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MATHEMATICS
R. A. SHAFIEV
ON THE METHOD OF TANGENT HYPERBOLAS
(Presented by Academician A. N. Kolmogorov on 31 X 1962)
- Let \(P(x)\) be a nonlinear operator acting from the ball
\(D(x_0,R)=\{x:\|x-x_0\|<R\}\) of a Banach space \(X\) into a Banach space \(Y\).
For the approximate solution of the equation
\[ P(x)=0 \tag{1} \]
various methods are known \((^{1-3})\). However, in many cases the method of tangent hyperbolas, proposed in \((^4)\), gives better convergence than other methods.
The successive approximations of the method of tangent hyperbolas are defined from the recurrence relations
\[ x_{n+1}=x_n-\theta_n\Gamma_n P(x_n),\qquad n=0,1,2,\ldots, \tag{2} \]
where
\(\theta_n=[I-\tfrac12\Gamma_n P''(x_n)\Gamma_n P(x_n)]^{-1}\),
\(\Gamma_n=[P'(x_n)]^{-1}\).
In the present paper, by means of the majorant principle, both new convergence conditions for the iterative process (2) and convergence conditions for the process considered by us,
\[ x'_{n+1}=x'_n-Q_n\Gamma_0 P(x'_n),\qquad n=0,1,2,\ldots, \tag{3} \]
are established, where
\(Q_n=[I-\tfrac12\Gamma_0 P''(x_0)\Gamma_0 P(x'_n)]^{-1}\), representing a modification of the method of tangent hyperbolas.
- Consider the equation
\[ \varphi(t)=0, \tag{4} \]
where \(\varphi(t)\) is a real function given on the interval \([t_0,t']\), \(t'=t_0+r<t_0+R\). We shall assume that the function \(\varphi(t)\) is three times continuously differentiable on \([t_0,t']\), and that the operator \(P(x)\) is three times continuously differentiable with respect to Gâteaux \((^5)\) in the closed ball \(\overline{D}(x_0,r)\).
We shall say that equation (4) majorizes equation (1) if the following conditions are satisfied:
\(1^\circ.\ \|P(x_0)\|\leq \varphi(t_0)\).
\(2^\circ.\) There exists the operator \(\Gamma_0=[P'(x_0)]^{-1}\), and moreover \(\varphi'(t_0)\neq 0\) and
\[
\|\Gamma_0\|\leq -\frac{1}{\varphi'(t_0)}.
\]
\(3^\circ.\ \|P''(x_0)\|\leq \varphi''(t_0)\) and
\(\|P'''(x)\|\leq \varphi'''(t)\), when
\(\|x-x_0\|\leq t-t_0\leq t'-t_0\).
For the real equation (4), the iterative processes (2) and (3) take, respectively, the form
\[ t_{n+1}=t_n-\frac{2\varphi'(t_n)\varphi(t_n)} {2[\varphi'(t_n)]^2-\varphi''(t_n)\varphi(t_n)},\qquad n=0,1,2,\ldots; \tag{5} \]
\[ t'_{n+1}=t'_n-\frac{2\varphi'(t_0)\varphi(t'_n)} {2[\varphi'(t_0)]^2-\varphi''(t_0)\varphi(t'_n)},\qquad n=0,1,2,\ldots. \tag{6} \]
- In this and in the following sections we shall establish two general theorems on the convergence of the processes (2) and (3). In doing so we shall assume-
that equation (4) has solutions belonging to the interval \((t_0, t')\), and let us denote the smallest of them by \(t^*\).
Theorem 1. Suppose the following conditions are satisfied:
1) equation (4) majorizes equation (1);
2) \(\varphi''(t)\varphi(t)[\varphi'(t)]^{-2}\leqslant \sigma < 2\) for \(t\in [t_0,t^*]\).
Then the following assertions hold:
1) there exists a solution \(x^*\) of equation (1), belonging to the ball \(\widetilde D(x_0,r)\), where \(r=t^*-t_0\), to which process (2) converges;
2) process (5) converges to the root \(t^*\);
3) the rate of convergence of process (2) is determined by the inequality
\[ \|x^*-x_n\|\leqslant t^*-t_n . \tag{7} \]
We give the plan of the proof. First it is established that
\[ \|\Gamma_0 P''(x_0)\Gamma_0 P(x_0)\|<2, \]
and hence there exists an operator \(\theta_0\), for which the estimate
\[ \|\theta_0\|\leqslant \frac{2[\varphi'(t_0)]^2}{2[\varphi'(t_0)]^2-\varphi''(t_0)\varphi(t_0)} \]
is derived.
This estimate leads to the inequality
\[ \|x_1-x_0\|\leqslant t_1-t_0, \tag{8} \]
where \(t_1\leqslant t^*\).
Using now an analogue of Taylor’s formula
\[ P(x_1)=P(x_0)+P'(x_0)(x_1-x_0)+\frac12 P''(x_0)(x_1-x_0)^2+ \]
\[ +\frac12\int_{x_0}^{x_1} P'''(x)(x_1-x)^2\,dx \]
(on integrals in a Banach space see, for example, \((^6,^7)\)), we find that
\[ P(x_1)=\frac14 P''(x_0)\Gamma_0 P''(x_0)\Gamma_0 P(x_0)(x_1-x_0)^2 +\frac12\int_{x_0}^{x_1} P'''(x)(x_1-x)^2\,dx . \]
Hence, and from other relations, it follows that
\[ \|P(x_1)\|\leqslant \varphi(t_1). \]
It is not difficult to see that \(\|\Gamma_0[P'(x_0)-P'(x_1)]\|<1\), whence it follows that the operator \(H=\Gamma_0P'(x_1)\) has an inverse \(H^{-1}\), for which the estimate
\[ \|H^{-1}\|\leqslant \frac{\varphi'(t_0)}{\varphi'(t_1)} \]
is valid.
Consequently, there exists the operator \(\Gamma_1=H^{-1}\Gamma_0\), and
\[ \|\Gamma_1\|\leqslant -[\varphi'(t_1)]^{-1}. \]
In view of (8), the ball \(D(x_1,\widetilde t-t_1)\subset D(x_0,\widetilde t-t_0)\), where \(\widetilde t\in[t_1,t^*]\), so that conditions \(1^\circ\)—\(3^\circ\) are satisfied when \(x_0\) is replaced by \(x_1\) and \(t_0\) by \(t_1\). By induction it is established that conditions \(1^\circ\)—\(3^\circ\) are satisfied when \(x_0\) is replaced by
on \(x_n\) and \(t_0\) to \(t_n\) for any natural \(n\). In this case it is obtained that \(t_n \leq t_{n+1}\), \(\varphi(t_n)\geq 0\), and
\[ \|P(x_n)\|\leq \varphi(t_n),\qquad n=0,1,2,\ldots \tag{9} \]
To complete the proof it is established that the sequence \(\{t_n\}\) converges to \(t^*\) and
\[ \|x_{n+p}-x_n\|\leq t_{n+p}-t_n, \tag{10} \]
i.e., that \(\{x_n\}\) is a fundamental sequence. Hence, by virtue of (9), the existence of a solution \(x^*\) of equation (1) follows, and from (10) the estimate (7) follows.
- The convergence of the modified process of tangent hyperbolas is established by the following proposition.
Theorem 2. Suppose that the following conditions are fulfilled:
1) equation (4) majorizes equation (1);
2) \(\varphi''(t_0)\varphi(t_0)[\varphi'(t_0)]^{-2}\leq \sigma < 2\).
Then the following assertions hold:
1) equation (1) has a solution \(x^*\in \widetilde D(x_0,r)\), where \(r=t^*-t_0\), to which process (3) converges;
2) process (6) converges to the root \(t^*\) of equation (4);
3) the rate of convergence of process (3) is determined by the inequality
\[ \|x^*-x'_n\|\leq t^*-t'_n. \]
In proving this theorem, approximately the same plan is used as in the proof of Theorem 1, but the computations differ. Essential changes are introduced in the proof of the inequality
\[ \|P(x'_2)\|\leq \varphi(t'_2). \tag{11} \]
By using an analogue of Taylor’s formula and the equalities
\[ P'(x'_1)(x'_2-x'_1) = P'(x_0)(x'_2-x'_1) + \int_{x_0}^{x'_1} P''(x)(x'_2-x'_1)\,dx, \]
\[ P''(x'_1)(x'_2-x'_1)^2 = P''(x_0)(x'_2-x'_1)^2 + \int_{x_0}^{x'_1} P'''(x)(x'_2-x'_1)^2\,dx, \]
one derives the formula
\[ P(x'_2) = \frac14 P''(x_0)\Gamma_0 P''(x_0)\Gamma_0 P(x'_1)(x'_2-x'_1)^2 + \]
\[ + \int_{x_0}^{x'_1} P''(x)(x'_2-x'_1)\,dx + \frac12\int_{x_0}^{x'_1} P'''(x)(x'_2-x'_1)^2\,dx + \frac12\int_{x'_1}^{x'_2} P'''(x)(x'_2-x)^2\,dx, \]
which leads to inequality (11).
- With the aid of Theorems 1 and 2 we establish the following propositions, convenient for applications.
Theorem 3. Suppose that at some point \(x_0\in X\) the following conditions are fulfilled:
1) \(\|P(x_0)\|\leq \delta\);
2) there exists an operator \(\Gamma=[P'(x_0)]^{-1}\), and \(\|\Gamma\|\leq B\);
3) in the domain \(G=\{x:\|x-x_0\|\leq t^*\}\), where \(t^*\) is the smallest positive solution of the equation
\[ \varphi(t)\equiv \frac16 Nt^3+\frac12 Mt^2-B^{-1}t+\delta=0, \]
the inequalities hold
\[ M \geqslant \|P''(x_0)\|,\qquad N \geqslant \sup_{x\in G}\|P'''(x)\|; \]
4) \(h=MB^2\delta \leqslant \dfrac{1}{2+\gamma}\), where \(\gamma=NB^{-1}M^{-2}\).
Then there exists a solution \(x^*\) of equation (1), to which process (2) converges; moreover, the rate of convergence is determined by the inequality
\(\|x^*-x_n\|\leqslant t^*-t_n\), where \(t_n\) is determined by process (5) with \(t_0=0\), converging to \(t^*\).
Theorem 4. Suppose that conditions 1), 2), 3) of Theorem 3 and the condition
\[ 4)\quad h=MB^2\delta \leqslant \frac{1}{3}\gamma^{-2}\bigl[(1+2\gamma)^{3/2}-(1+3\gamma)\bigr]. \]
are satisfied. Then there exists a solution \(x^*\) of equation (1), to which process (3) converges; moreover, the rate of convergence is determined by the inequality
\(\|x^*-x'_n\|\leqslant t^*-t'_n\), where \(t'_n\) is determined by process (6) with \(t_0=0\), converging to \(t^*\).
- Let us give the simplest example. For the equation
\[ x^5+x^4+x^3+x^2-10x+1=0, \]
choosing \(x_0=0\), we have \(\gamma=8.952,\ h=0.004,\ \dfrac{1}{2+\gamma}>0.091\). Hence, by Theorems 3 and 4, the implementability of processes (2) and (3) follows.
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
31 X 1962
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