\334bung 3.8 <Text-field layout="Heading 1" style="Heading 1">Aufgabe 5</Text-field> <Text-field layout="Heading 1" style="Heading 1">Nevillesche Methode</Text-field> a) restart: M:=5; NiM+SSJNRzYiIiIm x:=Vector[row]([ -2 , -1 , 0 , 1 , 2 ]); NiM+SSJ4RzYiLUknUlRBQkxFR0YlNiUiJy9EVC1JJ1ZFQ1RPUkdGJTYjNychIiMhIiIiIiEiIiIiIiMmSSdWZWN0b3JHNiRJKnByb3RlY3RlZEdGNkkoX3N5c2xpYkdGJTYjSSRyb3dHRiU= y:=Vector(M): for i from 1 by 1 while i <= M do
y[i]:=3^x[i];
end do: Q:=Matrix(M,M): for i from 1 by 1 while i <= M do
Q[i,1]:=y[i];
end do: Q: for k from 2 by 1 while k <= M do
for i from k by 1 while i <= M do
Q[i,k]:= ( (t-x[i-k+1])*Q[i,k-1] - (t-x[i])*Q[i-1,k-1] ) / ( x[i]-x[i-k+1] ) ;
end do;
end do; Q: P:=simplify( Q[M,M] ); NiM+SSJQRzYiLCwqJEkidEdGJSIiJSMiIiMiI0YqJEYoRisjIiM7RiwqJEYoIiIkIyIiKUYsRigjIiNHRiwiIiJGNg== plot(P , t=-2.5..2.5 , 0..10); 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 b) restart: M:=5; NiM+SSJNRzYiIiIm y:=Vector[row]([ 0 , 1 , 1.4 , 2 , 2.2 ]); NiM+SSJ5RzYiLUknUlRBQkxFR0YlNiUiKCEzMDctSSdWRUNUT1JHRiU2IzcnIiIhIiIiJCIjOSEiIiIiIyQiI0FGMiZJJ1ZlY3Rvckc2JEkqcHJvdGVjdGVkR0Y5SShfc3lzbGliR0YlNiNJJHJvd0dGJQ== x:=Vector[row]([ 0 , 1 , 1.96 , 4 , 4.84 ]); NiM+SSJ4RzYiLUknUlRBQkxFR0YlNiUiKUsqRyQ+LUknVkVDVE9SR0YlNiM3JyIiISIiIiQiJCc+ISIjIiIlJCIkJVtGMiZJJ1ZlY3Rvckc2JEkqcHJvdGVjdGVkR0Y5SShfc3lzbGliR0YlNiNJJHJvd0dGJQ== Q:=Matrix(M,M): for i from 1 by 1 while i <= M do
Q[i,1]:=y[i];
end do: Q: for k from 2 by 1 while k <= M do
for i from k by 1 while i <= M do
Q[i,k]:= ( (t-x[i-k+1])*Q[i,k-1] - (t-x[i])*Q[i-1,k-1] ) / ( x[i]-x[i-k+1] ) ;
end do;
end do; Q: P:=simplify( Q[M,M] ); NiM+SSJQRzYiLCoqJEkidEdGJSIiJSQhKzoqZTZAIiEjNiokRigiIiQkIitUKypbWyIhIzUqJEYoIiIjJCErW0pvWmxGMUYoJCIrKzRSPTohIio= plot( [P,sqrt(t)] , t=0..5 ); 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 <Text-field layout="Heading 1" style="Heading 1">Dividierte Differenzen</Text-field> restart: M:=5; NiM+SSJNRzYiIiIm #x:=Vector[row]([ -2 , -1 , 0 , 1 , 2 ]);
x:=Vector[row]([ 0 , 1 , 1.96 , 4 , 4.84 ]); NiM+SSJ4RzYiLUknUlRBQkxFR0YlNiUiJ0s+VC1JJ1ZFQ1RPUkdGJTYjNyciIiEiIiIkIiQnPiEiIyIiJSQiJCVbRjImSSdWZWN0b3JHNiRJKnByb3RlY3RlZEdGOUkoX3N5c2xpYkdGJTYjSSRyb3dHRiU= Q:=Matrix(M,M): for i from 1 by 1 while i <= M do
Q[i,1]:=3^x[i];
end do: for s from 2 by 1 while s <= M do
for z from s by 1 while z <= M do
Q[z,s]:=(Q[z,s-1]-Q[z-1,s-1])/(x[z]-x[z-s+1]);
end do:
end do:
Q;
NiMtSSdSVEFCTEVHNiI2JSInMydmKC1JJ01BVFJJWEdGJTYjNyc3JyIiIiIiIUYuRi5GLjcnIiIkIiIjRi5GLkYuNyckIistUDE4JykhIiokIitjOCVwJWVGNSQiKyM9REYnPkY1Ri5GLjcnIiMiKSQiK0kjeiRbTiEiKSQiKzguJip5KSpGNSQiKyRHYyF6PkY1Ri43JyQiKz9VSFE/ISIoJCIrQ11EaTlGRiQiK24jKT5YUUY+JCIrV1EqM1coRjUkIitOKXklRzZGNUknTWF0cml4RzYkSSpwcm90ZWN0ZWRHRlFJKF9zeXNsaWJHRiU= P:=0:
for i from 1 by 1 while i <= M do
s:=1;
for j from 1 by 1 while j < i do
s := s*(t-x[j]);
end do:
P := P+s*Q[i,i];
end do:
simplify(P); NiMsLCQiIiIiIiFGJUkidEc2IiQhK00qWzUkXCEiKiokRiciIiMkIiteRXhuNiEiKSokRiciIiQkISs0azp2ZUYrKiRGJyIiJSQiK04peSVHNkYr TTdSMApJM1JUQUJMRV9TQVZFLzQxMTkzMlgqJSlhbnl0aGluZ0c2IjYiW2dsISQlISEhIiYiJiIiISIiIiQiJCc+ISIjIiIlJCIkJVtGK0YmCg== TTdSMApJM1JUQUJMRV9TQVZFLzc1OTYwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzoiJiImIiIiIiIkJCIrLVAxOCcpISIqIiMiKSQiKz9VCkhRPyEiKCIiISIiIyQiK2M4JXAlZUYrJCIrSSN6JFtOISIpJCIrQ11EaTlGL0YwRjAkIisjPURGJz5GKyQiKzguJip5KSpGKyQiK24jKT4KWFFGNkYwRjBGMCQiKyRHYyF6PkYrJCIrV1EqM1coRitGMEYwRjBGMCQiK04peSVHNkYrRiYK