On some interpolatory iterative methods for the second degree polynomial operators (I)

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Abstract

In this note we consider the chord (secant) method and the Steffensen method for solving polynomial operators of degree 2 on Banach spaces, $F:X\rightarrow Y$.

The convergence conditions in this case are simplified, as the divided difference of order 3 is the null trilinear operator.

As particular cases, we study the eigenproblem for quadratic matrices and integral equations of Volterra type.

We obtain semilocal convergence results, which show the r-convergence orders of the iterates.

Authors

Emil Cătinaş

Emil Cătinaş

Ion Păvăloiu

Ion Păvăloiu

Keywords

chord/secant method; Steffensen method; polynomial operator equation of degree 2 on Banach spaces; divided differences; eigenproblem for quadratic matrices; integral equations of Volterra type; semilocal convergence; r-convergence order.

Cite this paper as:

E. Cătinaş, I. Păvăloiu, On some interpolatory iterative methods for the second degree polynomial operators (I), Rev. Anal. Numér. Théor. Approx., 27 (1998) no. 1, pp. 33-45.

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Journal

Revue d’Analyse Numerique et de Theorie de l’Approximation

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Editions de l’Academie Roumaine

Article on the journal website

https://ictp.acad.ro/jnaat/journal/article/view/1998-vol27-no1-art5

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1222-9024

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2457-8126

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ZBL

Google Scholar citations

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Paper (preprint) in HTML form

On Some Interpolatory Iterative Methods for the Second Degree Polynomial Operators (I)

On Some Interpolatory Iterative Methods
for the Second Degree Polynomial Operators (I)

Emil Cătinaş and Ion Păvăloiu

1991 AMS Subject Classification: 65J20, 65H17.

1. Introduction

The most known interpolatory iterative methods have been studied by several authors ([3], [4], [9], [13], [14], [15] [16] and [19]) also in the case of operator equations. These methods have the advantage of a higher efficiency when compared to the methods that use the Fréchet derivatives of the corresponding operators, and, moreover, they may be applied even when the Fréchet derivative vanishes at some points in the neighborhood of the solution [2], [16]. On the other hand, the construction of the finite difference operators may be difficult for a large class of general operators.

In this note we shall consider second degree polynomial operators; for these operators the divided differences at any four points are the null trilinear operators. These equations belong to an important class which has many applications in practice [2].

In Sections 2 and 3 we construct the divided differences of some particular operators, and the Lagrange interpolatory polynomial in the Newton form. In Section 4 we study the convergence of two interpolatory iterative methods, namely the chord method and the Steffensen method. We have considered that this study may present interest because of the conditions for the convergence, which are simpler than in the general case.

2. Divided differences

Let $X$ and $Y$ be two normed linear spaces and $F:X\rightarrow Y$ a mapping. Denote by $\mathcal{L}\left(X,Y\right)$ the set of linear continuous operators from $X$ into $Y$ and by $\mathcal{L}\left(X^{i},Y\right)$ the set of $i$ -linear continuous operators from $X^{i}$ into $Y .$

Definition 2.1.

[14] Given the distinct points $x_{1},x_{2}\in X,$ the mapping $\left[x_{1},x_{2};F\right]\in\mathcal{L}\left(X,Y\right)$ is called the first order divided difference of $F$ at the points $x_{1},x_{2},$ if

a) $\left[x_{1},x_{2};F\right]\left(x_{2}-x_{1}\right)=F\left(x_{2}\right)-F\left(% x_{1}\right);$

b) if $F$ is Fréchet differentiable at $x_{s}$ then $\left[x_{s},x_{s};F\right]=F^{\prime}\left(x_{s}\right).$

The higher order divided differences are constructed recursively.

Definition 2.2.

[14] Given the distinct points $x_{1},\ldots,x_{i+1}\in X,$ $i\geq 2$ , the mapping $\left[x_{1},\ldots,x_{i+1};F\right]\in\mathcal{L}\left(X^{i},Y\right)$ is called the $i$ -th order divided difference of $F$ at the points $x_{1},\ldots,x_{i+1}$ if

a) $\left[x_{1},\ldots,x_{i+1};F\right]\left(x_{i+1}-x_{1}\right)=\left[x_{2},% \ldots,x_{i+1};F\right]-\left[x_{1},\ldots,x_{i};F\right];$

b) if there exists the $i$ -th order Fréchet derivative of $F$ at $x_{s}\in X,$ then

\left[x_{s},\ldots,x_{s};F\right]=\tfrac{1}{i!}F^{\left(i\right)}\left(x_{s}% \right).

Definition 2.3.

[14] The divided difference $\left[x_{1},\ldots,x_{i+1};F\right]$ is called symmetric with respect to $x_{1},\ldots,x_{i+1}$ if the equalities

(1)

\left[x_{1},\ldots,x_{i+1};F\right]=\left[x_{k_{1}},\ldots,x_{k_{i+1}};F\right]

hold for any permutation $\left(k_{1},\ldots,k_{i+1}\right)$ of $\left(1,2,\ldots,i+1\right).$

Remark.

When $X=\mathbb{R}$ then it is well known that the equalities (1) hold. However these equalities do not generally hold for any normed space $X .$

Example 2.1.

Let $X=\mathbb{R}^{2},$ $Y=\mathbb{R}$ and $F:\mathbb{R}^{2}\rightarrow\mathbb{R}$ . Denote $x_{i}=\left(u_{i},v_{i}\right)\in\mathbb{R}^{2},$ $i=1,2,$ $u_{1}\neq u_{2}$ and $v_{1}\neq v_{2}.$ Any of the following two expressions define a first order divided difference of $F$ at $x_{1}$ and $x_{2}$ :

\left[x_{2},x_{1};F\right]=\left(\tfrac{F\left(u_{1},v_{1}\right)-F\left(u_{2}% ,v_{1}\right)}{u_{1}-u_{2}},\tfrac{F\left(u_{2},v_{1}\right)-F\left(u_{2},v_{2% }\right)}{v_{1}-v_{2}}\right)

\left[x_{1},x_{2};F\right]=\left(\tfrac{F\left(u_{2},v_{2}\right)-F\left(u_{1}% ,v_{2}\right)}{u_{2}-u_{1}},\tfrac{F\left(u_{1},v_{2}\right)-F\left(u_{1},v_{1% }\right)}{v_{2}-v_{1}}\right).

It is obvious that in general $\left[x_{1},x_{2};F\right]\neq\left[x_{2},x_{1};F\right],$ which means that these divided differences are not symmetric with respect to the points $x_{1},x_{2}.$

For symmetric divided differences we may consider in this case the expression

	$\displaystyle\left[x_{1},x_{2};F\right]$	$\displaystyle=$	$\displaystyle\tfrac{1}{2}\left(\tfrac{F\left(u_{1},v_{1}\right)-F\left(u_{2},v% _{1}\right)}{u_{1}-u_{2}}+\tfrac{F\left(u_{2},v_{2}\right)-F\left(u_{1},v_{2}% \right)}{u_{2}-u_{1}},\right.$
			$\displaystyle\qquad\left.\tfrac{F\left(u_{2},v_{1}\right)-F\left(u_{2},v_{2}% \right)}{v_{1}-v_{2}}+\tfrac{F\left(u_{1},v_{2}\right)-F\left(u_{1},v_{1}% \right)}{v_{2}-v_{1}}\right).$

Example 2.2.

Let $V$ be a Banach space over the field $\mathbb{K}$ ( $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ ) and consider $A\in\mathcal{L}\left(V\right)$ a linear continuous operator on $V .$ The scalar $\lambda\in\mathbb{K}$ is an eigenvalue of $A$ if the equation

Av-\lambda v=\theta

has at least a solution $v\neq\theta$ . In this case $v$ is an eigenvector of $A$ corresponding to $\lambda$ .

For determining an eigenpair $\left(v,\lambda\right)$ consider a linear continuous mapping $G:V\rightarrow\mathbb{K}$ and the Banach space $X=V\times\mathbb{K}$ with the norm $\left\|x\right\|=\max\left\{\left\|v\right\|,\left|\lambda\right|\right\},$ $x=\tbinom{v}{\lambda},$ $v\in V,$ $\lambda\in\mathbb{K}.$ Let $F:X\rightarrow X$ be defined

(2)

F\left(x\right)=\left(\begin{array}[]{c}Av-\lambda v\\ G(v)-1\end{array}\right).

Denoting by $\theta_{1}=\tbinom{\theta}{0}$ the null element of $X$ then the eigenpairs are the solutions of the equation

F\left(x\right)=\theta_{1}.

We shall construct for $F$ the divided differences, and we shall show that for $i\geq 3$ the $i$ -th order divided differences are the $i$ -linear null operators.

Remark.

When $V=\mathbb{K}^{n},$ there is usually considered the quadratic function $G\left(v\right)=\frac{1}{2}v^{t}v-1,$ instead of a linear one. We have shown in [3] and [4] that for the choice $G\left(v\right)=\frac{1}{2n}v^{t}v-1$ the norm of the second derivative has the constant value $2,$ smaller when compared to the constant value $n,$ which corresponds to the second quadratic function. ∎

The divided differences of the corresponding operators $F$ may be easily constructed.

Let $x_{i}=\tbinom{v_{i}}{\lambda_{i}}\in X,$ $i=\overline{1,3}$ and $k_{i}=\tbinom{h_{i}}{\alpha_{i}}\in X,\quad i=1,2.$

For determining the first order divided differences of $F$ we have that

F\left(x_{2}\right)-F\left(x_{1}\right)\!=\!\left(\!\!\begin{array}[]{c}A\left% (v_{2}-v_{1}\right)-\frac{1}{2}\left[\left(\lambda_{1}+\lambda_{2}\right)\left% (v_{2}-v_{1}\right)+\left(\lambda_{2}-\lambda_{1}\right)\left(v_{2}+v_{1}% \right)\right]\\ G\left(v_{2}-v_{1}\right)\end{array}\!\!\right)

whence it follows that

\left[x_{1},x_{2};F\right]k_{1}=\left(\begin{array}[]{c}Ah_{1}-\frac{1}{2}% \left(\left(\lambda_{1}+\lambda_{2}\right)h_{1}+\alpha_{1}\left(v_{2}+v_{1}% \right)\right)\\ Gh_{1}\end{array}\right)

Using the above formula we get for the second order divided differences of $F$ that

\left[x_{1},x_{2},x_{3};F\right]k_{1}k_{2}=\left(\begin{array}[]{c}-\frac{1}{2% }\left(\alpha_{2}h_{1}+\alpha_{1}h_{2}\right)\\ 0\end{array}\right).

Since the above divided difference does not depend on $x_{i},$ $i=\overline{1,3},$ it follows that the higher order divided differences are the null multilinear operators.

Consider now the matrix $A\in\mathcal{M}_{n}\left(\mathbb{K}\right).$ For $V=\mathbb{K}^{n}$ and $X=\mathbb{K}^{n+1},$ the operator (2) is given by the relations

(3)

F_{i}\left(x\right)=F_{i}\left(x^{1},\ldots,x^{n+1}\right),\quad i=\overline{1% ,n+1},

where

F_{i}\left(x\right)=a_{i1}x^{1}+\ldots+a_{i,i-1}x^{i-1}+\left(a_{ii}-x^{n+1}% \right)x^{i}+a_{i,i+1}x^{i+1}+\ldots+a_{in}x^{n},

$i=\overline{1,n}$ , and for $G\left(v\right)-1$ we may take

F_{n+1}\left(x\right)=x^{i_{0}}-1,

where $i_{0}\in\left\{1,\ldots,n\right\}$ is fixed. We have denoted $x=\left(x^{1},\ldots,x^{n+1}\right),$ $x^{n+1}$ being an unknown eigenvalue of $A,$ and $v=\left(x^{1},\ldots,x^{n}\right)$ a corresponding eigenvector.

The matrices associated to the first order divided differences of $F$ at the points $x_{1},x_{2}\in\mathbb{K}^{n+1}$ are

\left[x_{1},x_{2};F\right]=\left(\begin{array}[]{ccccccc}b_{11}&a_{12}&\cdots&% a_{1i_{0}}&\cdots&a_{1n}&a_{1,n+1}\\ a_{21}&b_{22}&\cdots&a_{2i_{0}}&\cdots&a_{2n}&a_{2,n+1}\\ \vdots&\vdots&&\vdots&&\vdots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{ni_{0}}&\cdots&b_{nn}&a_{n,n+1}\\ 0&0&\cdots&1&\cdots&0&0\end{array}\right)

with $b_{ii}=a_{ii}-\frac{1}{2}\left(x_{2}^{n+1}+x_{1}^{n+1}\right),$ $i=\overline{1,n}$ and $a_{i,n+1}^{1}=-\frac{1}{2}\left(x_{2}^{i}+x_{1}^{i}\right),$ $i=\overline{1,n}.$

Consider now $x_{3}\in\mathbb{K}^{n+1}$ and $h_{1}=\left(h_{1}^{1},\ldots,h_{1}^{n+1}\right),$ $h_{2}=\left(h_{2}^{1},\ldots,h_{2}^{n+1}\right)\in\mathbb{K}^{n+1}.$

The second order divided differences of $F$ at $x_{1},x_{2},x_{3}$ are

\left[x_{1},x_{2},x_{3};F\right]h_{1}h_{2}=\left(\!\!\begin{array}[]{ccccc}-% \frac{1}{2}h_{2}^{n+1}&0&\cdots&0&-\frac{1}{2}h_{2}^{1}\\ 0&-\frac{1}{2}h_{2}^{n+1}&\cdots&0&-\frac{1}{2}h_{2}^{2}\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-\frac{1}{2}h_{2}^{n+1}&-\frac{1}{2}h_{2}^{n}\\ 0&0&\cdots&0&0\end{array}\!\!\right)\!\left(\!\!\begin{array}[]{l}h_{1}^{1}\\ h_{1}^{2}\\ \vdots\\ h_{1}^{n}\\ h_{1}^{n+1}\end{array}\!\!\right)

for all $h_{1},h_{2}\in\mathbb{K}^{n+1}.$

It can be easily seen that the above finite differences are symmetric with respect to the considered points.

Example 2.3.

Let $T:X\rightarrow X$ be given by

T\left(x\right)=y+B_{1}\left(x\right)+B_{2}\left(x\right),

with $y\in X$ fixed, $B_{1}\in\mathcal{L}\left(X\right)$ and $B_{2}$ the restriction to the diagonal of $X\times X$ of a bilinear symmetric operator $\overline{B}_{2}\in\mathcal{L}\left(X^{2},X\right),$ i.e. $B_{2}\left(x\right)=\overline{B}_{2}\left(x,x\right).$

Let $x_{i}\in X,$ $i=\overline{1,3}$ be three distinct points. The first order divided differences of $T$ at $x_{1}$ and $x_{2}$ are

\left[x_{1},x_{2};T\right]h_{1}=B_{1}\left(h_{1}\right)+\overline{B}_{2}\left(% x_{1}+x_{2},h_{1}\right),\quad h_{1}\in X

while the second order ones are given by

\left[x_{1},x_{2},x_{3};T\right]h_{1}h_{2}=\overline{B}_{2}\left(h_{1},h_{2}% \right),\quad h_{1},h_{2}\in X.

It is clear that, in this case too, the higher order divided differences are the null multilinear operators.

Example 2.4.

Let $X=Y=C\left[0,1\right]$ be the Banach space of continuous functions on $\left[0,1\right],$ equipped with the $\max$ norm. We consider the mapping $F:C\left[0,1\right]\rightarrow C\left[0,1\right]$ given by

F\left(x\right)\left(s\right)=\int_{0}^{1}K\left(s,t,x\left(t\right)\right)dt

where $K:\left[0,1\right]\times\left[0,1\right]\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function.

The first order divided differences of $F$ at the points $x_{1},x_{2}\in C\left[0,1\right],$ when $x_{1}\left(t\right)\neq x_{2}\left(t\right),$ $t\in\left[0,1\right]$ is given by

\left[x_{1},x_{2};F\right]h=\int_{0}^{1}\frac{K\left(s,t,x_{2}\left(t\right)% \right)-K\left(s,t,x_{1}\left(t\right)\right)}{x_{2}\left(t\right)-x_{1}\left(% t\right)}h\left(t\right)dt,\quad h\in C\left[0,1\right].

Indeed, it can be seen that

	$\displaystyle\left[x_{1},x_{2};F\right]\left(x_{2}-x_{1}\right)$	$\displaystyle=$	$\displaystyle\int_{0}^{1}\tfrac{K\left(s,t,x_{2}\left(t\right)\right)-K\left(s% ,t,x_{1}\left(t\right)\right)}{x_{2}\left(t\right)-x_{1}\left(t\right)}\big(x_% {2}\left(t\right)-x_{1}\left(t\right)\big)dt$
		$\displaystyle=$	$\displaystyle F\left(x_{2}\right)-F\left(x_{1}\right),$

and so relation $a)$ from Definition 2.1 is satisfied.

If $K$ admits a partial derivative with respect to the third argument on $\mathbb{R}$ then relation $b)$ from Definition 2.1 is also verified for all $\left(s,t,x\right)\in\left[0,1\right]\times\left[0,1\right]\times\mathbb{R}.$

It can be easily seen that these divided differences are symmetric with respect to $x_{1}$ and $x_{2}.$

For the higher order divided differences we consider $x_{i}\in C\left[0,1\right]$ , $i=\overline{1,n+1}$ with $x_{i}\left(t\right)\neq x_{j}\left(t\right)$ for $i\neq j$ and $t\in\left[0,1\right].$

Denote

	$\displaystyle\left[x_{i},x_{i+1};K\right]$	$\displaystyle=\frac{K\left(s,t,x_{i+1}\left(t\right)\right)-K\left(s,t,x_{i}% \left(t\right)\right)}{x_{i+1}\left(t\right)-x_{i}\left(t\right)},\quad i=% \overline{1,n},$
	$\displaystyle\left[x_{i},x_{i+1},x_{i+2};K\right]$	$\displaystyle=\frac{\left[x_{i+1},x_{i+2};K\right]-\left[x_{i},x_{i+1};K\right% ]}{x_{i+2}\left(t\right)-x_{i}\left(t\right)},\quad i=\overline{1,n-1},$
	$\displaystyle\left[x_{i},x_{i+1},\ldots,x_{i+s};K\right]$	$\displaystyle=\frac{\left[x_{i+1},\ldots,x_{i+s};K\right]-\left[x_{i},\ldots,x% _{i+s-1};K\right]}{x_{i+s}\left(t\right)-x_{i}\left(t\right)},\ \ \!i=\!% \overline{1\!,\!n\!\!-\!\!s\!\!+\!\!1}.$

With these notations, we easily obtain that

\left[x_{i},x_{i+1},x_{i+2};F\right]h_{1}h_{2}=\int_{0}^{1}\left[x_{i},x_{i+1}% ,x_{i+2};K\right]h_{1}\left(t\right)h_{2}\left(t\right)dt,\quad i=\overline{1,% n-1}

and, in general,

\left[x_{i},\ldots,x_{i+s};F\right]h_{1}\ldots h_{s}=\int_{0}^{1}\left[x_{i},% \ldots,x_{i+s};K\right]h_{1}\left(t\right)\ldots h_{s}\left(t\right)dt,\ \ i\!% =\!\overline{1\!,\!n\!\!-\!\!s\!\!+\!\!1}.

3. Interpolation

Let $F:X\rightarrow Y$ be a mapping and $x_{i}\in X,$ $i=\overline{1,n+1},$ some distinct points. Consider the polynomial operator $L_{n}:X\rightarrow Y$ given by

(4)

L_{n}\left(x\right)=F\left(x_{1}\right)+\left[x_{1},x_{2};F\right]\left(x-x_{1% }\right)+\ldots+\left[x_{1},x_{2},\ldots,x_{n+1};F\right]\!\left(x\!\!-\!\!x_{% n}\right)\!\ldots\!\left(x\!\!-\!\!x_{1}\right).

The following result holds.

Theorem 3.1.

[14] If the mapping $F$ admits symmetric divided differences from the first order to the $n+1$ -th order with respect to the points $x_{i},x\in X,$ $i=\overline{1,n+1}$ then the following relations hold:

(5)

L_{n}\left(x_{i}\right)=F\left(x_{i}\right),\quad i=\overline{1,n+1}

(6)

F\left(x\right)-L_{n}\left(x\right)=\left[x,x_{1},\ldots,x_{n+1};F\right]\left% (x-x_{n+1}\right)\ldots\left(x-x_{1}\right).

Let $D\subseteq X$ be an open subset and suppose that the restriction $G=F|_{D}$ is a homeomorphism between $D$ and $F\left(D\right).$ The following result can be easily obtained:

Lemma 3.2.

[14] If, for some $x_{1},x_{2}\in D,$ the mapping $\left[x_{1},x_{2};G\right]\in\mathcal{L}\left(X,Y\right)$ is invertible, then

\left[y_{1},y_{2};G^{-1}\right]=\left[x_{1},x_{2};G\right]^{-1},

where $G^{-1}$ is the inverse of $G$ and $y_{i}=G\left(x_{i}\right),$ $i=1,2.$

4. Iterative methods for second degree polynomial operator equations.

In the following we shall consider second degree polynomial operator equations

(7)

F\left(x\right)=0,

with $F:X\rightarrow X$ which satisfies then

(8)

\left[x,y,z,t;F\right]=\theta_{3},\quad\forall x,y,z,t\in X

$\theta_{3}$ being the trilinear null operator.

It follows that $F$ also satisfies

(9)

F\left(x\right)=F\left(x_{1}\right)+\left[x_{1},x_{2};F\right]\left(x-x_{1}% \right)+\left[x_{1},x_{2},x;F\right]\left(x-x_{2}\right)\left(x-x_{1}\right)

where $\left[x_{1},x_{2},x;F\right]$ does not depend on $x_{1},x_{2}$ and $x$ (see also Examples 2.2 and 2.3).

We are interested in the convergence of the chord method and Steffensen method for this equation.

The chord method is given by the iteration

(10)

x_{k+1}=x_{k-1}-\left[x_{k-1},x_{k};F\right]^{-1}F\left(x_{k-1}\right),\quad k% =1,2\ldots,\;x_{0},x_{1}\in X

or, equivalently,

(11)

x_{k+1}=x_{k}-\left[x_{k-1},x_{k};F\right]^{-1}F\left(x_{k}\right),\quad k=1,2% ,\ldots,\;x_{0},x_{1}\in X.

For the Steffensen method, there is considered an auxiliary function $g:X\rightarrow X$ and the equation

(12)

x-g\left(x\right)=0,

equivalent to (7). The sequence $\left(x_{k}\right)_{k\geq 0}$ is then constructed by the iteration

(13)

x_{k+1}=x_{k}-\left[x_{k},g\left(x_{k}\right);F\right]^{-1}F\left(x_{k}\right)% ,\quad k=0,1,\ldots,\;x_{0}\in X

(14)

x_{k+1}=g\left(x_{k}\right)-\left[x_{k},g\left(x_{k}\right);F\right]^{-1}F% \left(g\left(x_{k}\right)\right),\quad k=0,1,\ldots,\;x_{0}\in X.

It can be easily verified that the two above relations define the same sequence.

Because of the special form of equation (7), we shall see that the conditions for the convergence of the sequences generated by (10)–(11) and (13)–(14) are much simplified compared to the general case of an arbitrary equation (7).

The convergence of the chord method. Let $\alpha=\left\|\left[x,y,z;F\right]\right\|$ and $x_{0}\in X.$ Denote $B\left(x_{0},r\right)=\left\{x\in X\;|\;\left\|x-x_{0}\right\|\leq r\right\},% \;r>0.$ Consider $x_{1}\in B\left(x_{0},r\right)$ and $d_{0}=\left\|x_{1}-x_{0}\right\|.$ The following result holds:

Theorem 4.1.

If the mapping $F$ and the elements $x_{0},x_{1}\in X,$ and $\alpha,$ $r,$ $b_{0}$ , $d_{0}\in\mathbb{R}_{+}$ satisfy

i. there exists $\left[x_{0},x_{1};F\right]^{-1}$

ii. $\quad b_{0}\,\alpha\left(2r+d_{0}\right)=q<1,$ with $b_{0}=\big\|\left[x_{0},x_{1};F\right]^{-1}\big\|$

iii. $\alpha b^{2}\left\|F\left(x_{0}\right)\right\|=\varepsilon_{0}<1,$ $\alpha b^{2}\left\|F\left(x_{1}\right)\right\|\leq\varepsilon_{0}^{l},$ where $b=\frac{b_{0}}{1-q}$ and $l=\frac{1+\sqrt{5}}{2}$

iv. $\displaystyle\frac{\varepsilon_{0}^{l}}{\alpha b\left(1-\varepsilon_{0}\right)% }+d_{0}\leq r,$
then the sequence $\left(x_{k}\right)_{k\geq 0}$ generated by $(\ref{f4.4})$ is well defined and its elements belong to $B\left(x_{0},r\right).$ Moreover, $\left(x_{k}\right)_{k\geq 0}$ is convergent, and denoting $x^{*}=\lim\limits_{k\rightarrow\infty}x_{k},$ then $F\left(x^{*}\right)=0.$ The following estimation also holds:

\left\|x^{*}-x_{k}\right\|\leq\frac{\varepsilon_{0}^{l^{k}}}{\alpha\,b\left(1-% \varepsilon_{0}^{l}\right)}.

Proof.

We shall first prove that if $x,y\in B\left(x_{0},r\right)$ then there exists $\left[x,y;F\right]^{-1}$ and

(15)

\big\|\left[x,y;F\right]^{-1}\big\|\leq b.

Let $T=\left[x_{0},x_{1};F\right]^{-1}\big(\left[x_{0},x_{1};F\right]-\left[x,y;F% \right]\big).$

It can be easily seen that

\left\|T\right\|\leq b_{0}\alpha\left(2r+d_{0}\right),

whence by $ii)$ it follows the existence of $\left(I-T\right)^{-1}$ and the inequality (15).

Now we show that $x_{2}\in B\left(x_{0},r\right).$ From $i)$ it follows that $x_{2}$ is well defined, and from (11), for $k=1$ we get

\left\|x_{2}-x_{1}\right\|\leq b\left\|F\left(x_{1}\right)\right\|=\frac{% \varepsilon_{0}^{l}}{\alpha b}.

Using this inequality and the hypothesis $iv)$ one obtains

\left\|x_{2}-x_{0}\right\|\leq\left\|x_{2}-x_{1}\right\|+\left\|x_{1}-x_{0}% \right\|\leq\frac{\varepsilon_{0}^{l}}{\alpha b\left(1-\varepsilon_{0}\right)}% +d_{0}\leq r,

i.e. $x_{2}\in B\left(x_{0},r\right).$

Now, using (9) with $x=x_{2}$ and taking into account (10)–(11) it follows that

\left\|F\left(x_{2}\right)\right\|\leq\alpha b^{2}\left\|F\left(x_{1}\right)% \right\|\cdot\left\|F\left(x_{0}\right)\right\|,

whence, by $iii),$

\left\|F\left(x_{2}\right)\right\|\leq\frac{1}{\alpha b^{2}}\varepsilon_{0}^{1% +l}=\frac{1}{\alpha b^{2}}\varepsilon_{0}^{l^{2}}.

Let $k\in\mathbb{N},$ and suppose that $x_{k-1},x_{k}\in B\left(x_{0},r\right),$ and

\left\|F\left(x_{k-1}\right)\right\|\leq\frac{1}{\alpha b^{2}}\varepsilon_{0}^% {l^{k-1}}

\left\|F\left(x_{k}\right)\right\|\leq\frac{1}{\alpha b^{2}}\varepsilon_{0}^{l% ^{k}}.

It easily follows that $x_{k+1}$ is well defined, that

\left\|x_{k+1}-x_{k}\right\|\leq b\left\|F\left(x_{k}\right)\right\|\leq\frac{% 1}{\alpha b}\varepsilon_{0}^{l^{k}}

and that

	$\displaystyle\left\\|x_{k+1}-x_{0}\right\\|$	$\displaystyle\leq\sum\limits_{i=0}^{k}\left\\|x_{i+1}-x_{i}\right\\|$
		$\displaystyle=\left\\|x_{1}-x_{0}\right\\|+\sum\limits_{i=1}^{k}\left\\|x_{i+1}-x% _{i}\right\\|$
		$\displaystyle\leq d_{0}+\tfrac{1}{\alpha b}\sum\limits_{i=1}^{k}\varepsilon_{0% }^{l^{i}}$
		$\displaystyle\leq d_{0}+\tfrac{\varepsilon_{0}^{l}}{\alpha b\left(1-% \varepsilon_{0}\right)}\leq r$

i.e. $x_{k+1}\in B\left(x_{0},r\right).$

Using (9) we get

\left\|F\left(x_{k+1}\right)\right\|\leq\alpha b^{2}\left\|F\left(x_{k}\right)% \right\|\cdot\left\|F\left(x_{k-1}\right)\right\|\leq\tfrac{1}{\alpha b^{2}}% \varepsilon_{0}^{\left(l+1\right)l^{k-1}}=\tfrac{1}{\alpha b^{2}}\varepsilon_{% 0}^{l^{k+1}}.

It can be easily shown that $\left(x_{k}\right)_{k\geq 0}$ is a Cauchy sequence, hence it converges and it satisfies the stated estimation.

The convergence of the Steffensen method. We shall consider that the mapping $g$ from (12) is given by

(16)

g\left(x\right)=x-\lambda F\left(x\right),

with $\lambda\in\mathbb{K}$ a fixed scalar.

In this assumption, the following relations are obvious:

(17)		$\displaystyle F\left(g\left(x\right)\right)-F\left(x\right)$	$\displaystyle=-\lambda\left[x,g\left(x\right);F\right]F\left(x\right)$
(18)		$\displaystyle\left[x,y;g\right]$	$\displaystyle=I-\lambda\left[x,y;F\right],$

$I$ being the identity mapping on $X .$

From (17) it follows that

(19)

\left\|F\left(g\left(x\right)\right)\right\|\leq\left\|I-\lambda\left[x,g\left% (x\right);F\right]\right\|\cdot\left\|F\left(x\right)\right\|.

Denote again by $\alpha$ the constant expression $\left\|\left[x,y,z;F\right]\right\|$ and let $x_{0}\in X.$

Assume that

(20)

\left\|I-\lambda\left[x,y;F\right]\right\|\leq\gamma,\quad\text{ for all }x,y% \in B\left(x_{0},r\right),

and some $\gamma>0.$ The following theorem hold.

Theorem 4.2.

If $x_{0}\in X,$ the mapping $F$ and the positive numbers $\alpha,\beta,\gamma,r$ are such that

i.

$g\left(x_{0}\right)\in B\left(x_{0},r\right)$ and there exists $\left[x_{0},g\left(x_{0}\right);F\right]^{-1};$
ii.

$p=\beta\alpha\left(\gamma+1\right)r<1,$ with $\beta=\big\|\left[x_{0},g\left(x_{0}\right);F\right]^{-1}\big\|;$
iii.

$\delta_{0}=\alpha\gamma\delta^{2}\left\|F\left(x_{0}\right)\right\|<1,$ with $\delta=\frac{\beta}{1-p};$
iv.

$\max\big\{\frac{\delta_{0}}{\alpha\gamma\delta\left(1-\delta_{0}\right)},\frac% {\delta_{0}}{\alpha\delta}\big(\frac{1}{1-\delta_{0}}+\frac{\left|\lambda% \right|}{\gamma\delta}\big)\big\}\leq r$

then the following relations hold:

j. the sequence $\left(x_{k}\right)_{k\geq 0}$ given by $(\ref{f4.7})$ is well defined and converges.

jj. denoting $x^{*}=\lim\limits_{k\rightarrow\infty}x_{k}$ then $F\left(x^{*}\right)=0.$

jjj. $\left\|x^{*}-x_{k}\right\|\leq\frac{\delta_{0}^{2^{k}}}{\alpha\delta\gamma% \left(1-\delta_{0}\right)},$ $k=0,1,\ldots$

Proof.

For $x,g\left(x\right)\in B\left(x_{0},r\right),$ using that

\big\|\left[x_{0},g\left(x_{0}\right);F\right]^{-1}\big(\left[x_{0},g\left(x_{% 0}\right);F\right]-\left[x,g\left(x\right);F\right]\big)\big\|\leq\beta\alpha% \left(\gamma+1\right)r=p<1

it follows that there exists $\left[x,g\left(x\right);F\right]^{-1}$ and

(21)

\big\|\left[x,g\left(x\right);F\right]^{-1}\big\|\leq\tfrac{\beta}{1-p}=\delta.

We shall use the identity

	$\displaystyle F\left(y\right)$	$\displaystyle=$	$\displaystyle F\left(x\right)+\left[x,g\left(x\right);F\right]\left(y-x\right)$
			$\displaystyle+\left[y,x,g\left(x\right);F\right]\left(y-x\right)\left(y-g\left% (x\right)\right),\;\forall x,y\in X.$

From (13) for $k=0$ we get

\left\|x_{1}-x_{0}\right\|\leq\beta\left\|F\left(x_{0}\right)\right\|\leq% \delta\left\|F\left(x_{0}\right)\right\|\leq\tfrac{\alpha\gamma\delta^{2}}{% \alpha\gamma\delta}\left\|F\left(x_{0}\right)\right\|\leq\tfrac{\delta_{0}}{% \alpha\gamma\delta\left(1-\delta_{0}\right)}\leq r,

i.e. $x_{1}\in B\left(x_{0},r\right).$

Further,

	$\displaystyle\left\\|g\left(x_{1}\right)-g\left(x_{0}\right)\right\\|$	$\displaystyle\leq\left\\|g\left(x_{1}\right)-g\left(x_{0}\right)\right\\|+\left% \\|g\left(x_{0}\right)-x_{0}\right\\|$
		$\displaystyle\leq\gamma\left\\|x_{1}-x_{0}\right\\|+\left\|\lambda\right\|\left\\|F% \left(x_{0}\right)\right\\|$
		$\displaystyle\leq\tfrac{\delta_{0}}{\alpha\delta\left(1-\delta_{0}\right)}+% \tfrac{\left\|\lambda\right\|}{\gamma\alpha\delta^{2}}\delta_{0}$
		$\displaystyle\leq\tfrac{\delta_{0}}{\alpha\delta}\left(\tfrac{1}{1-\delta_{0}}% +\tfrac{\left\|\lambda\right\|}{\gamma\delta}\right)\leq r$

which shows that $g\left(x_{1}\right)\in B\left(x_{0},r\right).$ Taking into account (12) it results that

F\left(x_{1}\right)\!=\!F\left(x_{0}\right)+\left[x_{0},g\left(x_{0}\right);F% \right]\!\left(x_{1}\!-\!x_{0}\right)+\left[x_{1},x_{0},g\left(x_{0}\right);F% \right]\!\left(x_{1}\!-\!x_{0}\right)\!\left(x_{1}\!-\!g\left(x_{0}\right)\right)

and by (13), (14) and (17) we get

\left\|F\left(x_{1}\right)\right\|\leq\alpha\delta^{2}\gamma\left\|F\left(x_{0% }\right)\right\|^{2},

whence for $\delta_{1}=\alpha\delta^{2}\gamma\left\|F\left(x_{1}\right)\right\|$ we obtain

\delta_{1}\leq\delta_{0}^{2}.

Suppose now that $x_{k},$ $g\left(x_{k}\right)\in B\left(x_{0},r\right)$ and that

\left\|F\left(x^{i}\right)\right\|\leq\frac{\delta_{0}^{2^{i}}}{\alpha\delta^{% 2}\gamma},\qquad i=\overline{1,k}.

Then obviously $x_{k+1}$ is well defined and

\left\|x_{k+1}-x_{k}\right\|\leq\delta\left\|F\left(x_{k}\right)\right\|,

and

\left\|x_{k+1}-g\left(x_{k}\right)\right\|\leq\delta\gamma\left\|F\left(x_{k}% \right)\right\|.

For proving that $x_{k+1},$ $g\left(x_{k+1}\right)\in B\left(x_{0},r\right),$ first we have for $x_{k+1}$ that

	$\displaystyle\left\\|x_{k+1}-x_{0}\right\\|$	$\displaystyle\leq$	$\displaystyle\sum\limits_{i=0}^{k}\left\\|x_{i+1}-x_{i}\right\\|\leq\sum_{i=0}^{% k}\delta\left\\|F\left(x_{i}\right)\right\\|\leq\sum_{i=0}^{k}\delta\frac{\delta% _{0}^{2^{i}}}{\alpha\delta^{2}\gamma}\leq$
		$\displaystyle\leq$	$\displaystyle\frac{\delta_{0}}{\alpha\delta\gamma\left(1-\delta_{0}\right)}% \leq r,$

and for $g\left(x_{k+1}\right)$ we obtain

$\displaystyle\left\\|g\left(x_{k+1}\right)-x_{0}\right\\|$	$\displaystyle\leq$	$\displaystyle\left\\|g\left(x_{k+1}\right)-g\left(x_{0}\right)\right\\|+\left\\|g% \left(x_{0}\right)-x_{0}\right\\|$
	$\displaystyle\leq$	$\displaystyle\left\\|\left[x_{0},x_{k+1};g\right]\right\\|\cdot\left\\|\left[x_{k% +1}-x_{0}\right]\right\\|+\left\|\lambda\right\|\cdot\left\\|F\left(x_{0}\right)\right\\|$
	$\displaystyle\leq$	$\displaystyle\tfrac{\delta_{0}}{\alpha\delta\left(1-\delta_{0}\right)}+\tfrac{% \left\|\lambda\right\|\delta_{0}}{\alpha\delta^{2}\gamma}=\tfrac{\delta_{0}}{% \alpha\delta}\left(\tfrac{1}{1-\delta_{0}}+\tfrac{\left\|\lambda\right\|}{\delta% \gamma}\right)\leq r.$

It can be easily seen that $\left(x_{k}\right)_{k\geq 0}$ is a Cauchy sequence, and hence it converges.

The stated evaluation of the error can be obtained from

\left\|x_{k+m}-x_{k}\right\|\leq\frac{\delta_{0}^{2^{k}}}{\alpha\delta\gamma% \left(1-\delta_{0}\right)}

for $m\rightarrow\infty$ . The theorem is proved. ∎

References

[1] M.P. Anselone, L.B. Rall, The solution of characteristic value-vector problems by Newton method, Numer. Math. 11 (1968), pp. 38–45.
[2] I.K. Argyros, Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Austral. Math. Soc. 38 (1988), pp. 275–292.
[3] E. Cătinaş, I. Păvăloiu, On the Chebyshev method for approximating the eigenvalues of linear operators, Rev. Anal. Numér. Théor. Approx. 25 (1996) nos. 1–2, pp. 43–56.
[4] E. Cătinaş, I. Păvăloiu, On a Chebyshev-type method for approximating the solutions of polynomial operator equations of degree 2, Proceedings of International Conference on Approximation and Optimization, Cluj-Napoca, July 29 - August 1, 1996, vol. 1, pp. 219–226.
[5] F. Chatelin, Valeurs propres de matrices, Mason, Paris, 1988.
[6] P.G. Ciarlet, Introduction à l’analyse numérique matricielle et à l’optimisation, Mason, Paris, 1990.
[7] L. Collatz, Functionalanalysis und Numerische Mathematik, Springer-Verlag, Berlin, 1964.
[8] A. Diaconu, On the convergence of an iterative method of Chebyshev type, Rev. Anal. Numér. Théor. Approx. 24 (1995) nos. 1–2, pp. 91–102.
[9] A. Diaconu, I. Păvăloiu, Sur quelque méthodes itératives pour la résolution des équations opérationelles, Rev. Anal. Numér. Théor. Approx. 1 (1972) no. 1, pp. 45–61.
[10] V.S. Kartîşov, F.L. Iuhno, O nekotorîh Modifikaţiah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci, J. Vîcisl. matem. i matem. fiz. (33) (1973) 9, pp. 1403–1409.
[11] I. Lazǎr, On a Newton type method, Rev. Anal. Numér. Théor. Approx. 23 (1994) no. 2, pp. 167–174.
[12] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
[13] I. Păvăloiu, Sur les procédés itératifs à un ordre élevé de convergence, Mathematica (Cluj) 12 (35) (1970) no. 2, pp. 309–324.
[14] I. Păvăloiu, Introduction in the Approximation Theory for the Solutions of Equations, Ed. Dacia, Cluj-Napoca, 1986 (in Romanian).
[15] I. Păvăloiu, Observations concerning some approximation methods for the solutions of operator equations, Rev. Anal. Numér. Théor. Approx. 23 (1994) no. 2, pp. 185–196.
[16] I. Păvăloiu, Approximation of the root of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32 (1995) nos 1–2, pp. 69–82.
[17] R.A. Tapia, L.D. Whitley, The projected Newton method has order $1+\sqrt{2}$ for the symmetric eigenvalue problem, SIAM J. Numer. Anal. 25 (1988) no. 6, pp. 1376–1382.
[18] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1964.
[19] S. Ul’m, On the iterative method with simultaneous approximation of the inverse of the operator, Izv. Acad. Nauk. Estonskoi S.S.R. 16 (1967) no. 4, pp. 403–411.
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Received 15.08.1996

”T. Popoviciu” Institute of Numerical Analysis

Romanian Academy

P.O. Box 68, 3400 Cluj-Napoca 1

Romania

e-mail: ecatinas@ictp.acad.ro

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	$\displaystyle\left\\|x_{k+1}-x_{0}\right\\|$	$\displaystyle\leq\sum\limits_{i=0}^{k}\left\\|x_{i+1}-x_{i}\right\\|$
		$\displaystyle=\left\\|x_{1}-x_{0}\right\\|+\sum\limits_{i=1}^{k}\left\\|x_{i+1}-x% _{i}\right\\|$
		$\displaystyle\leq d_{0}+\tfrac{1}{\alpha b}\sum\limits_{i=1}^{k}\varepsilon_{0% }^{l^{i}}$
		$\displaystyle\leq d_{0}+\tfrac{\varepsilon_{0}^{l}}{\alpha b\left(1-% \varepsilon_{0}\right)}\leq r$

	$\displaystyle\left\\|g\left(x_{1}\right)-g\left(x_{0}\right)\right\\|$	$\displaystyle\leq\left\\|g\left(x_{1}\right)-g\left(x_{0}\right)\right\\|+\left% \\|g\left(x_{0}\right)-x_{0}\right\\|$
		$\displaystyle\leq\gamma\left\\|x_{1}-x_{0}\right\\|+\left\|\lambda\right\|\left\\|F% \left(x_{0}\right)\right\\|$
		$\displaystyle\leq\tfrac{\delta_{0}}{\alpha\delta\left(1-\delta_{0}\right)}+% \tfrac{\left\|\lambda\right\|}{\gamma\alpha\delta^{2}}\delta_{0}$
		$\displaystyle\leq\tfrac{\delta_{0}}{\alpha\delta}\left(\tfrac{1}{1-\delta_{0}}% +\tfrac{\left\|\lambda\right\|}{\gamma\delta}\right)\leq r$

$\displaystyle\left\\|g\left(x_{k+1}\right)-x_{0}\right\\|$	$\displaystyle\leq$	$\displaystyle\left\\|g\left(x_{k+1}\right)-g\left(x_{0}\right)\right\\|+\left\\|g% \left(x_{0}\right)-x_{0}\right\\|$
	$\displaystyle\leq$	$\displaystyle\left\\|\left[x_{0},x_{k+1};g\right]\right\\|\cdot\left\\|\left[x_{k% +1}-x_{0}\right]\right\\|+\left\|\lambda\right\|\cdot\left\\|F\left(x_{0}\right)\right\\|$
	$\displaystyle\leq$	$\displaystyle\tfrac{\delta_{0}}{\alpha\delta\left(1-\delta_{0}\right)}+\tfrac{% \left\|\lambda\right\|\delta_{0}}{\alpha\delta^{2}\gamma}=\tfrac{\delta_{0}}{% \alpha\delta}\left(\tfrac{1}{1-\delta_{0}}+\tfrac{\left\|\lambda\right\|}{\delta% \gamma}\right)\leq r.$

On some interpolatory iterative methods for the second degree polynomial operators (I)

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On Some Interpolatory Iterative Methods for the Second Degree Polynomial Operators (I)

1. Introduction

2. Divided differences

Definition 2.1.

Definition 2.2.

Definition 2.3.

Remark.

Example 2.1.

Example 2.2.

Remark.

Example 2.3.

Example 2.4.

3. Interpolation

Theorem 3.1.

Lemma 3.2.

4. Iterative methods for second degree polynomial operator equations.

Theorem 4.1.

Proof.

Theorem 4.2.

Proof.

References

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On Some Interpolatory Iterative Methods
for the Second Degree Polynomial Operators (I)