Gauss-Seidel Iteration

• Even though we have $newx^1$ available, we do not use it to compute $newx^2$.
• In nearly all cases the new values are better than the old and ought to be used instead.
• When this done, the procedure known as Gauss-Seidel iteration.
• We proceed to improve each $x$-value in turn, using always the most recent approximations of the other variables.
  

  Table 3: Successive estimates of solution (Gauss-Seidel method)

  	First	Second	Third	Fourth	Fifth	Sixth
  $x_1$	0	1.833	2.069	1.998	1.999	2.000
  $x_2$	0	1.238	1.002	0.995	1.000	1.000
  $x_3$	0	1.062	1.015	0.998	1.000	1.000

  
  

• These values were computed by using this iterative scheme:
  
  $$\begin{array}{l} x_1^{(n+ 1)} = 1.8333 + 0.3333x_2^{(n)}- 0.... ...= 0.2000 + 0.2000x_1^{(n+1)} + 0.4000x_2^{(n+1)}\\ \end{array}$$
  
  
  beginning with $x^{(1)} = (0,0,0)^T$
• The rate of convergence is more rapid than for the Jacobi method (see Table 3).