Uebung 12with(plots):
with(linalg):
#with(stats):1)a)rec1a:=proc(n,xx)
local f0, f1, i:
#f0:=xx:
#f1:=xx/2;
if n=0 then return xx:
elif n=1 then return xx/2:
else return (3*rec1a(n-1,xx)+rec1a(n-2,xx)/2):
end if:
return xx;
end proc:
rec1a(3,2);f17_7:=rec1a(17,7);
evalf(%);b)fib:=proc(n)
option remember:
local f0, f1, i:
#f0:=xx:
#f1:=xx/2;
if n=1 then return 1:
elif n=2 then return 2:
else return (fib(n-1)+fib(n-2)):
end if:
return xx;
end proc:fib23:=fib(23);
fib29:=fib(29);
with(combinat, fibonacci);
kontrolle23:=seq(fibonacci(i), i=1..24);
kontrolle29:=fibonacci(30);2)a)restart:
with(linalg):
with(plots):hilbert(3);f:=x->x*ln(x);
a:=1;
b:=4;#hilbert-Matrix (kubisch):
m:=matrix([[b-a, (b^2-a^2)/2, (b^3-a^3)/3], [(b^2-a^2)/2, (b^3-a^3)/3, (b^4-a^4)/4], [(b^3-a^3)/3, (b^4-a^4)/4, (b^5-a^5)/5]]);
m4:=matrix([[b-a, (b^2-a^2)/2, (b^3-a^3)/3, (b^4-a^4)/4], [(b^2-a^2)/2, (b^3-a^3)/3, (b^4-a^4)/4, (b^5-a^5)/5], [(b^3-a^3)/3, (b^4-a^4)/4, (b^5-a^5)/5, (b^6-a^6)/6],[(b^4-a^4)/4, (b^5-a^5)/5, (b^6-a^6)/6, (b^7-a^7)/7] ]);#L\366sungsvektor:
v:=vector([int(f(x),x=a..b), int(f(x)*x,x=a..b), int(f(x)*x^2,x=a..b)]);
v4:=vector([int(f(x),x=a..b), int(f(x)*x,x=a..b), int(f(x)*x^2,x=a..b), int(f(x)*x^3,x=a..b)]);v2:=vector([evalf(int(f(x),x=a..b)), evalf(int(f(x)*x,x=a..b)), evalf(int(f(x)*x^2,x=a..b))]);
v4:=vector([int(f(x),x=a..b), int(f(x)*x,x=a..b), int(f(x)*x^2,x=a..b), int(f(x)*x^3,x=a..b)]);loesung:=linsolve(m,v2);
loesung4:=linsolve(m4,v4);poly:=loesung[1]+loesung[2]*x+loesung[3]*x^2;
poly4:=loesung4[1]+loesung4[2]*x+loesung4[3]*x^2+loesung4[4]*x^3;
evalf(%);plot1:=plot(f(x), x=-5..5, color=black):
plot2:=plot(poly4(x), x=-5..5, color=red):
display(plot1,plot2);b)restart:
with(stats);
#with(linalg):
#with(combinat):
#with(plots):
f:=x->x*ln(x);
x_i:=[1.0,2.0,3.0,4.0];
f_x:=map(f,x_i);ff := fit[leastsquare[[x,y], y=a*x^3+b*x^2+c*x+d, {a,b,c,d}]]([x_i,f_x]);fn :=unapply(rhs(ff), x);#importiert:
b := -0.3057502361e-1*x^3+.4450742333*x^2+.2650968069*x-.6795960083;plot1:=plot(f, -5..5,color=black):
plot2:=plot(fn, -5..5, color=red):
display(plot1,plot2);c)f:=x->x*ln(x);
a:=poly4;
#importiert:
b := -0.3057502361e-1*x^3+.4450742333*x^2+.2650968069*x-.6795960083;
c:=evalf(abs(a-b));smartplot(c);3)a)restart:
Length := proc(ll) return nops(ll); end proc:
Last := proc(ll) return op(-1, ll); end proc:
Append := proc(ll, x) return [op(ll), x]; end proc:
Member := proc(ll, p) return op(p, ll); end proc:
Take := proc(ll, n)
if n > 0 then return [ op(1..n, ll) ]
elif n < 0 then return [ op(n..-1, ll) ]
end if;
end proc:
with(plots):f:=x->x*ln(x);lagrPol := proc(xl, fl)
local i, j, l, ll, pol;
ll := []:
pol := 0:
i := 1:
while i <= Length(xl) do
l := 1;
j := 1;
while j <= Length(xl) do
if j <> i then
l := l*(x-Member(xl, j))/(Member(xl, i) - Member(xl, j));
end if;
j := j+1;
end do;
ll := Append(ll, l);
pol := pol + l*Member(fl, i);
i := i+1;
end do:
return( sort( expand( pol ) ) );
end proc:x_i:=[1,2,3,4];
f_x:=map(f,x_i);l:=lagrPol(x_i,f_x);poly:=unapply(l,x);plot(poly,1..4);b)nst4:=k->evalf( cos( (2*k-1)*Pi/8) );
ltr:=x->0.5*( (4-1)*x +4+1);