and

 "11 = 0

 a12 = -AII-1A12 XI



 "In = -Anl-1Aln

Thus, the first row of a can be calculated in terms of A. It will now

bw shown that any row of a can be calculated from A and elements of

above the row in question, according to formulas 1) to 6).

 For a Gauss-Seidel partition,

 A11.x = -AilxI-Ai2x2 -A. -1x -A i,1xi+ 1

 -A. x +b XII
 in n i

which becomes, in successive substitution form,

 x. -Li..-(Ailx1+Ai2x2 +A. + x +U x. .i+

 +A. x )+B. X IT
 in n i

Now we must substitute for all x P.<i, in terms of x's.

From (II),
 n
 x = a k xk XIV
 v ^k k
 k=-

Substituting (XIV) into (XIII) gives
 n n
 xy = -Lii-'[(Ail I alkk)+(Ai2 ~C ) k )
 n k=l k=1
 +(A. T a. x )+U ..x.-+A x. .
 +(Ai ,i-l1 i-1,k k k)+U ii 1,+Ai,+1 +"

 +A. x ]+.i XV
 Ainn n i

Collecting terms,

 for j>i,
 i-i
 a.. = -L -[A ij.+ A aik
 3 11 Li-j k1 ik