A21x +A22x2-b2 = g 6-15

Substituting 6-14 into 6-15 we have

 A2l[-Li-~1(Uli!x-Al2x2+bl)]-'-A222-b2y = g 6-16

Expanding,

 -A21Li1-zU11tl+(A92-A21Lli-AI2)X2-A2iLi-lbl-b2 = g 6-1/

 X2 = X2+Ax2 6-18

 AX2 = -2J g 6-19

Differentiating 6-17 with respect to x2 gives

 32 = (A22-A21L11-A12) 6-20
 x2

Substituting 6-17 and 6-20 into 6-19, and 6-19 into 6-18 gives

 X2 = X2-(A22-A21Lli-1A12)-[-AILz-L11U1l+(A22-A21Ll-1-AI2)x2
 +A21L1-Ib1-bh2] 6-21

 = X2-x2-(A22-A2L 1-A12)- [-A[2 Ll-1UL 1 +A21L -1b]-b]

which indicates that the value of the Newton-Raphson variables are

independent of their guessed values, which is indeed the case for

linear equations.

 We could thus write


 -l L i ? --
 1 x (12 X1
 =-22
 X9 C ;. 1 0 !._

for the set of equations in eqn. 6-12.

 What is desired, then, is a means of calculating the partitions of

the matrix u.. From these the entire matrix (C can be constructed andI its

eigenvalues calculated. In this way, the convergence properties of a

modified solution procedure can be estimated. The following theorem

indicates how those partitions can be calculated.