714 Methods for Solving Grid Equationssince the operators Ai and A2 are commuting. As a final result we get
n
(8) Zn = Tn Zo , Tn = II sj ,
j=i
where Tn is a self-adjoint operator (T~ = Tn) as a product of commutative
self-adjoint operators involved.
From (8) the estimate II Z 71 II < II Tn II · II z 0 II immediately follows, in
which the quantity 11 T;, 11 depends on the paran1eters rp l and rj^2 l. Roughly
speaking, a proper choice of such parameters is stipulated by the minimum
condition for the norm II T,, II in connection with a minimal nun1ber of the
necessary iterations. To be more specific, when making a substantiated
choice, we have at our disposal two collections of parameters r? l, ri^1 l,
... , r,\il and r 1 C^2 l, ri^2 ), ... , r~^2 ) with a prescribed number n = n(c) that
are known to us in advance:n^1 in II T,, II= qn.
{T(l) ,T(2)}
J J- A proper choice of iteration parameters by Jordan's rule. First of all,
observe that the spectra of the operators Ai and A2 are located, because
of (6), on different segments DCY <,(ACY)< ~CY with 01 =/= 02 and ~ 1 =/= ~ 2 •
One trick we have encountered is to replace Ai and A2 by the newly formed
operators A~ and A~ with coinciding bounds:
17E SA~ SE, C\'=l,2, 17 > 0.
This can be done using the decompositions
(9) Ai= (q E - r A~)-i(A~ - pE), A2 = (q E + r A~)-i(A~ + pE),
where the numbers r, q, pare free to be chosen in any convenient way and
the new parameters w(ll, w(ll are taken to be
(i) - Tl I) - r
(10) w - q-rllJp " ,vVith these, we arrive at
5cil=(E+w(iJAJ J^1 i )-i(E-wCJ^2 lA^1 i ) ,