596 
MATHEMATICS: A. A. BENNETT 
Newton's Method consists in the following steps. Choose any Xo in 
the domain and then select /jq as a solution of ao+&o^o = 0. Put Xi = 
Xo+ho, and repeat. Thus ai-^-bih^ = 0, and = ^,-+ i = 0, 1, . . . , 
where in general a^, bi, Ci mean a{xi), h{x^, c{xi) respectively. If lim,.= co 
exists uniquely, ( = x'), and is in the domain, then x' is a root of 
/(«) = 0. 
In order that Newton's Method may be applied, we note that the 
equations a^- + = 0 must be solved uniquely for hi, so that b(x) must 
be non-singular in the domain. We shall write || ai\\ =ai, \\ bj^ || = l/jS,-, 
II II = ti' Taking the norm of both members of equation (1), and 
recalling that a^^-b^ k = 0, we have || /"fe + || = !| c{Q h] \\ ^ 7,- !1 h \\ \ 
From /j,- = we have 
II k II S (l/ft)«.-, - (2) 
Thus we may write 
^i+i ^ 7,-(lM-)'a:- or {aiyi/^])ai. 
If the sequence of approximations x^ is to yield a root x' as a limit, it is 
necessary that the sequence of values of || / || , viz., q;,-_}.i = || || = 
II f(Xi-\-hi) II approach zero as a limit, and hence necessary although 
not sufficient that the limit of {aiyj^]) shall not exceed unity. A re- 
finement of these considerations suggests the theorems of the follow- 
ing section. 
Justification of Newton's Method. — Theorem 1. Let Xq be given and 
let D be a positive number such that fix) is expansible in the form (1) for 
every x and x-{-h {if either be denoted by u) in the domain \\ u—Xo \\ < D. 
Let f {x) be non-singular in the domain, and B be a positive number 
such that \\f~^{x) II < 1/B in the domain, and C a positive number such 
that II J f" {x) II < C in the domain. Then Newton^ s Method yields a root 
x' in the domain, whenever (CD/B) <p[C \\ f(xo) \\ where <p (X) = 
X + + . . . +X2*H- .... 
An idea of the behavior of ^(X) may be had by noting that ^(X) has 
the unit circle in the complex plane for natural boundary, so that ^(X) 
is not defined for real positive X's except when X is not greater than 
one. Obviously (p (X) increases with positive X's. Some of the numer- 
ical values of (p(\) are given by the following table. 
X 
X 
V5(X) 
0. 
0. 
0.566126 
1.000000 
0.1 
0.110100 
0.6 
1.106678 
0.2 
0.241603 
0.7 
1.491062 
0.3 
0.398165 
0.8 
2.046305 
0.356497 
0.500000 
0.9 
3.017540 
0.4 
0.586255 
1. 
00 
0.5 
0.816421 
<P[1/ (k-hl)] 
|< l/y^,y^>0. 
