Using MATLAB to realize: Jacobi, Gauss Seidel iteration

Example

Equations $\{\begin{array}{l}5x_1+2x_2+x_3=-12\\ -x_1+4x_2+2x_3=20\\ 2x_1-3x_2+10x_3=3\end{array}$ solve , When $max|x_i^{(k+1)}-x_i^{(k)}|\le 10^{-5}$ When iteration ends .

The following solution process , Superscript indicates the number of iterations , Subscript indicates serial number .

Jacobi iteration

Defining variables ：
$$\begin{gathered} D=diag(a_{11},a_{22},...,a_{nn}), \\ L=\left[\begin{array}{cccccc}0& & & & & \\ -a_{21}& 0& & & & \\ ...& & & & & \\ -a_{i1}& ...& -a_{i,i-1}& 0& & \\ ...& & & & & \\ -a_{n1}& ...& -a_{n,i-1}& ...& -a_{n,n-1}& 0\end{array}\right], \\ U=\left[\begin{array}{cccccc}0& -a_{12}& ...& -a_{1,i}& ...& -a_{1,n}\\ ...& & & & & \\ & & 0& -a_{i-1,i}& ...& -a_{i-1,n}\\ ...& & & & & \\ & & & & 0& -a_{n-1,n}\\ ...& & & & & \\ & & & & & 0\end{array}\right] \end{gathered}$$

Its matrix iteration form is ：
$$\begin{gathered} x^{(k+1)}=B_J\cdot x^{(k)}+f_J \\ {B}_J=D^{-1}\cdot (L+U),f_J=D^{-1}\cdot b \end{gathered}$$.

Write the component form ：

$\{\begin{array}{l}{x}_1^{(k+1)}=\frac{1}{5}(-12-2x_2^{(k)}-x_3^{(k)})\\ x_2^{(k+1)}=\frac{1}{4}(20+x_1^{(k)}-2x_3^{(k)})\\ x_3^{(k+1)}=\frac{1}{10}(3-2x_1^{(k)}+3x_2^{(k)})\end{array}$

In matrix form ：

$\left[\begin{array}{c}{x}_1^{(k+1)}\\ x_2^{(k+1)}\\ x_3^{(k+1)}\end{array}\right]=\left[\begin{array}{ccc}0& -0.4& -0.2\\ 0.25& 0& -0.5\\ -0.2& 0.3& 0\end{array}\right]\cdot \left[\begin{array}{c}{x}_1^{(k)}\\ x_2^{(k)}\\ x_3^{(k)}\end{array}\right]+\left[\begin{array}{c}-2.4\\ 5\\ 0.3\end{array}\right]$

Take the initial vector ：$x^0=(0,0,0{)}^T$, Iterate according to the above formula in turn . Use Matlab Program to solve .

a=[0,-0.4,-0.2;0.25,0,-0.5;-0.2,0.3,0];
b = [-2.4;5;0.3];
x = [0;0;0];
xx = a * x + b;
i = 0;
while norm(x - xx, inf) >= 1e-5
	x = xx;
	xx = a * x + b;
	i = i +1;
end

Above code , Final $x=x^i,xx=x^{(i+1)}$ , Final iteration order $i+1$ Time , If you need to see a longer decimal position , You can use the following Matlab Code , Said the use of 15 Bit floating-point or fixed-point number .

format long g

The running result is ：

That is, the exact solution is $x=(-4,3,2{)}^T$ .

Gauss-Seidel iteration

Its matrix iteration form is ：
$$\begin{gathered} x^{(k+1)}=B_G\cdot x^{(k)}+f_G \\ {B}_G=(D-L{)}^{-1}\cdot U,f_G=(D-L{)}^{-1}\cdot b \end{gathered}$$

Use Matlab Programming to solve ：

d = [5,0,0;0,4,0;0,0,10];
l = [0,0,0;1,0,0;-2,3,0];
u = [0,-2,-1;0,0,-2;0,0,0];
b = [-12;20;3];
t = inv(d - l);
bg = t * u;
fg = t * b;
x = [0;0;0];
xx = [-2.4;4.4;2.1];
i = 0;
while norm(x - xx, inf) >= 1e-5
	x = xx;
	xx = bg * x + fg;
	i = i +1;
end

The running result is ：

The exact solution is also obtained as $x=(-4,3,2{)}^T$ .