660 Methods for Solving Grid Equations- The explicit scheme with optimal set of Chebyshev's parameters. In
what follows the intention is to use the explicit scheme (10) without concern
for how the parameters T 1 , T 2 ,.,. , Tn will be chosen in trying to minimize
the total number of iterations n = n(c:). Also, under the agreement that the
operator A is self-adjoint and positive we operate with its smallest / 1 > 0
and greatest /, eigenvalues:
(15) A= A*> 0, / 1 E <A< / 2 E, /1 > 0'
The meaning of this is that we should have/ 1 IIx11^2 <(Ax, x) < / 2 11x11^2 for any x EH.
If the paran1eter T = canst is independent of the subscript k, that is,
T 1 = T 2 = · · · = T 11 = T, schen1e (10) is called the si1nple iteration scherne:( 16) Yk+1 = Yk -T(Ayk - f).
In Section 1 of the present chapter we have established that the residual
rk = A Yk - f satisfies the homogeneous equation( 17) k=0,1,2, ... , r 0 =Ay 0 -fEH,
( 18)Here T, 1 is the resolving operator being a polynomial of degree n with
respect to the operator A:so that rn = Pn(A)r 0 • On this basis the residual r 11 obeys the estimate
(20)The next step is to evaluate the quantity 11Pn(A)11 of interest in terms
of / 1 and / 2 , making it possible to extract those parameters T 1 , T 2 , ... , T 11 ,
for which the minimal value of qn = 11Pn(A)11 is attained. The preceding
polynomial
n n
Pn (A) = IT (E - Tm A) = I: Ck Ak, C 0 = 1, Pn(O) = 1,
m=l k=O