Jacobi Method

The iterative methods depend on the rearrangement of the equations in this manner:

$$x_i=\frac{b_i}{a_{ii}}- \sum_{j=1,j\neq i}^n \frac{a_{ij}}{a_{ii}... ...\ldots, n,\mapsto x_1=\frac{11}{6}-\left(\frac{-2}{6}x_2+\frac{1}{6}x_3\right)$$

Each equation now solved for the variables in succession:

$$\begin{array}{l} x_1 = 1.8333 + 0.3333x_2 - 0.1667x_3 \\ x_... ...857x_3 \\ x_3 = 0.2000 + 0.2000x_1 + 0.4000x_2 \\ \end{array}$$

• We begin with some initial approximation to the value of the variables. (Each component might be taken equal to zero if no better initial estimates are at hand.)
• The new values are substituted in the right-hand sides to generate a second approximation, and the process is repeated until successive values of each of the variables are sufficiently alike.

  $$\begin{array}{l} x_1^{(n+ 1)} = 1.8333 + 0.3333x_2^{(n)}- 0.l... ... 1)} = 0.2000 + 0.2000x_1^{(n)} + 0.4000x_2^{(n)} \end{array}$$ (4.1)

  
  
  Starting with an initial vector of $x^{(0)}=(0,0,0,)$, we get
  

  Table 4.2: Successive estimates of solution (Jacobi method)

  	First	Second	Third	Fourth	Fifth	Sixth	$\ldots$	Ninth
  $x_1$	0	1.833	2.038	2.085	2.004	1.994	$\ldots$	2.000
  $x_2$	0	0.7l4	1.181	l.053	l.001	0.990	$\ldots$	l.000
  $x_3$	0	0.200	0.852	1.080	l.038	l.00l	$\ldots$	1.000

  
  
• Note that this method is exactly the same as the method of fixed-point iteration for a single equation that was discussed in Section 3.6, but it is now applied to a set of equations; we see this if we write Eq. 4.1 in the form of
  $$x^{(n+1)}=G(x^{(n)})= b' - Bx^{n}$$
  which is identical to $x_{n+1}= g(x_n)$ as used in Section 3.6.
• In the present context, $x^{(n)}$ and $x^ {(n+1)}$ refer to the $n$th and $(n+1)$st iterates of a vector rather than a simple variable, and $g$ is a linear transformation rather than a nonlinear function.
  $$Ax=b, \left[ \begin{array}{rrr} 6 &-2 & 1 \\ -2 & 7 & 2 \... ...left[ \begin{array}{r} 11\\ 5\\ -1\\ \end{array}\right]$$
  Now, let $A = L + D + U$, where
  $$L=\left[ \begin{array}{rrr} 0 & 0 & 0 \\ -2 & 0 & 0 \\ ... ...&-2 & 1 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \\ \end{array} \right]$$
  rewritten as
  $$\begin{array}{l} Ax = (L + D + U)x= b \\ Dx = -(L + U)x + b\\ x = -D^{-1}(L + U)x + D^{-1}b\\ \end{array}$$
  From this we have, identifying $x$ on the left as the new iterate,
  $$x^{(n+1)} = -D^{-1}(L + U)x^{(n)} + D^{-1}b$$
  In Eq. 4.1,
  $$b'=D^{-1}b=\left[ \begin{array}{l} 1.8333 \\ 0.7143 \\ 0.2000 \\ \end{array} \right]$$
  $$D^{-1}(L + U)=\left[ \begin{array}{rrr} 0 &-0.3333 & 0.1667... ... 0 & 0.2857 \\ -0.2000 & -0.4000 & 0 \\ \end{array} \right]$$

• The procedure we have just described is known as the Jacobi method, also called ''the method of simultaneous displacements'', because each of the equations is simultaneously changed by using the most recent set of $x$-values. See Table 4.2.