value of x. The process is repeated until there is no change in x.

Equations 6-5, 6-6, and 6-7 are combined to give:

 x = kx 3- If(x ) 6-8
 -k+1 -k T\ -k

which expresses the variable values at iteration k+1 in terms of the

variable values at iteration k.

 Let x* be the solution point and expand f to the first order

around x*.

 f(x) f(x*)+ (x-x*) g(x) 6-9
 lxJx*
then,

 x, = X -a 9 6-10


 -+ 2k t L x,

 X -



 X
 n x*) = O

SO,

 Sx -f f- (x -x*) 6 11
 --k+ ,. T .-k


 -k -XX
This indicates that Newton-Raphson converges in on iteration when the

first order expansion is exact (i.e., for linear equations), which is

known to be the case.

 This section has illustrated means of estima .ing th.e convergence

characteristics of both Gauss-Seidel and Newton-Raphsoin solution

procedures. These characteristics will be extended in the third section

to more complex systems.