1549301742-The_Theory_of_Difference_Schemes__Samarskii

Two-layer iteration schenies 671

7. Re-ordering of iteration para111eters. With regard to scheme (14) one
   interesting problem arises in connection with re-ordering of the iteration
   parameters { Tk} so as to minimize as much as possible the influence of
   rounding errors and to avoid large intermediate values dependent on n.
   The niain goal of subsequent considerations is to constitute a "stable
   collection" Jl/1 11 of paran1eters T 1 , T 2 , ..• , T 11 for which scheme ( 14) bPcon1es
   stable and then show the way this result is used in practical in1plementa-
   tions. We improve our chances of ordering the set

Mn = { - cos (3; , i= 1,2, ... ,n}

if a sequence of odd integers will be available such as

for i = 1, 2, ... , n.

With the aid of its members the parameters { Tk} are calculated by the

formula

( 43) Tk = ----To

l+poGk

Gk = - cos [ _!!___ G 11 ( k)] , k = 1, 2 , ... , n.

2n

Thus, the further composition of the set consisting of n numbers G,, may

be of help in achieving these aims.

The traditional tool for carrying out this work is step-by-step transi-

tions fr0111 the sets Gm to the sets G 2 m and from the sets G 2 m to the sets

G 2 m+l. The intention in this direction is to use the formulas

( 44)

£ = 1, 2, ... , in,

or, what amounts to the sa111e,

( 45)

i= 1,2, ... ,m

for the first operation in passing from the sets Gm to the sets G 2 ,.,, with
placing the extra. member G 2 m+l (2m + 1) for the second one in passing
from the sets G 2 m to the sets G 2 m+l:

( 46) i=l,2, ... ,2m,

G2m+ 1 (2m + 1) = 2m + 1.