1549301742-The_Theory_of_Difference_Schemes__Samarskii

Two-layer iteration schemes^669

The iteration number k rs recorded along the first column and the

quantity of interest

is placed along the second one. From here it is easily seen that the iteration
process is divergent, thus causing abnormal tern1ination for k=64.
By obvious rearranging of the paran1eters Tk in reverse order in con-
formity with forn1ula. ( 42) the inherent instability of this process is more
significant clue to the fact that abnornial termination occurs very fastly for
k=l2 (see Table 4).
In this regard, rounding errors can be treated as possible perturbations
of the right-hand side of equation (1) at every step. The iteration scheme
( 14) with parameters (29), ( 41) or ( 42) becomes unstable with respect to
the right-hand side by exactly the san1e reasoning as before: the norm of
the operator 5k = E - T1.;A for the transition fronl the (k - l)th iteration
to the kth iteration may exceed l for negative values oftk> since

II^5 k II

__ Po (1 + lt"I) ,. O

10r tk < '

1 - Po lt1.: I

115k 11>1 for p 0 (1+21tk1)>1.

By the same token,

II '-'" c,, II= Po 1 +Po (1 + i1.: td <^1 ,. ior t I.: > ,^0

The preceding formulas need certain clarification. Since 51.: = 5~, we obtain

II S\, II= sup 1(51.:x, .r)I.

11:i;11=1

By virtue of the relations / 1 E < A < / 2 E we find that

( T1.; ll - 1) E < T1.; A - E < ( Tk /2 - 1) E,

which are followed by

Po (1 - th,)

Tk/2 -1 = ----

1 +Po i1.: