then solved by Newton-Raphson. The process is repeated until conver-

gence. Notice that the variables y and z are implicit outputs and that

in fact neither of them occurs in the function h.


6.3 Convergence Properties of Combined Gauss-Seidel and Newton-Raphson

 When a modification such as the one considered in the last section

 is made, it is necessary to estimate the convergence properties of the

 new solution procedure.

 For a linearization of the equations we can represent the

equations by


 All A12 X1 b,
 = 6-12
 A21 A22 X2 b2


In this section the lower case letters will refer to vectors and upper

case letters to matrices. We call the matrix made up of the partitions

All, A12, A21 and A22 the coefficient matrix. The coefficient matrix is

the Jacobian matrix in the case of non-linear equations. The variables

x, are those to be solved by Gauss-Seidel and x2 are those to be solved

by Newton-Raphson. We desire a relationship of the form:

 x = _x+B 6-13

where "^" indicates "from the previous iteration." The eigenvalues of

g., called the convergence matrix, could then be evaluated to determine

if the proposed solution procedure is locally convergent.

 For the Gauss-Seidel variables,

 xI = -L11-1[U1xix-A12 2+b1] 6-14

where L and U are as defined in the first section. For the Newton-

Raphson variables we write