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aple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 
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{SECT 0 {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 0 107 "unassign('X','Y');n:=5;
\nfor i from 1 to n do\nX[i]:=i; Y[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$&%\"yGF'%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "for j from 1 to n do\ncitatel[j]:=simplify(product(x-
X[i],i=1..n)/(x-X[j])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "jmenovat
el[j]:=subs(x=X[j],citatel[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
P[j]:=citatel[j]/jmenovatel[j]*Y[j];\nod;" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "A tedy hledan\375 polynom je:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "Lagrange:=(sum(P[i],i=1..n));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%)LagrangeG,**,$\"+,`Cb6!#5\"\"\",&%\"xGF*F*!\"\"F*,&F
,F*\"\"$F-F*,&F,F*\"\"%F-F*,&F,F*\"\"&F-F*F-*,$\"+A2`YFF)F*F+F*,&F,F*
\"\"#F-F*F0F*F2F*F**,$\"+-1\\5BF)F*F+F*F7F*F.F*F2F*F-*,$\"++8*fq'!#6F*
F+F*F7F*F.F*F0F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "neboli" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 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 polyno
m vytvo\370\355. " }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "Lagrange:=pr
oc(X,Y) # Vstup: pole bodu a pole hodnot. Vystup: Lagrangeuv polynom (
funkce)\nlocal C,J,N,x,i,j,substitut,xxx;\ni:='i';\nXpom:=convert(X,li
st);\nN:=nops(Xpom);\nfor j from 1 to N do\nC[j]:=product((x-X[i]),i=1
..N)/(x-X[j]);\nJ[j]:=subs(x=X[j],C[j]);\n#print(C[j],J[j]);\nod;\ni:=
'i';\nxxx:=sum(C[i]/J[i]*Y[i],i=1..N);\nunapply(xxx,x)\nend;" }}{PARA 
7 "" 1 "" {TEXT -1 69 "Warning, `Xpom` is implicitly declared local to
 procedure `Lagrange`\n" }}{PARA 0 "" 0 "" {XPPMATH 20 "6#>%)LagrangeG
j+6$%\"XG%\"YG6+%\"CG%\"JG%\"NG%\"xG%\"iG%\"jG%*substitutG%$xxxG%%Xpom
G6\"F3C)>8(.F6>8,-%(convertG6$9$%%listG>8&-%%nopsG6#F9?(8)\"\"\"FFF@%%
trueGC$>&8$6#FE*&-%(productG6$,&8'FF&F=6#F6!\"\"/F6;FFF@FF,&FRFF&F=FLF
UFU>&8%FL-%%subsG6$/FRFYFJ>F6F7>8+-%$sumG6$*(&FKFTFF&FfnFTFU&9%FTFFFV-
%(unapplyG6$F]oFRF3F3F36$\"\"!Fjo" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "x:='x
';i:='i';\nN:=2;\nP:=sum(a[i]*x^i,i=0..N);\nP:=unapply(P,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"PG,(&%\"aG6#\"\"!\"\"\"*&&F'6#F*F*%\"xGF*F**
&&F'6#\"\"#F*)F.F2F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGj+6#%
\"xG6\"6$%)operatorG%&arrowGF(,(&%\"aG6#\"\"!\"\"\"*&&F.6#F1F19$F1F1*&
&F.6#\"\"#F1)F5F9F1F1F(F(F(6#F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "for i from 1 to 2 do\nrce[i]:=P(x[i])=y[i];\nod;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"\"/,(&%\"aG6#\"\"!F'*&&F+F&F'&%\"x
GF&F'F'*&&F+6#\"\"#F')F0F5F'F'&%\"yGF&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%$rceG6#\"\"#/,(&%\"aG6#\"\"!\"\"\"*&&F+6#F.F.&%\"xGF&F.F.*&&F
+F&F.)F2F'F.F.&%\"yGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r
ce[3]:=D(P)(x[1])=Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"
$/,&&%\"aG6#\"\"\"F-*(\"\"#F-&F+6#F/F-&%\"xGF,F-F-%\"QG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "i:='i';\nsolve(\{seq(rce[i],i=1..3)
\},\{a[i] $i=0..2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/&%\"aG6#\"\"#*&,**&&%\"xGF'\"\"\"%
\"QGF.!\"\"*&&F-6#F.F.F/F.F.&%\"yGF3F0&F5F'F.F.,(*$)F2F(F.F.*$)F,F(F.F
.*(F(F.F,F.F2F.F0F0/&F&6#\"\"!*&,,**F(F.F,F.F2F.F4F.F0*(F9F.F,F.F/F.F.
*&F9F.F6F.F.*&F;F.F4F.F.*(F;F.F2F.F/F.F0F.F7F0/&F&F3,$*&,**&F9F.F/F.F.
*(F(F.F2F.F4F.F0*(F(F.F2F.F6F.F.*&F/F.F;F.F0F.F7F0F0" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 7 "Splainy" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 368 "Nev\375hoou Lagrangeova polynomu je to, \236e m\341me-li mnoho
 bod\371, dostabneme polznom p\370\355li\232 vysok\351ho stzupn\354. N
ev\375hodou po \350\341stech afinn\355\355 aproximace je v tom, \236e \+
dostaneme funkci, kter\341 nem\341 v bodech, jimi\236 ji prokl\341d
\341me derivaci. Najdeme aproximaci po \350\341stech polynomialn\355 f
unkc\355, kter\341 bude m\355t v ka\236d\351m bod\354 derivaci. Jako p
\370\355klad uvedeme po \350\341stech kvadratick\375 pol\375nom:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "unssign('a','b','c','x');" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(unssignG6&%\"aG%\"bG%\"cG%\"xG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:=a*x^2+b*x+c; # obecn\375
 tvar polznomu druh\351ho stupn\354" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*&%\"aG\"\"\")%\"xG\"\"
#F(F(*&%\"bGF(F*F(F(%\"cGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "P
olynom druh\351ho stupn\354 zavis\355 na t\370ech parametrech. Hled
\341me takov\351 hodnoty parmetr\371, aby m\354l polynom p\370edepsan
\351 hodnoty  v bodech " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[j+1];" "6#&%\"xG6#,&%\"jG\"\"\"F(F(
" }{TEXT -1 32 ", a p\370edepsanou derivaci v bod\354 " }{XPPEDIT 18 
0 "x[j];" "6#&%\"xG6#%\"jG" }{TEXT -1 49 ".  Polynom mus\355 spl\362ov
at n\341sleduj\355c\355 t\370i rovnice:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "ddd:=diff(p,x);\nSoustavaRovnic:=\nsubs(x=x[j],ddd)=
der,\nsubs(x=x[j],p)=y[j],\nsubs(x=x[j+1],p)=y[j+1];" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$dddG,&*(\"\"#\"\"\"%\"aGF(%\"xGF(F(%\"bGF(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/SoustavaRovnicG6%/,&*(\"\"#\"\"\"%
\"aGF*&%\"xG6#\"\"'F*F*%\"bGF*%$derG/,(*&F+F*)F,F)F*F**&F0F*F,F*F*%\"c
GF*&%\"yGF./,(*&F+F*)&F-6#\"\"(F)F*F**&F0F*F>F*F*F7F*&F9F?" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "Jejich \370e\232en\355 je" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(\{SoustavaRovnic\},\{a,b,c\})
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"aG,$*&,(*$)&%\"xG6#\"\"'\"
\"#\"\"\"F0*$)&F,6#\"\"(F/F0F0*(F/F0F3F0F+F0!\"\"F7,**&F+F0%$derGF0F7&
%\"yGF-F0*&F3F0F:F0F0&F<F4F7F0F7/%\"cG,$*&,,*(F*F0F3F0F:F0F7*&F*F0F>F0
F7*(F+F0F:F0F2F0F0*&F;F0F2F0F7**F/F0F;F0F3F0F+F0F0F0F(F7F7/%\"bG*&,**&
F*F0F:F0F7*(F/F0F+F0F;F0F0*(F/F0F+F0F>F0F7*&F:F0F2F0F0F0F(F7" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Polznom, kter\375proch\341z\355 t
\370emi zadan\375mi body zase spl\362uje rovnice" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 84 "PrvniSoustavaRovnic:=\nsubs(x=x[1],p)=y[1],\n
subs(x=x[2],p)=y[2],\nsubs(x=x[3],p)=y[3];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%4PrvniSoustavaRovnicG6%/,(*&%\"aG\"\"\")&%\"xG6#F*\"
\"#F*F**&%\"bGF*F,F*F*%\"cGF*&%\"yGF./,(*&F)F*)&F-6#F/F/F*F**&F1F*F9F*
F*F2F*&F4F:/,(*&F)F*)&F-6#\"\"$F/F*F**&F1F*FAF*F*F2F*&F4FB" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 18 "kter\351 maj\355 \370e\232en\355:" }
{MPLTEXT 1 0 38 "\nsolve(\{PrvniSoustavaRovnic\},\{a,b,c\});" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#<%/%\"aG*&,.*&&%\"xG6#\"\"\"F,&%\"yG6#\"\"$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,F
1*&F)F,)F6F8F,F,*&F3F,FEF,F1F1/%\"bG,$*&,.*&FAF,F4F,F,*&FAF,F;F,F1*&FE
F,F4F,F1*&F-F,FEF,F,*&F-F,F?F,F1*&F;F,F?F,F,F,F=F1F1/%\"cG*&,.*(FAF,F6
F,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 trema body\n
\nQ:=z<x[3],p[1];\n\nfor i from 2 to K do\n  j:=i+1; # cislo prvniho z
 obou bodu, kterym prochazi dalsi parabola\n\n#print(i,j,`|`,`|`,p[i-1
]);\n#print(x[i],y[j]);\n#print(x[i+1],y[j+1]);\n\n  der:=2*a*x[j]+b; \+
# derivace posledniho polynomu v poslednim bode\n  b := -(-2*x[j]*y[j]
+2*x[j]*y[j+1]+x[j]^2*der-der*x[j+1]^2)/(x[j]^2+x[j+1]^2-2*x[j+1]*x[j]
); \n  a := (-y[j]-x[j+1]*der+y[j+1]+x[j]*der)/(x[j]^2+x[j+1]^2-2*x[j+
1]*x[j]);   c := (x[j]^2*x[j+1]*der+x[j]^2*y[j+1]-x[j]*der*x[j+1]^2+y[
j]*x[j+1]^2-2*y[j]*x[j+1]*x[j])/(x[j]^2+x[j+1]^2-2*x[j+1]*x[j]);\np[i]
:=a*z^2+b*z+c;\n\nif i<K then\n   Q:=Q,z<x[j+1],p[i];\n  else\n   Q:=Q
,p[i]\nfi\n\n\nod; #vytvori se postupne vsechny oplynomy\n\nQ:=piecewi
se(Q);\nunapply(Q,z)\nend;\n" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, \+
`p` is implicitly declared local to procedure `Spl2`\n" }}{PARA 7 "" 
1 "" {TEXT -1 62 "Warning, `i` is implicitly declared local to procedu
re `Spl2`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, `der` is implicit
ly declared local to procedure `Spl2`\n" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%%Spl2Gj+6$%\"xG%\"yG6-%\"NG%\"KG%\"jG%\"aG%\"bG%\"cG%\"QG%\"zG
%\"pG%\"iG%$derG6\"F5C,>8$-%%nopsG6#9$>8%,&F8\"\"\"\"\"#!\"\">8',$*&,.
*&&F<6#\"\"$F@&9%6#F@F@F@*&&F<FNF@&FMFJF@FB*&&F<6#FAF@FLF@FB*&FSF@FQF@
F@*&FIF@&FMFTF@FB*&FPF@FWF@F@F@,.*&FSF@)FIFAF@FB*&)FPFAF@FSF@F@*&FPF@F
enF@F@*&FIF@)FSFAF@F@*&FIF@FgnF@FB*&FPF@FjnF@FBFBFB>8(*&,.*&FgnF@FQF@F
B*&FgnF@FWF@F@*&FjnF@FQF@F@*&FLF@FenF@F@*&FLF@FjnF@FB*&FWF@FenF@FBF@FY
FB>8)*&,.*(FgnF@FSF@FQF@F@*(FgnF@FIF@FWF@FB*(FjnF@FPF@FQF@FB*(FWF@FPF@
FenF@F@*(FLF@FSF@FenF@FB*(FjnF@FIF@FLF@F@F@FYFB>&8,FN,(*&FDF@)8+FAF@F@
*&F^oF@FgpF@F@FhoF@>8*6$2FgpFIFbp?(8-FAF@F>%%trueGC)>8&,&F^qF@F@F@>8.,
&*(FAF@FDF@&F<6#FbqF@F@F^oF@>F^o,$*&,**(FAF@FhqF@&FMFiqF@FB*(FAF@FhqF@
&FM6#,&FbqF@F@F@F@F@*&)FhqFAF@FeqF@F@*&FeqF@)&F<FbrFAF@FBF@,(*$FerF@F@
*$FgrF@F@*(FAF@FhrF@FhqF@FBFBFB>FD*&,*F_rFB*&FhrF@FeqF@FBFarF@*&FhqF@F
eqF@F@F@FirFB>Fho*&,,*(FerF@FhrF@FeqF@F@*&FerF@FarF@F@*(FhqF@FeqF@FgrF
@FB*&F_rF@FgrF@F@**FAF@F_rF@FhrF@FhqF@FBF@FirFB>&Fcp6#F^qFdp@%2F^qF>>F
jp6%Fjp2FgpFhrF[t>Fjp6$FjpF[t>Fjp-%*piecewiseG6#Fjp-%(unapplyG6$FjpFgp
F5F5F56$\"\"!F\\u" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Priklad:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "with(plots):\nN:=7;f:=x->ln
(x)*sin(x);\nx:=seq(round(2*i+sin(i)^2),i=1..N);\ny:=seq(evalf(f(x[i])
),i=1..N);x[N];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGj+6#%\"xG6\"6$%)operatorG%&arrow
GF(*&-%#lnG6#9$\"\"\"-%$sinGF/F1F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xG6)\"\"$\"\"&\"\"'\"\"*\"#6\"#7\"#9" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"yG6)$\"+^<O]:!#5$!+#3HLa\"!\"*$!+ZOX1]F($\"+
['o^0*F($!+*yryR#F+$!+7OLL8F+$\"+.'pUh#F+" }}{PARA 11 "" 1 "" 
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