742 Methods for Solving Grid EquationsLet now R = R* > 0 be a regularizer such that c 1 R < A < c 2 R,
c 1 > 0. Knowing wk> it is plain to find the (k + l)th iteration Yk+l in line
with (43), so there is some reason to be concerned about this.
The operator Bk can be specified in a number of different ways and,
in particular, in explicit forn1. One way of covering this is( 44)( 45)where R 1 , R2 are economical operators and wi^1 >, wi^2 ) are iteration parame-
ters. vVe dealt with the factorized operator (45) during the course of ATM
with Ri = R;, w~l) = wi^2 ) = w as well as of ADJ!I with R~ = Rn > 0,
Ct= 1, 2, Ri R2 = R2 Ri.
a) A direct method for determination of the coJ'l'ection. Let Bk = R
and one of the available direct methods will be employed for solving the
system of algebraic equations Rw = rk. We refer the users to Section
1 (items 2-3) in which such methods have been designed for elliptic grid
equations. Among them the decomposition inethod with / 1 = c 1 and / 2 =
c 2 is highly recommended for practical implen1entations. By the selection
rule for Chebyshev's parameters r 1 , T 2 , .•. , Tn we derive the estimate for
the number of the necessary iterations
In this rega.rd, it should be noted that for the difference elliptic prob-
lems posed in Section 3, item 7, no(E) is independent of the grid step. vVha.t
is more, when G is a rectangle and h 1 = h 2 = h, the work in doing this is
(^0) (/f;-^2 -^1 In - In^1 - 2).
c 1 h^2 h [
b) An iterative method for determination of the correction. The start-
ing point for subsequent considerations is the equation Rwk = r,., which can
be solved by means of some (internal) iterative 1nethod, whose use pern1its
us to find w(m) =wk> where m is the number of the internal iteration.
Further reduction of a two-layer iteration scheme with the accompa-
nying self-adjoint operators to an explicit one by replacing the variable w
by R^112 w leads to