V. A. KURCHATOV

ON A METHOD FOR SOLVING NONLINEAR FUNCTIONAL EQUATIONS

(Presented by Academician M. A. Lavrent’ev on 24 III 1969)

1°. Let \(P(x)=0\) be a nonlinear functional equation having a unique root \(x^*\) in some domain \(R=\{\|x-x^*\|\leq \eta\}\), \(\eta=\mathrm{const}>0\), and let the operation \(P(x)\) map elements of a normed space \(X\) of type \(B\) into elements of a normed space \(Y\) of the same type. We assume that \(P(x)\) is differentiable (in the Fréchet sense, \((^1)\)) \(k+1\) times and that the inverse operation \([P'(x)]^{-1}\) exists. We seek a method of successive approximations in the form

\[ x_{n+1}=x_n+\sum_{i=1}^{k} C_i(x_n)P^i(x_n)=F(x_n), \tag{1.1} \]

where \(C_i(x)\) are certain operations. We shall choose them so that method (1.1) converges to the root of the equation.

Differentiating \(F(x)\) with respect to \(x\),

\[ F'(x)=I+\sum_{i=1}^{k}\{C_i'(x)P^i(x)+iC_i(x)P^{i-1}(x)P'(x)\} \]

or

\[ F'(x)=\sum_{i=0}^{k-1}\{C_i'(x)+(i+1)C_{i+1}(x)P'(x)\}P^i(x)+C_k'(x)P^k(x); \qquad C_0(x)=x. \]

We choose \(C_i(x)\) so that

\[ C_0(x)\equiv x,\qquad C_i'(x)+(i+1)C_{i+1}(x)P'(x)=0,\quad i=1,\ldots,k-1. \tag{1.2} \]

Solving system (1.2) with respect to \(C_i(x)\), we obtain

\[ C_i(x)=-\,C_{i-1}'(x)/iP'(x),\qquad C_0(x)\equiv x,\quad i=1,\ldots,k. \tag{1.3} \]

Now method (1.2) can be written in the form

\[ x_{n+1}=x_n+\sum_{i=1}^{k} C_i(x_n)P^i(x_n),\qquad C_i(x_n)=-\,\frac{C_{i-1}'(x_n)}{iP'(x_n)},\qquad C_0(x)\equiv x. \tag{1.4} \]

Theorem. The method of successive approximations (1.4), for an initial approximation \(x_0\) sufficiently close to \(x^*\), converges with a rate characterized by the inequality

\[ \|x_{n+1}-x_n\|\leq q^{[(k+1)^n-1]/k}\|x_0-x^*\|,\qquad 0<q<1. \tag{1.5} \]

Proof. By virtue of (1.3) we have

\[ F'(x)=C_k'(x)P^k(x), \]

whence it follows that

\[ F'(x^*) = F''(x^*) = \ldots = F^{(k)}(x^*) = 0,\qquad F^{(k+1)}(x^*) = k! C_k'(x^*) P'^k(x^*). \tag{1.6} \]

Without loss of generality in the proof we assume that \(F^{(k+1)}(x^*) \ne 0\). Using an analogue of Taylor’s formula for the case of nonlinear functional equations \((^1)\), we obtain

\[ \left\| F(x_n) - F(x^*) - (x_n - x^*)F'(x^*) - \ldots - \frac{(x_n - x^*)^k}{k!} F^{(k)}(x^*) \right\| \le \]

\[ \le \frac{\|x_n - x^*\|^{k+1}}{(k+1)!} \sup_{\substack{\widetilde{x}=x_n+\theta_n(x_n-x^*)\\ 0\le \theta_n\le 1}} \left\| F^{(k+1)}(\widetilde{x}) \right\|. \]

Hence, in view of (1.6), we have

\[ \|x_{n+1}-x^*\| \le \frac{M_{k+1}}{(k+1)!}\|x_n-x^*\|^{k+1} \quad \left( M_{k+1}= \sup_{\substack{x\in \rho,\ \|x-x^*\|\le \eta_0\\ \eta_0=\|x_0-x^*\|<\eta}} \left\|F^{(k+1)}(x)\right\| \right), \]

and after obvious transformations, for \(\|x_0-x^*\|<1\),

\[ \frac{M_{k+1}}{(k+1)!}\|x_0-x^*\|=q<1, \]

we obtain the validity of (1.5), which indicates a very high rate of convergence of the method under consideration. The theorem is proved.

It follows from what has been obtained that method (1.4) is an iterative method of order \((k+1)\).

\(2^0\). Let us consider some special cases of method (1.4).

1. Setting \(k=1\) in (1.4), we obtain the iterative method of second order

\[ x_{n+1}=x_n-P(x_n)/P'(x_n), \]

which is widely known as Newton’s method for solving functional equations.

1. Setting \(k=2\) in method (1.4), we obtain the iterative method of third order

\[ x_{n+1}=x_n-P(x_n)/P'(x_n)-P''(x_n)P^2(x_n)/2P'^3(x_n). \tag{2.1} \]

\(3^0\). Let us consider some applications of method (2.1).

a) Let \(P(x)=0\) be an algebraic or transcendental equation. Then the successive approximations will be determined by formula (2.1), where \(P(x)\) is some function of a real or complex variable.

b) We now apply method (2.1) to the solution of \(m\) algebraic (transcendental) equations with \(m\) unknowns

\[ f_k(\xi_1,\ldots,\xi_m)=0,\qquad k=1,\ldots,m. \tag{3.1} \]

We rewrite method (2.1) in the form *

\[ P_n'^3(x_{n+1}-x_n)=-(P_n'^2-\tfrac12 P_n''P_n)P_n. \]

Taking into account the meaning of the operations \(P'(x)\), \(P''(x)\) in the case of systems of algebraic (transcendental) equations \((^1)\), we determine the successive approximations from the system of equations for the corrections

\[ \sum_{j=1}^{m} a_{ij}^{(n)}\left(\xi_j^{(n+1)}-\xi_j^{(n)}\right)=-b_i^{(n)},\qquad i=1,\ldots,m, \]

\[ \text{* Notation: } P_n=P(x_n),\quad P_n'=P'(x_n),\quad P_n''=P''(x_n). \]

\[ a_{ij}^{(n)}=\sum_{s,r=1}^{m} \left(\frac{\partial f_i}{\partial \xi_s}\right)_n \left(\frac{\partial f_s}{\partial \xi_r}\right)_n \left(\frac{\partial f_r}{\partial \xi_j}\right)_n, \]

\[ b_i^{(n)}=\sum_{\nu,\mu=1}^{m}(f_\nu)_n \left\{ \left(\frac{\partial f_i}{\partial \xi_\mu}\right)_n \left(\frac{\partial f_\mu}{\partial \xi_\nu}\right)_n +\frac{1}{2}(f_\mu)_n \left(\frac{\partial^2 f_i}{\partial \xi_\nu \partial \xi_\mu}\right)_n \right\}. \]

As is evident, when solving system (3.1) by method (2.1), as in the case of Newton’s method, at each iteration step it is necessary to solve a system of linear algebraic equations of order \(m\); however, the convergence rate of method (2.1) is substantially higher than the convergence rate of Newton’s method. Thus, for example, if \(q<10^{-1}\) and \(\|x_0-x^*\|<1\), then in the case \(k=1\) (Newton’s method) we have

\[ \|x_1-x^*\|<10^{-1}, \qquad \|x_2-x^*\|<10^{-3}, \]

\[ \|x_3-x^*\|<10^{-7}, \qquad \|x_4-x^*\|<10^{-15}, \]

whereas for \(k=2\) (method 2.1),

\[ \|x_1-x^*\|<10^{-1}, \qquad \|x_2-x^*\|<10^{-4}, \]

\[ \|x_3-x^*\|<10^{-13}, \qquad \|x_4-x^*\|<10^{-40}. \]

As is evident, the number of correct decimal digits in the approximations obtained by method (2.1), in comparison with the approximations obtained by Newton’s method, grows very rapidly.

Example. Let us refine an approximate solution \(\xi_1^{(0)}=0.28,\ \xi_2^{(0)}=0.77\) of the system
\(f_1(\xi_1,\xi_2)=\xi_1^2+\xi_2^3+\xi_1-0.902=0,\)
\(f_2(\xi_1,\xi_2)=\xi_1^3+\xi_2^2+\xi_2-1.467=0.\)
The first approximation by method (2.1) gives
\(\xi_1^{(1)}=0.3012,\ \xi_2^{(1)}=0.8004.\)
The true value of the root is
\(\xi_1^*=0.3,\ \xi_2^*=0.8.\)

Received
24 III 1969

REFERENCES

1. L. V. Kantorovich, UMN, 3, no. 6 (1948).