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
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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ON SOME STEFFENSEN-TYPE ITERATIVE METHODS FOR A CLASS OF NONLINEAR EQUATIONS
1. INTRODUCTION
Consider a Banach space XX and the equation
{:(1)F(x)+G(x)=0:}\begin{equation*}
F(x)+G(x)=0 \tag{1}
\end{equation*}
where F,G:X rarr XF, G: X \rightarrow X are nonlinear operators, FF being Fréchet differentiable and GG being continuously but nondifferentiable. This is the case when we study an equation H(x)=0H(x)=0, with H:X rarr XH: X \rightarrow X a nondifferentiable operator to which we cannot apply Newton's method. HH 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:
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)))","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*}
and, more generally,
{:(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*}
where AA is a linear operator approximating F^(')F^{\prime}.
In [1] is studied the secant-type method
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)))","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*}
[x,y;F][x, y ; F] denoting the first order divided difference of FF at the points x,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)))","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*}
having the convergence order (1+sqrt5)/(2)~~1.618\frac{1+\sqrt{5}}{2} \approx 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),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 X, where varphi:X rarr X\varphi: X \rightarrow 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 X, where varphi:X rarr X\varphi: X \rightarrow X,
varphi(x)=x-lambda(F(x)+G(x)),\varphi(x)=x-\lambda(F(x)+G(x)),
lambda\lambda 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]\left[x_{0}, y_{0} ; G\right] belonging to the space L(X,X)\mathcal{L}(X, X) (the Banach space of the linear and bounded operators from XX to XX ) is called the first order divided difference of the operator G:X rarr XG: X \rightarrow X at the points x_(0)x_{0}, y_(0)in Xy_{0} \in X if the following properties hold:
a) [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), for x_(0)!=y_(0)x_{0} \neq y_{0};
b) if GG is Fréchet differentiable at x_(0)x_{0}, then
a) [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), for x_(0)!=y_(0)x_{0} \neq y_{0};
b) if GG is Fréchet differentiable at x_(0)x_{0}, then
[x_(0),x_(0);G]=G^(')(x_(0))\left[x_{0}, x_{0} ; G\right]=G^{\prime}\left(x_{0}\right)
Definition 2. An operator belonging to the space L(X,L(X,X))\mathcal{L}(X, \mathcal{L}(X, X)), denoted by [x_(0),y_(0),z_(0);G]\left[x_{0}, y_{0}, z_{0} ; G\right], is called the second-order divided difference of the operator G:X rarr XG: X \rightarrow X at the points x_(0),y_(0),z_(0)in Xx_{0}, y_{0}, z_{0} \in 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]\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], for the distinct points x_(0),y_(0),z_(0)in Xx_{0}, y_{0}, z_{0} \in X;
b) if GG is two times Fréchet differentiable at x_(0)in Xx_{0} \in X, then
a) [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], for the distinct points x_(0),y_(0),z_(0)in Xx_{0}, y_{0}, z_{0} \in X;
b) if GG is two times Fréchet differentiable at x_(0)in Xx_{0} \in X, then
[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)
We shall denote by 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\} the open ball having the center at x_(0)in Xx_{0} \in X and the radius r > 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)in Xx_{0} \in X, and the positive real numbers K,l,epsi,M,rK, l, \varepsilon, M, r such that:
i) GG is continuous on B_(r)(x_(0))B_{r}\left(x_{0}\right);
ii) FF is Fréchet differentiable on B_(r)(x_(0))B_{r}\left(x_{0}\right), with the Fréchet derivative satisfying the Lipschitz condition
i) GG is continuous on B_(r)(x_(0))B_{r}\left(x_{0}\right);
ii) FF is Fréchet differentiable on B_(r)(x_(0))B_{r}\left(x_{0}\right), with the Fréchet derivative satisfying the Lipschitz condition
||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) ;
iii) The second-order divided difference of GG is uniformly bounded on B_(r)(x_(0))B_{r}\left(x_{0}\right) :
||[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) ;
iv) The operators F^(')(x)+[x,varphi(x);G]F^{\prime}(x)+[x, \varphi(x) ; G] are invertible, with the inverses uniformly bounded: AA x inB_(r)(x_(0))\forall x \in B_{r}\left(x_{0}\right) with varphi(x)inB_(r)(x_(0))\varphi(x) \in B_{r}\left(x_{0}\right) there exists (F^(')(x)+[x,varphi(x);G])^(-1)\left(F^{\prime}(x)+[x, \varphi(x) ; G]\right)^{-1} and
||(F^(')(x)+[x,varphi(x);G])^(-1)|| <= M\left\|\left(F^{\prime}(x)+[x, \varphi(x) ; G]\right)^{-1}\right\| \leq M
v) lambda\lambda is chosen such that lambda <= M\lambda \leq M;
vi) q:=M^(2)epsi((l)/(2)+2K) < 1q:=M^{2} \varepsilon\left(\frac{l}{2}+2 K\right)<1 and the radius is given by
vi) q:=M^(2)epsi((l)/(2)+2K) < 1q:=M^{2} \varepsilon\left(\frac{l}{2}+2 K\right)<1 and the radius is given by
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}}
then
j) The sequence (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0} given by (6) is well defined and (x_(n))_(n >= 0)subB_(r)(x_(0));\left(x_{n}\right)_{n \geq 0} \subset B_{r}\left(x_{0}\right) ;
jj) (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0} converges to some x^(**)in bar(B_(r)(x_(0)))x^{*} \in \overline{B_{r}\left(x_{0}\right)}, which is a solution of equation (1);
jjj) The following estimation holds:
j) The sequence (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0} given by (6) is well defined and (x_(n))_(n >= 0)subB_(r)(x_(0));\left(x_{n}\right)_{n \geq 0} \subset B_{r}\left(x_{0}\right) ;
jj) (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0} converges to some x^(**)in bar(B_(r)(x_(0)))x^{*} \in \overline{B_{r}\left(x_{0}\right)}, which is a solution of equation (1);
jjj) The following estimation holds:
||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)}
Proof. From the hypothesis ii ) concerning FF it is known [6] that we get
{:(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*}
From the definitions of the divided differences we obtain
{:(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*}
Indeed,
{:[[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}
We shall prove by induction that
{:[(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*}
From the above inequality it can be easily deduced by (6) that EEx_(k+1)\exists x_{k+1} and
{:[(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*}
For k=0k=0 we have:
{:[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}
which imply that
{:[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} .
Suppose now that 9 is true for k= bar(1,n-1)k=\overline{1, n-1}. By it follows that EEx_(n)\exists x_{n}, and we have that x_(n)inB_(r)(x_(0))x_{n} \in B_{r}\left(x_{0}\right). Indeed,
||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
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),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}
{:[ <= (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}
It remains to show that varphi(x_(n))inB_(r)(x_(0))\varphi\left(x_{n}\right) \in B_{r}\left(x_{0}\right) :
{:[||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}
The induction (9) is proved.
Now we prove that the sequence (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0} is a Cauchy sequence, hence it converges to some element x^(**)in bar(B_(r)(x_(0)))x^{*} \in \overline{B_{r}\left(x_{0}\right)} :
Now we prove that the sequence (x_(n))_(n >= 0)\left(x_{n}\right)_{n \geq 0} is a Cauchy sequence, hence it converges to some element x^(**)in bar(B_(r)(x_(0)))x^{*} \in \overline{B_{r}\left(x_{0}\right)} :
{:[||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}
Passing to limit for n rarr oon \rightarrow \infty in relation (6) and taking into account the hypotheses concerning FF and GG, we get that x^(**)x^{*} is a solution of (1).
The estimation jjjj j j ) is obtained from the above relation, for p rarr oop \rightarrow \infty.
3. NUMERICAL EXAMPLES
Given the system
{[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.
we shall consider X=(R^(2),||*||_(oo))X=\left(\mathbb{R}^{2},\|\cdot\|_{\infty}\right) and 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), with 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|, g_(2)(x,y)=|y|g_{2}(x, y)=|y|.
We shall take [x,y;G]inM_(2)(R)[x, y ; G] \in \mathbb{M}_{2}(\mathbb{R}) given by
[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}}
i=1,2i=1,2.
Using the method (2) with x_(0)=(1,0)x_{0}=(1,0), we obtain
Using the method (2) with x_(0)=(1,0)x_{0}=(1,0), we obtain
| nn | x_(n)^(1)x_{n}^{1} | x_(n)^(2)x_{n}^{2} | ||x_(n)-x_(n-1)||\left\|x_{n}-x_{n-1}\right\| |
| 0 | 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0} | 0.00000000000000*10^(+0)0.00000000000000 \cdot 10^{+0} | |
| 1 | 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0} | 3.33333333333333*10^(-1)3.33333333333333 \cdot 10^{-1} | 3.33*10^(-01)3.33 \cdot 10^{-01} |
| 2 | 9.06550218340611*10^(-1)9.06550218340611 \cdot 10^{-1} | 3.54002911208151*10^(-1)3.54002911208151 \cdot 10^{-1} | 9.344*10^(-02)9.344 \cdot 10^{-02} |
| 3 | 8.85328400663412*10^(-1)8.85328400663412 \cdot 10^{-1} | 3.38027276361332*10^(-1)3.38027276361332 \cdot 10^{-1} | 2.122*10^(-02)2.122 \cdot 10^{-02} |
| 4 | 8.91329556832800*10^(-1)8.91329556832800 \cdot 10^{-1} | 3.26613976593566*10^(-1)3.26613976593566 \cdot 10^{-1} | 1.141*10^(-02)1.141 \cdot 10^{-02} |
| 39 | 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1} | 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1} | 5.149*10^(-19)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) we get:
| nn | x_(n)^(1)x_{n}^{1} | x_(n)^(2)x_{n}^{2} | ||x_(n)-x_(n-1)||\left\|x_{n}-x_{n-1}\right\| |
| 0 | 2.00000000000000*10^(+0)2.00000000000000 \cdot 10^{+0} | 2.00000000000000*10^(+0)2.00000000000000 \cdot 10^{+0} | |
| 1 | 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0} | 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0} | 1.000*10^(+00)1.000 \cdot 10^{+00} |
| 2 | 3.33333333333333*10^(-1)3.33333333333333 \cdot 10^{-1} | 1.33333333333333*10^(+0)1.33333333333333 \cdot 10^{+0} | 6.666*10^(+01)6.666 \cdot 10^{+01} |
| 3 | 9.62025316455696*10^(-1)9.62025316455696 \cdot 10^{-1} | 3.54430379746835*10^(-1)3.54430379746835 \cdot 10^{-1} | 9.789*10^(-01)9.789 \cdot 10^{-01} |
| 4 | 9.00696217156264*10^(-1)9.00696217156264 \cdot 10^{-1} | 3.30465935597986*10^(-1)3.30465935597986 \cdot 10^{-1} | 6.132*10^(-02)6.132 \cdot 10^{-02} |
| 5 | 8.94706409693425*10^(-1)8.94706409693425 \cdot 10^{-1} | 3.27855252188766*10^(-1)3.27855252188766 \cdot 10^{-1} | 5.989*10^(-03)5.989 \cdot 10^{-03} |
| 6 | 8.94655376809408*10^(-1)8.94655376809408 \cdot 10^{-1} | 3.27826524565125*10^(-1)3.27826524565125 \cdot 10^{-1} | 5.103*10^(-05)5.103 \cdot 10^{-05} |
| 7 | 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1} | 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1} | 3.474*10^(-09)3.474 \cdot 10^{-09} |
| 8 | 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1} | 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1} | 2.003*10^(-17)2.003 \cdot 10^{-17} |
| 9 | 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1} | 3.27826521746298*10^(-1)3.27826521746298 \cdot 10^{-1} | 2.710*10^(-20)2.710 \cdot 10^{-20} |
Using method (6) with lambda=0.5\lambda=0.5 and x_(0)=(1,1)x_{0}=(1,1) we get:
| nn | x_(n)^(1)x_{n}^{1} | x_(n)^(2)x_{n}^{2} | ||x_(n)-x_(n-1)||\left\|x_{n}-x_{n-1}\right\| |
| 0 | 1.00000000000000*10^(+0)1.00000000000000 \cdot 10^{+0} | 0.00000000000000*10^(+00)0.00000000000000 \cdot 10^{+00} | |
| 1 | 1.40000000000000*10^(+0)1.40000000000000 \cdot 10^{+0} | 0.00000000000000*10^(+00)0.00000000000000 \cdot 10^{+00} | 1.000*10^(+00)1.000 \cdot 10^{+00} |
| 2 | 1.15421294962624*10^(+0)1.15421294962624 \cdot 10^{+0} | 1.43841335097579*10^(-01)1.43841335097579 \cdot 10^{-01} | 2.457*10^(-01)2.457 \cdot 10^{-01} |
| 3 | 1.01057150046324*10^(+1)1.01057150046324 \cdot 10^{+1} | 2.69169893550861*10^(-01)2.69169893550861 \cdot 10^{-01} | 1.436*10^(-01)1.436 \cdot 10^{-01} |
| 4 | 8.99073392876452*10^(-1)8.99073392876452 \cdot 10^{-1} | 3.76267383109311*10^(-01)3.76267383109311 \cdot 10^{-01} | 1.114*10^(-01)1.114 \cdot 10^{-01} |
| 5 | 8.95022505657807*10^(-1)8.95022505657807 \cdot 10^{-1} | 3.28815382034089*10^(-01)3.28815382034089 \cdot 10^{-01} | 4.745*10^(-02)4.745 \cdot 10^{-02} |
| 6 | 8.94655504144107*10^(-1)8.94655504144107 \cdot 10^{-1} | 3.27827488746546*10^(-01)3.27827488746546 \cdot 10^{-01} | 9.878*10^(-04)9.878 \cdot 10^{-04} |
| 7 | 8.94655373334787*10^(-1)8.94655373334787 \cdot 10^{-1} | 3.27826521746806*10^(-01)3.27826521746806 \cdot 10^{-01} | 9.669*10^(-07)9.669 \cdot 10^{-07} |
| 8 | 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1} | 3.27826521746298*10^(-01)3.27826521746298 \cdot 10^{-01} | 5.086*10^(-13)5.086 \cdot 10^{-13} |
| 9 | 8.94655373334687*10^(-1)8.94655373334687 \cdot 10^{-1} | 3.27826521746298*10^(-01)3.27826521746298 \cdot 10^{-01} | 2.710*10^(-20)2.710 \cdot 10^{-20} |
It seems that the best results are not obtained here for lambda\lambda taken too small, because the divided differences cannot be computed in this case for ||x_(n)-x_(n-1)|| <= 1.0*10^(-16)\left\|x_{n}-x_{n-1}\right\| \leq 1.0 \cdot 10^{-16}.
REFERENCES
[1] I.K. Argyros, On the secant method and the Pták error estimates, Rev. Anal. Numér. Théor. Approx., 24 (1995) nos. 1-2, pp. 3-14. ㄸ
[2] M. Balázs, A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numér. Théor. Approx., 21 (1992) no. 2, pp. 111-117. ㄸ
[3] E. Cătinaş, On some iterative methods for solving nonlinear equations, Rev. Anal. Numér. Théor. Approx., 23 (1994) no. 1, pp. 47-53. «
[4] G. Goldner, M. Balázs, Asupra metodei coardei si a unei modificari a ei pentru rezolvarea ecuatiilor operationale neliniar, Stud. Cerc. Mat., 20 (1968), pp. 981-990. [English title: On the method of chord and on its modification for solving the nonlinear operatorial equations]
[5] G. Goldner, M. Balázs, Observații asupra diferențelor divizate și asupra metodei coardei, Revista de Analiză Numerică şi Teoria Aproximaţiei, 3 (1974) no. 1, pp. 19-30. [English title: Remarks on divided differences and method of chord] «
[6] L.V. Kantorovici, G.P. Akilov, Functional Analysis, Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1986 (in Romanian).
[7] I. Păvăloiu, On the monotonicity of the sequences of approximations obtained by Steffensen's method, Mathematica, 35(58) (1993) no. 1, pp. 71-76. ㄸ
[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. ㄸ
[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. ㄸ
[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. ㄸ
[2] M. Balázs, A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numér. Théor. Approx., 21 (1992) no. 2, pp. 111-117. ㄸ
[3] E. Cătinaş, On some iterative methods for solving nonlinear equations, Rev. Anal. Numér. Théor. Approx., 23 (1994) no. 1, pp. 47-53. «
[4] G. Goldner, M. Balázs, Asupra metodei coardei si a unei modificari a ei pentru rezolvarea ecuatiilor operationale neliniar, Stud. Cerc. Mat., 20 (1968), pp. 981-990. [English title: On the method of chord and on its modification for solving the nonlinear operatorial equations]
[5] G. Goldner, M. Balázs, Observații asupra diferențelor divizate și asupra metodei coardei, Revista de Analiză Numerică şi Teoria Aproximaţiei, 3 (1974) no. 1, pp. 19-30. [English title: Remarks on divided differences and method of chord] «
[6] L.V. Kantorovici, G.P. Akilov, Functional Analysis, Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1986 (in Romanian).
[7] I. Păvăloiu, On the monotonicity of the sequences of approximations obtained by Steffensen's method, Mathematica, 35(58) (1993) no. 1, pp. 71-76. ㄸ
[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. ㄸ
[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. ㄸ
[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. ㄸ
Recevied: December 1, 1994
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