Iterative alternating direction rnethods 713Also, we take for granted that(5) 1)(6) 2)- the operators Ai and A2 are commuting: Ai A2 = A2 Ai ,
and develop the iterative alternating direction method with these members
in just the same way as was clone before for scheme (2):(7)
Yj+i/2 - Yj
(i)
Tj +lgiven Yo EH, j = 0, 1,2, ...
For the error zJ+i = Yj+l - u, we must solve the homogeneous equations
Z 0 =Yo - u E H, j = 0, 1, 2, ....
Having completed the elimination of zj +i; 2 , we find that
by observing thatSJi) = (E + rf l Ai)-i (E - r}^2 l Ai),
sj2l = (E + r)^2 l A2)-^1 (E - rj1l A2),