MATHEMATICS

S. Yu. UL’M

ITERATIVE METHODS WITH SECOND-ORDER DIVIDED DIFFERENCES

(Presented by Academician A. A. Dorodnitsyn on 18 III 1964)

1. Let the equation be given

\[ \mathcal{P}(x)=0, \tag{1} \]

where \(\mathcal{P}(x)\) is a nonlinear operator mapping the linear normed space \(X\) into a space \(Y\) of the same type. In what follows we use the following notation: \(\mathcal{P}(x',x'')\), \(\mathcal{P}(x',x'',x''')\) are analogues of divided differences \((^1)\), respectively of the first and second orders, for the operator \(\mathcal{P}(x)\); \(\Omega_n=[\mathcal{P}(x_n,x_{n-1})]^{-1}\); \(U_n=\Omega_n\mathcal{P}(x_n,x_{n-1},x_{n-2})\); \(\widetilde{x}_{n+1}-x_n=-\Omega_n\mathcal{P}(x)\); \(E\) is the identity operator of the space \(X\).

To introduce a class of iterative methods, suppose that we know three approximations \(x_n, x_{n-1}, x_{n-2}\) to the solution \(x^*\) of equation (1). We use an analogue of Newton’s interpolation formula \((^1)\):

\[ \mathcal{P}(x)=\mathcal{P}(x_n)+\mathcal{P}(x_n,x_{n-1})(x-x_n)+ \]
\[ +\mathcal{P}(x_n,x_{n-1},x_{n-2})(x-x_n)(x-x_{n-1})+R_n, \tag{2} \]

where

\[ R_n=[\mathcal{P}(x,x_n,x_{n-1})-\mathcal{P}(x_n,x_{n-1},x_{n-2})](x-x_n)(x-x_{n-1}). \tag{3} \]

Discarding the remainder term \(R_n\) in formula (2), we consider, instead of equation (1), the approximate equation

\[ \mathcal{P}(x_n)+\mathcal{P}(x_n,x_{n-1})(x-x_n)+\mathcal{P}(x_n,x_{n-1},x_{n-2})(x-x_n)(x-x_{n-1})=0. \tag{4} \]

Introduce into equation (4) a real parameter \(\alpha\) and write this equation in the form

\[ \mathcal{P}(x_n)+[\mathcal{P}(x_n,x_{n-1})-\alpha\mathcal{P}(x_n,x_{n-1},x_{n-2})(x-x_n)](x-x_n)+ \]
\[ +(1+\alpha)\mathcal{P}(x_n,x_{n-1},x_{n-2})(x-x_n)^2+ \]
\[ +\mathcal{P}(x_n,x_{n-1},x_{n-2})(x-x_n)(x_n-x_{n-1})=0. \tag{5} \]

Since, by the chord method \((^1)\),

\[ x^*-x_n\approx \Omega_n\mathcal{P}(x_n)=\widetilde{x}_{n+1}-x_n, \tag{6} \]

we replace in equation (5) the element \(x-x_n\) in square brackets and in the third and fourth terms by the element \(\widetilde{x}_{n+1}-x_n\). Then, instead of equation (1), we obtain the approximate linear equation

\[ \mathcal{P}(x_n)+[\mathcal{P}(x_n,x_{n-1})-\alpha\mathcal{P}(x_n,x_{n-1},x_{n-2})(\widetilde{x}_{n+1}-x_n)](x-x_n)+ \]
\[ +(1+\alpha)\mathcal{P}(x_n,x_{n-1},x_{n-2})(\widetilde{x}_{n+1}-x_n)^2+ \]
\[ +\mathcal{P}(x_n,x_{n-1},x_{n-2})(\widetilde{x}_{n+1}-x_n)(x_n-x_{n-1})=0. \tag{7} \]

Solving equation (7) with respect to \(x\), we take the solution found as the new approximation \(x_{n+1}\) to the solution \(x^*\) of equation (1). Thus, we arrive at the following class of iterative methods for the approximate solution of equation (1):

\[ x_{n+1}=x_n+[E-\alpha U_n(\widetilde{x}_{n+1}-x_n)]^{-1}[E-(1+\alpha)U_n(\widetilde{x}_{n+1}-x_n)- \]
\[ -\,U_n(x_n-x_{n-1})](\widetilde{x}_{n+1}-x_n), \tag{8} \]

where \(n=2,3,\ldots;\ x_0,\ x_1,\ x_2\) are three initial approximations to the solution \(x^*\) of equation (1).

2. Theorem. Suppose:

\(1^\circ.\) Equation (1) has a solution \(x^*\), and

\[ \max\{\|x^*-x_0\|,\ \|x^*-x_1\|,\ \|x^*-x_2\|\}\leqslant d . \]

\(2^\circ.\) For every \(x', x'', x''', x^{\mathrm{IV}}\) from the sphere \(\|x-x^*\|\leqslant d\), the estimates hold:

a) \(\|[\mathcal P(x',x'')]^{-1}\|\leqslant B;\)

b) \(\|\mathcal P(x',x'',x''')\|\leqslant H;\)

c) \(\|\mathcal P(x',x'',x''')-\mathcal P(x'',x''',x^{\mathrm{IV}})\|\leqslant K\|x'-x^{\mathrm{IV}}\|\).

\(3^\circ.\)

\[ |\alpha|BHd+\bigl[BK+(1+|\alpha|+|1+\alpha|)B^2H^2\bigr]d^2 +|1+\alpha|B^3H^3d^3<1. \]

Then the sequence (8) converges to the solution \(x^*\) of equation (1) with rate

\[ \|x^*-x_n\|\leqslant \frac{1}{M}(Md)^{t_n}\qquad (n=0,1,\ldots), \tag{9} \]

where

\[ M=\left[ \frac{BK+B^2H^2+|1+\alpha|B^2H^2(1+BHd)^{7/2}} {1-|\alpha|BHd(1+BHd)} \right] \]

and the numbers \(t_n\) are generalized Fibonacci numbers \((t_0=t_1=t_2=1;\ t_{n+1}=t_n+t_{n-1}+t_{n-2};\ n=2,3,\ldots)\).

Proof. We use the principle of complete induction. On the basis of condition \(1^\circ\), the estimates (9) are valid for \(n=0,1,2\). By formula (8),

\[ x^*-x_{n+1} = [E-\alpha U_n(\widetilde{x}_{n+1}-x_n)]^{-1} [x^*-\widetilde{x}_{n+1} -(1+\alpha)U_n(\widetilde{x}_{n+1}-x_n)(x^*-\widetilde{x}_{n+1}) +U_n(x^*-x_{n-1})(\widetilde{x}_{n+1}-x_n)] . \tag{10} \]

On the basis of Newton’s interpolation formula,

\[ x^*-\widetilde{x}_{n+1} = -Q_n^* = -U_n(x^*-x_n)(x^*-x_{n-1})-R_n^*, \tag{11} \]

\[ \widetilde{x}_{n+1}-x_n=x^*-x_n+Q_n^*, \tag{12} \]

where

\[ R_n^* = \Omega_n[\mathcal P(x^*,x_n,x_{n-1})-\mathcal P(x_n,x_{n-1},x_{n-2})](x^*-x_n)(x^*-x_{n-1}), \tag{13} \]

\[ Q_n^* = \Omega_n\mathcal P(x^*,x_n,x_{n-1})(x^*-x_n)(x^*-x_{n-1}). \tag{14} \]

Replacing in formula (10) \(x^*-\widetilde{x}_{n+1}\) and \(\widetilde{x}_{n+1}-x_n\) by formulas (11) and (12), we obtain:

\[ x^*-x_{n+1} = [E-\alpha U_n'(x^*-x_n)-\alpha U_nQ_n^*]^{-1} [-R_n^* +(1+\alpha)U_n(x^*-x_n)Q_n^* +(1+\alpha)U_nQ_n^{*2} +U_n(x^*-x_{n-1})Q_n^*]. \tag{15} \]

Denote \(d_i=\|x^*-x_i\|\). By the Banach theorem,

\[ \|[E-\alpha U_n(x^*-x_n)-\alpha U_nQ_n^*]^{-1}\| \leqslant (1-|\alpha|BHd-|\alpha|B^2H^2d^2)^{-1} = A(\alpha). \]

Consequently, on the basis of formula (15),

\[ d_{n+1}\leqslant M^{-3}\bigl[ M_1^2(Md)^{t_n+t_{n-1}+t_{n-2}} + M_2^2(Md)^{2t_n+t_{n-1}} + M_3^2(Md)^{t_n+2t_{n-1}} \bigr], \tag{16} \]

where

\[ M_1^2=BKA(\alpha),\qquad M_2^2=[1+\alpha]B^2H^2(1+BHd)A(\alpha),\qquad M_3^2=B^2H^2A(\alpha). \]

Since \(Md<1\) (condition \(3^\circ\)) and \(2t_n+t_{n-1}\geqslant t_n+2t_{n-1}\geqslant t_n+t_{n-1}+t_{n-2}\), it follows that

\[ d_{n+1}\leqslant \frac{1}{M}(Md)^{t_{n+1}}. \tag{17} \]

Passing to the limit in formula (17) as \((n \to \infty)\), we find that \(x_n \to x^*\). The theorem is proved.

Remark. The sphere \(\|x^* - x\| \leqslant d\) in condition \(2^\circ\) of the theorem may be replaced, for example, by the sphere \(\|x - x_0\| \leqslant 2d\). Indeed, if \(x\) is an element of the first sphere, then

\[ \|x - x_0\| \leqslant \|x - x^*\| + \|x^* - x_0\| \leqslant 2d, \]

i.e., \(x\) also belongs to the second sphere. If the estimates \(2^\circ\) a, b, c are found in the sphere \(\|x - x_0\| \leqslant 2d\), then it is easy to see that the solution \(x^*\) is unique in the sphere \(\|x - x_0\| \leqslant d\).

1. Let us note that the class of iterative methods (8) is an interpolation analogue of the class of differential methods studied in the works of R. Ludwig \((^2)\) and Yu. Ya. Kaazik \((^3)\). From the estimates (9) it is seen that the methods of class (8) converge to the solution of equation (1) faster than the chord method, for which the exponents \(t_n\) are the ordinary Fibonacci numbers \((^1)\). The methods (8) are effective for solving such nonlinear transcendental equations for which finding derivatives of the functions is difficult. For the approximate solution of algebraic and transcendental equations

\[ f(x) = 0 \tag{18} \]

the methods (8) have the form:

\[ x_{n+1}=x_n+\{[f(x_n,x_{n-1})-(1+\alpha)f(x_n,x_{n-1},x_{n-2})(\widetilde{x}_{n+1}-x_n)- \]
\[ -f(x_n,x_{n-1},x_{n-2})(x_n-x_{n-1})]/[f(x_n,x_{n-1})- \]
\[ -\alpha f(x_n,x_{n-1},x_{n-2})(\widetilde{x}_{n+1}-x_n)]\}(\widetilde{x}_{n+1}-x_n), \tag{19} \]

where \(\widetilde{x}_{n+1}-x_n=-f(x_n)/f(x_n,x_{n-1})\), \(n=2,3,\ldots\).

Taking in formula (8) (or (19)) \(\alpha=0\) and \(\alpha=-1\), we obtain the iterative methods considered by the author \((^4)\) and which are, respectively, interpolation analogues of the methods of tangent parabolas \((^5)\) and tangent hyperbolas \((^6)\).

Institute of Cybernetics
Academy of Sciences of the Estonian SSR

Received
17 III 1964

CITED LITERATURE

\(^1\) A. S. Sergeev, Sibirsk. matem. zhurn., 2, No. 2, 282 (1961).
\(^2\) R. Ludwig, ZAMM, 34, No. 6, 210 (1954).
\(^3\) Yu. Ya. Kaazik, DAN, 112, No. 4, 579 (1957).
\(^4\) S. U. Ulm, Izv. AN Estonian SSR, ser. fiz.-matem. i tekhn. nauk, 12, No. 1, 24 (1963).
\(^5\) V. E. Mirakov, UMN, 11, No. 3, 171 (1956).
\(^6\) M. A. Mertvetsova, DAN, 88, No. 4, 611 (1953).