546 
MATHEMATICS: H. B. FINE 
ON NEWTON'S METHOD OF APPROXIMATION 
By Henry B. Fine 
DEPARTMENT OF MATHEMATICS. PRINCETON UNIVERSITY 
Received by the Academy, August 8, 1916 
In the first of the following theorems^ a condition is given under which 
Newton's method of approximation for computing a real root of an 
equation f(x)=0 and the extension of this method used in computing a real 
solution of a system of equations X2, . . . , Xn) =0, (i=l, 2, . . ,n), 
will with certainty lead to such a root or solution. The condition relates 
to the absolute value of f{x) or of the functions fi(xi, X2, . . . , Xn) cor- 
responding to the initial values of x and of the variables 
respectively. No assumption is made as to the existence of a solution. 
On the contrary it is proved that under the condition to which reference 
has been made one and but one solution exists in a certain designated 
neighborhood of the initial x or (xi, X2, . . . , Xn). The second theorem 
is the extension of the first (for n=l) to the case of a complex root of an 
analytic equation f(z) = 0. 
Theorem 1. Let fi{xi, X2, . . . , Xn), (i = 1,2, . . . , n), be a system 
of real functions of the real variables Xi, X2, . . • , Xn which have continuous 
first and second derivatives in the region R, (x[^\ X2^\ • • • ^n^) ci set of 
values of Xi, X2, . . . , Xn belonging to this region, and ^1, ^2, • • • , in the 
set of numbers determined by the equations. 
and let S denote the interval, circle, sphere, or hypersphere whose center is 
supposed to belong to R. 
Suppose also that in S the functional determinant F of the functions U 
does not vanish, fjL{< co) is the upper bound of the absolute values of the frac- 
tions whose denominators are F and whose numerators are the several first 
{i= 1,2, . . , , n) 
{x^i^ -h ^1, . • . , xf' -\- Q and whose radius 
minors of F, and v{<co) is the upper bound of the absolute values of the 
second derivatives of the functions ft. 
Then, if 
n 
1 
< 
n' 
,7/2 2 
{a) 
_ i= 1 
