Two-layer iteration schemes 657- proper choices of the para1neter Tk+ 1 and the operators Bk are
caused only by the necessity of convergence of the iterations and the
economy requiren1ents in trying to solve the original problem with
a prescribed accuracy, while the restrictions on the steps for non-
stationary problems are connected with the approxi1nations which
do arise in such matters.
Let now Q(c:) be the total number of arithmetic operations necessary
for obtaining a solution to equation (1) with a prescribed accuracy c; > 0
regardless of the initial approximation in the iteration scheme (3). Its
ingredients Bk and Tk should be so chosen as to mini1nize the quantity
Q(c:). If the desirable accuracy can be attained in a n1inimal nun1ber of the
iterations n = n(c:), then
n( e)
Q(c:) = L Qk = Qnn,
k=l
where Qk is the nmnber of the necessary actions during the course of kth
iteration. Thus, the 111ini1num proble1n for Q ( c;) reduces to the n1ininrn111
problems for n(c:) and the nmnber Qki which depends on B1.:.
In this context, if Bk = E is the identity operator, then scheme (3)
refers to explicit iteration schemes of the structure( 10) k=0,1,2, ... ' for any Yo E H.
If Bk f. E, then scheme (3) is termed an i1nplicit iteration schen1e.
- A stationary scheme. The inain theore1n on the convergence of itera-
tions. Quite often, the iteration sche1nes such as
( 3') B Yk+l - Yk +A Yk = f,
Tk=O,l, .. .,
with a constant operator B and constant T are callee! stationary meth-
ods of iterations. In particular, the upper relaxation method and Seidel
1nethod fall within the category of such 1nethods. ln that case equation (9)
related to the error of approximation z1; = Yk - 1l takes the fonn( 9')z z
B k+l - k +A zk = 0, k = 0, 1, ... , Z 0 =Yo - ll,
T
and it remains valid for the correction wk = B-^1 (Ayk - f). The operator B
is, generally speaking, non-self-adjoint and possesses the own inverse B-^1.
This type of situation is covered by the following assertion.