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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 7 "Splainy" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 29 "M\341me zad\341nu posloupnost bod\371 " }
{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT -1 23 " a \372posloupno
st hodnot " }{XPPEDIT 18 0 "y[i];" "6#&%\"yG6#%\"iG" }{TEXT -1 31 ". H
led\341me posloupnost polynom\371 " }{XPPEDIT 18 0 "P[i];" "6#&%\"PG6#
%\"iG" }{TEXT -1 22 " s t\354mito vlastnostmi:" }}}{EXCHG {PARA 15 "" 
0 "" {TEXT -1 36 "i-t\375 polynom m\341 v i-t\351 skupin\354 bodu " }
{XPPEDIT 18 0 "x[i*kappa+iota];" "6#&%\"xG6#,&*&%\"iG\"\"\"%&kappaGF)F
)%%iotaGF)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "iota = 1 .. kappa;" "6#/%
%iotaG;\"\"\"%&kappaG" }{TEXT -1 9 " hodnoty " }{XPPEDIT 18 0 "P(x[i*k
appa+iota]) = y([i*kappa+iota]);" "6#/-%\"PG6#&%\"xG6#,&*&%\"iG\"\"\"%
&kappaGF-F-%%iotaGF--%\"yG6#7#,&*&F,F-F.F-F-F/F-" }}{PARA 15 "" 0 "" 
{XPPEDIT 18 0 "`@@`(D,s)(P[i])(x[(i+1)*kappa]) = `@@`(D,s)(P[i+1])(x[(
i+1)*kappa]);" "6#/---%#@@G6$%\"DG%\"sG6#&%\"PG6#%\"iG6#&%\"xG6#*&,&F/
\"\"\"F6F6F6%&kappaGF6---%#@@G6$%\"DG%\"sG6#&F-6#,&F/F6F6F66#&F26#*&,&
F/F6F6F6F6F7F6" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "s = 1 .. lambda;" "6#
/%\"sG;\"\"\"%'lambdaG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Nejjedn
odu\236\232\355 p\370\355pad, " }{XPPEDIT 18 0 "kappa = 2,lambda = 1;
" "6$/%&kappaG\"\"#/%'lambdaG\"\"\"" }{TEXT -1 141 ":\nHled\341me kvad
ratick\351 polynomy z nich\236 ka\236d\375 proch\341z\355 dv\354ma zad
an\375mi body a v bod\354, ve kter\351m se dva polynomy st\375kaj\355,
 maj\355 stejnou derivaci." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "rho:=sum(a[i]*x^i,i=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r
hoG,(*&&%\"aG6#\"\"\"F*%\"xGF*F**&&F(6#\"\"#F*)F+F/F*F**&&F(6#\"\"$F*)
F+F4F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rho:=unapply(rh
o,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoGj+6#%\"xG6\"6$%)operat
orG%&arrowGF(,(*&&%\"aG6#\"\"\"F19$F1F1*&&F/6#\"\"#F1)F2F6F1F1*&&F/6#
\"\"$F1)F2F;F1F1F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "B:=[x[i],y[i]] $i=0..1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"B
G6$7$&%\"xG6#\"\"!&%\"yGF)7$&F(6#\"\"\"&F,F/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 54 "for i from 1 to 2 do\nrce[i]:=rho(B[i][1])=B[i][
2];\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"\"/,(*&&%\"a
GF&F'&%\"xG6#\"\"!F'F'*&&F,6#\"\"#F')F-F4F'F'*&&F,6#\"\"$F')F-F9F'F'&%
\"yGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"#/,(*&&%\"aG6#
\"\"\"F.&%\"xGF-F.F.*&&F,F&F.)F/F'F.F.*&&F,6#\"\"$F.)F/F7F.F.&%\"yGF-
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rce[3]:=D(rho)(x[0])=Rh
o;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$rceG6#\"\"$/,(&%\"aG6#\"\"\"
F-*(\"\"#F-&F+6#F/F-&%\"xG6#\"\"!F-F-*(F'F-&F+F&F-)F2F/F-F-%$RhoG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "params:=solve(\{seq(rce[i],i
=1..3)\},\{seq(a[i],i=1..3)\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'
paramsG<%/&%\"aG6#\"\"$,$**,,**\"\"#\"\"\"&%\"xG6#F0F0&F26#\"\"!F0&%\"
yGF5F0F0*()F4F/F0F1F0%$RhoGF0!\"\"*()F1F/F0F4F0F;F0F0*&F:F0&F8F3F0F<*&
F>F0F7F0F<F0F4!\"#F1F<,(*$F:F0F0*$F>F0F0*(F/F0F1F0F4F0F<F<F</&F(6#F/**
,,**F*F0F1F0F:F0F7F0F0*()F4F*F0F1F0F;F0F<*(F/F0FNF0F@F0F<*(F4F0F;F0)F1
F*F0F0*&F7F0FQF0F<F0F4FBF1F<FCF</&F(F3,$**,,*&FNF0F@F0F<FPF0*(F/F0F7F0
FQF0F<*(F>F0F:F0F;F0F<**F*F0F4F0F>F0F7F0F0F0F4F<F1F<FCF<F<" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xxx:=subs(params,rho(x));" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%$xxxG,(*,,,*&)&%\"xG6#\"\"!\"\"$\"\"\"&%\"
yG6#F/F/!\"\"*(F*F/%$RhoGF/)&F+F2F.F/F/*(\"\"#F/&F1F,F/F6F/F3*()F7F9F/
)F*F9F/F5F/F3**F.F/F*F/F<F/F:F/F/F/F*F3F7F3,(*$F=F/F/*$F<F/F/*(F9F/F7F
/F*F/F3F3F+F/F3*,,,**F.F/F7F/F=F/F:F/F/*(F)F/F7F/F5F/F3*(F9F/F)F/F0F/F
3F4F/*&F:F/F6F/F3F/F*!\"#F7F3F?F3F+F9F/*,,,**F9F/F7F/F*F/F:F/F/*(F=F/F
7F/F5F/F3*(F<F/F*F/F5F/F/*&F=F/F0F/F3*&F<F/F:F/F3F/F*FIF7F3F?F3F+F.F3
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 29 "je polynom, kter\375 m\341 v bod\354  " }{XPPEDIT 
18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 9 " hodnotu " }{XPPEDIT 18 0 
"y[0];" "6#&%\"yG6#\"\"!" }{TEXT -1 9 ", v bod\354 " }{XPPEDIT 18 0 "x
[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 9 " hodnotu " }{XPPEDIT 18 0 "y[1];
" "6#&%\"yG6#\"\"\"" }{TEXT -1 24 " a v bod\354 x[0] derivaci " }
{XPPEDIT 18 0 "Rho;" "6#%$RhoG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "F:=unapply(xxx,x,y,Rho)" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"FGj+6%%\"xG%\"yG%$RhoG6\"6$%)operatorG%&arrowGF*,(*
,,,*&)&9$6#\"\"!\"\"$\"\"\"&9%6#F8F8!\"\"*(F3F89&F8)&F4F;F7F8F8*(\"\"#
F8&F:F5F8F?F8F<*()F@FBF8)F3FBF8F>F8F<**F7F8F3F8FEF8FCF8F8F8F3F<F@F<,(*
$FFF8F8*$FEF8F8*(FBF8F@F8F3F8F<F<F4F8F<*,,,**F7F8F@F8FFF8FCF8F8*(F2F8F
@F8F>F8F<*(FBF8F2F8F9F8F<F=F8*&FCF8F?F8F<F8F3!\"#F@F<FHF<F4FBF8*,,,**F
BF8F@F8F3F8FCF8F8*(FFF8F@F8F>F8F<*(FEF8F3F8F>F8F8*&FFF8F9F8F<*&FEF8FCF
8F<F8F3FRF@F<FHF<F4F7F<F*F*F*6%F6F6F6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1289 "SP:=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 pol
ynom 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, k
terym prochazi dalsi parabola\n\n  der:=2*a*x[j]+b; # derivace posledn
iho 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\n
if 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:=piecewise(Q);\nunapply(Q,
z)\nend;\n" }}{PARA 7 "" 1 "" {TEXT -1 60 "Warning, `p` is implicitly \+
declared local to procedure `SP`\n" }}{PARA 7 "" 1 "" {TEXT -1 60 "War
ning, `i` is implicitly declared local to procedure `SP`\n" }}{PARA 7 
"" 1 "" {TEXT -1 62 "Warning, `der` is implicitly declared local to pr
ocedure `SP`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#SPGj+6$%\"xG%\"yG
6-%\"NG%\"KG%\"jG%\"aG%\"bG%\"cG%\"QG%\"zG%\"pG%\"iG%$derG6\"F5C,>8$-%
%nopsG6#9$>8%,&F8\"\"\"\"\"#!\"\">8',$*&,.*&&F<6#\"\"$F@&9%6#F@F@F@*&&
F<FNF@&FMFJF@FB*&FLF@&F<6#FAF@FB*&FSF@FQF@F@*&FIF@&FMFTF@FB*&FPF@FWF@F
@F@,.*&FSF@)FIFAF@FB*&)FPFAF@FSF@F@*&FPF@FenF@F@*&FIF@)FSFAF@F@*&FIF@F
gnF@FB*&FPF@FjnF@FBFBFB>8(*&,.*&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@FB*(FWF@FPF@FenF@F@*(FLF@FSF@FenF@FB*(Fj
nF@FIF@FLF@F@F@FYFB>&8,FN,(*&FDF@)8+FAF@F@*&FgpF@F^oF@F@FhoF@>8*6$2Fgp
FIFbp?(8-FAF@F>%%trueGC)>8&,&F^qF@F@F@>8.,&*(FAF@FDF@&F<6#FbqF@F@F^oF@
>F^o,$*&,**(FAF@FhqF@&FMFiqF@FB*(FAF@FhqF@&FM6#,&FbqF@F@F@F@F@*&)FhqFA
F@FeqF@F@*&FeqF@)&F<FbrFAF@FBF@,(*$FerF@F@*$FgrF@F@*(FAF@FhrF@FhqF@FBF
BFB>FD*&,*F_rFB*&FhrF@FeqF@FBFarF@*&FhqF@FeqF@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>>Fjp6%Fjp2FgpFhrF[t>Fjp6$FjpF[
t>Fjp-%*piecewiseG6#Fjp-%(unapplyG6$FjpFgpF5F5F56$\"\"!F\\u" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Priklad:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 109 "with(plots):\nN:=7;f:=x->ln(x)*sin(x);\nx:=seq(ro
und(2*i+sin(i)^2),i=1..N);\ny:=seq(evalf(f(x[i])),i=1..N);x[N];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGj+6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%#lnG6#9$\"
\"\"-%$sinGF/F1F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG
6)\"\"$\"\"&\"\"'\"\"*\"#6\"#7\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"yG6)$\"+^<O]:!#5$!+#3HLa\"!\"*$!+ZOX1]F($\"+['o^0*F($!+*yryR#F+$!+
7OLL8F+$\"+.'pUh#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "Body:=pointplot([seq([x[i],
y[i]],i=1..N)], color=NAVY,symbolsize=15):\nAproximace:=plot(SP([x],[y
])(z),z=x[1]-1..x[N]+1,color=RED):\nSP([x],[y])(z);\nFce:=plot(f(z),z=
x[1]-1..x[N]+1,color=BLUE):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIEC
EWISEG6'7$,(*&$\"+`6A1j!#5\"\"\")%\"zG\"\"#F,F,*&$\"+[&fT*e!\"*F,F.F,!
\"\"$\"+!e\">;7!\")F,2F.\"\"'7$,(*&$\"+WOG:SF+F,F-F,F4*&$\"+/ik\"\\'F3
F,F.F,F,$\"+_,b*\\#F7F42F.\"\"*7$,(*&$\"+b)\\\"zXF+F,F-F,F4*&$\"+!Q0m]
(F3F,F.F,F,$\"+v<GcHF7F42F.\"#67$,(*&$\"*[i?j$F7F,F-F,F,*&$\"*))*GZ#)!
\"(F,F.F,F4$\"+[XC`YFZF,2F.\"#77$,(*&$\"+e&*Rh8F3F,F-F,F4*&$\"+0!>qt$F
7F,F.F,F,$\"+s2MPDFZF4%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "display(\{Body,Aproximace,Fce\});" }}{PARA 13 "" 1 "
" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6'-%'POINTSG6+7$$\"\"$\"\"!$\"+^<
O]:!#57$$\"\"&F)$!+#3HLa\"!\"*7$$\"\"'F)$!+ZOX1]F,7$$\"\"*F)$\"+['o^0*
F,7$$\"#6F)$!+*yryR#F27$$\"#7F)$!+7OLL8F27$$\"#9F)$\"+.'pUh#F2-%'COLOU
RG6&%$RGBG$\")!\\DP\"!\")FP$\")viobFR-%'SYMBOLG6$%(DEFAULTG\"#:-%'CURV
ESG6$7]q7$$\"\"#F)$\"3!))***>l^3'*G!#<7$$\"3ILL3-:oT@F]o$\"3,G`ZF1/JCF
]o7$$\"3gmm;/IO$G#F]o$\"3+e)=D'QJ\"*>F]o7$$\"3$**\\i:KRmS#F]o$\"3b!\\
\")4)zIH;F]o7$$\"3FL$e*Qc\"*HDF]o$\"3Mb>i\"HpkG\"F]o7$$\"3w*\\P4v_&oEF
]o$\"3&))GJ9*z4Q#*!#=7$$\"3smm\"H')*=2GF]o$\"3%42mfM<R&eFgp7$$\"3wmmm^
fuYHF]o$\"3OpLFyS8#p#Fgp7$$\"3ymmTS?I'3$F]o$!3S@\\d-]5SA!#>7$$\"3'**\\
PM%[>DKF]o$!3S)e*y0\\S#)GFgp7$$\"39L$ekk(3kLF]o$!3+pU'*G+\\(H&Fgp7$$\"
3Tm;Hi/j@OF]o$!37CHigj\"=8*Fgp7$$\"3%***\\iI6I))QF]o$!3evhakx/A7F]o7$$
\"3%om\"H<B4kTF]o$!3a.\"o*yD8Z9F]o7$$\"3'HL3xnX:I%F]o$!3;^%)RdN\\B:F]o
7$$\"3'***\\7Q!***QWF]o$!3%oZC4JDgd\"F]o7$$\"3r#3-j(Gp4XF]o$!3UU30jNw$
f\"F]o7$$\"3Mm\"zWr'Q!e%F]o$!3,'>atf)>0;F]o7$$\"3)*\\il_03^YF]o$!3A[X$
QTI.h\"F]o7$$\"3hLL$3Ru<s%F]o$!3_&*=\\7!f\"4;F]o7$$\"3K+]7L(3j%[F]o$!3
/s/9*Rj<f\"F]o7$$\"3/nmTvI%3(\\F]o$!3)p*eA)R2[b\"F]o7$$\"3%QLe*[#R56&F
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mFgp7$$\"3mKLL)>'*e8(F]o$\"32S3Q3E!)>))Fgp7$$\"3h**\\iNZF;uF]o$\"3CCI7
jg\"Q1\"F]o7$$\"3%****\\72o(\\vF]o$\"3)Qrs'4t?G6F]o7$$\"3C+](oShKo(F]o
$\"3qz9N2xGy6F]o7$$\"32n\"H#)pZD#yF]o$\"3=qp7%\\&G:7F]o7$$\"3+LLe*)R$=
'zF]o$\"35WaaTMqO7F]o7$$\"3'H$3_&GU(*3)F]o$\"3^yx\\2!\\EC\"F]o7$$\"3%H
Le9e]w@)F]o$\"3!o`/!QhXN7F]o7$$\"3_*\\P%ephb$)F]o$\"3ASxG())oH@\"F]o7$
$\"35mmTNLe$\\)F]o$\"3'o5\\cn&>v6F]o7$$\"32nTNE;*oj)F]o$\"3Ub(=6su(>6F
]o7$$\"3Em;H<**>!y)F]o$\"3MXStw6'y/\"F]o7$$\"3m**\\P%\\+(H!*F]o$\"3[++
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