Metoda celor mai mici patrate

Dandu-se punctele

(0, -4), (1, 0), (2, 4), (3, -2)

determinati polinomul de gradul I corespunzator acestor date prin metoda celor mai mici patrate.

x=0:3;

y=[-4,0,4,-2];

Construim design matrix si determinam coeficientii

X=[ones(4,1),x'];

c=X\y'

c = 2x1 double
-2.0000 1.0000

Reprezentam grafic punctele si aproximanta

t=linspace(-1,4,10);

ty=c(1)+c(2)*t;

plot(x,y,'o',t,ty,'MarkerFaceColor','blue')

Ecuatiile normale

B=X'*X

B = 2x2 double
4 6 6 14

X'*y'

ans = 2x1 double
-2 2

[Q,R]=qr(X)

Q = 4x4 double
-0.5000 0.6708 0.0236 0.5472 -0.5000 0.2236 -0.4393 -0.7120 -0.5000 -0.2236 0.8079 -0.2176 -0.5000 -0.6708 -0.3921 0.3824
R = 4x2 double
-2.0000 -3.0000 0 -2.2361 0 0 0 0

[Q2,R2]=qr(X,0)

Q2 = 4x2 double
-0.5000 0.6708 -0.5000 0.2236 -0.5000 -0.2236 -0.5000 -0.6708
R2 = 2x2 double
-2.0000 -3.0000 0 -2.2361