<Text-field layout="Heading 1" style="Heading 1">Uebung 11</Text-field><Text-field layout="Heading 2" style="Heading 2">Aufgabe 1</Text-field>restart;Digits:=50;r := proc( x , i )
Digits:50;
if i = 0 then
return evalf(x);
else
return evalf(1/(r(x,i-1)-a(x,i-1)));
end if;
end proc:a := proc( x , i )
Digits:50;
return trunc(r(x,i));
end proc:p := proc( x , i )
if i = -2 then
return 0;
elif i = -1 then
return 1;
elif i = 0 then
return a(x,0);
else
return (a(x,i)*p(x,i-1)+p(x,i-2));
end if;
end proc:q := proc( x , i )
Digits:50;
if i = -2 then
return 1;
elif i = -1 then
return 0;
elif i = 0 then
return 1;
else
return (a(x,i)*q(x,i-1)+q(x,i-2));
end if;end proc:reell_zu_rational := proc ( x , i )
Digits:50;
return p(x,i)/q(x,i);
end proc:reell_zu_rational(Pi,10);reell_zu_rational(exp(1),15);evalf(exp(1))-evalf(%);<Text-field layout="Heading 2" style="Heading 2">Aufgabe 2</Text-field>restart;newton := proc(p0,N,f)
local converging,fprime,i,pold,p;
Digits := 50;
fprime := (x) -> D(f)(x);
i := 0;
pold := p0;
while i < N do
i := i+1;
p := pold-f(pold)/fprime(pold);
pold := p;
end do;
return evalf(pold);
end proc:approx_irr := proc(a,p,q,n)
local f;
f:=(x)->x^q-a;
return newton(a,n,f)^p;
end proc:approx_irr(2,1,2,10);<Text-field layout="Heading 2" style="Heading 2">Aufgabe 3</Text-field>restart;Digits:=500;sum(1/16^k*(4/(8*k+1)-2/(8*k+4)-1/(8*k+5)-1/(8*k+6)),k=0..161);evalf(evalf(%)-Pi);sum(1/(k)!,k=0..120);evalf(evalf(%)-exp(1));<Text-field layout="Heading 2" style="Heading 2">Aufgabe 4</Text-field><Text-field layout="Heading 3" style="Heading 3">b)</Text-field>restart;with(stats);fit[leastsquare[[x,y]]]([[-2,-1,1,3,3.5,5,7,10],[8.1,5.2,-1.1,-7.3,-8.1,-13.2,-18.7,-27.5]]);<Text-field layout="Heading 3" style="Heading 3">c)</Text-field>restart;with(stats);l1:=[-Pi/4,-Pi/8,0,Pi/8,Pi/4];l2:=map(cos,l1);fit[leastsquare[[x,y], y=a*x^2+b*x+c, {a, b, c}]]([l1,l2]);f:=unapply(rhs(%),x);plot({cos(x),f(x)},x=-Pi/2..Pi/2,y=0..1.5);