Abstract
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MATHEMATICS
S. N. SLUGIN
ON THE THEORY OF THE METHODS OF NEWTON AND CHAPLYGIN
(Presented by Academician S. L. Sobolev, 14 I 1958)
- In an analogue of Newton’s method \((^1)\) for the approximate solution of the equation \(P(x)=0\), the algorithm
\[ x_{n+1}=x_n-\Gamma_n^{-1}P(x_n), \tag{1} \]
is used, where \(\Gamma_x=P'(x_n)\) or \(\Gamma_n=P'(x_0)\). Instead of the derivative operators \(\Gamma_n\) in (1), one may take an operator \(L\), having an inverse \(L^{-1}\), “close” to \(P'(x)\) in the Banach norm \((^2)\) (our notation). We may impose weaker restrictions on the operator \(L\) and choose a simpler (not necessarily invertible) operator convenient for computations.
Let \(X\) be a space of type \((B_K)\) \((^3)\). We shall use the notation:
\[ V^n(z)=V[V^{n-1}(z)], \qquad V^0=I, \qquad I(x)\equiv x, \qquad \Delta U=U(x+\Delta x)-U(x). \]
Theorem 1. Suppose that on the set \(G\subset X\): 1) \(|\Delta U|\le V(|\Delta x|)\), where \(U=I-LP\); 2) the operator \(V\) is monotonically increasing; 3) \(G\) contains the bounded sphere \((x_0,R)\):
\[ |x-x_0|\le R=\sum_{k=0}^{\infty}V^k(z), \qquad z=|LP(x_0)|; \tag{2} \]
4) the operator \(LP\) is \((bk)\)-continuous; 5) the equation \(L(y)=0\) has the unique zero solution.
Then the algorithm
\[ x_{n+1}=x_n-LP(x_n)\qquad (n=0,1,2,\ldots) \tag{3} \]
converges in the sphere \((x_0,R)\) to the solution \(x^*\) of the equation \(P(x)=0\) with rate
\[ |x_n-x^*|\le \sum_{k=n}^{\infty}V^k(z). \tag{4} \]
For condition 2) it is sufficient that the operator \(V\) be additive; for 4), \((o)\)-continuity of \(V\) is sufficient; for 5), additivity and invertibility of the operator \(L\) are sufficient. If \(V\) is not additive, then in the inequalities (2), (4) one may take \(|V_k|\), and, in general, a stronger estimate is obtained than when using the ordinarily employed (in Banach spaces) power \(\|V\|^k\) or even the norm \(\|V^k\|\) of the iteration.
The operator \(V\) is called Volterra if \((I-V)^{-1}=\sum_{k=0}^{\infty}V^k\). In this case \(R=(I-V)^{-1}(z)\). If \(V\) is additive, then \(|x_n-x^*|\le V^n(R)\).
- In what follows \(X\) is a \(K\)-space \((^3)\). For practical estimates of a solution it is sometimes necessary to establish an operator analogue of Chaplygin’s theorem—
Lygina [^4] on differential inequalities, i.e., the establishment of conditions sufficient for the inequality \(P(x)\geq 0\) to imply \(x\geq x^*\), where \(P(x^*)=0\). For an additive invertible operator \(P\) this means its positive invertibility, i.e., \(P^{-1}>0\).
Theorem 2 (comparison of additive operators). If the operators \(\Gamma\) and \(\Lambda\) are additive and invertible, \(\Gamma>\Lambda\), \(\Gamma^{-1}>0\), \((I-\Gamma^{-1}\Lambda)^n(x)\overset{(o)}{\to}0\) \((n\to\infty)\), then \(\Lambda^{-1}>0\).
Theorem 3 (on operator inequalities). Suppose that on the set \(G\subset X\) the following conditions hold: \(P(x_0)>0\) (or \(<0\)); \(\Gamma(\Delta x)\geq \Delta P\geq \Lambda(\Delta x)\) for \(\Delta x>0\); the operators \(\Gamma\) and \(\Lambda\) are additive and positively invertible; the operator \(\Gamma^{-1}P\) is monotonically continuous; the set \(G\) contains \([x_1,x_0]\) (or \([x_0,x_1]\)), where \(x_1=x_0-\Lambda^{-1}P(x_0)\).
Then on this interval there exists a unique solution of the equation \(P(x)=0\).
From this, in particular, it is not difficult to obtain the comparison theorem from the paper [^5].
- Let us construct an algorithm of Chaplygin type on the basis of replacing the operator \(\Gamma_n^{-1}\) in the algorithm of Theorem 1 of the paper [^6], similarly to how this was done above.
Theorem 4. Suppose that on each \([a,b]\subseteq[x_0,\bar{x}_0]\) there exist additive positively invertible operators \(\underline{\Gamma}(a,b)\) and \(\bar{\Gamma}(a,b)\) such that throughout \([a,b]\)
\[ \underline{\Gamma}(a,b)(x-a)\geq P(x)-P(a),\qquad \bar{\Gamma}(a,b)(b-x)\geq P(b)-P(x). \]
Suppose there exists an additive positively invertible operator \(\Gamma\geq\underline{\Gamma}(a,b)\), \(\Gamma\geq\bar{\Gamma}(a,b)\) on all \([a,b]\), the operator \(\Gamma^{-1}P\) is monotonically continuous, and \(P(x_0)\leq 0\leq P(\bar{x}_0)\).
Define the algorithm
\[ \underline{x}_{n+1}=\underline{x}_n-\underline{L}_n(\underline{z}_n),\qquad \bar{x}_{n+1}=\bar{x}_n-\bar{L}_n(\bar{z}_n), \tag{5} \]
where \(P(\underline{x}_n)\leq \underline{z}_n\leq 0\leq \bar{z}_n\leq P(\bar{x}_n)\), the operators \(L_n\) are positive and homogeneous,
\[ \underline{\Gamma}_n\underline{L}_n\leq I,\qquad \bar{\Gamma}_n\bar{L}_n\leq I,\qquad \underline{\Gamma}_n=\underline{\Gamma}(\underline{x}_n,\bar{x}_n),\quad \bar{\Gamma}_n=\bar{\Gamma}(\underline{x}_n,\bar{x}_n). \]
Then the algorithm determines sequences \(x_n\) satisfying the inequalities
\[ \underline{x}_n\leq \underline{x}_{n+1}\leq \underline{x}\leq \bar{x}\leq \bar{x}_{n+1}\leq \bar{x}_n, \]
where \(\underline{x},\bar{x}\) are the least and greatest solutions on \([x_0,\bar{x}_0]\) of the equation \(P(x)=0\).
For uniqueness of the solution it is sufficient that there exist an additive positively invertible operator \(\Lambda\) such that \(\Delta P\geq \Lambda(\Delta x)\) for \(\Delta x>0\).
If it is known in advance that the solution is unique, then the condition \(x_0\leq \bar{x}_0\) need not be imposed. Instead of the conditions imposed on \(L_n\), it is sufficient to require that \(\underline{\Gamma}_n\underline{L}_n(\underline{z}_n)\geq \underline{z}_n\), \(\bar{\Gamma}_n\bar{L}_n(\bar{z}_n)\leq \bar{z}_n\). Under some additional assumptions concerning \(L_n(z_n)\), the algorithm converges to the solution. Instead of continuity of the operator \(\Gamma^{-1}P\), one may require continuity of the operators \(\Gamma\) and \(P\). This remark also applies to Theorem 3.
Putting \(\Gamma_n=\bar{\Gamma}_n\), \(L_n=\Gamma_n^{-1}\), we obtain the algorithms of the paper [^7].
Theorem 5. If, under the conditions of Theorem 4, the operators \(I-\Gamma_n\) are positive Volterra operators, then one may take as \(L_n\)
\[ \sum_{k=0}^{m}(I-\Gamma_n)^k \]
where \(m\) is arbitrary.
Hence, for \(z_n=P(x_n)\), acting in the space of collections of derivatives, one can, in particular, obtain (concrete) algorithms of work\({}^{8}\). They occupy an intermediate place between the algorithms
\[ x_{n+1}=x_n-P(x_n) \]
and the more complicated, but more rapidly convergent,
\[ x_{n+1}=x_n-\Gamma_n^{-1}P(x_n). \]
- The additivity and positive invertibility condition on the operator \(\Gamma'\), usually used in analogues of Chaplygin’s method\({}^{6,7}\), can be replaced by another one.
Theorem 6. Let the following conditions be fulfilled on the set \(G\subset X\): \(P(x_0)>0\) (or \(<0\)), \(\Gamma(\Delta x)\geq \Delta P\) for \(\Delta x>0\); there exists an operator \(L\) such that \(L>0\), \(\Gamma L\leq I\) (or \(L(y)<0\leq \Gamma L(y)-y\) for \(y<0\)); the equation \(L(y)=0\) has the unique zero solution; the operator \(LP\) is monotonically continuous; \(\Delta U\leq \Delta V\) for \(\Delta x>0\), where \(U=I-LP\); the operator \(V\) is monotonically increasing; the set \(G\) contains the bounded interval
\[ [x_0-R,x_0]\quad (\text{or }[x_0,x_0+R]), \]
where
\[ R=\sum_{k=0}^{\infty}V^k(z),\qquad z=|LP(x_0)|. \]
Then the equation \(P(x)=0\) has on this interval a solution \(x^*\), which can be obtained by algorithm (3), and moreover \(x_n\searrow x^*\) (or \(x_n\nearrow x^*\)) with rate
\[ |x_n-x^*|\leq \sum_{k=n}^{\infty}V^k(z). \]
A consequence of this is Theorem 3 under the additional condition: the operator \(V=I-\Gamma^{-1}\Lambda\) is Volterra. The theorem can be modified by replacing algorithm (3) with an algorithm of the form (5).
Kazan State
University
Received
13 I 1958
CITED LITERATURE
\({}^{1}\) L. V. Kantorovich, DAN, 76, No. 1, 17 (1951).
\({}^{2}\) B. A. Vertgeim, Uspekhi Mat. Nauk, 12, issue 1 (73), 166 (1957).
\({}^{3}\) L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semi-Ordered Spaces, Moscow–Leningrad, 1950.
\({}^{4}\) S. A. Chaplygin, A New Method of Approximate Integration of Differential Equations, Moscow–Leningrad, 1950.
\({}^{5}\) N. V. Azbelev, DAN, 89, No. 4, 589 (1953).
\({}^{6}\) S. N. Slugin, DAN, 103, No. 4, 565 (1955).
\({}^{7}\) A. N. Baluev, Vestn. LGU, No. 13, issue 3, 27 (1956).
\({}^{8}\) G. A. Artemov, DAN, 101, No. 2, 197 (1955).