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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 17 "Aufgabe (3
6) (b):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Es muss
 gelten:   " }{XPPEDIT 18 0 "f(i^k) = (k-2)^2;" "6#/-%\"fG6#)%\"iG%\"k
G*$,&F)\"\"\"\"\"#!\"\"F-" }{TEXT -1 7 "  f\374r  " }{XPPEDIT 18 0 "k \+
= 0 .. 2;" "6#/%\"kG;\"\"!\"\"#" }{TEXT -1 3 "  ." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 64 "restart:b:=[4,1,0]:M:=matrix(2,4,[[x,1,i,-1]
,[` f `(x),op(b)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGK%'matri
xG6#7$7&%\"xG\"\"\"%\"iG!\"\"7&-%$~f~G6#F*\"\"%F+\"\"!Q(pprint16\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "m1,m2,m3:=x-1,x^2+1,x+1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%#m1G%#m2G%#m3G6%,&%\"xG\"\"\"F+!
\"\",&*$)F*\"\"#F+F+F+F+,&F*F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "m12,m13,m23:=collect(m1*m2,x),collect(m1*m3,x),collec
t(m2*m3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%$m12G%$m13G%$m23G6%
,*\"\"\"!\"\"*$)%\"xG\"\"$F*F**$)F.\"\"#F*F+F.F*,&F*F+F0F*,*F*F*F,F*F0
F*F.F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Gewinnung der B\351zout
identit\344ten:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "gcdex(m
1,m23,x,'u1','v1'):gcdex(m2,m13,x,'u2','v2'):gcdex(m3,m12,x,'u3','v3')
:\nf1,f2,f3:=collect(m23*v1,x),collect(m13*v2,x),collect(m12*v3,x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%#f1G%#f2G%#f3G6%,*#\"\"\"\"\"%F+*
&F*F+*$)%\"xG\"\"$F+F+F+*&F*F+*$)F0\"\"#F+F+F+*&F*F+F0F+F+,&#F+F5F+*&F
5!\"\"F0F5F:,*F*F+*&#F+F,F+F.F+F:*&F*F+F3F+F+*&#F+F,F+F0F+F:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=sort(add(f||k*M[2,k+1],k=
1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**$)%\"xG\"\"$\"\"\"
F**&#F*\"\"#F**$)F(F-F*F*F*F(F*#F)F-F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 82 "Das Gleiche mit dem Newton-Verfahren entlang meines Skrip
ts zu \247 14, Seite 4 oben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "z1:=b[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G\"\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Berechnung von  " }{XPPEDIT 18 0 "
mu1;" "6#%$mu1G" }{TEXT -1 3 "  :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "gcdex(m1,m2,x,'mu1','lambda1'):'mu1'=mu1;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "z2:=z1+mu1*(b[2]-z1)*m1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%$mu1G,&#\"\"\"\"\"#!\"\"*&F(F)%\"xGF'F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#z2G,&\"\"%\"\"\"*(\"\"$F',&#F'\"\"#!\"\"*
&F,F-%\"xGF'F-F',&F/F'F'F-F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 40 "z2:=sort(simplify(z1+mu1*(b[2]-z1)*m1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#z2G,&*(\"\"$\"\"\"\"\"#!\"\"%\"xGF)F(#\"\"&F)F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "gcdex(m1*m2,m3,x,'mu2','lamb
da2'):'mu2'=mu2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$mu2G#!\"\"\"\"%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z3:=z2+mu2*(b[3]-z2)*m1
*m2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z3G,(*(\"\"$\"\"\"\"\"#!\"
\"%\"xGF)F(#\"\"&F)F(**\"\"%F*,&#F-F)F**(F'F(F)F*F+F)F*F(,&F+F(F(F*F(,
&*$)F+F)F(F(F(F(F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "z3:
=sort(simplify(z2+mu2*(b[3]-z2)*m1*m2));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#z3G,.*&#\"\"$\"\")\"\"\"*$)%\"xG\"\"&F*F*F**&#F(F)F**$)F-\"\"
%F*F*!\"\"*$)F-F(F*F**&#F*\"\"#F**$)F-F9F*F*F**&#F.F)F*F-F*F*#\"#:F)F*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Die L\366sung ist nur eindeut
ig modulo  " }{XPPEDIT 18 0 "m1*m2*m3;" "6#*(%#m1G\"\"\"%#m2GF%%#m3GF%
" }{TEXT -1 88 ",  daher kann man das Ergebnis noch reduzieren (durch \+
geschicktere Anordnung der Module " }{XPPEDIT 18 0 "m1,m2,m3;" "6%%#m1
G%#m2G%#m3G" }{TEXT -1 55 " (n\344mlich wie ?) l\344sst sich die Reduk
tion vermeiden !):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=rem(z3,m1*m2*m3,x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"fG,**$)%\"xG\"\"$\"\"\"F**&#F*\"\"#F**$)F(F
-F*F*F*F(F*#F)F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "subs(x=1,f),subs(x=I,f),subs
(x=-1,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%\"\"\"\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Zum Vegleich das Ergebnis der gew
\366hnlichen Interpolation, die hier zwangsl\344ufig ein komplexes Pol
ynom liefert." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "interp([1,
I,-1],[4,1,0],t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#!\"\"%\"
tGF%\"\"\"*&F%F(F'F(F(*&)F'F%F(^#F(F(F(^$#\"\"$F%F&F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 257 11 "Bemerkung: " }{TEXT -1 90 "Es handelt si
ch hier nur um ein recht einfaches \334bungsbeispiel, da nur eines der
 Polynome " }{XPPEDIT 18 0 "m1,m2,m3;" "6%%#m1G%#m2G%#m3G" }{TEXT -1 
80 " nichtlinear und dann auch nur vom Grad zwei ist. Siehe auch Beisp
iel 14.10 (b)." }}}}{MARK "0 0 1" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }