5) j>i
 i-i i-1
 i j = -[ Aikaki+Aii]-'[Aij+ Y Aikakj]
 k=1 k=1


 6) j<i
 i-i i-1
 aij = -[ X AikCki+Aii1-I[ Aikakj]
 k=1 k=1

Proof: xl must be either a Gauss-Seidel or a Newton-Raphson type

variable. First consider that it is of Gauss-Seidel type.

 AllX1 = -A12X2-AI3X3 -Alnx,

from eqn. I.

Rearranging into a successive substitution Gauss-Seidel structure,

 Llixi = -Ux111-A12X2-Ai3x3 -A1nxn

 xl = L1i '[-U11x1-A12x2-A[)33 -Ainxn]


Thus,


all =

ul =


 -1
-LI1 U11

-L I]1 A12


Now suppose that xi is to be solved by Newton-Raphson. Let

 g = AxliX +A12X 2 Ainxn-b


 x, = x- axi g


X1T


x 1 = -Ar- L[Al il+A A2 x -A i -A '


VII


VI1i


C'In = ILIL-1Aln