For j<i,
 i-i
 t -L Aikakj
 .i. = -L 1 A. ]
 k=-l
 For ii term,
 i-i
 a.. = -L. EU..[ A. a, .]
 -ii 11 i- +ii Aik kki
 k=l

Thus 1), 2), and 3) are proved.

Now suppose that partition i is to be solved by Newton-Raphson.

 gi = Aiilx+ +Aii -x +Ai..x .i+A ,i+x. + .
 i ,1-1 -1 1,+1 i,2+1 i+l
 +A. x -b. XVI
 in n i

The iterative Newton-Raphson solution is defined by


 x = XVII


Substituting (XIV) into (XVI) and differentiating gives


 i -= A+ ) A XVI[I
 T ii )- k kk

Now, substituting (XVIII) and (XVI) into (XVII) gives

 -1 i-1 n
 X. = x .-[A .+ + .A. u.]- '[ \j Ai.( i 'i +A
 k= k=1 j=1

 + }] A. x -b ]
 k-i+

SO,

 for j<i,
 i-i i-i
 a.. = -[Ai. + A. a ]-. [ C A.,a. ]
 Yij i 1 A ik ki k=- i kj

 for j=i,

 l. = 0
 1!