On some Steffensen-type iterative methods for a class of nonlinear equations

Emil Cătinaş
(original), Newton method, nonlinear equations in Banach spaces, Numerical Analysis, paper, semilocal convergence, solving F(x)=0 iteratively, Steffenssen methods

Abstract

Consider the nonlinear equations \(H(x):=F(x)+G(x)=0\), with \(F\) differentiable and \(G\) continuous, where \(F,G,H:X \rightarrow X\) are nonlinear operators and \(X\) is a Banach space. 

The Newton method for solving the nonlinear equation \(H(x)=0\) cannot be applied, and we propose an iterative method for solving this equation by combining the Newton method with the Steffensen method: \[x_{k+1} = \big(F^\prime(x_k)+[x_k,\varphi(x_k);G]\big)^{-1}(F(x_k)+G(x_k)),\] where \(\varphi(x)=x-\lambda (F(x)+G(x))\), \(\lambda >0\) fixed.

The method is obtained by combining the Newton method for the differentiable part with the Steffensen method for the nondifferentiable part.

We show that the R-convergence order of this method is 2, the same as of the Newton method.

We provide some numerical examples and compare different methods for a nonlinear system in \(\mathbb{R}^2\).

Authors

E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

nonlinear equation; Banach space; Newton method; Steffensen method; combined method; nondifferentiable mapping; nonsmooth mapping; r-convergence order.

Cite this paper as:

E. Cătinaş, On some Steffensen-type iterative methods for a class of nonlinear equations, Rev. Anal. Numér. Théor. Approx., 24 (1995) nos. 1-2, pp. 37-43.

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Revue d’Analyse Numérique et de Théorie de l’Approximation

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Editions de l’Academie Roumaine

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https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art4

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2457-6794

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citations of this paper

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[3] E. Catinas, On some iterative methods for solving nonlinear equations, Rev. Anal. Numer. Theor. Approx., 23 (1994) no. 1, pp. 47–53.
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[4] G. Goldner, M. Balazs, Asupra metodei coardei si a unei modificari a ei pentru rezolvarea ecuațiilor operationale neliniare, Stud. Cerc. Mat., 20 (1968), pp. 981–990. [English title: On the method of chord and on its modification for solving the nonlinear operator equations]

[5] G. Goldner, M. Balazs, Observații asupra diferențelor divizate și asupra metodei coardei, Revista de Analiza Numerica si Teoria Aproximatiei, 3 (1974) no. 1, pp. 19–30 (in Romanian).
[English title: Remarks on divided differences and method of chords]
article on journal website

[6] L.V. Kantorovici, G.P. Akilov, Functional Analysis, Editura Stiintifica si Enciclopedica, Bucuresti, 1986 (in Romanian).

[7] I. Pavaloiu, On the monotonicity of the sequences of approximations obtained by Steffensen’s method, Mathematica, 35(58) (1993) no. 1, pp. 71–76.
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[8] T. Yamamoto, A note on a posteriori error bound of Zabrejko and Nguen for Zincenko’s iteration, Numer. Funct. Anal. Optimiz., 9 (1987) nos. 9&10, pp. 987–994.
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[9] T. Yamamoto, Ball convergence theorems and error estimates for certain iterative methods for nonlinear equations, Japan Journal of Applied Mathematics, 7 (1990) no. 1, pp. 131–143.
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[10] X. Chen, T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optimiz., 10 (1989) 1&2, pp. 37–48.
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On-some-Steffensen-type-iterative-methods-for-a-class-of-nonlinear-equations

ON SOME STEFFENSEN-TYPE ITERATIVE METHODS FOR A CLASS OF NONLINEAR EQUATIONS

EMIL CĂTINAŞ(Cluj-Napoca)

1. INTRODUCTION

Consider a Banach space X X XXX and the equation
(1) F ( x ) + G ( x ) = 0 (1) F ( x ) + G ( x ) = 0 {:(1)F(x)+G(x)=0:}\begin{equation*} F(x)+G(x)=0 \tag{1} \end{equation*}(1)F(x)+G(x)=0
where F , G : X → X F , G : X → X F,G:X rarr XF, G: X \rightarrow XF,G:X→X are nonlinear operators, F F FFF being Fréchet differentiable and G G GGG being continuously but nondifferentiable. This is the case when we study an equation H ( x ) = 0 H ( x ) = 0 H(x)=0H(x)=0H(x)=0, with H : X → X H : X → X H:X rarr XH: X \rightarrow XH:X→X a nondifferentiable operator to which we cannot apply Newton's method. H H HHH is then split into two parts: a differentiable part and a nondifferentiable one.
Various methods have been proposed for solving these kind of problems.
In [8, 9, 10] are considered the Newton-like methods:
(2) x n + 1 = x n − F ′ ( x n ) − 1 ( F ( x n ) + G ( x n ) ) , n = 0 , 1 , … , x 0 ∈ X , (2) x n + 1 = x n − F ′ x n − 1 F x n + G x n , n = 0 , 1 , … , x 0 ∈ X , {:(2)x_(n+1)=x_(n)-F^(')(x_(n))^(-1)(F(x_(n))+G(x_(n)))","quad n=0","1","dots","x_(0)in X",":}\begin{equation*} x_{n+1}=x_{n}-F^{\prime}\left(x_{n}\right)^{-1}\left(F\left(x_{n}\right)+G\left(x_{n}\right)\right), \quad n=0,1, \ldots, x_{0} \in X, \tag{2} \end{equation*}(2)xn+1=xn−F′(xn)−1(F(xn)+G(xn)),n=0,1,…,x0∈X,
and, more generally,
(3) x n + 1 = x n − A ( x n ) − 1 ( F ( x n ) + G ( x n ) ) , n = 0 , 1 , … , x 0 ∈ X (3) x n + 1 = x n − A x n − 1 F x n + G x n , n = 0 , 1 , … , x 0 ∈ X {:(3)x_(n+1)=x_(n)-A(x_(n))^(-1)(F(x_(n))+G(x_(n)))","quad n=0","1","dots","x_(0)in X:}\begin{equation*} x_{n+1}=x_{n}-A\left(x_{n}\right)^{-1}\left(F\left(x_{n}\right)+G\left(x_{n}\right)\right), \quad n=0,1, \ldots, x_{0} \in X \tag{3} \end{equation*}(3)xn+1=xn−A(xn)−1(F(xn)+G(xn)),n=0,1,…,x0∈X
where A A AAA is a linear operator approximating F ′ F ′ F^(')F^{\prime}F′.
In [1] is studied the secant-type method
(4) x n + 1 = x n − [ x n − 1 , x n ; F ] − 1 ( F ( x n ) + G ( x n ) ) , n = 1 , 2 , … , x 0 , x 1 ∈ X (4) x n + 1 = x n − x n − 1 , x n ; F − 1 F x n + G x n , n = 1 , 2 , … , x 0 , x 1 ∈ X {:(4)x_(n+1)=x_(n)-[x_(n-1),x_(n);F]^(-1)(F(x_(n))+G(x_(n)))","quad n=1","2","dots","x_(0)","x_(1)in X:}\begin{equation*} x_{n+1}=x_{n}-\left[x_{n-1}, x_{n} ; F\right]^{-1}\left(F\left(x_{n}\right)+G\left(x_{n}\right)\right), \quad n=1,2, \ldots, x_{0}, x_{1} \in X \tag{4} \end{equation*}(4)xn+1=xn−[xn−1,xn;F]−1(F(xn)+G(xn)),n=1,2,…,x0,x1∈X
[ x , y ; F ] [ x , y ; F ] [x,y;F][x, y ; F][x,y;F] denoting the first order divided difference of F F FFF at the points x , y x , y x,yx, yx,y. The convergence order of these sequences is linear (as it can be also seen in the numerical example).
In [3] is considered a combination of Newton's method and the secant method:
(5) x n + 1 = x n − ( F ′ ( x n ) + [ x n − 1 , x n ; G ] ) − 1 ( F ( x n ) + G ( x n ) ) , n = 1 , 2 , … , x 0 , x 1 ∈ X , (5) x n + 1 = x n − F ′ x n + x n − 1 , x n ; G − 1 F x n + G x n , n = 1 , 2 , … , x 0 , x 1 ∈ X , {:(5)x_(n+1)=x_(n)-(F^(')(x_(n))+[x_(n-1),x_(n);G])^(-1)(F(x_(n))+G(x_(n)))","quad n=1","2","dots","x_(0)","x_(1)in X",":}\begin{equation*} x_{n+1}=x_{n}-\left(F^{\prime}\left(x_{n}\right)+\left[x_{n-1}, x_{n} ; G\right]\right)^{-1}\left(F\left(x_{n}\right)+G\left(x_{n}\right)\right), \quad n=1,2, \ldots, x_{0}, x_{1} \in X, \tag{5} \end{equation*}(5)xn+1=xn−(F′(xn)+[xn−1,xn;G])−1(F(xn)+G(xn)),n=1,2,…,x0,x1∈X,
having the convergence order 1 + 5 2 ≈ 1.618 1 + 5 2 ≈ 1.618 (1+sqrt5)/(2)~~1.618\frac{1+\sqrt{5}}{2} \approx 1.6181+52≈1.618, i.e., the convergence order of the secant method.
In the present paper we propose a method based on Steffensen's method and Newton's method, having quadratic convergence:
x n + 1 = x n − ( F ′ ( x n ) + [ x n , φ ( x n ) ; G ] ) − 1 ( F ( x n ) + G ( x n ) ) , n = 0 , 1 , … , x 0 ∈ X x n + 1 = x n − F ′ x n + x n , φ x n ; G − 1 F x n + G x n , n = 0 , 1 , … , x 0 ∈ X x_(n+1)=x_(n)-(F^(')(x_(n))+[x_(n),varphi(x_(n));G])^(-1)(F(x_(n))+G(x_(n))),quad n=0,1,dots,x_(0)in Xx_{n+1}=x_{n}-\left(F^{\prime}\left(x_{n}\right)+\left[x_{n}, \varphi\left(x_{n}\right) ; G\right]\right)^{-1}\left(F\left(x_{n}\right)+G\left(x_{n}\right)\right), \quad n=0,1, \ldots, x_{0} \in Xxn+1=xn−(F′(xn)+[xn,φ(xn);G])−1(F(xn)+G(xn)),n=0,1,…,x0∈X, where φ : X → X φ : X → X varphi:X rarr X\varphi: X \rightarrow Xφ:X→X,
φ ( x ) = x − λ ( F ( x ) + G ( x ) ) , φ ( x ) = x − λ ( F ( x ) + G ( x ) ) , varphi(x)=x-lambda(F(x)+G(x)),\varphi(x)=x-\lambda(F(x)+G(x)),φ(x)=x−λ(F(x)+G(x)),
λ λ lambda\lambdaλ being a fixed positive number.

2. THE CONVERGENCE OF THE METHOD

We shall use, as in [4, 5] the known definitions for the divided differences of an operator:
Definition 1. An operator [ x 0 , y 0 ; G ] x 0 , y 0 ; G [x_(0),y_(0);G]\left[x_{0}, y_{0} ; G\right][x0,y0;G] belonging to the space L ( X , X ) L ( X , X ) L(X,X)\mathcal{L}(X, X)L(X,X) (the Banach space of the linear and bounded operators from X X XXX to X X XXX ) is called the first order divided difference of the operator G : X → X G : X → X G:X rarr XG: X \rightarrow XG:X→X at the points x 0 x 0 x_(0)x_{0}x0, y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X if the following properties hold:
a) [ x 0 , y 0 ; G ] ( y 0 − x 0 ) = G ( y 0 ) − G ( x 0 ) x 0 , y 0 ; G y 0 − x 0 = G y 0 − G x 0 [x_(0),y_(0);G](y_(0)-x_(0))=G(y_(0))-G(x_(0))\left[x_{0}, y_{0} ; G\right]\left(y_{0}-x_{0}\right)=G\left(y_{0}\right)-G\left(x_{0}\right)[x0,y0;G](y0−x0)=G(y0)−G(x0), for x 0 ≠ y 0 x 0 ≠ y 0 x_(0)!=y_(0)x_{0} \neq y_{0}x0≠y0;
b) if G G GGG is Fréchet differentiable at x 0 x 0 x_(0)x_{0}x0, then
[ x 0 , x 0 ; G ] = G ′ ( x 0 ) x 0 , x 0 ; G = G ′ x 0 [x_(0),x_(0);G]=G^(')(x_(0))\left[x_{0}, x_{0} ; G\right]=G^{\prime}\left(x_{0}\right)[x0,x0;G]=G′(x0)
Definition 2. An operator belonging to the space L ( X , L ( X , X ) ) L ( X , L ( X , X ) ) L(X,L(X,X))\mathcal{L}(X, \mathcal{L}(X, X))L(X,L(X,X)), denoted by [ x 0 , y 0 , z 0 ; G ] x 0 , y 0 , z 0 ; G [x_(0),y_(0),z_(0);G]\left[x_{0}, y_{0}, z_{0} ; G\right][x0,y0,z0;G], is called the second-order divided difference of the operator G : X → X G : X → X G:X rarr XG: X \rightarrow XG:X→X at the points x 0 , y 0 , z 0 ∈ X x 0 , y 0 , z 0 ∈ X x_(0),y_(0),z_(0)in Xx_{0}, y_{0}, z_{0} \in Xx0,y0,z0∈X if the following properties hold:
a) [ x 0 , y 0 , z 0 ; G ] ( z 0 − x 0 ) = [ y 0 , z 0 ; G ] − [ x 0 , y 0 ; G ] x 0 , y 0 , z 0 ; G z 0 − x 0 = y 0 , z 0 ; G − x 0 , y 0 ; G [x_(0),y_(0),z_(0);G](z_(0)-x_(0))=[y_(0),z_(0);G]-[x_(0),y_(0);G]\left[x_{0}, y_{0}, z_{0} ; G\right]\left(z_{0}-x_{0}\right)=\left[y_{0}, z_{0} ; G\right]-\left[x_{0}, y_{0} ; G\right][x0,y0,z0;G](z0−x0)=[y0,z0;G]−[x0,y0;G], for the distinct points x 0 , y 0 , z 0 ∈ X x 0 , y 0 , z 0 ∈ X x_(0),y_(0),z_(0)in Xx_{0}, y_{0}, z_{0} \in Xx0,y0,z0∈X;
b) if G G GGG is two times Fréchet differentiable at x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, then
[ x 0 , x 0 , x 0 ; G ] = 1 2 G ′ ′ ( x 0 ) x 0 , x 0 , x 0 ; G = 1 2 G ′ ′ x 0 [x_(0),x_(0),x_(0);G]=(1)/(2)G^('')(x_(0))\left[x_{0}, x_{0}, x_{0} ; G\right]=\frac{1}{2} G^{\prime \prime}\left(x_{0}\right)[x0,x0,x0;G]=12G′′(x0)
We shall denote by B r ( x 0 ) = { x ∈ X ∣ ‖ x − x 0 ‖ < r } B r x 0 = x ∈ X ∣ x − x 0 < r B_(r)(x_(0))={x in X∣||x-x_(0)|| < r}B_{r}\left(x_{0}\right)=\left\{x \in X \mid\left\|x-x_{0}\right\|<r\right\}Br(x0)={x∈X∣‖x−x0‖<r} the open ball having the center at x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X and the radius r > 0 r > 0 r > 0r>0r>0.
Concerning the convergence of the iterative process (6) we shall prove the following theorem:
Theorem 3. If there exists the element x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, and the positive real numbers K , l , ε , M , r K , l , ε , M , r K,l,epsi,M,rK, l, \varepsilon, M, rK,l,ε,M,r such that:
i) G G GGG is continuous on B r ( x 0 ) B r x 0 B_(r)(x_(0))B_{r}\left(x_{0}\right)Br(x0);
ii) F F FFF is Fréchet differentiable on B r ( x 0 ) B r x 0 B_(r)(x_(0))B_{r}\left(x_{0}\right)Br(x0), with the Fréchet derivative satisfying the Lipschitz condition
‖ F ′ ( x ) − F ′ ( y ) ‖ ≤ l ‖ x − y ‖ , ∀ x , y ∈ B r ( x 0 ) ; F ′ ( x ) − F ′ ( y ) ≤ l ‖ x − y ‖ , ∀ x , y ∈ B r x 0 ; ||F^(')(x)-F^(')(y)|| <= l||x-y||,quad AA x,y inB_(r)(x_(0));\left\|F^{\prime}(x)-F^{\prime}(y)\right\| \leq l\|x-y\|, \quad \forall x, y \in B_{r}\left(x_{0}\right) ;‖F′(x)−F′(y)‖≤l‖x−y‖,∀x,y∈Br(x0);
iii) The second-order divided difference of G G GGG is uniformly bounded on B r ( x 0 ) B r x 0 B_(r)(x_(0))B_{r}\left(x_{0}\right)Br(x0) :
‖ [ x , y , z ; G ] ‖ ≤ K , ∀ x , y , z ∈ B r ( x 0 ) ; ‖ [ x , y , z ; G ] ‖ ≤ K , ∀ x , y , z ∈ B r x 0 ; ||[x,y,z;G]|| <= K,quad AA x,y,z inB_(r)(x_(0));\|[x, y, z ; G]\| \leq K, \quad \forall x, y, z \in B_{r}\left(x_{0}\right) ;‖[x,y,z;G]‖≤K,∀x,y,z∈Br(x0);
iv) The operators F ′ ( x ) + [ x , φ ( x ) ; G ] F ′ ( x ) + [ x , φ ( x ) ; G ] F^(')(x)+[x,varphi(x);G]F^{\prime}(x)+[x, \varphi(x) ; G]F′(x)+[x,φ(x);G] are invertible, with the inverses uniformly bounded: ∀ x ∈ B r ( x 0 ) ∀ x ∈ B r x 0 AA x inB_(r)(x_(0))\forall x \in B_{r}\left(x_{0}\right)∀x∈Br(x0) with φ ( x ) ∈ B r ( x 0 ) φ ( x ) ∈ B r x 0 varphi(x)inB_(r)(x_(0))\varphi(x) \in B_{r}\left(x_{0}\right)φ(x)∈Br(x0) there exists ( F ′ ( x ) + [ x , φ ( x ) ; G ] ) − 1 F ′ ( x ) + [ x , φ ( x ) ; G ] − 1 (F^(')(x)+[x,varphi(x);G])^(-1)\left(F^{\prime}(x)+[x, \varphi(x) ; G]\right)^{-1}(F′(x)+[x,φ(x);G])−1 and
‖ ( F ′ ( x ) + [ x , φ ( x ) ; G ] ) − 1 ‖ ≤ M F ′ ( x ) + [ x , φ ( x ) ; G ] − 1 ≤ M ||(F^(')(x)+[x,varphi(x);G])^(-1)|| <= M\left\|\left(F^{\prime}(x)+[x, \varphi(x) ; G]\right)^{-1}\right\| \leq M‖(F′(x)+[x,φ(x);G])−1‖≤M
v) λ λ lambda\lambdaλ is chosen such that λ ≤ M λ ≤ M lambda <= M\lambda \leq Mλ≤M;
vi) q := M 2 ε ( l 2 + 2 K ) < 1 q := M 2 ε l 2 + 2 K < 1 q:=M^(2)epsi((l)/(2)+2K) < 1q:=M^{2} \varepsilon\left(\frac{l}{2}+2 K\right)<1q:=M2ε(l2+2K)<1 and the radius is given by
r := 1 M ( l 2 + 2 K ) ∑ k = 0 ∞ q 2 k r := 1 M l 2 + 2 K ∑ k = 0 ∞   q 2 k r:=(1)/(M((l)/(2)+2K))sum_(k=0)^(oo)q^(2^(k))r:=\frac{1}{M\left(\frac{l}{2}+2 K\right)} \sum_{k=0}^{\infty} q^{2^{k}}r:=1M(l2+2K)∑k=0∞q2k
then
j) The sequence ( x n ) n ≥ 0 x n n ≥ 0 (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0}(xn)n≥0 given by (6) is well defined and ( x n ) n ≥ 0 ⊂ B r ( x 0 ) ; x n n ≥ 0 ⊂ B r x 0 ; (x_(n))_(n >= 0)subB_(r)(x_(0));\left(x_{n}\right)_{n \geq 0} \subset B_{r}\left(x_{0}\right) ;(xn)n≥0⊂Br(x0);
jj) ( x n ) n ≥ 0 x n n ≥ 0 (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0}(xn)n≥0 converges to some x ∗ ∈ B r ( x 0 ) ― x ∗ ∈ B r x 0 ¯ x^(**)in bar(B_(r)(x_(0)))x^{*} \in \overline{B_{r}\left(x_{0}\right)}x∗∈Br(x0)―, which is a solution of equation (1);
jjj) The following estimation holds:
‖ x ∗ − x n ‖ ≤ q 2 n M ( l 2 + 2 K ) ( 1 − q 2 n ) x ∗ − x n ≤ q 2 n M l 2 + 2 K 1 − q 2 n ||x^(**)-x_(n)|| <= (q^(2^(n)))/(M((l)/(2)+2K)(1-q^(2^(n))))\left\|x^{*}-x_{n}\right\| \leq \frac{q^{2^{n}}}{M\left(\frac{l}{2}+2 K\right)\left(1-q^{2^{n}}\right)}‖x∗−xn‖≤q2nM(l2+2K)(1−q2n)
Proof. From the hypothesis i i iii ) concerning F F FFF it is known [6] that we get
(7) ‖ F ( y ) − F ( x ) − F ′ ( x ) ( y − x ) ‖ ≤ l 2 ‖ y − x ‖ 2 . (7) F ( y ) − F ( x ) − F ′ ( x ) ( y − x ) ≤ l 2 ‖ y − x ‖ 2 . {:(7)||F(y)-F(x)-F^(')(x)(y-x)|| <= (l)/(2)||y-x||^(2).:}\begin{equation*} \left\|F(y)-F(x)-F^{\prime}(x)(y-x)\right\| \leq \frac{l}{2}\|y-x\|^{2} . \tag{7} \end{equation*}(7)‖F(y)−F(x)−F′(x)(y−x)‖≤l2‖y−x‖2.
From the definitions of the divided differences we obtain
(8) G ( y ) − G ( x ) − [ x , φ ( x ) ; G ] ( y − x ) = [ x , φ ( x ) , y ; G ] ( y − φ ( x ) ) ( y − x ) (8) G ( y ) − G ( x ) − [ x , φ ( x ) ; G ] ( y − x ) = [ x , φ ( x ) , y ; G ] ( y − φ ( x ) ) ( y − x ) {:(8)G(y)-G(x)-[x","varphi(x);G](y-x)=[x","varphi(x)","y;G](y-varphi(x))(y-x):}\begin{equation*} G(y)-G(x)-[x, \varphi(x) ; G](y-x)=[x, \varphi(x), y ; G](y-\varphi(x))(y-x) \tag{8} \end{equation*}(8)G(y)−G(x)−[x,φ(x);G](y−x)=[x,φ(x),y;G](y−φ(x))(y−x)
Indeed,
[ x , φ ( x ) , y ; G ] ( y − φ ( x ) ) ( y − x ) = = [ φ ( x ) , y ; G ] ( y − φ ( x ) ) − [ x , φ ( x ) ; G ] ( y − φ ( x ) ) = G ( y ) − G ( φ ( x ) ) + [ x , φ ( x ) ; G ] ( φ ( x ) − x ) − [ x , φ ( x ) ; G ] ( y − x ) = G ( y ) − G ( φ ( x ) ) + G ( φ ( x ) ) − G ( x ) − [ x , φ ( x ) ; G ] ( y − x ) = G ( y ) − G ( x ) − [ x , φ ( x ) ; G ] ( y − x ) [ x , φ ( x ) , y ; G ] ( y − φ ( x ) ) ( y − x ) = = [ φ ( x ) , y ; G ] ( y − φ ( x ) ) − [ x , φ ( x ) ; G ] ( y − φ ( x ) ) = G ( y ) − G ( φ ( x ) ) + [ x , φ ( x ) ; G ] ( φ ( x ) − x ) − [ x , φ ( x ) ; G ] ( y − x ) = G ( y ) − G ( φ ( x ) ) + G ( φ ( x ) ) − G ( x ) − [ x , φ ( x ) ; G ] ( y − x ) = G ( y ) − G ( x ) − [ x , φ ( x ) ; G ] ( y − x ) {:[[x","varphi(x)","y;G](y-varphi(x))(y-x)=],[=[varphi(x)","y;G](y-varphi(x))-[x","varphi(x);G](y-varphi(x))],[=G(y)-G(varphi(x))+[x","varphi(x);G](varphi(x)-x)-[x","varphi(x);G](y-x)],[=G(y)-G(varphi(x))+G(varphi(x))-G(x)-[x","varphi(x);G](y-x)],[=G(y)-G(x)-[x","varphi(x);G](y-x)]:}\begin{aligned} & {[x, \varphi(x), y ; G](y-\varphi(x))(y-x)=} \\ & =[\varphi(x), y ; G](y-\varphi(x))-[x, \varphi(x) ; G](y-\varphi(x)) \\ & =G(y)-G(\varphi(x))+[x, \varphi(x) ; G](\varphi(x)-x)-[x, \varphi(x) ; G](y-x) \\ & =G(y)-G(\varphi(x))+G(\varphi(x))-G(x)-[x, \varphi(x) ; G](y-x) \\ & =G(y)-G(x)-[x, \varphi(x) ; G](y-x) \end{aligned}[x,φ(x),y;G](y−φ(x))(y−x)==[φ(x),y;G](y−φ(x))−[x,φ(x);G](y−φ(x))=G(y)−G(φ(x))+[x,φ(x);G](φ(x)−x)−[x,φ(x);G](y−x)=G(y)−G(φ(x))+G(φ(x))−G(x)−[x,φ(x);G](y−x)=G(y)−G(x)−[x,φ(x);G](y−x)
We shall prove by induction that
(9) x k , φ ( x k ) ∈ B r ( x 0 ) , k ∈ N ‖ F ( x k ) + G ( x k ) ‖ ≤ M − 2 ( l 2 + 2 K ) − 1 q 2 k , k ∈ N (9) x k , φ x k ∈ B r x 0 , k ∈ N F x k + G x k ≤ M − 2 l 2 + 2 K − 1 q 2 k , k ∈ N {:[(9)x_(k)","varphi(x_(k)) inB_(r)(x_(0))","quad k inN],[||F(x_(k))+G(x_(k))|| <= M^(-2)((l)/(2)+2K)^(-1)q^(2^(k))","quad k inN]:}\begin{align*} x_{k}, \varphi\left(x_{k}\right) & \in B_{r}\left(x_{0}\right), \quad k \in \mathbb{N} \tag{9}\\ \left\|F\left(x_{k}\right)+G\left(x_{k}\right)\right\| & \leq M^{-2}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{k}}, \quad k \in \mathbb{N} \end{align*}(9)xk,φ(xk)∈Br(x0),k∈N‖F(xk)+G(xk)‖≤M−2(l2+2K)−1q2k,k∈N
From the above inequality it can be easily deduced by (6) that ∃ x k + 1 ∃ x k + 1 EEx_(k+1)\exists x_{k+1}∃xk+1 and
(10) ‖ x k + 1 − x k ‖ = ‖ ( F ′ ( x k ) + [ x k , φ ( x k ) ; G ] ) − 1 ( F ( x k ) + G ( x k ) ) ‖ ≤ M − 1 ( l 2 + 2 K ) − 1 q 2 k (10) x k + 1 − x k = F ′ x k + x k , φ x k ; G − 1 F x k + G x k ≤ M − 1 l 2 + 2 K − 1 q 2 k {:[(10)||x_(k+1)-x_(k)||=||(F^(')(x_(k))+[x_(k),varphi(x_(k));G])^(-1)(F(x_(k))+G(x_(k)))||],[ <= M^(-1)((l)/(2)+2K)^(-1)q^(2^(k))]:}\begin{align*} \left\|x_{k+1}-x_{k}\right\| & =\left\|\left(F^{\prime}\left(x_{k}\right)+\left[x_{k}, \varphi\left(x_{k}\right) ; G\right]\right)^{-1}\left(F\left(x_{k}\right)+G\left(x_{k}\right)\right)\right\| \tag{10}\\ & \leq M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{k}} \end{align*}(10)‖xk+1−xk‖=‖(F′(xk)+[xk,φ(xk);G])−1(F(xk)+G(xk))‖≤M−1(l2+2K)−1q2k
For k = 0 k = 0 k=0k=0k=0 we have:
x 0 ∈ B r ( x 0 ) ; ‖ x 0 − φ ( x 0 ) ‖ = ‖ x 0 − x 0 + λ ( F ( x 0 ) + G ( x 0 ) ) ‖ ≤ λ ε ≤ M ε < r , x 0 ∈ B r x 0 ; x 0 − φ x 0 = x 0 − x 0 + λ F x 0 + G x 0 ≤ λ ε ≤ M ε < r , {:[x_(0) inB_(r)(x_(0));],[||x_(0)-varphi(x_(0))||=||x_(0)-x_(0)+lambda(F(x_(0))+G(x_(0)))|| <= lambda epsi <= M epsi < r","]:}\begin{aligned} x_{0} & \in B_{r}\left(x_{0}\right) ; \\ \left\|x_{0}-\varphi\left(x_{0}\right)\right\| & =\left\|x_{0}-x_{0}+\lambda\left(F\left(x_{0}\right)+G\left(x_{0}\right)\right)\right\| \leq \lambda \varepsilon \leq M \varepsilon<r, \end{aligned}x0∈Br(x0);‖x0−φ(x0)‖=‖x0−x0+λ(F(x0)+G(x0))‖≤λε≤Mε<r,
which imply that
φ ( x 0 ) ∈ B r ( x 0 ) ‖ F ( x 0 ) + G ( x 0 ) ‖ ≤ ε = M − 2 ( l 2 + 2 K ) − 1 q 2 0 . φ x 0 ∈ B r x 0 F x 0 + G x 0 ≤ ε = M − 2 l 2 + 2 K − 1 q 2 0 . {:[varphi(x_(0)) inB_(r)(x_(0))],[||F(x_(0))+G(x_(0))|| <= epsi=M^(-2)((l)/(2)+2K)^(-1)q^(2^(0))]:}.\begin{aligned} \varphi\left(x_{0}\right) & \in B_{r}\left(x_{0}\right) \\ \left\|F\left(x_{0}\right)+G\left(x_{0}\right)\right\| & \leq \varepsilon=M^{-2}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{0}} \end{aligned} .φ(x0)∈Br(x0)‖F(x0)+G(x0)‖≤ε=M−2(l2+2K)−1q20.
Suppose now that 9 is true for k = 1 , n − 1 ― k = 1 , n − 1 ¯ k= bar(1,n-1)k=\overline{1, n-1}k=1,n−1―. By it follows that ∃ x n ∃ x n EEx_(n)\exists x_{n}∃xn, and we have that x n ∈ B r ( x 0 ) x n ∈ B r x 0 x_(n)inB_(r)(x_(0))x_{n} \in B_{r}\left(x_{0}\right)xn∈Br(x0). Indeed,
‖ x n − x 0 ‖ ≤ ‖ x 1 − x 0 ‖ + ⋯ + ‖ x n − x n − 1 ‖ ≤ M − 1 ( l 2 + 2 K ) − 1 ∑ k = 0 n − 1 q 2 k < r x n − x 0 ≤ x 1 − x 0 + ⋯ + x n − x n − 1 ≤ M − 1 l 2 + 2 K − 1 ∑ k = 0 n − 1   q 2 k < r ||x_(n)-x_(0)|| <= ||x_(1)-x_(0)||+cdots+||x_(n)-x_(n-1)|| <= M^(-1)((l)/(2)+2K)^(-1)sum_(k=0)^(n-1)q^(2^(k)) < r\left\|x_{n}-x_{0}\right\| \leq\left\|x_{1}-x_{0}\right\|+\cdots+\left\|x_{n}-x_{n-1}\right\| \leq M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} \sum_{k=0}^{n-1} q^{2^{k}}<r‖xn−x0‖≤‖x1−x0‖+⋯+‖xn−xn−1‖≤M−1(l2+2K)−1∑k=0n−1q2k<r
Then, using (6), (7), (8) and (9),
‖ F ( x n ) + G ( x n ) ‖ ≤ ‖ F ( x n ) − F ( x n − 1 ) − F ′ ( x n − 1 ) ( x n − x n − 1 ) ‖ + ‖ G ( x n ) − G ( x n − 1 ) − [ x n − 1 , φ ( x n − 1 ) ; G ] ( x n − x n − 1 ) ‖ ≤ l 2 ‖ x n − x n − 1 ‖ 2 + K ‖ x n − x n − 1 ‖ ⋅ ‖ x n − φ ( x n − 1 ) ‖ ≤ F x n + G x n ≤ F x n − F x n − 1 − F ′ x n − 1 x n − x n − 1 + G x n − G x n − 1 − x n − 1 , φ x n − 1 ; G x n − x n − 1 ≤ l 2 x n − x n − 1 2 + K x n − x n − 1 ⋅ x n − φ x n − 1 ≤ {:[||F(x_(n))+G(x_(n))|| <= ||F(x_(n))-F(x_(n-1))-F^(')(x_(n-1))(x_(n)-x_(n-1))||],[+||G(x_(n))-G(x_(n-1))-[x_(n-1),varphi(x_(n-1));G](x_(n)-x_(n-1))||],[ <= (l)/(2)||x_(n)-x_(n-1)||^(2)+K||x_(n)-x_(n-1)||*||x_(n)-varphi(x_(n-1))|| <= ]:}\begin{aligned} \left\|F\left(x_{n}\right)+G\left(x_{n}\right)\right\| \leq & \left\|F\left(x_{n}\right)-F\left(x_{n-1}\right)-F^{\prime}\left(x_{n-1}\right)\left(x_{n}-x_{n-1}\right)\right\| \\ & +\left\|G\left(x_{n}\right)-G\left(x_{n-1}\right)-\left[x_{n-1}, \varphi\left(x_{n-1}\right) ; G\right]\left(x_{n}-x_{n-1}\right)\right\| \\ \leq & \frac{l}{2}\left\|x_{n}-x_{n-1}\right\|^{2}+K\left\|x_{n}-x_{n-1}\right\| \cdot\left\|x_{n}-\varphi\left(x_{n-1}\right)\right\| \leq \end{aligned}‖F(xn)+G(xn)‖≤‖F(xn)−F(xn−1)−F′(xn−1)(xn−xn−1)‖+‖G(xn)−G(xn−1)−[xn−1,φ(xn−1);G](xn−xn−1)‖≤l2‖xn−xn−1‖2+K‖xn−xn−1‖⋅‖xn−φ(xn−1)‖≤
≤ l 2 M − 2 ( l 2 + 2 K ) − 2 q 2 n + K M − 1 ( l 2 + 2 K ) − 1 q 2 n − 1 ( M − 1 ( l 2 + 2 K ) − 1 q 2 n − 1 + λ M 2 ( l 2 + 2 K ) − 1 q 2 n − 1 ) ≤ M − 2 ( l 2 + 2 K ) − 1 q 2 n ≤ l 2 M − 2 l 2 + 2 K − 2 q 2 n + K M − 1 l 2 + 2 K − 1 q 2 n − 1 M − 1 l 2 + 2 K − 1 q 2 n − 1 + λ M 2 l 2 + 2 K − 1 q 2 n − 1 ≤ M − 2 l 2 + 2 K − 1 q 2 n {:[ <= (l)/(2)M^(-2)((l)/(2)+2K)^(-2)q^(2^(n))],[+KM^(-1)((l)/(2)+2K)^(-1)q^(2^(n-1))(M^(-1)((l)/(2)+2K)^(-1)q^(2^(n-1))+lambdaM^(2)((l)/(2)+2K)^(-1)q^(2^(n-1)))],[ <= M^(-2)((l)/(2)+2K)^(-1)q^(2^(n))]:}\begin{aligned} \leq & \frac{l}{2} M^{-2}\left(\frac{l}{2}+2 K\right)^{-2} q^{2^{n}} \\ & +K M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n-1}}\left(M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n-1}}+\lambda M^{2}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n-1}}\right) \\ \leq & M^{-2}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n}} \end{aligned}≤l2M−2(l2+2K)−2q2n+KM−1(l2+2K)−1q2n−1(M−1(l2+2K)−1q2n−1+λM2(l2+2K)−1q2n−1)≤M−2(l2+2K)−1q2n
It remains to show that φ ( x n ) ∈ B r ( x 0 ) φ x n ∈ B r x 0 varphi(x_(n))inB_(r)(x_(0))\varphi\left(x_{n}\right) \in B_{r}\left(x_{0}\right)φ(xn)∈Br(x0) :
‖ x 0 − φ ( x n ) ‖ ≤ ‖ x 0 − x n ‖ + λ ‖ F ( x n ) + G ( x n ) ‖ ≤ M − 1 ( l 2 + 2 K ) − 1 ∑ k = 0 n − 1 q 2 k + M − 1 ( l 2 + 2 K ) − 1 q 2 n < r . x 0 − φ x n ≤ x 0 − x n + λ F x n + G x n ≤ M − 1 l 2 + 2 K − 1 ∑ k = 0 n − 1   q 2 k + M − 1 l 2 + 2 K − 1 q 2 n < r . {:[||x_(0)-varphi(x_(n))|| <= ||x_(0)-x_(n)||+lambda||F(x_(n))+G(x_(n))||],[ <= M^(-1)((l)/(2)+2K)^(-1)sum_(k=0)^(n-1)q^(2^(k))+M^(-1)((l)/(2)+2K)^(-1)q^(2^(n)) < r.]:}\begin{aligned} \left\|x_{0}-\varphi\left(x_{n}\right)\right\| & \leq\left\|x_{0}-x_{n}\right\|+\lambda\left\|F\left(x_{n}\right)+G\left(x_{n}\right)\right\| \\ & \leq M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} \sum_{k=0}^{n-1} q^{2^{k}}+M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n}}<r . \end{aligned}‖x0−φ(xn)‖≤‖x0−xn‖+λ‖F(xn)+G(xn)‖≤M−1(l2+2K)−1∑k=0n−1q2k+M−1(l2+2K)−1q2n<r.
The induction (9) is proved.
Now we prove that the sequence ( x n ) n ≥ 0 x n n ≥ 0 (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0}(xn)n≥0 is a Cauchy sequence, hence it converges to some element x ∗ ∈ B r ( x 0 ) ― x ∗ ∈ B r x 0 ¯ x^(**)in bar(B_(r)(x_(0)))x^{*} \in \overline{B_{r}\left(x_{0}\right)}x∗∈Br(x0)― :
‖ x n + p − x n ‖ ≤ ‖ x n + p − x n + p − 1 ‖ + ⋯ + ‖ x n + 1 − x n ‖ ≤ M − 1 ( l 2 + 2 K ) − 1 ∑ k = n n + p − 1 q 2 k = M − 1 ( l 2 + 2 K ) − 1 q 2 n ∑ k = n n + p − 1 q 2 k − 2 n ≤ M − 1 ( l 2 + 2 K ) − 1 q 2 n ( 1 + q 2 ⋅ 2 n − 2 n + q 4 ⋅ 2 n − 2 n + … ) ≤ M − 1 ( l 2 + 2 K ) − 1 q 2 n 1 − q 2 n x n + p − x n ≤ x n + p − x n + p − 1 + ⋯ + x n + 1 − x n ≤ M − 1 l 2 + 2 K − 1 ∑ k = n n + p − 1   q 2 k = M − 1 l 2 + 2 K − 1 q 2 n ∑ k = n n + p − 1   q 2 k − 2 n ≤ M − 1 l 2 + 2 K − 1 q 2 n 1 + q 2 ⋅ 2 n − 2 n + q 4 ⋅ 2 n − 2 n + … ≤ M − 1 l 2 + 2 K − 1 q 2 n 1 − q 2 n {:[||x_(n+p)-x_(n)|| <= ||x_(n+p)-x_(n+p-1)||+cdots+||x_(n+1)-x_(n)||],[ <= M^(-1)((l)/(2)+2K)^(-1)sum_(k=n)^(n+p-1)q^(2^(k))],[=M^(-1)((l)/(2)+2K)^(-1)q^(2^(n))sum_(k=n)^(n+p-1)q^(2^(k)-2^(n))],[ <= M^(-1)((l)/(2)+2K)^(-1)q^(2^(n))(1+q^(2*2^(n)-2^(n))+q^(4*2^(n)-2^(n))+dots)],[ <= M^(-1)((l)/(2)+2K)^(-1)(q^(2^(n)))/(1-q^(2^(n)))]:}\begin{aligned} \left\|x_{n+p}-x_{n}\right\| & \leq\left\|x_{n+p}-x_{n+p-1}\right\|+\cdots+\left\|x_{n+1}-x_{n}\right\| \\ & \leq M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} \sum_{k=n}^{n+p-1} q^{2^{k}} \\ & =M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n}} \sum_{k=n}^{n+p-1} q^{2^{k}-2^{n}} \\ & \leq M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} q^{2^{n}}\left(1+q^{2 \cdot 2^{n}-2^{n}}+q^{4 \cdot 2^{n}-2^{n}}+\ldots\right) \\ & \leq M^{-1}\left(\frac{l}{2}+2 K\right)^{-1} \frac{q^{2^{n}}}{1-q^{2^{n}}} \end{aligned}‖xn+p−xn‖≤‖xn+p−xn+p−1‖+⋯+‖xn+1−xn‖≤M−1(l2+2K)−1∑k=nn+p−1q2k=M−1(l2+2K)−1q2n∑k=nn+p−1q2k−2n≤M−1(l2+2K)−1q2n(1+q2⋅2n−2n+q4⋅2n−2n+…)≤M−1(l2+2K)−1q2n1−q2n
Passing to limit for n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞ in relation (6) and taking into account the hypotheses concerning F F FFF and G G GGG, we get that x ∗ x ∗ x^(**)x^{*}x∗ is a solution of (1).
The estimation j j j j j j jjjj j jjjj ) is obtained from the above relation, for p → ∞ p → ∞ p rarr oop \rightarrow \inftyp→∞.

3. NUMERICAL EXAMPLES

Given the system
{ 3 x 2 y + y 2 − 1 + | x − 1 | = 0 x 4 + x y 3 − 1 + | y | = 0 3 x 2 y + y 2 − 1 + | x − 1 | = 0 x 4 + x y 3 − 1 + | y | = 0 {[3x^(2)y+y^(2)-1+|x-1|=0],[x^(4)+xy^(3)-1+|y|=0]:}\left\{\begin{array}{c} 3 x^{2} y+y^{2}-1+|x-1|=0 \\ x^{4}+x y^{3}-1+|y|=0 \end{array}\right.{3x2y+y2−1+|x−1|=0x4+xy3−1+|y|=0
we shall consider X = ( R 2 , ‖ ⋅ ‖ ∞ ) X = R 2 , ‖ ⋅ ‖ ∞ X=(R^(2),||*||_(oo))X=\left(\mathbb{R}^{2},\|\cdot\|_{\infty}\right)X=(R2,‖⋅‖∞) and F , G : X → X , F = ( f 1 , f 2 ) , G = ( g 1 , g 2 ) F , G : X → X , F = f 1 , f 2 , G = g 1 , g 2 F,G:X rarr X,F=(f_(1),f_(2)),G=(g_(1),g_(2))F, G: X \rightarrow X, F=\left(f_{1}, f_{2}\right), G=\left(g_{1}, g_{2}\right)F,G:X→X,F=(f1,f2),G=(g1,g2), with f 1 ( x , y ) = 3 x 2 y + y 2 − 1 , f 2 ( x , y ) = x 4 + x y 3 − 1 , g 1 ( x , y ) = | x − 1 | f 1 ( x , y ) = 3 x 2 y + y 2 − 1 , f 2 ( x , y ) = x 4 + x y 3 − 1 , g 1 ( x , y ) = | x − 1 | f_(1)(x,y)=3x^(2)y+y^(2)-1,f_(2)(x,y)=x^(4)+xy^(3)-1,g_(1)(x,y)=|x-1|f_{1}(x, y)=3 x^{2} y+y^{2}-1, f_{2}(x, y)=x^{4}+x y^{3}-1, g_{1}(x, y)=|x-1|f1(x,y)=3x2y+y2−1,f2(x,y)=x4+xy3−1,g1(x,y)=|x−1|, g 2 ( x , y ) = | y | g 2 ( x , y ) = | y | g_(2)(x,y)=|y|g_{2}(x, y)=|y|g2(x,y)=|y|.
We shall take [ x , y ; G ] ∈ M 2 ( R ) [ x , y ; G ] ∈ M 2 ( R ) [x,y;G]inM_(2)(R)[x, y ; G] \in \mathbb{M}_{2}(\mathbb{R})[x,y;G]∈M2(R) given by
[ x , y ; G ] ( i , 1 ) = g i ( y 1 , y 2 ) − g ( x 1 , y 2 ) y 1 − x 1 , [ x , y ; G ] ( i , 2 ) = g i ( x 1 , y 2 ) − g i ( x 1 , x 2 ) y 2 − x 2 [ x , y ; G ] ( i , 1 ) = g i y 1 , y 2 − g x 1 , y 2 y 1 − x 1 , [ x , y ; G ] ( i , 2 ) = g i x 1 , y 2 − g i x 1 , x 2 y 2 − x 2 [x,y;G](i,1)=(g_(i)(y^(1),y^(2))-g(x^(1),y^(2)))/(y^(1)-x^(1)),quad[x,y;G](i,2)=(g_(i)(x^(1),y^(2))-g_(i)(x^(1),x^(2)))/(y^(2)-x^(2))[x, y ; G](i, 1)=\frac{g_{i}\left(y^{1}, y^{2}\right)-g\left(x^{1}, y^{2}\right)}{y^{1}-x^{1}}, \quad[x, y ; G](i, 2)=\frac{g_{i}\left(x^{1}, y^{2}\right)-g_{i}\left(x^{1}, x^{2}\right)}{y^{2}-x^{2}}[x,y;G](i,1)=gi(y1,y2)−g(x1,y2)y1−x1,[x,y;G](i,2)=gi(x1,y2)−gi(x1,x2)y2−x2
i = 1 , 2 i = 1 , 2 i=1,2i=1,2i=1,2.
Using the method (2) with x 0 = ( 1 , 0 ) x 0 = ( 1 , 0 ) x_(0)=(1,0)x_{0}=(1,0)x0=(1,0), we obtain
n n nnn x n 1 x n 1 x_(n)^(1)x_{n}^{1}xn1 x n 2 x n 2 x_(n)^(2)x_{n}^{2}xn2 ‖ x n − x n − 1 ‖ x n − x n − 1 ||x_(n)-x_(n-1)||\left\|x_{n}-x_{n-1}\right\|‖xn−xn−1‖
0 1.00000000000000 ⋅ 10 + 0 1.00000000000000 ⋅ 10 + 0 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0}1.00000000000000⋅10+0 0.00000000000000 ⋅ 10 + 0 0.00000000000000 ⋅ 10 + 0 0.00000000000000*10^(+0)0.00000000000000 \cdot 10^{+0}0.00000000000000⋅10+0
1 1.00000000000000 ⋅ 10 + 0 1.00000000000000 ⋅ 10 + 0 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0}1.00000000000000⋅10+0 3.33333333333333 ⋅ 10 − 1 3.33333333333333 ⋅ 10 − 1 3.33333333333333*10^(-1)3.33333333333333 \cdot 10^{-1}3.33333333333333⋅10−1 3.33 ⋅ 10 − 01 3.33 ⋅ 10 − 01 3.33*10^(-01)3.33 \cdot 10^{-01}3.33⋅10−01
2 9.06550218340611 ⋅ 10 − 1 9.06550218340611 ⋅ 10 − 1 9.06550218340611*10^(-1)9.06550218340611 \cdot 10^{-1}9.06550218340611⋅10−1 3.54002911208151 ⋅ 10 − 1 3.54002911208151 ⋅ 10 − 1 3.54002911208151*10^(-1)3.54002911208151 \cdot 10^{-1}3.54002911208151⋅10−1 9.344 ⋅ 10 − 02 9.344 ⋅ 10 − 02 9.344*10^(-02)9.344 \cdot 10^{-02}9.344⋅10−02
3 8.85328400663412 ⋅ 10 − 1 8.85328400663412 ⋅ 10 − 1 8.85328400663412*10^(-1)8.85328400663412 \cdot 10^{-1}8.85328400663412⋅10−1 3.38027276361332 ⋅ 10 − 1 3.38027276361332 ⋅ 10 − 1 3.38027276361332*10^(-1)3.38027276361332 \cdot 10^{-1}3.38027276361332⋅10−1 2.122 ⋅ 10 − 02 2.122 ⋅ 10 − 02 2.122*10^(-02)2.122 \cdot 10^{-02}2.122⋅10−02
4 8.91329556832800 ⋅ 10 − 1 8.91329556832800 ⋅ 10 − 1 8.91329556832800*10^(-1)8.91329556832800 \cdot 10^{-1}8.91329556832800⋅10−1 3.26613976593566 ⋅ 10 − 1 3.26613976593566 ⋅ 10 − 1 3.26613976593566*10^(-1)3.26613976593566 \cdot 10^{-1}3.26613976593566⋅10−1 1.141 ⋅ 10 − 02 1.141 ⋅ 10 − 02 1.141*10^(-02)1.141 \cdot 10^{-02}1.141⋅10−02
39 8.94655373334687 ⋅ 10 − 1 8.94655373334687 ⋅ 10 − 1 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1}8.94655373334687⋅10−1 3.27826521746298 ⋅ 10 − 1 3.27826521746298 ⋅ 10 − 1 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1}3.27826521746298⋅10−1 5.149 ⋅ 10 − 19 5.149 ⋅ 10 − 19 5.149*10^(-19)5.149 \cdot 10^{-19}5.149⋅10−19
n x_(n)^(1) x_(n)^(2) ||x_(n)-x_(n-1)|| 0 1.00000000000000*10^(+0) 0.00000000000000*10^(+0) 1 1.00000000000000*10^(+0) 3.33333333333333*10^(-1) 3.33*10^(-01) 2 9.06550218340611*10^(-1) 3.54002911208151*10^(-1) 9.344*10^(-02) 3 8.85328400663412*10^(-1) 3.38027276361332*10^(-1) 2.122*10^(-02) 4 8.91329556832800*10^(-1) 3.26613976593566*10^(-1) 1.141*10^(-02) 39 8.94655373334687*10^(-1) 3.27826521746298*10^(-1) 5.149*10^(-19)| $n$ | $x_{n}^{1}$ | $x_{n}^{2}$ | $\left\\|x_{n}-x_{n-1}\right\\|$ | | :--- | :--- | :--- | :--- | | 0 | $1.00000000000000 \cdot 10^{+0}$ | $0.00000000000000 \cdot 10^{+0}$ | | | 1 | $1.00000000000000 \cdot 10^{+0}$ | $3.33333333333333 \cdot 10^{-1}$ | $3.33 \cdot 10^{-01}$ | | 2 | $9.06550218340611 \cdot 10^{-1}$ | $3.54002911208151 \cdot 10^{-1}$ | $9.344 \cdot 10^{-02}$ | | 3 | $8.85328400663412 \cdot 10^{-1}$ | $3.38027276361332 \cdot 10^{-1}$ | $2.122 \cdot 10^{-02}$ | | 4 | $8.91329556832800 \cdot 10^{-1}$ | $3.26613976593566 \cdot 10^{-1}$ | $1.141 \cdot 10^{-02}$ | | 39 | $8.94655373334687 \cdot 10^{-1}$ | $3.27826521746298 \cdot 10^{-1}$ | $5.149 \cdot 10^{-19}$ |
Using the method ( 5 ) with x 0 = ( 1 , 1 ) , x 1 = ( 2 , 2 ) x 0 = ( 1 , 1 ) , x 1 = ( 2 , 2 ) x_(0)=(1,1),x_(1)=(2,2)x_{0}=(1,1), x_{1}=(2,2)x0=(1,1),x1=(2,2) we get:
n n nnn x n 1 x n 1 x_(n)^(1)x_{n}^{1}xn1 x n 2 x n 2 x_(n)^(2)x_{n}^{2}xn2 ‖ x n − x n − 1 ‖ x n − x n − 1 ||x_(n)-x_(n-1)||\left\|x_{n}-x_{n-1}\right\|‖xn−xn−1‖
0 2.00000000000000 ⋅ 10 + 0 2.00000000000000 ⋅ 10 + 0 2.00000000000000*10^(+0)2.00000000000000 \cdot 10^{+0}2.00000000000000⋅10+0 2.00000000000000 ⋅ 10 + 0 2.00000000000000 ⋅ 10 + 0 2.00000000000000*10^(+0)2.00000000000000 \cdot 10^{+0}2.00000000000000⋅10+0
1 1.00000000000000 ⋅ 10 + 0 1.00000000000000 ⋅ 10 + 0 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0}1.00000000000000⋅10+0 1.00000000000000 ⋅ 10 + 0 1.00000000000000 ⋅ 10 + 0 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0}1.00000000000000⋅10+0 1.000 ⋅ 10 + 00 1.000 ⋅ 10 + 00 1.000*10^(+00)1.000 \cdot 10^{+00}1.000⋅10+00
2 3.33333333333333 ⋅ 10 − 1 3.33333333333333 ⋅ 10 − 1 3.33333333333333*10^(-1)3.33333333333333 \cdot 10^{-1}3.33333333333333⋅10−1 1.33333333333333 ⋅ 10 + 0 1.33333333333333 ⋅ 10 + 0 1.33333333333333*10^(+0)1.33333333333333 \cdot 10^{+0}1.33333333333333⋅10+0 6.666 ⋅ 10 + 01 6.666 ⋅ 10 + 01 6.666*10^(+01)6.666 \cdot 10^{+01}6.666⋅10+01
3 9.62025316455696 ⋅ 10 − 1 9.62025316455696 ⋅ 10 − 1 9.62025316455696*10^(-1)9.62025316455696 \cdot 10^{-1}9.62025316455696⋅10−1 3.54430379746835 ⋅ 10 − 1 3.54430379746835 ⋅ 10 − 1 3.54430379746835*10^(-1)3.54430379746835 \cdot 10^{-1}3.54430379746835⋅10−1 9.789 ⋅ 10 − 01 9.789 ⋅ 10 − 01 9.789*10^(-01)9.789 \cdot 10^{-01}9.789⋅10−01
4 9.00696217156264 ⋅ 10 − 1 9.00696217156264 ⋅ 10 − 1 9.00696217156264*10^(-1)9.00696217156264 \cdot 10^{-1}9.00696217156264⋅10−1 3.30465935597986 ⋅ 10 − 1 3.30465935597986 ⋅ 10 − 1 3.30465935597986*10^(-1)3.30465935597986 \cdot 10^{-1}3.30465935597986⋅10−1 6.132 ⋅ 10 − 02 6.132 ⋅ 10 − 02 6.132*10^(-02)6.132 \cdot 10^{-02}6.132⋅10−02
5 8.94706409693425 ⋅ 10 − 1 8.94706409693425 ⋅ 10 − 1 8.94706409693425*10^(-1)8.94706409693425 \cdot 10^{-1}8.94706409693425⋅10−1 3.27855252188766 ⋅ 10 − 1 3.27855252188766 ⋅ 10 − 1 3.27855252188766*10^(-1)3.27855252188766 \cdot 10^{-1}3.27855252188766⋅10−1 5.989 ⋅ 10 − 03 5.989 ⋅ 10 − 03 5.989*10^(-03)5.989 \cdot 10^{-03}5.989⋅10−03
6 8.94655376809408 ⋅ 10 − 1 8.94655376809408 ⋅ 10 − 1 8.94655376809408*10^(-1)8.94655376809408 \cdot 10^{-1}8.94655376809408⋅10−1 3.27826524565125 ⋅ 10 − 1 3.27826524565125 ⋅ 10 − 1 3.27826524565125*10^(-1)3.27826524565125 \cdot 10^{-1}3.27826524565125⋅10−1 5.103 ⋅ 10 − 05 5.103 ⋅ 10 − 05 5.103*10^(-05)5.103 \cdot 10^{-05}5.103⋅10−05
7 8.94655373334687 ⋅ 10 − 1 8.94655373334687 ⋅ 10 − 1 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1}8.94655373334687⋅10−1 3.27826521746298 ⋅ 10 − 1 3.27826521746298 ⋅ 10 − 1 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1}3.27826521746298⋅10−1 3.474 ⋅ 10 − 09 3.474 ⋅ 10 − 09 3.474*10^(-09)3.474 \cdot 10^{-09}3.474⋅10−09
8 8.94655373334687 ⋅ 10 − 1 8.94655373334687 ⋅ 10 − 1 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1}8.94655373334687⋅10−1 3.27826521746298 ⋅ 10 − 1 3.27826521746298 ⋅ 10 − 1 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1}3.27826521746298⋅10−1 2.003 ⋅ 10 − 17 2.003 ⋅ 10 − 17 2.003*10^(-17)2.003 \cdot 10^{-17}2.003⋅10−17
9 8.94655373334687 ⋅ 10 − 1 8.94655373334687 ⋅ 10 − 1 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1}8.94655373334687⋅10−1 3.27826521746298 ⋅ 10 − 1 3.27826521746298 ⋅ 10 − 1 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1}3.27826521746298⋅10−1 2.710 ⋅ 10 − 20 2.710 ⋅ 10 − 20 2.710*10^(-20)2.710 \cdot 10^{-20}2.710⋅10−20
n x_(n)^(1) x_(n)^(2) ||x_(n)-x_(n-1)|| 0 2.00000000000000*10^(+0) 2.00000000000000*10^(+0) 1 1.00000000000000*10^(+0) 1.00000000000000*10^(+0) 1.000*10^(+00) 2 3.33333333333333*10^(-1) 1.33333333333333*10^(+0) 6.666*10^(+01) 3 9.62025316455696*10^(-1) 3.54430379746835*10^(-1) 9.789*10^(-01) 4 9.00696217156264*10^(-1) 3.30465935597986*10^(-1) 6.132*10^(-02) 5 8.94706409693425*10^(-1) 3.27855252188766*10^(-1) 5.989*10^(-03) 6 8.94655376809408*10^(-1) 3.27826524565125*10^(-1) 5.103*10^(-05) 7 8.94655373334687*10^(-1) 3.27826521746298*10^(-1) 3.474*10^(-09) 8 8.94655373334687*10^(-1) 3.27826521746298*10^(-1) 2.003*10^(-17) 9 8.94655373334687*10^(-1) 3.27826521746298*10^(-1) 2.710*10^(-20)| $n$ | $x_{n}^{1}$ | $x_{n}^{2}$ | $\left\\|x_{n}-x_{n-1}\right\\|$ | | :--- | :--- | :--- | :--- | | 0 | $2.00000000000000 \cdot 10^{+0}$ | $2.00000000000000 \cdot 10^{+0}$ | | | 1 | $1.00000000000000 \cdot 10^{+0}$ | $1.00000000000000 \cdot 10^{+0}$ | $1.000 \cdot 10^{+00}$ | | 2 | $3.33333333333333 \cdot 10^{-1}$ | $1.33333333333333 \cdot 10^{+0}$ | $6.666 \cdot 10^{+01}$ | | 3 | $9.62025316455696 \cdot 10^{-1}$ | $3.54430379746835 \cdot 10^{-1}$ | $9.789 \cdot 10^{-01}$ | | 4 | $9.00696217156264 \cdot 10^{-1}$ | $3.30465935597986 \cdot 10^{-1}$ | $6.132 \cdot 10^{-02}$ | | 5 | $8.94706409693425 \cdot 10^{-1}$ | $3.27855252188766 \cdot 10^{-1}$ | $5.989 \cdot 10^{-03}$ | | 6 | $8.94655376809408 \cdot 10^{-1}$ | $3.27826524565125 \cdot 10^{-1}$ | $5.103 \cdot 10^{-05}$ | | 7 | $8.94655373334687 \cdot 10^{-1}$ | $3.27826521746298 \cdot 10^{-1}$ | $3.474 \cdot 10^{-09}$ | | 8 | $8.94655373334687 \cdot 10^{-1}$ | $3.27826521746298 \cdot 10^{-1}$ | $2.003 \cdot 10^{-17}$ | | 9 | $8.94655373334687 \cdot 10^{-1}$ | $3.27826521746298 \cdot 10^{-1}$ | $2.710 \cdot 10^{-20}$ |
Using method (6) with λ = 0.5 λ = 0.5 lambda=0.5\lambda=0.5λ=0.5 and x 0 = ( 1 , 1 ) x 0 = ( 1 , 1 ) x_(0)=(1,1)x_{0}=(1,1)x0=(1,1) we get:
n n nnn x n 1 x n 1 x_(n)^(1)x_{n}^{1}xn1 x n 2 x n 2 x_(n)^(2)x_{n}^{2}xn2 ‖ x n − x n − 1 ‖ x n − x n − 1 ||x_(n)-x_(n-1)||\left\|x_{n}-x_{n-1}\right\|‖xn−xn−1‖
0 1.00000000000000 ⋅ 10 + 0 1.00000000000000 ⋅ 10 + 0 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0}1.00000000000000⋅10+0 0.00000000000000 ⋅ 10 + 00 0.00000000000000 ⋅ 10 + 00 0.00000000000000*10^(+00)0.00000000000000 \cdot 10^{+00}0.00000000000000⋅10+00
1 1.40000000000000 ⋅ 10 + 0 1.40000000000000 ⋅ 10 + 0 1.40000000000000*10^(+0)1.40000000000000 \cdot 10^{+0}1.40000000000000⋅10+0 0.00000000000000 ⋅ 10 + 00 0.00000000000000 ⋅ 10 + 00 0.00000000000000*10^(+00)0.00000000000000 \cdot 10^{+00}0.00000000000000⋅10+00 1.000 ⋅ 10 + 00 1.000 ⋅ 10 + 00 1.000*10^(+00)1.000 \cdot 10^{+00}1.000⋅10+00
2 1.15421294962624 ⋅ 10 + 0 1.15421294962624 ⋅ 10 + 0 1.15421294962624*10^(+0)1.15421294962624 \cdot 10^{+0}1.15421294962624⋅10+0 1.43841335097579 ⋅ 10 − 01 1.43841335097579 ⋅ 10 − 01 1.43841335097579*10^(-01)1.43841335097579 \cdot 10^{-01}1.43841335097579⋅10−01 2.457 ⋅ 10 − 01 2.457 ⋅ 10 − 01 2.457*10^(-01)2.457 \cdot 10^{-01}2.457⋅10−01
3 1.01057150046324 ⋅ 10 + 1 1.01057150046324 ⋅ 10 + 1 1.01057150046324*10^(+1)1.01057150046324 \cdot 10^{+1}1.01057150046324⋅10+1 2.69169893550861 ⋅ 10 − 01 2.69169893550861 ⋅ 10 − 01 2.69169893550861*10^(-01)2.69169893550861 \cdot 10^{-01}2.69169893550861⋅10−01 1.436 ⋅ 10 − 01 1.436 ⋅ 10 − 01 1.436*10^(-01)1.436 \cdot 10^{-01}1.436⋅10−01
4 8.99073392876452 ⋅ 10 − 1 8.99073392876452 ⋅ 10 − 1 8.99073392876452*10^(-1)8.99073392876452 \cdot 10^{-1}8.99073392876452⋅10−1 3.76267383109311 ⋅ 10 − 01 3.76267383109311 ⋅ 10 − 01 3.76267383109311*10^(-01)3.76267383109311 \cdot 10^{-01}3.76267383109311⋅10−01 1.114 ⋅ 10 − 01 1.114 ⋅ 10 − 01 1.114*10^(-01)1.114 \cdot 10^{-01}1.114⋅10−01
5 8.95022505657807 ⋅ 10 − 1 8.95022505657807 ⋅ 10 − 1 8.95022505657807*10^(-1)8.95022505657807 \cdot 10^{-1}8.95022505657807⋅10−1 3.28815382034089 ⋅ 10 − 01 3.28815382034089 ⋅ 10 − 01 3.28815382034089*10^(-01)3.28815382034089 \cdot 10^{-01}3.28815382034089⋅10−01 4.745 ⋅ 10 − 02 4.745 ⋅ 10 − 02 4.745*10^(-02)4.745 \cdot 10^{-02}4.745⋅10−02
6 8.94655504144107 ⋅ 10 − 1 8.94655504144107 ⋅ 10 − 1 8.94655504144107*10^(-1)8.94655504144107 \cdot 10^{-1}8.94655504144107⋅10−1 3.27827488746546 ⋅ 10 − 01 3.27827488746546 ⋅ 10 − 01 3.27827488746546*10^(-01)3.27827488746546 \cdot 10^{-01}3.27827488746546⋅10−01 9.878 ⋅ 10 − 04 9.878 ⋅ 10 − 04 9.878*10^(-04)9.878 \cdot 10^{-04}9.878⋅10−04
7 8.94655373334787 ⋅ 10 − 1 8.94655373334787 ⋅ 10 − 1 8.94655373334787*10^(-1)8.94655373334787 \cdot 10^{-1}8.94655373334787⋅10−1 3.27826521746806 ⋅ 10 − 01 3.27826521746806 ⋅ 10 − 01 3.27826521746806*10^(-01)3.27826521746806 \cdot 10^{-01}3.27826521746806⋅10−01 9.669 ⋅ 10 − 07 9.669 ⋅ 10 − 07 9.669*10^(-07)9.669 \cdot 10^{-07}9.669⋅10−07
8 8.94655373334687 ⋅ 10 − 1 8.94655373334687 ⋅ 10 − 1 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1}8.94655373334687⋅10−1 3.27826521746298 ⋅ 10 − 01 3.27826521746298 ⋅ 10 − 01 3.27826521746298*10^(-01)3.27826521746298 \cdot 10^{-01}3.27826521746298⋅10−01 5.086 ⋅ 10 − 13 5.086 ⋅ 10 − 13 5.086*10^(-13)5.086 \cdot 10^{-13}5.086⋅10−13
9 8.94655373334687 ⋅ 10 − 1 8.94655373334687 ⋅ 10 − 1 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1}8.94655373334687⋅10−1 3.27826521746298 ⋅ 10 − 01 3.27826521746298 ⋅ 10 − 01 3.27826521746298*10^(-01)3.27826521746298 \cdot 10^{-01}3.27826521746298⋅10−01 2.710 ⋅ 10 − 20 2.710 ⋅ 10 − 20 2.710*10^(-20)2.710 \cdot 10^{-20}2.710⋅10−20
n x_(n)^(1) x_(n)^(2) ||x_(n)-x_(n-1)|| 0 1.00000000000000*10^(+0) 0.00000000000000*10^(+00) 1 1.40000000000000*10^(+0) 0.00000000000000*10^(+00) 1.000*10^(+00) 2 1.15421294962624*10^(+0) 1.43841335097579*10^(-01) 2.457*10^(-01) 3 1.01057150046324*10^(+1) 2.69169893550861*10^(-01) 1.436*10^(-01) 4 8.99073392876452*10^(-1) 3.76267383109311*10^(-01) 1.114*10^(-01) 5 8.95022505657807*10^(-1) 3.28815382034089*10^(-01) 4.745*10^(-02) 6 8.94655504144107*10^(-1) 3.27827488746546*10^(-01) 9.878*10^(-04) 7 8.94655373334787*10^(-1) 3.27826521746806*10^(-01) 9.669*10^(-07) 8 8.94655373334687*10^(-1) 3.27826521746298*10^(-01) 5.086*10^(-13) 9 8.94655373334687*10^(-1) 3.27826521746298*10^(-01) 2.710*10^(-20)| $n$ | $x_{n}^{1}$ | $x_{n}^{2}$ | $\left\\|x_{n}-x_{n-1}\right\\|$ | | :--- | :--- | :--- | :--- | | 0 | $1.00000000000000 \cdot 10^{+0}$ | $0.00000000000000 \cdot 10^{+00}$ | | | 1 | $1.40000000000000 \cdot 10^{+0}$ | $0.00000000000000 \cdot 10^{+00}$ | $1.000 \cdot 10^{+00}$ | | 2 | $1.15421294962624 \cdot 10^{+0}$ | $1.43841335097579 \cdot 10^{-01}$ | $2.457 \cdot 10^{-01}$ | | 3 | $1.01057150046324 \cdot 10^{+1}$ | $2.69169893550861 \cdot 10^{-01}$ | $1.436 \cdot 10^{-01}$ | | 4 | $8.99073392876452 \cdot 10^{-1}$ | $3.76267383109311 \cdot 10^{-01}$ | $1.114 \cdot 10^{-01}$ | | 5 | $8.95022505657807 \cdot 10^{-1}$ | $3.28815382034089 \cdot 10^{-01}$ | $4.745 \cdot 10^{-02}$ | | 6 | $8.94655504144107 \cdot 10^{-1}$ | $3.27827488746546 \cdot 10^{-01}$ | $9.878 \cdot 10^{-04}$ | | 7 | $8.94655373334787 \cdot 10^{-1}$ | $3.27826521746806 \cdot 10^{-01}$ | $9.669 \cdot 10^{-07}$ | | 8 | $8.94655373334687 \cdot 10^{-1}$ | $3.27826521746298 \cdot 10^{-01}$ | $5.086 \cdot 10^{-13}$ | | 9 | $8.94655373334687 \cdot 10^{-1}$ | $3.27826521746298 \cdot 10^{-01}$ | $2.710 \cdot 10^{-20}$ |
It seems that the best results are not obtained here for λ λ lambda\lambdaλ taken too small, because the divided differences cannot be computed in this case for ‖ x n − x n − 1 ‖ ≤ 1.0 ⋅ 10 − 16 x n − x n − 1 ≤ 1.0 ⋅ 10 − 16 ||x_(n)-x_(n-1)|| <= 1.0*10^(-16)\left\|x_{n}-x_{n-1}\right\| \leq 1.0 \cdot 10^{-16}‖xn−xn−1‖≤1.0⋅10−16.

REFERENCES

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Recevied: December 1, 1994
Academia Română
Institutul de Calcul "Tiberiu Popoviciu"
P.O. Box 68
3400 Cluj-Napoca 1
Romania
1995

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