724 Methods for Solving Grid EquationsWhen the operator R happens to be a regularizer for the operator A,
that is, c 1 R < A < c 2 R, the constants of equivalence for the operators A
and B such that fl B <A < f 2 B become(^0 0)
fl = C1 f 1 1 f2 = C2 f^2.
The intention is to use such a factorized operator 111 the explicit
method with optimal set of Chebyshev's para1neters:
B Yk+l - Yk +A Yk = f, k " = , ' 1 2 ... , n ' Yo E f ,
Tk+l
which requires no less than n 0 (o-) iterations:
n (E) = In (2/s)
0 2/[, ,
~ = 2 /Ti c^1.
1 + T) c 2
In complete agreement with (22), T/ is expressed through the parameters c5 1 ,
c5 2 , f::..1 and f::..2 of the operators R1 and R2 involved.
It is worth noting here that the san1e estimate for n 0 ( E) was established
before for ATivI with optin1al set of Chebyshev's parameters, but other
formulas were used to specify T/ in terms of c5a and l:..a. If R = -A, where
A is the difference Laplace operator, and the Dirichlet problem is posed on
a square grid in a unit square, then
Jr h
11 = o It:.. = tg2 2
for both cases: ATM and the factorizPd schen1e (26)-(28) relating to ADivI.
Just for this reason these methods require the same nun1ber of iterations. As
a 1natter of experience, ATM is more econ01nical and preferable, since one
iteration necessitates perforn1ing a sn1aller number of arith1netic operations.
S01ne consensus of opinion here is to accept B = Rand then find Yk+l from
the equation
by one of the available direct methods, say by the decomposition method
or Fourier fast transform method.
Sumn1arizing, we are somewhat uncertain in which situations scheme
(26)-(28) would be more better than, for example, ATM.