1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other iterative inethocls 737

yielding

11 s 112 ~ 1 - 2 T fl + T2 f1 f2.

Having stipulated the 1ninimum condition for the preceding trinomial, the

parameter T is found to be T = l/f 2 , so that II S 112 < ( 1 - fl f'h) or, what

amounts to the same,

(32)

In an atten1pt to generalize the results obtained to the case of three

parameters fl, f 2 , fJ, the operator A arranges itself as a sum

Here Ao is a sy1nmetric operator and A 1 is a. sketch-syn11netric operator,

so that

meaning (Ax, x) = ( A 0 x, x). Let the the members of the operator A be in

line with the conditions

(34) II Ai II ~ f3,

where f 2 >fl > 0 and f 3 2:: 0 are known numbers.

An alternative form of the equation zk+i = (E - rA)zk for the enor

zk+i = Yk+i - u Is

(:3.5) zk+l = (E - T Ao - T Ai) zk =(BE - T Ao) zk + [(1 - B) E - T Ai) zk,

where the number 0 < B < 1 is free to be chosen in any convenient way.
The main goal of further developn1ent is to 1ninirnize the norn1

II S II = II E - T (Ao+ Ai) II

by observing that on account of the triangle inequality

To this end, the nmnbers T and B are so chosen as to satisfy the mininrnm
condition that we have 1nentioned above. As far as flE <Ao < f 2 E for a
self-adjoint operator Ao, we 1night have

(37) r;ijr llE - ~Ao II =Po for

T 2

• =To= '
  e fl+ f2