1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction methods 723

We learn from Section 4 the values of paran1eters w 1 and w 2 for which

the minimum p of the norm 11 S( w) 11 is attained:

= ( 1 - J77)2

p l+J17

1-~

1 + ~'

111in llSll = p,

Wl' W2

where 17, w 1 and w 2 are given by fon11ulas (14), (LS) and (25). Knowing (J,

0 0

w 1 and w 2 , it is sin1ple to compute the constants of equivalence-, 1 and "1 2

for the operators B and A such that

(31)

which will be needed in subsequent discussions of the operator B frmn a

viewpoint of the possible general theory.

A necessary and sufficient condition for the p-stabity of schen1e (26)

IS

T

with assigned values of p and T = w 1 + w 2. The outcome of this is

( 32)

0 1 - p

l1 = ---

wl +w2

0 1 + p

l2 = ---

wl +w2

2

A keystone in the design of an operator of the type (27) is the possible

structure of an auxiliary operator R = R* > 0 being a sum of two operators

R1 and R2 such that

Ra= R~, o:=l,2,

by means of which the factorized operator in question reveals to be

(33)
0 0
This serves to 1notivate instead of (31) the relations/ 1B < R < / 2B with
0 0
constants / 1 and/ 2 arising from (32).