Letter to the Editor

In my paper [1], the study of convergence of the process

\[ x_{n+1}=x_n-\Delta(x_n) \tag{1} \]

for the approximate solution of the real equation

\[ \varphi(x)=0 \tag{2} \]

is reduced to the study of the convergence of Newton’s method.

As N. A. Khalitova correctly noted in [2], the conditions of the theorem formulated in [1] ensure the convergence of process (1) only to a root of the equation

\[ \Delta(x)=0. \tag{3} \]

It is necessary to require an additional condition: in the domain used, all solutions of equation (3) are solutions of equation (2). However, unless additional assumptions are made about the form of the function \(\Delta(x)\) (the relation of \(\Delta(x)\) to \(\varphi(x)\)), this condition cannot be formulated in analytic form.

Therefore, in the theorem of paper [1], all references to equation (1) should be replaced by references to equation (3).

Cited Literature

1. M. I. Nechepurenko, DAN, 109, No. 4, 704 (1956).
2. N. A. Khalitova, Uch. zap. Kazan Univ., 117, book 2, 79 (1957).

M. Nechepurenko