SC-90-09.pdf - ZIB

and by using property C 0 we obtain
Denote the Kantorovitch quantity by
From (2.18) and (2.22), it follows that
«i = KII = IK--'i«l 0 such
that h) < 1/2 for j > j*. It follows that
°° a*
Pi ••= «i E(^) 2 ' -1 < TZT7 < 2«* < 2^,- .
«=o
l
Now, applying Theorem 1.5 to Newton's process (2.3) with starting point x'j, we
deduce the existence of a solution x*- € S'j(njX*, p*-) which is the unique solution
of (2.1) in the region
"•j
oo
D*=[jS j (x k j,2/u j )nD.
Jt=0
In particular x^ is the unique solution of (2.1) in the region (2.21).
Next consider Newton's process (2.3) with some starting point x° 6 Dj and let
aj(x'j) be given by
«;(*") == m^^n • ( 2 -23)
if
M*J) = a i (xJ)a; i /2 < 1 (2.24)
then we may define
oo
«(*?):= 2ME[M*J)] a '- ( 2 - 25 )
t'=0
If besides (2.24) we also have
M x M*ty cD *
(2 - 26)
then according to Theorem 1.5 and Lemma 2.1 the sequence {x^}^L Q given
by (2.3) will converge to a solution of the equation (2.1). In general it may
converge to a solution that is different from the solution x'j whose existence
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