ABSTRACT
An initial solution vector x(O) is chosen. The superscript in parentheses denotes the iteration number, with zero denoting the initial solution vector. The initial solution vector x(O) is substituted into Eq. (1.196) to yield the first improved solution vector x(l). Thus,
X~I) = ~ (bi - 'I: ai.;X}°) - t a;.jxjO)) au j=1 /=i+1
(1.198)0= 1,2, ... , n)
This procedure is repeated (i.e., iterated) until some convergence criterion is satisfied. The Jacobi algorithm for the general iteration step (k) is:
(1.199)0= 1,2, ... , n)
An equivalent, but more convenient, form of Eq. (1.198) can be obtained by adding and subtracting x~k) from the right-hand side of Eq. (1.198) to yield
(HI) _ (k) +~ (b _~ (k»)Xi - Xi aiL. ai.jx/ u j=1
Equation (1.199) is generally written in the form
(i = 1,2, ... , n) (1.200a)
(i = 1,2, ... , n) (1.200b)
where the term R~k) is called the residual of equation i. The residuals R~k) are simply the net values of the equations evaluated for the approximate solution vector X(k).
