by

Abstract. In this paper we apply the metod of v. PTAK ([4], [5]) to the study of the coinvergance of a modified secant method. We prove that the rate of convargence of this method is of the form

where $a,d,H$$a,d,H$a,d,Ha, d, H$a,d,H$ and $r$$r$rr$r$ are positive mumbers depending on the initial conditions. We also give sharp estimates for the distance $\Vert x_n-x^{\ast }\Vert$$‖x_n-x^{\ast }‖$||x_(n)-x^(**)||\left\|x_{n}-x^{*}\right\|$\Vert x_n-x^{\ast }\Vert$, $n=1$$n=1$n=1n=1$n=1$, 2, ... where ${\left(x_n\right)}_{n=1}^{\mathrm{\infty }}$${\left(x_n\right)}_{n=1}^{\mathrm{\infty }}$(x_(n))_(n=1)^(oo)\left(x_{n}\right)_{n=1}^{\infty}${\left(x_n\right)}_{n=1}^{\mathrm{\infty }}$ is the sequence obtained by the modified secant method and $x$$x$xx$x$ * is its limit.

The method of Nondiscrete Mathematical Induction, introduced by V. PiAK [4], has allowed a new approach in the study of the convergence of iterative procedures. An important role in this approach is played by the notion of the rate of convergance $[5],[6]$$[5],[6]$[5],[6][5],[6]$[5],[6]$. Let $T$$T$TT$T$ be an interval of the form $T=\{r\in \mathit{R};0<r\leqq r_0\}$$T=\left\{r\in \mathit{R};0<r\leqq r_0\right\}$T={r in R;0 < r <= r_(0)}T=\left\{r \in \boldsymbol{R} ; 0<r \leqq r_{0}\right\}$T=\{r\in \mathit{R};0<r\leqq r_0\}$, for some positive $r_0(1.e.T^{-}]0,r_0])$$\left.\left.r_0\left(1.e.T^{-}\right]0,r_0\right]\right)${:r_(0)(1.e.T^(-)]0,r_(0)])\left.\left.r_{0}\left(1 . e . T^{-}\right] 0, r_{0}\right]\right)$r_0(1.e.T^{-}]0,r_0])$. Let $\omega$$\omega$omega\omega$\omega$ be a finction defined on $T$$T$TT$T$. We define by reccurence:

DEFINITION 1.1. Tha function o, defined on $T$$T$TT$T$, is called a rate of convergence, if it satisfies the folbowing properties:
(1) $\omega$$\omega$omega\omega$\omega$ maps $T$$T$TT$T$ into itself;
(2) for each $r\in T$$r\in T$r in Tr \in T$r\in T$ the series $\sum _{n=0}^{\mathrm{\infty }}{\omega }^n(r)$$\sum _{n=0}^{\mathrm{\infty }} {\omega }^n(r)$sum_(n=0)^(oo)omega^(n)(r)\sum_{n=0}^{\infty} \omega^{n}(r)$\sum _{n=0}^{\mathrm{\infty }}{\omega }^n(r)$ is convergent.

The sum of the above series, $\sigma (r)=\sum _{n=0}^{\mathrm{\infty }}{\omega }^n(r)$$\sigma (r)=\sum _{n=0}^{\mathrm{\infty }} {\omega }^n(r)$sigma(r)=sum_(n=0)^(oo)omega^(n)(r)\sigma(r)=\sum_{n=0}^{\infty} \omega^{n}(r)$\sigma (r)=\sum _{n=0}^{\mathrm{\infty }}{\omega }^n(r)$, obviously satisfies the following functional equation:

We shall justify the name of "rate of convergence", given to the function $\omega$$\omega$omega\omega$\omega$, after stating the Induction Theorem.

Let $(X,d)$$(X,d)$(X,d)(X, d)$(X,d)$ be a complete metric space. If $A$$A$AA$A$ is a subset of $X$$X$XX$X$, and $x$$x$xx$x$ an element of $X$$X$XX$X$, we shall denote by $d(x,A)$$d(x,A)$d(x,A)d(x, A)$d(x,A)$ the g.1.b. of the set $\{d(x,y)$$\{d(x,y)${d(x,y)\{d(x, y)$\{d(x,y)$; $y\in A\}$$y\in A\}$y in A}y \in A\}$y\in A\}$. For any positive number $r$$r$rr$r$ we shall denote by $U(A,r)$$U(A,r)$U(A,r)U(A, r)$U(A,r)$ the set $\{x\in X;d(x,A)\leqq r\}$$\{x\in X;d(x,A)\leqq r\}${x in X;d(x,A) <= r}\{x \in X ; d(x, A) \leqq r\}$\{x\in X;d(x,A)\leqq r\}$. If $x$$x$xx$x$ is an element of $X$$X$XX$X$, we shall write for simplicity $U(x,r)$$U(x,r)$U(x,r)U(x, r)$U(x,r)$ instead of $U(\{x\},r)$$U(\{x\},r)$U({x},r)U(\{x\}, r)$U(\{x\},r)$.

Let us denote by $T$$T$TT$T$ the interval $]0,r_0]$$\left.]0,r_0\right]${:]0,r_(0)]\left.] 0, r_{0}\right]$]0,r_0]$ of the real line, and for each $r\in T$$r\in T$r in Tr \in T$r\in T$, let $Z(r)$$Z(r)$Z(r)Z(r)$Z(r)$ represent a certain subset of $X$$X$XX$X$. We shall use the following notation for the limit of the family $Z($$Z($Z(Z($Z($.$).$$).$).) .$).$

Now, we can state the Induction Theorem [4].
theorem 1.1. If
(5)

for each $r\in T$$r\in T$r in Tr \in T$r\in T$, then

for each $r\in T$$r\in T$r in Tr \in T$r\in T$.
We shall sketch below how the method of nondiscrete mathematical induction can be applied to the study of the convergence of iterative procedures. Let $F$$F$FF$F$ be a mapping of the complete metric space $X$$X$XX$X$ into itself, and let $x_0$$x_0$x_(0)x_{0}$x_0$ be an element of $X$$X$XX$X$. Suppose that we can attach to the pair ( $F$$F$FF$F$, $x_0$$x_0$x_(0)x_{0}$x_0$ ) a rate of convergence $\omega$$\omega$omega\omega$\omega$ on the interval $T=]0,r_0$$\left.T=\right]0,r_0${:T=]0,r_(0)\left.T=\right] 0, r_{0}$T=]0,r_0$ ], and a family of sets $\{Z(r){\}}_{,\in T}$$\{Z(r){\}}_{,\in T}${Z(r)}_(,in T)\{Z(r)\}_{, \in T}$\{Z(r){\}}_{,\in T}$, such that the following relations be fulfiled:

Then the Induction Theorem assures the fact that $Z(0)\ne \varnothing$$Z(0)\ne \varnothing$Z(0)!=O/Z(0) \neq \varnothing$Z(0)\ne \varnothing$. On the other hand (8) implies that each element $\xi$$\xi$xi\xi$\xi$ of $Z(0)$$Z(0)$Z(0)Z(0)$Z(0)$ is a fixed element of the mapping $F$$F$FF$F$ i.e. $F(\xi )=\xi$$F(\xi )=\xi$F(xi)=xiF(\xi)=\xi$F(\xi )=\xi$. It also follows that via the iterative procedure:

We obtain al sequence ${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$(x_(n))_(n=0)^(oo)\left(x_{n}\right)_{n=0}^{\infty}${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$ which converges to an element $x^{\ast }\in Z(0)$$x^{\ast }\in Z(0)$x^(**)in Z(0)x^{*} \in Z(0)$x^{\ast }\in Z(0)$, such that the following inequalities are satisfied:

From (10) one obtains the following estimates of the distance between the $n^{\prime }$$n^{\prime }$n^(')n^{\prime}$n^{\prime }$ th iterate $x_n$$x_n$x_(n)x_{n}$x_n$ and the "starting point" $x_0$$x_0$x_(0)x_{0}$x_0$ :

The relation (11) will be called an apriori estimate fo the distance between the $n^{\prime }$$n^{\prime }$n^(')n^{\prime}$n^{\prime }$ th iterate given by the procedure (9) and the fixed point $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$. The name ,apriori estimate" is justified by the fact that one can compute this estimate before performing the iterative procedure.

Suppose, that for a certain $n\in \{1,2,\dots \}$$n\in \{1,2,\dots \}$n in{1,2,dots}n \in\{1,2, \ldots\}$n\in \{1,2,\dots \}$, one has already computed $x_1,x_2,\dots ,x_n$$x_1,x_2,\dots ,x_n$x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}$x_1,x_2,\dots ,x_n$. If

then it can easily be proved that the following inequality is satisfied:

The above estimate will be called an „aposteriori estimate“, because it can be computed only after performing the iterative procedure (9). The aposteriori estimates are generally better than the apriori ones.

Summing up what we have stated above, we get the following:
Corollary. If the conditions (7) and (8) are satisfied, then by the iterative procedure (9) one obtains a sequence ${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$(x_(n))_(n=0)^(oo)\left(x_{n}\right)_{n=0}^{\infty}${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$ which converges to a fixed point $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ of the mapping $F$$F$FF$F$, and for each $n\in \{0,1,2,\dots \}$$n\in \{0,1,2,\dots \}$n in{0,1,2,dots}n \in\{0,1,2, \ldots\}$n\in \{0,1,2,\dots \}$ the inequalities (10)-(12) are fulfiled. Moreover, if for a certain $n\in \{1,2,3,\dots$$n\in \{1,2,3,\dots$n in{1,2,3,dotsn \in\{1,2,3, \ldots$n\in \{1,2,3,\dots$. the condition (13) is satisfied, then for this $n$$n$nn$n$, the inequality (14) is also fulfiled.

The above corollary will be the basis of the prof of the Theorem 3.1, concerning the convergence of the modified secant method, which will be given in Section 3.

The notion of divided difference of a (nonlinear) operator is an extension of the usual notion of divided difference of a function, in the same sense in which the Fréchet derivative of an operator is an extension of the classical notion of the derivative of a function. This notion was introduced by J. schroder [8] and was used by A. sergeev [9] and J. schmidt [7] to the extension of the secant method for the iterative solution of the nonlinear operatorial equations in Banach spacis.

Let $E$$E$EE$E$ and $F$$F$FF$F$ be two Banach spaces. We shall denote by $L(E,F)$$L(E,F)$L(E,F)L(E, F)$L(E,F)$ the Banach space of all linear and bounded operators, from $E$$E$EE$E$ into $F$$F$FF$F$. Let $f$$f$ff$f$ be a (nonlinear) operator from $E$$E$EE$E$ into $F$$F$FF$F$, and let $x$$x$xx$x$ and $y$$y$yy$y$ be two different points of the domain of $f$$f$ff$f$.

DEFINITION 2.1. A bounded linear operator $A\in L(E,F)$$A\in L(E,F)$A in L(E,F)A \in L(E, F)$A\in L(E,F)$ is called a divided difference of the operator $f$$f$ff$f$ on the points $x$$x$xx$x$ and $y$$y$yy$y$, if :

In the scalar case the divided difference of a function is unique, but in the general case this assertion is not true. Let us examine as an illustration the case where $E=F=R^2$$E=F=R^2$E=F=R^(2)E=F=R^{2}$E=F=R^2$. In this case, a nonlinear operator $f$$f$ff$f$ is characterized by two real functions of two real variables $f_1$$f_1$f_(1)f_{1}$f_1$ and $f_2$$f_2$f_(2)f_{2}$f_2$ i.e.

Then each of the linear operators $A_1$$A_1$A_(1)A_{1}$A_1$ and $A_2$$A_2$A_(2)A_{2}$A_2$ given by the following two matrices satisfy (15):

If $f$$f$ff$f$ is differentiable and its Fréchet derivatives $f^{\prime }$$f^{\prime }$f^(')f^{\prime}$f^{\prime }$ is continuous on the segment $[x,y]=\{tx+(1-t)y;t\in [0,1]\}$$[x,y]=\{tx+(1-t)y;t\in [0,1]\}$[x,y]={tx+(1-t)y;t in[0,1]}[x, y]=\{t x+(1-t) y ; t \in[0,1]\}$[x,y]=\{tx+(1-t)y;t\in [0,1]\}$, then the linear operator given by

also satisfies (15). That means that any of the three linear operators $A_1$$A_1$A_(1)A_{1}$A_1$, $A_2,A_3$$A_2,A_3$A_(2),A_(3)A_{2}, A_{3}$A_2,A_3$, are devided differences of the operator $f$$f$ff$f$ on the points $x$$x$xx$x$ and $y$$y$yy$y$. Moreover, any convex combination of $A_1,A_2$$A_1,A_2$A_(1),A_(2)A_{1}, A_{2}$A_1,A_2$ and $A_3$$A_3$A_(3)A_{3}$A_3$ is also a divided difference of $f$$f$ff$f$ on the points $x$$x$xx$x$ and $y$$y$yy$y$. It we have two divided differences of $f$$f$ff$f$ on the points $x$$x$xx$x$ and $y$$y$yy$y$, represented by the matrices $A$$A$AA$A$ and $B$$B$BB$B$, then the matrix, $C$$C$CC$C$, having the first line equal to the first line of $A$$A$AA$A$, and the second line equal to the second line of $B$$B$BB$B$, also represents a divided difference of $f$$f$ff$f$ on the points $x$$x$xx$x$ and $y$$y$yy$y$.

Let us now return to the general case. Concerning the existence of the divided differences see [1]. Concerning other examples in some concrete spaces see $[10]$$[10]$[10][10]$[10]$. Let us suppose that the closed sphere $U=U(x_0,m)$$U=U\left(x_0,m\right)$U=U(x_(0),m)U=U\left(x_{0}, m\right)$U=U(x_0,m)$ is included into the domain of the operator $f$$f$ff$f$, and let us denote by $D$$D$DD$D$ the set $D=\{(x,y)\in U^{\prime }\times U;x\ne y\}$$D=\left\{(x,y)\in U^{\prime }\times U;x\ne y\right\}$D={(x,y)inU^(')xx U;x!=y}D=\left\{(x, y) \in U^{\prime} \times U ; x \neq y\right\}$D=\{(x,y)\in U^{\prime }\times U;x\ne y\}$. We consider the mapping:

where, for any pair $(x,y)\in D$$(x,y)\in D$(x,y)in D(x, y) \in D$(x,y)\in D$, the linear operator $[x,y;f]$$[x,y;f]$[x,y;f][x, y ; f]$[x,y;f]$ is a divided difference of $f$$f$ff$f$ on the points $x$$x$xx$x$ and $y$$y$yy$y$ i.e. :

In [9] one assumes that the mapping $(x,y)\to [x,y;f]$$(x,y)\to [x,y;f]$(x,y)rarr[x,y;f](x, y) \rightarrow[x, y ; f]$(x,y)\to [x,y;f]$ is symmetric i.e. $[x,y;f]=[y,x;f]$$[x,y;f]=[y,x;f]$[x,y;f]=[y,x;f][x, y ; f]=[y, x ; f]$[x,y;f]=[y,x;f]$. In [7] this condition is no longer required. Let us remark that in our example $A_1$$A_1$A_(1)A_{1}$A_1$ and $A_2$$A_2$A_(2)A_{2}$A_2$ are not symmetric, while $A_3$$A_3$A_(3)A_{3}$A_3$ and $\frac{1}{2}{A}_1+\frac{1}{2}{A}_2$$\frac{1}{2}{A}_1+\frac{1}{2}{A}_2$(1)/(2)A_(1)+(1)/(2)A_(2)\frac{1}{2} A_{1}+\frac{1}{2} A_{2}$\frac{1}{2}{A}_1+\frac{1}{2}{A}_2$ are.

In both of the above cited papers, one supposes, in order to assure sufficient conditions for the convergence of the secant method, that the mapping $(x,y)\to [x,y;f]$$(x,y)\to [x,y;f]$(x,y)rarr[x,y;f](x, y) \rightarrow[x, y ; f]$(x,y)\to [x,y;f]$ satisfies a Lipschitz condition at least. We shall write this condition under the form:

It is easy to prove that if the above inequality is fulfiled for all $x,y$$x,y$x,yx, y$x,y$, $u,v\in U=U(x_0,m)$$u,v\in U=U\left(x_0,m\right)$u,v in U=U(x_(0),m)u, v \in U=U\left(x_{0}, m\right)$u,v\in U=U(x_0,m)$, with $x\ne y$$x\ne y$x!=yx \neq y$x\ne y$ and $u\ne v$$u\ne v$u!=vu \neq v$u\ne v$, then for each $x\in U$$x\in U$x in Ux \in U$x\in U$ there exists the limit $\underset{y\to x}{lim}[x,y;f]$$\underset{y\to x}{lim} [x,y;f]$lim_(y rarr x)[x,y;f]\lim _{y \rightarrow x}[x, y ; f]$\underset{y\to x}{lim}[x,y;f]$, and it equals the Fréchet derivative $f^{\prime }(x)$$f^{\prime }(x)$f^(')(x)f^{\prime}(x)$f^{\prime }(x)$. We have then:

The above remark allows us to take by definition $[x,x;f]=f^{\prime }(x)$$[x,x;f]=f^{\prime }(x)$[x,x;f]=f^(')(x)[x, x ; f]=f^{\prime}(x)$[x,x;f]=f^{\prime }(x)$ for each $x\in X$$x\in X$x in Xx \in X$x\in X$. Thus (18) implies (17).

Reversely, if the operator $f$$f$ff$f$ is Fréchet differentiable for each $x\in U$$x\in U$x in Ux \in U$x\in U$, and if (18) is satisfied, then there exists a mapping $U\times U\ni (x,y)\to \to [x,y;f]\in L(E,F)$$U\times U\ni (x,y)\to \to [x,y;f]\in L(E,F)$U xx U∋(x,y)rarr rarr[x,y;f]in L(E,F)U \times U \ni(x, y) \rightarrow \rightarrow[x, y ; f] \in L(E, F)$U\times U\ni (x,y)\to \to [x,y;f]\in L(E,F)$ which satisfies(16) and (17). We can take, for example,

This remark will be used to obtain the theorem concerning the convergence of the modified Newton's process [3] as a consequence of the theorem concerning the convergence of the modified secant method wich will be proved in the next section.

The same as in the preceding section, let $f$$f$ff$f$ be a nonlinear operator from the Banach space $E$$E$EE$E$ into the Banach space $F$$F$FF$F$, and let the sphere $U==U(x_0,m)$$U==U\left(x_0,m\right)$U==U(x_(0),m)U= =U\left(x_{0}, m\right)$U==U(x_0,m)$ be included into its domain of definition. We suppose that there exists a mapping.

which satisfies (16) and (17). Let ${\overline{x}}_0$${\overline{x}}_0$bar(x)_(0)\bar{x}_{0}${\overline{x}}_0$ be a point of $U$$U$UU$U$, for which the linear operator $[x_0,{\overline{x}}_0;f]$$\left[x_0,{\overline{x}}_0;f\right]$[x_(0), bar(x)_(0);f]\left[x_{0}, \bar{x}_{0} ; f\right]$[x_0,{\overline{x}}_0;f]$ is boundedly invertible. The modified secant method, we are going to study, consists of the following interative procedure:

For the study of the convergence of the sequence ${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$(x_(n))_(n=0)^(oo)\left(x_{n}\right)_{n=0}^{\infty}${\left(x_n\right)}_{n=0}^{\mathrm{\infty }}$ yielded by (19), we need some results concerning the behaviour of such a sequence in the particular case where $f$$f$ff$f$ is a certain real quadratic polinomial.
lemma 3.1. If $d,H,q_0$$d,H,q_0$d,H,q_(0)d, H, q_{0}$d,H,q_0$ and $r_0$$r_0$r_(0)r_{0}$r_0$ are positive numbers satisfying the conditions