT 19 1 "ph
i(v[i],v[j]):=piecewise(i=j,0,1);" "6#>-%$phiG6$&%\"vG6#%\"iG&F(6#%\"j
G-%*piecewiseG6%/F*F-\"\"!\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
78 "M\240me-li naj\241t vektory, kter\202 generuj\241 stejn\354 prosto
r, jako nez\240visl\202 vektory  " }{XPPEDIT 18 0 "v[1]...v[n];" "6#;&
%\"vG6#\"\"\"&F%6#%\"nG" }{TEXT -1 75 " , m\205\247eme vz\241t vektory
 takov\202, \247e prvn\241 bude ten, kter\354 z\241sk\240me normov\240
n\241m " }{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#\"\"\"*&&%\"vG
6#F'F'-%$phiG6$&F*6#F'&F*6#F'!\"\"" }{TEXT -1 32 ", druh\354 bude line
\240rn\241 kombinac\241 " }{XPPEDIT 18 0 "u[1]" "6#&%\"uG6#\"\"\"" }
{TEXT -1 3 " a " }{XPPEDIT 18 0 "v[2]" "6#&%\"vG6#\"\"#" }{TEXT -1 86 
". Abychom zajistili, \247e vektory budou nad\240le line\240rn\330 nez
\240visl\202, zvol\241me koeficient u " }{XPPEDIT 18 0 "v[2]" "6#&%\"v
G6#\"\"#" }{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\237\241t\240me tak, \247e " }{XPPEDIT 18 0 "rce[1]" "6#&%
$rceG6#\"\"\"" }{TEXT -1 21 " vyn\240sob\241me skal\240rn\330 " }
{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]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "xxx:=i
solate(rce[1],c[1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "d\240le r
ekursivn\330. 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\241sk\240
me " }{XPPEDIT 18 0 " n-1 " "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 12 " ro
vnic 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\347echny vektory vyd\330l\241me jej
ich velikost\241." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Pozn\240mky:
" }}{PARA 15 "" 0 "" {TEXT -1 109 "Pokud jsou vektory n-tice \237\241s
el, je n\240soben\241 vektoru skal\240rem (zna\237eno dosud hv\330zdi
\237kou) prov\240d\330no p\375\241kazem  " }{HYPERLNK 17 "scalarmul  \+
" 2 "scalarmul" "" }{TEXT -1 35 "a skal\240rn\241 sou\237in vektor\205
 p\375\241kazem " }{HYPERLNK 17 "multiply" 2 "linalg,multiply" "" }
{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 101 "V takov\202m p\243\375
\241pad\330 lze pou\247\241t k nalezen\241 ortogon\240ln\241ho syst
\202mu )vektory nejsou ale normovan\202) p\375\241kaz " }{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\240d\241 se kontrola toho, zda jsou v\347echny vektory normovan\202
, pak toho, zda jsou ka\247d\202 dva r\205zn\202 vektory ortogon\240ln
\241 a nakonec toho, zda vektory generuj\241 tent\354\247 prostor, jak
o zadan\202 vektory" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Pro \237ty
\375i vektory z " }{XPPEDIT 18 0 "Re^n" "6#)%#ReG%\"nG" }{TEXT -1 91 "
 se zkoum\240 hodnost matice, kter\240 vznikne p\375id\240n\241m  zada
n\354ch vektor\205 k vypo\237\241tan\354m vektor\205m:" }}{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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
109 "Pokud jsou vektory 4 funkce, zkoum\240 se, zda je mo\247no line
\240rn\241 kombinac\241 vyj\240d\375it hodnoty ve \237ty\375ech 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 ne
generuji 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\240rn\241 kombinac\241 funkc\241" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "\351loha aproximovat funkci line
\240rn\241 kombinac\241 zadan\354ch funkc\241:\nPokud jsme o fourierov
u \375adu, nap\375\241klad, vol\241me p\375\241slu\347n\354 ortonorm
\240ln\241 syst\202m. Koeficienty pak m\205\247eme po\237\241tat p\375
\241mo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 683 "Fourier:=proc(f
,n,beta) #vstup: funkce v promenne x (vzorecek) a prirozene cislo n. V
ystup: prvnich n scitancu fourierovy rady (pocitaji se i nulove). -a[0
]+...+a[(n-1)/2]*sin/cos (x*((n-1)/2))- n intervalu 0..beta.\nlocal i,
N;\nif n mod 2 = 1 then N:=(n+1)/2\n  else\n    N:=n/2; \nfi;i:='i';\n
\nfor i from 0 to N do\n  a[i]:=evalf(2/beta*evalf(int(f(x)*cos(2*i*Pi
*x/beta),x=0..beta)));\n  b[i]:=evalf(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]*cos(2*i*Pi*x/beta)+b[i]*sin(2*i*Pi*x
/beta),i=1..N-1)+a[N]*cos(N*2*Pi*x/beta),x);\nfi\nend;\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 230 "\351loha aproximovat hodnoty hodnotami l
ine\240rn\241 kombinace funkc\241:\nPokud nechceme nejprve syst\202m f
unkc\241 respektive vektory jejich hodnot nahradit ortonorm\240ln\241m
 syst\202mem, mus\241me poka\247d\202 \375e\347it soustavu line\240rn
\241ch rovnic pro koeficienty:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 601 "Aproximace:=proc(X,Y,F) #vstup: pole bodu, pole hodnot, pole \+
funkci. Vystup: linearni kombinace zadanych funkci, ktera nejlepe apro
ximuje zadane hodnoty.\nlocal i,j,k,PX,PF,v,Koeficienty,MaticeSoustavy
,PraveStrany\n;\nPF:=nops(F); # Pocet funkci\nPX:=nops(X); # Pocet bod
u\ni:='i';j:='j';k:='k';\nv:=evalf\n([ unapply(F[i],x)(X[j])  $j=1..PX
] $i=1..PF);\nMaticeSoustavy:=\nmatrix(PF,PF,\n[[sum((v[i][k]*v[j][k])
,k=1..PX) $j=1..PF] $i=1..PF]\n);\nPraveStrany:=vector([sum(Y[j]*v[i][
j],j=1..PX) $i=1..PF]);\ni:='i';\nKoeficienty:=linsolve(MaticeSoustavy
,PraveStrany):\nunapply(sum(Koeficienty[i]*F[i],i=1..PF),x);\nend;" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Metoda nejmen\347\241ch \237tve
rc\205" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Hled\240me p\375\241mku
, kter\240 m\240 nejmen\347\241\202 sou\237et \237tverc\205 vzd\240kle
ost\241 od zadan\354ch bvod\205. Vzd\240lenosti m\330\375\241me  ve sv
isl\202m sm\330ru. Jde o extrem\240ln\241 \243lohu, kteru vy\375e\347
\241me standatn\241 metodou infinitezim\240ln\241ho po\237tu." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "RegresniPrimka:=proc(X,Y)\n
local Primka,x,p,q;\nPrimka:=x->p*x+q; #obecna rovnice primky\nSoucetC
tvercu:=sum(((Y[i]-Primka(X[i]))^2),i=1..nops(X)); #soucet druhych moc
nin odchylek\nRce:=diff(SoucetCtvercu,p)=0,diff(SoucetCtvercu,q)=0; # \+
hledame minimum, tedy hledame parametry p a q tak, aby derivace `Souce
tCtvercu` podle p i podle q byla 0\nParam:=solve(\{Rce\},\{p,q\}); # p
arametry najdeme resenim vynikle rovnice\nunapply(subs(Param,Primka(x)
),x); #vystup je ve tvaru funkce\nend;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "V maple je vestavena funkce " }{HYPERLNK 17 "leastsquare
" 2 "leastsquare" "" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 17 "Racion\240ln\241 postup" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 340 "Tato \237\240st je zde pro ty, kte\375\241 nechct\330j\241 pou
\247\241t program matlab (v n\330m\247 jste s podobn\354mi funkcemi, j
ako jsou u\247ity zde dob\375e obezn\240meni), ale experimentuj\241 s \+
programem Maple. C\241lem je nazna\237it, jak lze s co nejmen\347\241 \+
n\240mahou na\237\241st data a nov\240 data vytvo\375it a ulo\247it. J
istou roli zde hraj\241 r\243zn\202 datov\202 tzpz, co\247 je specifik
um ka\247d\202ho jazyka." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"unassign('x','y','z');" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "P\375
\241klad:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "P\375edpokl\240dejme
, \247e m\240me za \243kol line\240rn\330 aproximovat hodnoty " }
{XPPEDIT 18 0 "z[i]" "6#&%\"zG6#%\"iG" }{TEXT -1 15 " pomoc\241 hodnot
 " }{XPPEDIT 18 0 "y[i]" "6#&%\"yG6#%\"iG" }{TEXT -1 22 " nam\330\375e
n\354ch v bodech  " }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 
55 ". P\375edpokl\240dejme, \247e zadan\202 hodnotz dostaneme ve tvaru
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "xxx:=[\nx[1]=1,x[2]=3
,x[3]=12,x[4]=21,x[5]=23,\ny[1]=27,y[2]=26,y[3]=23,y[4]=21,y[5]=19,\nz
[1]=2,z[2]=4,z[3]=5,z[4]=6,z[5]=22\n];" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Prvn\241 probl\202m, j
ak efektivn\330 p\375i\375adit data prom\330nn\354m v na\347em p\375
\241pad\330 vy\375e\347\241me funkc\241 " }{HYPERLNK 17 "assign" 2 "as
sign" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
x[1];x[2];print(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assi
gn(xxx);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x[1];x[2];print
(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "x je nzn9 tabulka. konver
tujeme ji na seznam p\375\241kazem convert:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 57 "x:=convert(x,list);y:=convert(y,list);z:=convert(z
,list);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Nyn\241 pou\247ijeme funkci Plin, \+
kterou jsme si j\247 p\375ipravili:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "Phi:=Plin(x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
64 "Hodnoty, kter\202 jsou \375e\347en\241m jsou hodnotami t\202to fun
kce v bodech " }{XPPEDIT 18 0 "z[i]" "6#&%\"zG6#%\"iG" }{TEXT -1 1 ":
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "reseni:= Phi(z[i]) $i=1
..nops(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Nyni chceme v\354sl
edky ulozit do souboru. Otev\375emeSoubor SouborSVysledky a nazveme je
j file:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "file:=fopen(`Sou
borSVysledky`,WRITE); #soubor otevren pro psani" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 33 "v\354sledky konvertujeme na \375et\330zec:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Vysledky:=convert([reseni],s
tring);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "a vynech\240me prvn
\241 a posledn\241 znak (hranat\202 z\240vorky)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "Vysledky:=substring(Vysledky,2..-2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Zap\241\347eme je na \375\240dek v
\354stupn\241ho souboru:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"writeline(file,Vysledky);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "a s
oubor uzav\375eme (po zaps\240n\241 v\347ech v\354sledk\205)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fclose(file);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 221 "Pokud chceme ps\240t soubor i s cestopu,
 mus\241me ka\247d\354 backslash zdvojit, tedy nap\375: `a.\\\\MyDir\\
\\MyFile.txt`. Soubor pos\241lejte v t\330le e-mailu, ne jako p\375
\241lohu. Pokud chcete do souboru je\347t\330 n\330co p\375ipsat, otev
\375ete jej v modu  " }{HYPERLNK 17 "append" 2 "fopen" "" }{TEXT -1 2 
"\010." }}}}}{MARK "2 1 20 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }