1549301742-The_Theory_of_Difference_Schemes__Samarskii

(jair2018) #1
Two-layer iteration schenies 671


  1. 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.

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