Theorem 6-1: Consider the linearization of a set of equations:


 All A An xl bl

 A2 A22 A2n X2 b2
 I


 Ani An2 Ann xn bn


where the partitions of x and A are such that within each partition all

variables are either solved by Gauss-Seidel or Newton-Raphson. For a

blockwise Gauss-Seidel solution, the equations can be expressed as


 X1 a1! ,12 n X1

 X2 c21 "22 c2n X2
 .+ 1TT


 Xn nl Cin2 nn Xn


For a solution procedure composed of blocks of Gauss-Seidel and Newton-

Raphson convergence schemes, solved by Gauss-Seidel in a blockwise

fashion, the following formulas indicate how to calculate the c--

partitions from the coefficient matrix A.

 For Gauss-Seidel partitions:
 i-i
 1) aii = -Lii-1[Uii+ Aika"ki]
 k=1
 2) i<j
 i-1
 cij = -Li-1l[A + I Aikakj]
 k=l

 3) i>j
 i-1
 aij = -Lii-l[ 1 Aikakj]
 k=1

 For Newton-Raphson partitions:

 4) aii = 0