x = -L- Ux+L-1b 6-3

Values for x would be guessed and substituted on the right hand side to

calculate new values for x on the left hand side, then these newly

calculated values are substituted back on the left hand side. The

process is repeated until the values for x don't change. The necessary

and sufficient condition for convergence is well known (Edie, 1970). It

is that the eigenvalues of L-1U be less than unity in magnitude. This,

then, gives an indication of the convergence properties of the set of

equations.

 Suppose that the equations to be solved are non-linear. Equation

6-3 still describes the Gauss-Seidel solution procedure with the

modification that the A matrix is the Jacobian matrix of the non-linear

equations linearized about a guessed solution point. Because the

equations are non-linear, the Jacobian will change value from iteration

to -iteration. This means that the eigenvalues will change from

iteration to iteration. Hence, the set of equations can only be

determined to be locally convergent or non-convergent.

 A second iterative method of solving a set of equations is by

Newton-Raphson. For the equations,

 f(x) = 0 6-4

a Newton-Raphson iteration is as follows:

 f = f(x ) 6-5

 Ax f 6-6
 -k


 S = X +Ax 6-7
 -X+1 -k -k
Values for xl are guessed and used to calculate fl. Using eqn. 6-6,

Axk, the change in x, is calculated. This is used to calculate a new