Demonstratie - metoda lui Jacobi

Contents

Consideram sistemul

$$\left[ \begin{array} [c]{cc}
2 & 1\\
5 & 7
\end{array} \right]x = \left[
\begin{array}
[c]{c}
4\\
19
\end{array}
\right]
  $$

Initializare

A = [2,1;5,7];
b = [4;19];
xn = [2;1];

Pregatirea matricelor metodei

$$M = diag(A), N = M - A$$

M = diag(diag(A))
N = M -A
M =
     2     0
     0     7
N =
     0    -1
    -5     0

Prima iteratie

xv = xn;
xn = M\(N*xv+b);
ea = norm(xn-xv,inf);
xn, ea
xn =
   1.500000000000000
   1.285714285714286
ea =
   0.500000000000000

A doua iteratie

xv = xn;
xn = M\(N*xv+b);
ea = norm(xn-xv,inf);
xn, ea
xn =
   1.357142857142857
   1.642857142857143
ea =
   0.357142857142857

Dupa 25 de iteratii

for k=1:23
    xv = xn;
    xn = M\(N*xv+b);
    ea = norm(xn-xv,inf);
end
xn
ea
er =ea/norm(xn,inf)
xn =
   1.000002153146738
   1.999996924076088
ea =
     2.153146738237410e-06
er =
     1.076575024850136e-06