>> aufgabe_4

t = (0.1:0.1:1.0)'

t =

    0.1000
    0.2000
    0.3000
    0.4000
    0.5000
    0.6000
    0.7000
    0.8000
    0.9000
    1.0000

f = [100 34 17 12 9 6 5 4 4 2]'

f =

   100
    34
    17
    12
     9
     6
     5
     4
     4
     2

A = [1./t 1./t.^2 exp(-(t-1)) exp(-2*(t-1))]

A =

   10.0000  100.0000    2.4596    6.0496
    5.0000   25.0000    2.2255    4.9530
    3.3333   11.1111    2.0138    4.0552
    2.5000    6.2500    1.8221    3.3201
    2.0000    4.0000    1.6487    2.7183
    1.6667    2.7778    1.4918    2.2255
    1.4286    2.0408    1.3499    1.8221
    1.2500    1.5625    1.2214    1.4918
    1.1111    1.2346    1.1052    1.2214
    1.0000    1.0000    1.0000    1.0000


%grad = 0 mit Gauss
gamma_g = (A'*A)\(A'*f)

gamma_g =

    4.0591
    0.6140
   -2.5315
    0.7058


%qr-zerlegung
[Q R_s] = qr(A);
[m n] = size(A);

R = R_s(1:n,1:n);
U = Q(:,1:n);

gamma_qr = R\(U'*f)

gamma_qr =

    4.0591
    0.6140
   -2.5315
    0.7058


%konditionszahlen
cond(A)

ans =

  339.4047

cond(A'*A)

ans =

   1.1520e+05


%differenz gamma_g - gamma_qr
gamma_g - gamma_qr

ans =

   1.0e-12 *

   -0.6404
    0.0393
    0.1492
    0.3544


t_f = linspace(0.1,1.0,1000)';
f_g = [1./t_f 1./t_f.^2 exp(-(t_f-1)) exp(-2*(t_f-1))] * gamma_g;
f_qr = [1./t_f 1./t_f.^2 exp(-(t_f-1)) exp(-2*(t_f-1))] * gamma_g;

semilogy(t,f,'O',t_f,f_g,'blue',t_f,f_qr,'red')
title('Aufgabe 4 Serie 8')
xlabel('t')
ylabel('f(t)')
>>