Optimal Steffensen type iterative methods obtained by inverse interpolation

Ion Păvăloiu
(original), html, paper

Abstract

Let \(f:I\subset \mathbb{R\rightarrow R}\) be a nonlinear mapping and the equation \(f\left( x\right) =0\) with solution \(x^{\ast}\); consider the equivalent equations \(\varphi_{i}\left( x\right) =x\), \(i=1,…,n+1\). Given \(x_{k}\) an approximation to \(x^{\ast}\), we consider the following nodes for the Hermite interpolation polynomial \[x_{k}^{1}=\varphi_{1}(x_{k}) ,\ \ x_{k}^{2}=\varphi_{2}(x_{k} ^{1}), \ldots, \ \ x_{k}^{n+1}=\varphi_{n+1}(x_{k}^{n}). \] Assume that the nodes \(x_{k}^{i},i=1,…,n+1\) have the multiplicity orders resp. \(\alpha_{i},i=1,…,n+1\). Moreover, the convergence orders of the successive iterations for \(\varphi_{i}\) are resp. \(p_{i},p_{i}\in \mathbb{N},p_{i}\geq1,i=1,…,n+1\). The iterative method obtained from the inverse interpolation Hermite polynomial is a Steffensen type method. If we permute the multiplicity orders of the nodes and the assumed convergence orders, we obtain class of iterative methods. Among this class we determine the methods with the highest convergence orders.

Authors

Crăciun Iancu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ioan Şerb
(Tiberiu Popoviciu Institute of Numerical Analysis)

Title

Original title (in French)

Méthodes itératives optimales de type Steffensen obtenues par interpolation invèrse

English translation of the title

Optimal Steffensen type iterative methods obtained by inverse interpolation

Keywords

Hermite inverse interpolation; Steffensen type methods; iterative methods; nonlinear equations in R; convergence order

PDF

Scanned paper.

PDF-Latex version of the paper.

Cite this paper as:

C. Iancu, I. Păvăloiu, I. Şerb, Méthodes itératives optimales de type Steffensen obtenues par interpolation invèrse, Seminar on functional analysis and numerical methods, Preprint no. 1 (1983), pp. 81-88 (in French).

About this paper

Journal

Seminar on functional analysis and numerical methods,
Preprint

Publisher Name

“Babes-Bolyai” University
Faculty of Mathematics and Physics
Research Seminars

DOI

Not available yet.

References

[1] C. Iancu, I. Pavaloiu, La resolution des equations par interpolation inverse de type Hemite. Seminar of Functional and Numerical Methods, “Babes-Bolyai” University, Faculty de Matematica, Research Seminaries, Preprint Nr. 4 (1981), 72-84.

[2] I. Pavaloiu, Rezolvarea ecuatiilor prin interpolare. Editura Dacia, Cluj-Napoca, 1981.

[3] S. Popa, Asupra unei probleme a lui E. Erdos si G. Weiss, Studii si cercetari matematice, 33, 5 (1981), 539-542.

Paper (preprint) in HTML form

Optimal-Steffensen-type-iterative-methods-obtained-by-inverse-interpolation
Original text
Rate this translation
Your feedback will be used to help improve Google Translate

Optimal Steffensen type iterative methods obtained by inverse interpolation


In the work [1] we proceeded to the study of a class of methods for the resolution of questions of the form:
(2)
f ( x ) = 0 , f ( x ) = 0 , f(x)=0,f(x)=0,f(x)=0,
hey x ; x → R , x x ; x → R ,  x x;x rarrR_(", ")xx ; x \rightarrow R_{\text {, }} xx;x→R, x s. x x xxxund fonction réello de vaxiable róello et T T TTTis an interval of the real axis. We arrive at this class of methods using the inverse interpolation method, l ′ l ′ l^(')l^{\prime}l′alde of the Hersite interpolation polynomial.
The order of convergence of the methods studied by Dows [1] depends on the number of nodes and their multiplicity orders.
In this work we will demonstrate that in the class of methods studied in [1], a group of Steffensen-type methods stands out for which the order of convergence is the largest. In what follows, the interpolation nodes are no longer chosen arbitrarily, they will be generated on the basis of iterative functions.
Let us designate by f 1 ∗ I → I , 1 = 1 , 2 , … , n + 1 , n + 1 f 1 ∗ I → I , 1 = 1 , 2 , … , n + 1 , n + 1 varphi_(1)**I rarr I,1=1,2,dots,n+1,n+1\varphi_{1} * I \rightarrow I, 1=1,2, \ldots, n+1, n+1f1∗I→I,1=1,2,…,n+1,n+1iterative functions, i.e. functions whose fixed points coincide with the roots of equation (1) in the interval I I III.
We also assume that the functions f i ; 1 : 1 , n + 1 ― f i ; 1 : 1 , n + 1 ¯ varphi_(i);1: bar(1,n+1)\varphi_{i} ; 1: \overline{1, n+1}fi;1:1,n+1―And f f fffverify the following conditions:
(2) | f ( f i ( x ) ) | ≦ r i | f ( x ) | p 1 ; i = 1 , 2 , … , n + 1 f f i ( x ) ≦ r i | f ( x ) | p 1 ; i = 1 , 2 , … , n + 1 quad|f(varphi_(i)(x))| <= rho_(i)|f(x)|^(p_(1));i=1,2,dots,n+1\quad\left|f\left(\varphi_{i}(x)\right)\right| \leqq \rho_{i}|f(x)|^{p_{1}} ; i=1,2, \ldots, n+1|f(fi(x))|≦ri|f(x)|p1;i=1,2,…,n+1,
Or
∫ 1 , i = 1 , 2 , … , n + 1 , are real and positive numbers ∫ 1   , i = 1 , 2 , … , n + 1 ,  are real and positive numbers  int_(1),i=1,2,dots,n+1," sont des nombres réels et positifs "\int_{1}, i=1,2, \ldots, n+1, \text { sont des nombres réels et positifs }∫1,i=1,2,…,n+1, are real and positive numbers 

Design by u 0 ∈ I u 0 ∈ I u_(0)in Iu_{0} \in Iin0∈I, and an approximation of the racing Ta Ta equation (1). We construct the elaboration nodes x 1 1 x 1 1 x_(1)^(1)x_{1}^{1}x11, 1. Ω 1 , 2 , … , a + 1 Ω 1 , 2 , … , a + 1 Omega1,2,dots,a+1\Omega 1,2, \ldots, a+1Oh1,2,…,a+1, of the following manner:
(3)
x 1 1 = φ 1 ( u 0 ) x 1 + 1 1 = φ 1 + 1 ( x 1 1 ) , i = 1 , 2 1 … , … , x 1 1 = φ 1 u 0 x 1 + 1 1 = φ 1 + 1 x 1 1 , i = 1 , 2 1 … , … , {:[x_(1)^(1)=varphi_(1)(u_(0))],[x_(1+1)^(1)=varphi_(1+1)(x_(1)^(1))","i=1","2_(1)dots","dots","]:}\begin{aligned} & x_{1}^{1}=\varphi_{1}\left(u_{0}\right) \\ & x_{1+1}^{1}=\varphi_{1+1}\left(x_{1}^{1}\right), i=1,2_{1} \ldots, \ldots, \end{aligned}x11=f1(in0)x1+11=f1+1(x11),i=1,21…,…,
We can y i = y i 1 ≈ f ( x 1 1 ) , 1 = 1 , 2 , … , 2 … 1 y i = y i 1 ≈ f x 1 1 , 1 = 1 , 2 , … , 2 … 1 y_(i)=y_(i)^(1)~~f(x_(1)^(1)),1=1,2,dots,2dots1y_{i}=y_{i}^{1} \approx f\left(x_{1}^{1}\right), 1=1,2, \ldots, 2 \ldots 1andi=andi1≈f(x11),1=1,2,…,2…1and we damago nons by K 1 , C 2 , … , C n + 1 , n + 1 K 1 , C 2 , … , C n + 1 , n + 1 K_(1),C_(2),dots,C_(n+1),n+1\mathscr{K}_{1}, \mathscr{C}_{2}, \ldots, \mathscr{C}_{n+1}, n+1K1,C2,…,Cn+1,n+1nomines aature lo donaés, for which
(4) α 1 + α 2 + … + α n + 1 = m + 1 , M ∈ N α 1 + α 2 + … + α n + 1 = m + 1 , M ∈ N quadalpha_(1)+alpha_(2)+dots+alpha_(n+1)=m+1,quad M inN\quad \alpha_{1}+\alpha_{2}+\ldots+\alpha_{n+1}=m+1, \quad M \in \mathbb{N}a1+a2+…+an+1=m+1,M∈N.
Now suppose that the anointing L L L\mathcal{L}L, admits in l 1 l 1 l^(1)l^{1}l1intores vallo I of the derivatives up to order m +2 y ooupris ob I ′ ( x ) ∉ 0 I ′ ( x ) ∉ 0 I^(')(x)!in0I^{\prime}(x) \notin 0I′(x)∉0, for each x ∈ I x ∈ I x in Ix \in Ix∈I.
In I hypothesize the function I , I → F I , I → F I,I rarr FI, I \rightarrow FI,I→F, on I I III s I ( I ) I ( I ) I(I)I(I)I(I)is bijective and therefore 11 exists f − 1 ; f → I f − 1 ; f → I f^(-1);f rarr If^{-1} ; f \rightarrow If−1;f→I. To calculate the successive derivatives of the function f − 1 f − 1 f^(-1)f^{-1}f−1we will use the foxule established in 2nd work [2] ।
(5) ( f − 1 ) ( k ) ( y ) = ∑ ( 2 k − 2 − i 1 ) I ( − 1 ) k − 1 + i 1 i 2 1 … i k I ( f ′ ( x ) ) 2 k − 1 ( f ′ ( x ) 1 ! ) 1 … f ( k ) ( x ) 1 k k ! f − 1 ( k ) ( y ) = ∑ 2 k − 2 − i 1 I ( − 1 ) k − 1 + i 1 i 2 1 … i k I f ′ ( x ) 2 k − 1 f ′ ( x ) 1 ! 1 … f ( k ) ( x ) 1 k k ! quad(f^(-1))^((k))(y)=sum((2k-2-i_(1))I(-1)^(k-1+i_(1)))/(i_(2)1dotsi_(k)I(f^(')(x))^(2k-1))((f^(')(x))/(1!))^(1)dots(f^((k))(x)^(1)k)/(k!)\quad\left(f^{-1}\right)^{(k)}(y)=\sum \frac{\left(2 k-2-i_{1}\right) I(-1)^{k-1+i_{1}}}{i_{2} 1 \ldots i_{k} I\left(f^{\prime}(x)\right)^{2 k-1}}\left(\frac{f^{\prime}(x)}{1!}\right)^{1} \ldots \frac{f^{(k)}(x){ }^{1} k}{k!}(f−1)(k)(and)=∑(2k−2−i1)I(−1)k−1+i1i21…ikI(f′(x))2k−1(f′(x)1!)1…f(k)(x)1kk!
Or y = f ( x ) y = f ( x ) y=f(x)y=f(x)and=f(x), and the sum below extends to all integer and non-negative solutions of the system of equations:
(6)
1 2 + 21 3 + … + ( k = 1 ) 1 k = k = 1 i 2 + i 3 + … + i k = k − 1 1 2 + 21 3 + … + ( k = 1 ) 1 k = k = 1 i 2 + i 3 + … + i k = k − 1 {:[1_(2)+21_(3)+dots+(k=1)1_(k)=k=1],[i_(2)+i_(3)+dots+i_(k)=k-1]:}\begin{aligned} & 1_{2}+21_{3}+\ldots+(k=1) 1_{k}=k=1 \\ & i_{2}+i_{3}+\ldots+i_{k}=k-1 \end{aligned}12+213+…+(k=1)1k=k=1i2+i3+…+ik=k−1
Given the above notations, the inverse Hernate inexplicative polynomial relative to the nodes if having the mule orders
in 2 o work [2]:
where
( θ ) M ( F ) = ∏ d = 2 ∞ ( y − y d ) C ( θ ) M ( F ) = ∏ d = 2 ∞   y − y d C {:(theta")"M(F)=prod_(d=2)^(oo)(y-y_(d))^(C):}\begin{equation*} M(F)=\prod_{d=2}^{\infty}\left(y-y_{d}\right)^{C} \tag{$\theta$} \end{equation*}(i)M(F)=∏d=2∞(and−andd)C

(9) f m − 1 ( ξ ) − f ( ξ ) = ( e − 1 ) ( m + 1 ) ( f ) ( m + 1 ) ! ω ( ξ ) , ψ ⇔ φ 0 (9) f m − 1 ( ξ ) − f ( ξ ) = e − 1 ( m + 1 ) ( f ) ( m + 1 ) ! ω ( ξ ) , ψ ⇔ φ 0 {:(9)f^(m-1)(xi)-f(xi)=((e^(-1))^((m+1))(f))/((m+1)!)omega(xi)","quad psi<=>varphi_(0):}\begin{equation*} f^{m-1}(\xi)-f(\xi)=\frac{\left(e^{-1}\right)^{(m+1)}(f)}{(m+1)!} \omega(\xi), \quad \psi \Leftrightarrow \varphi_{0} \tag{9} \end{equation*}(9)fm−1(x)−f(x)=(and−1)(m+1)(f)(m+1)!oh(x),ψ⇔f0


৪ய Wanto 8
(10)
u 2 = P ( 0 ) u 2 = P ( 0 ) u_(2)=P(0)u_{2}=P(0)in2=P(0)
| z − u 1 | ≦ 1 ( m + 1 ) ∣ | cos ⁡ ( 0 ) | , z − u 1 ≦ 1 ( m + 1 ) ∣ | cos ⁡ ( 0 ) | , |z-u_(1)| <= (1)/((m+1)∣)|cos(0)|,\left|\mathbb{z}-u_{1}\right| \leqq \frac{1}{(m+1) \mid}|\cos (0)|,|With−in1|≦1(m+1)∣|cos⁡(0)|,
or M = sim y ∈ Y | ( x ∞ ) ( x + 1 ) ( y ) | M = sim y ∈ Y x ∞ ( x + 1 ) ( y ) M=sim_(y in Y)|(x^(oo))(x+1)(y)|M=\operatorname{sim}_{y \in Y}\left|\left(x^{\infty}\right)(x+1)(y)\right|M=tryand∈AND|(x∞)(x+1)(and)|.
In the following we give a description for | ω ( 0 ) | | ω ( 0 ) | |omega(0)||\omega(0)||oh(0)|and using (2) from (3). Here are a few:
| f ( x 1 1 ) | : | f ( φ 2 ( u 0 ) ) | ≦ ρ 1 | f ( u 0 ) | p 1 | f ( x 2 1 ) | =∣ f ( φ 2 ( x 1 1 ) | ≦ ρ 2 | f ( x 1 1 ) | p 2 ≦ ρ 2 ρ 1 p 2 | f ( u 0 ) | p 1 p 2 | f ( x 3 1 ) | = | f ( φ 3 ( x 2 1 ) ) | ≦ ρ 3 | f ( x 2 1 ) | p 3 ≦ ρ 3 ∫ 2 p 3 ∫ 1 p 2 p 3 | f ( u 0 ) | p 1 p 2 p 3 f x 1 1 : f φ 2 u 0 ≦ ρ 1 f u 0 p 1 f x 2 1 =∣ f φ 2 x 1 1 ≦ ρ 2 f x 1 1 p 2 ≦ ρ 2 ρ 1 p 2 f u 0 p 1 p 2 f x 3 1 = f φ 3 x 2 1 ≦ ρ 3 f x 2 1 p 3 ≦ ρ 3 ∫ 2 p 3   ∫ 1 p 2 p 3   f u 0 p 1 p 2 p 3 {:[|f(x_(1)^(1))|:|f(varphi_(2)(u_(0)))| <= rho_(1)|f(u_(0))|^(p_(1))],[|f(x_(2)^(1))|=∣f(varphi_(2)(x_(1)^(1))| <= rho_(2)|f(x_(1)^(1))|^(p_(2)) <= rho_(2)rho_(1)^(p_(2))|f(u_(0))|^(p_(1)p_(2)):}],[|f(x_(3)^(1))|=|f(varphi_(3)(x_(2)^(1)))| <= rho_(3)|f(x_(2)^(1))|^(p_(3)) <= ],[rho_(3)int_(2)^(p_(3))int_(1)^(p_(2)p_(3))|f(u_(0))|^(p_(1)p_(2)p_(3))]:}\begin{gathered} \left|f\left(x_{1}^{1}\right)\right|:\left|f\left(\varphi_{2}\left(u_{0}\right)\right)\right| \leqq \rho_{1}\left|f\left(u_{0}\right)\right|^{p_{1}} \\ \left|f\left(x_{2}^{1}\right)\right|=\mid f\left(\left.\varphi_{2}\left(x_{1}^{1}\right)\left|\leqq \rho_{2}\right| f\left(x_{1}^{1}\right)\right|^{p_{2}} \leqq \rho_{2} \rho_{1}^{p_{2}}\left|f\left(u_{0}\right)\right|^{p_{1} p_{2}}\right. \\ \left|f\left(x_{3}^{1}\right)\right|=\left|f\left(\varphi_{3}\left(x_{2}^{1}\right)\right)\right| \leqq \rho_{3}\left|f\left(x_{2}^{1}\right)\right|^{p_{3}} \leqq \\ \rho_{3} \int_{2}^{p_{3}} \int_{1}^{p_{2} p_{3}}\left|f\left(u_{0}\right)\right|^{p_{1} p_{2} p_{3}} \end{gathered}|f(x11)|:|f(f2(in0))|≦r1|f(in0)|p1|f(x21)|=∣f(f2(x11)|≦r2|f(x11)|p2≦r2r1p2|f(in0)|p1p2|f(x31)|=|f(f3(x21))|≦r3|f(x21)|p3≦r3∫2p3∫1p2p3|f(in0)|p1p2p3
of in a general way
(12) | x ( x i + 1 1 ) | ≤ ρ i + 1 ρ i p i + 1 … ρ 1 p 2 p 3 … p i + 1 | x ( u 0 ) | p 1 = x p i + 1 , 1 = 1 , 2 , … , u 0 De ( C ) ол abdult :  (12)  x x i + 1 1 ≤ ρ i + 1 ρ i p i + 1 … ρ 1 p 2 p 3 … p i + 1 x u 0 p 1 = x p i + 1 , 1 = 1 , 2 , … , u 0  De (  C  ) ол abdult :  {:[" (12) "|x(x_(i+1)^(1))| <= rho_(i+1)rho_(i)^(p_(i+1))dotsrho_(1)^(p_(2)p_(3)dotsp_(i+1))|x(u_(0))|p_(1)=xp_(i+1)","],[1=1","2","dots","u_(0)" De ( "C" ) ол abdult : "]:}\begin{aligned} & \text { (12) }\left|x\left(x_{i+1}^{1}\right)\right| \leq \rho_{i+1} \rho_{i}^{p_{i+1}} \ldots \rho_{1}^{p_{2} p_{3} \ldots p_{i+1}}\left|x\left(u_{0}\right)\right| p_{1}=x p_{i+1}, \\ & 1=1,2, \ldots, u_{0} \text { De ( } \mathrm{C} \text { ) ол abdult : } \end{aligned} (12) |x(xi+11)|≤ri+1ripi+1…r1p2p3…pi+1|x(in0)|p1=xpi+1,1=1,2,…,in0 Of ( C ) he is abdult : 
| ω ( 0 ) | ∏ i = 1 π i | x ( x i i ) | σ i | ω ( 0 ) | ∏ i = 1 π i   x x i i σ i |omega(0)|prod_(i=1)^(pi i)|x(x_(i)^(i))|^(sigma_(i))|\omega(0)| \prod_{i=1}^{\pi i}\left|x\left(x_{i}^{i}\right)\right|^{\sigma_{i}}|oh(0)|∏i=1pi|x(xii)|si
Bow 1 levels (2) La口 Winpadu
(14) | ω ( 0 ) | ≦ ρ | t ( a 0 ) | α . (14) | ω ( 0 ) | ≦ ρ t a 0 α . {:(14)|omega(0)| <= rho|t(a_(0))|^(alpha).:}\begin{equation*} |\omega(0)| \leqq \rho\left|t\left(a_{0}\right)\right|^{\alpha} . \tag{14} \end{equation*}(14)|oh(0)|≦r|t(a0)|a.
(où) ρ = ∑ i = 1 n + 1 ∮ α 1 + ∑ j = i = 1 n ∞ α j = i + 1 ∞ ∏ [ k + 1 j (où) ρ = ∑ i = 1 n + 1   ∮ α 1 + ∑ j = i = 1 n ∞   α j = i + 1 ∞ ∏ [ k + 1 j   {:(où)rho=sum_(i=1)^(n+1)oint^(alpha_(1))+sum_(j=i=1)^(n oo)alpha_(j=i+1)^(oo)prod_([k+1)^(j):}\begin{equation*} \rho=\sum_{i=1}^{n+1} \oint^{\alpha_{1}}+\sum_{j=i=1}^{n \infty} \alpha_{j=i+1}^{\infty} \prod_{[k+1}^{j} \tag{où} \end{equation*}(Or)r=∑i=1n+1∮a1+∑j=i=1n∞aj=i+1∞∏[k+1j
(16) α = ∑ i = 1 n = 1 α i ∏ j = 3 1 g j . (16) α = ∑ i = 1 n = 1   α i ∏ j = 3 1   g j . {:(16)alpha=sum_(i=1)^(n=1)alpha_(i)prod_(j=3)^(1)g_(j).:}\begin{equation*} \alpha=\sum_{i=1}^{n=1} \alpha_{i} \prod_{j=3}^{1} g_{j} . \tag{16} \end{equation*}(16)a=∑i=1n=1ai∏j=31gj.
From the above inequality st do (21) on dódust lungazites ext. vante:
(1.7) | x ¨ − u 1 | ⩽ M ρ ( m + 1 ) ! | r ( u 0 ) | 2 (1.7) x ¨ − u 1 ⩽ M ρ ( m + 1 ) ! r u 0 2 {:(1.7)|(x^(¨))-u_(1)| <= (M rho)/((m+1)!)|r(u_(0))|^(2):}\begin{equation*} \left|\ddot{x}-u_{1}\right| \leqslant \frac{M \rho}{(m+1)!}\left|r\left(u_{0}\right)\right|^{2} \tag{1.7} \end{equation*}(1.7)|x¨−in1|⩽Mr(m+1)!|r(in0)|2

The taterpolation knots x i k , 1 = I i ― I i ― + I i ― x i k , 1 = I i ¯ I i ¯ + I i ¯ x_(i)^(k),1= bar(I_(i)) bar(I_(i))+ bar(I_(i))x_{i}^{k}, 1=\overline{I_{i}} \overline{I_{i}}+\overline{I_{i}}xik,1=Ii―Ii―+Ii―which corrogramant an pas pulvant si obtienneat d'une manders analogous to asilo mployon EU promiar pus en employant las relationa :
x 1 5 ≈ φ 3 ( u i ∈ ∞ ) x 1 5 ≈ φ 3 u i ∈ ∞ x_(1)^(5)~~varphi_(3)(u_(i in oo))x_{1}^{5} \approx \varphi_{3}\left(u_{i \in \infty}\right)x15≈f3(ini∈∞)
(28)
x i + 1 k = φ i + 2 ( x i k ) ; i = 1 , 2 i 0 … n , k ⩾ 2 0 x i + 1 k = φ i + 2 x i k ; i = 1 , 2 i 0 … n , k ⩾ 2 0 x_(i+1)^(k)=varphi_(i+2)(x_(i)^(k));i=1,2_(i0dots n),k >= 2_(0)x_{i+1}^{k}=\varphi_{i+2}\left(x_{i}^{k}\right) ; i=1,2_{i 0 \ldots n}, k \geqslant 2_{0}xi+1k=fi+2(xik);i=1,2i0…n,k⩾20
Ba proceeding conne to the promaer not we obtain the trigendut
(29) | x ¯ = u k + 1 | = m ( m + 1 ) ! m ρ | f ( u k ) | σ , k an d ϵ e θ … θ (29) x ¯ = u k + 1 = m ( m + 1 ) ! m ρ f u k σ , k  an  d ϵ e θ … θ {:(29)|( bar(x))=u_(k+1)|=(m)/((m+1)!)^(m rho)|f(u_(k))|^(sigma)","quad k" an "d_(epsilon)e_(theta)dots theta:}\begin{equation*} \left|\bar{x}=u_{k+1}\right| \stackrel{m \rho}{=\frac{m}{(m+1)!}}\left|f\left(u_{k}\right)\right|^{\sigma}, \quad k \text { an } d_{\epsilon} e_{\theta} \ldots \theta \tag{29} \end{equation*}(29)|x¯=ink+1|=m(m+1)!mr|f(ink)|s,k an dϵandi…i
51 we pose
(20)
β = sup x ∈ Φ | x ′ ( x ) | β = sup x ∈ Φ   x ′ ( x ) beta=s u p_(x in Phi)|x^(')(x)|\beta=\sup _{x \in \Phi}\left|x^{\prime}(x)\right|b=supx∈F|x′(x)|
then the inequalities (19) deviemont:
(21)
| x ¯ − u k + 1 | ⩽ m ¯ ρ β α ( m + 2 ) ! | x ¯ − u k | α x ¯ − u k + 1 ⩽ m ¯ ρ β α ( m + 2 ) ! x ¯ − u k α |( bar(x))-u_(k+1)| <= (( bar(m))rhobeta^(alpha))/((m+2)!)|( bar(x))-u_(k)|^(alpha)\left|\bar{x}-u_{k+1}\right| \leqslant \frac{\bar{m} \rho \beta^{\alpha}}{(m+2)!}\left|\bar{x}-u_{k}\right|^{\alpha}|x¯−ink+1|⩽m¯rba(m+2)!|x¯−ink|a

(22) | x ¯ − u k + 1 | ≤ 0 1 x ( 0 1 x − x | x ¯ − u 0 | ) k + 1 x ¯ − u k + 1 ≤ 0 1 x 0 1 x − x x ¯ − u 0 k + 1 |( bar(x))-u_(k+1)| <= 0^((1)/(sqrtx))(0^((1)/(x-x))|( bar(x))-u_(0)|)^(k+1)quad\left|\bar{x}-u_{k+1}\right| \leq 0^{\frac{1}{\sqrt{x}}}\left(0^{\frac{1}{x-x}}\left|\bar{x}-u_{0}\right|\right)^{k+1} \quad|x¯−ink+1|≤01x(01x−x|x¯−in0|)k+1. Or C = M ρ β α / ( m + 1 ) 1 C = M ρ β α / ( m + 1 ) 1 C=M rhobeta^(alpha)//(m+1)1C=M \rho \beta^{\alpha} /(m+1) 1C=Mrba/(m+1)1.
B4 on జuppose
(23) 0 1 x − 1 | x ¯ − n 0 | < 1 (23) 0 1 x − 1 x ¯ − n 0 < 1 {:(23)0^((1)/(x-1))|( bar(x))-n_(0)| < 1:}\begin{equation*} 0^{\frac{1}{x-1}}\left|\bar{x}-n_{0}\right|<1 \tag{23} \end{equation*}(23)01x−1|x¯−n0|<1
do (22) 11 results:
(24)
lim x → ∞ u x = x ¯ . lim x → ∞   u x = x ¯ . lim_(x rarr oo)u_(x)= bar(x).\lim _{x \rightarrow \infty} u_{x}=\bar{x} .limx→∞inx=x¯.
Dóвigonas by. k 1 , k 2 , … , k n + 1 k 1 , k 2 , … , k n + 1 k_(1),k_(2),dots,k_(n+1)k_{1}, k_{2}, \ldots, k_{n+1}k1,k2,…,kn+1And j 1 , j 2 , … , j n + 1 j 1 , j 2 , … , j n + 1 j_(1),j_(2),dots,j_(n+1)j_{1}, j_{2}, \ldots, j_{n+1}j1,j2,…,jn+1of arbitrary permutations of numbers 1 , 2 , … , n + 1 1 , 2 , … , n + 1 1,2,dots,n+11,2, \ldots, \mathrm{n}+11,2,…,n+1, and by
H ( y k 1 1 , α 1 1 , y k 2 1 , α j 2 ; … , y k n + 1 1 , α j n + 1 / x ) , H y k 1 1 , α 1 1 , y k 2 1 , α j 2 ; … , y k n + 1 1 , α j n + 1 / x , H(y_(k_(1))^(1),alpha_(1_(1)),y_(k_(2))^(1),alpha_(j_(2));dots,y_(k_(n+1))^(1),alpha_(j_(n+1))//x),H\left(y_{k_{1}}^{1}, \alpha_{1_{1}}, y_{k_{2}}^{1}, \alpha_{j_{2}} ; \ldots, y_{k_{n+1}}^{1}, \alpha_{j_{n+1}} / x\right),H(andk11,a11,andk21,aj2;…,andkn+11,ajn+1/x),
1* inverse interpolation goljnóne of Hemitte of the form (7), at the interpolation nodes z k 1 1 , … , y k n + 1 1 z k 1 1 , … , y k n + 1 1 z_(k_(1))^(1),dots,y_(k_(n+1))^(1)z_{k_{1}}^{1}, \ldots, y_{k_{n+1}}^{1}Withk11,…,andkn+11having the orders of multipl aity α 1 … , α 1 … α 1 … , α 1 … alpha_(1)dots,alpha_(1)dots\alpha_{1} \ldots, \alpha_{1} \ldotsa1…,a1…respectively, Using the above notations we consider the following set of iterative methods
(25) u 8 = H ( y k 1 8 , α j 1 , … , y k n + 1 8 , α j n + 1 / 0 ) , s = 1 , 2 , … , (25) u 8 = H y k 1 8 , α j 1 , … , y k n + 1 8 , α j n + 1 / 0 , s = 1 , 2 , … , {:(25)u_(8)=H(y_(k_(1))^(8),alpha_(j_(1)),dots,y_(k_(n+1))^(8),alpha_(j_(n+1))//0)","s=1","2","dots",":}\begin{equation*} u_{8}=H\left(y_{k_{1}}^{8}, \alpha_{j_{1}}, \ldots, y_{k_{n+1}}^{8}, \alpha_{j_{n+1}} / 0\right), s=1,2, \ldots, \tag{25} \end{equation*}(25)in8=H(andk18,aj1,…,andkn+18,ajn+1/0),s=1,2,…,
Or
(26) y k 1 a = f ( x k 1 a ) , 1 = 1 , 2 , … , n + 1 , s = 1 , 2 , … (26) y k 1 a = f x k 1 a , 1 = 1 , 2 , … , n + 1 , s = 1 , 2 , … {:(26)y_(k_(1))^(a)=f(x_(k_(1))^(a))","1=1","2","dots","n+1","s=1","2","dots:}\begin{equation*} \mathrm{y}_{\mathrm{k}_{1}}^{\mathrm{a}}=\mathrm{f}\left(\mathrm{x}_{\mathrm{k}_{1}}^{\mathrm{a}}\right), 1=1,2, \ldots, \mathrm{n}+1, \mathrm{~s}=1,2, \ldots \tag{26} \end{equation*}(26)andk1a=f(xk1a),1=1,2,…,n+1, s=1,2,…
And
x k 1 3 = φ k 1 ( u β − 1 ) x k 1 3 = φ k 1 u β − 1 x_(k_(1))^(3)=varphi_(k_(1))(u_(beta-1))x_{k_{1}}^{3}=\varphi_{k_{1}}\left(u_{\beta-1}\right)xk13=fk1(inb−1)
(27)
x k 1 ∞ = φ k 1 ( x k 1 − 1 ∞ ) , 1 = 2 , 3 , … , n + 1 , σ = 1 , 2 , … . x k 1 ∞ = φ k 1 x k 1 − 1 ∞ , 1 = 2 , 3 , … , n + 1 , σ = 1 , 2 , … . x_(k_(1))^(oo)=varphi_(k_(1))(x_(k_(1-1))^(oo)),1=2,3,dots,n+1,sigma=1,2,dots.x_{k_{1}}^{\infty}=\varphi_{k_{1}}\left(x_{k_{1-1}}^{\infty}\right), 1=2,3, \ldots, n+1, \sigma=1,2, \ldots .xk1∞=fk1(xk1−1∞),1=2,3,…,n+1,s=1,2,….
To each pair of perautetions k 1 , k 2 , … , x n + 1 k 1 , k 2 , … , x n + 1 k_(1),k_(2),dots,x_(n+1)k_{1}, k_{2}, \ldots, x_{n+1}k1,k2,…,xn+1 at i 1 , j 2 i 1 , j 2 i_(1),j_(2)i_{1}, j_{2}i1,j2 : ⋯ , j n + 1 ⋯ , j n + 1 cdots,j_(n+1)\cdots, j_{n+1}⋯,jn+1numbers 1 , 2 , … , n + 1 1 , 2 , … , n + 1 1,2,dots,n+11,2, \ldots, n+11,2,…,n+1corresponds to a rative method of the form (25) - (27). In this way, we obtain in total ( n + 1 ) 1 ( n + 1 ) 1 (n+1)1(n+1) 1(n+1)1iterative methods of catte fcrme. These methods generally have different ones.

Le plus grand ordra de convergenct ec，orive donó per（ 76 ）．

that the following relationships are verified
(28) p 1 ≥ p 2 ≥ 0.0 ± p 211 ; x 2 ≤ x 2 ≤ 0 ≤ x 241 . (28) p 1 ≥ p 2 ≥ 0.0 ± p 211 ; x 2 ≤ x 2 ≤ 0 ≤ x 241 . {:(28)p_(1) >= p_(2) >= 0.0+-p_(211);x_(2) <= x_(2) <= 0 <= x_(241).:}\begin{equation*} p_{1} \geq p_{2} \geq 0.0 \pm p_{211} ; x_{2} \leq x_{2} \leq 0 \leq x_{241} . \tag{28} \end{equation*}(28)p1≥p2≥0.0±p211;x2≤x2≤0≤x241.

Do all the numbers in the form:

（29） ∝ 2 = ∝ j 2 p k 1 + ∝ j 2 p k 1 p k 2 + ⋯ ∞ + ∝ j k + 1 B k 1 p k 2 ∝ 2 = ∝ j 2 p k 1 + ∝ j 2 p k 1 p k 2 + ⋯ ∞ + ∝ j k + 1 B k 1 p k 2 prop^(2)=prop_(j_(2))p_(k_(1))+prop_(j_(2))p_(k_(1))p_(k_(2))+cdots oo+prop_(j_(k+1))B_(k_(1))p_(k_(2))\propto^{2}=\propto_{j_{2}} p_{k_{1}}+\propto_{j_{2}} p_{k_{1}} p_{k_{2}}+\cdots \infty+\propto_{j_{k+1}} B_{k_{1}} p_{k_{2}}∝2=∝j2pk1+∝j2pk1pk2+⋯∞+∝jk+1Bk1pk2 ．．． P k n + 1 P k n + 1 P_(k_(n+1))P_{k_{n+1}}Pkn+1Or j 1 , j 2 , … , j 24 j 1 , j 2 , … , j 24 j_(1),j_(2),dots,j_(24)j_{1}, j_{2}, \ldots, j_{24}j1,j2,…,j24And k 1 , k 2 , … , k n ∗ k 1 , k 2 , … , k n ∗ k_(1),k_(2),dots,k_(n**)k_{1}, k_{2}, \ldots, k_{n *}k1,k2,…,kn∗are two arbitrary permutations of the numbers 1 , 2 , … , n + 1 , 1 e 1 , 2 , … , n + 1 , 1 e 1,2,dots,n+1,1e1,2, \ldots, n+1,1 e1,2,…,n+1,1and plus srand ests
（30） α n k y = α 1 p 1 + α 2 p 1 p 2 + … + α n + 1 p 1 p 2 … p n + 1 α n k y = α 1 p 1 + α 2 p 1 p 2 + … + α n + 1 p 1 p 2 … p n + 1 quadalpha_(nky)=alpha_(1)p_(1)+alpha_(2)p_(1)p_(2)+dots+alpha_(n+1)p_(1)p_(2)dotsp_(n+1)\quad \alpha_{n k y}=\alpha_{1} p_{1}+\alpha_{2} p_{1} p_{2}+\ldots+\alpha_{n+1} p_{1} p_{2} \ldots p_{n+1}ankand=a1p1+a2p1p2+…+an+1p1p2…pn+1．
Proof．From the first set of inequalities (28) 11 results:
（31） ∝ j 1 p k 1 + ∝ j 2 p k 1 p k 2 + … + ∝ j n + 1 p k 1 p k 2 … p k n + 1 ≜ ∝ j 1 p k 1 + ∝ j 2 p k 1 p k 2 + … + ∝ j n + 1 p k 1 p k 2 … p k n + 1 ≜ prop_(j_(1))p_(k_(1))+prop_(j_(2))p_(k_(1))p_(k_(2))+dots+prop_(j_(n+1))p_(k_(1))p_(k_(2))dotsp_(k_(n+1))≜\propto_{j_{1}} p_{k_{1}}+\propto_{j_{2}} p_{k_{1}} p_{k_{2}}+\ldots+\propto_{j_{n+1}} p_{k_{1}} p_{k_{2}} \ldots p_{k_{n+1}} \triangleq∝j1pk1+∝j2pk1pk2+…+∝jn+1pk1pk2…pkn+1≜
α j 1 p 1 + α j 2 p 1 p 2 + … + α j n + 1 p 1 p 2 … p n + 1 α j 1 p 1 + α j 2 p 1 p 2 + … + α j n + 1 p 1 p 2 … p n + 1 alpha_(j_(1))p_(1)+alpha_(j_(2))p_(1)p_(2)+dots+alpha_(j_(n+1))p_(1)p_(2)dotsp_(n+1)\alpha_{j_{1}} p_{1}+\alpha_{j_{2}} p_{1} p_{2}+\ldots+\alpha_{j_{n+1}} p_{1} p_{2} \ldots p_{n+1}aj1p1+aj2p1p2+…+ajn+1p1p2…pn+1
for each permutation j 1 , j 2 , … , j n + 1 j 1 , j 2 , … , j n + 1 j_(1),j_(2),dots,j_(n+1)j_{1}, j_{2}, \ldots, j_{n+1}j1,j2,…,jn+1And k 1 , k 2 , … , k n + 1 k 1 , k 2 , … , k n + 1 k_(1),k_(2),dots,k_(n+1)k_{1}, k_{2}, \ldots, k_{n+1}k1,k2,…,kn+1．
We pose
(32) b 1 = p 1 p 2 … p 1 , 1 = 1 , 2 , … , n + 1 , (32) b 1 = p 1 p 2 … p 1 , 1 = 1 , 2 , … , n + 1 , {:(32)b_(1)=p_(1)p_(2)dotsp_(1)","1=1","2","dots","n+1",":}\begin{equation*} b_{1}=p_{1} p_{2} \ldots p_{1}, 1=1,2, \ldots, n+1, \tag{32} \end{equation*}(32)b1=p1p2…p1,1=1,2,…,n+1,
and we propose to prove the following inequality i
(33) α j 1 b 1 + α j 2 b 2 + … + α j n + 1 b n + 1 ≤ α 1 b 1 α 2 b 2 + … + α n + 1 b n + 1 . (33) α j 1 b 1 + α j 2 b 2 + … + α j n + 1 b n + 1 ≤ α 1 b 1 α 2 b 2 + … + α n + 1 b n + 1 . {:[(33)alpha_(j_(1))b_(1)+alpha_(j_(2))b_(2)+dots+alpha_(j_(n+1))b_(n+1) <= ],[alpha_(1)b_(1)alpha_(2)b_(2)+dots+alpha_(n+1)b_(n+1).]:}\begin{gather*} \alpha_{j_{1}} b_{1}+\alpha_{j_{2}} b_{2}+\ldots+\alpha_{j_{n+1}} b_{n+1} \leq \tag{33}\\ \alpha_{1} b_{1} \alpha_{2} b_{2}+\ldots+\alpha_{n+1} b_{n+1} . \end{gather*}(33)aj1b1+aj2b2+…+ajn+1bn+1≤a1b1a2b2+…+an+1bn+1.
for each permutation 1 2 , j 2 , … , j n t 1 2 , j 2 , … , j n t 1_(2),j_(2),dots,j_(nt)1_{2}, j_{2}, \ldots, j_{n t}12,j2,…,jnt．
vades for i pass of numbers ；（ X 2 , b 2 X 2 , b 2 X_(2),b_(2)\mathscr{X}_{2}, b_{2}X2,b2 ），on，（ X n , b n X n , b n X_(n),b_(n)\mathscr{X}_{n}, b_{n}Xn,bn), c'est. - by one man, one pound, one pound, ten pound
(34)
a 1 b 1 + α j 2 b 2 + … … α n b n ≦ α 1 a 2 + α 2 b 2 + … + α n b n a 1 b 1 + α j 2 b 2 + … … α n b n ≦ α 1 a 2 + α 2 b 2 + … + α n b n {:[a_(1)b_(1)+alpha_(j_(2))b_(2)+dots dotsalpha_(n)b_(n) <= ],[alpha_(1)a_(2)+alpha_(2)b_(2)+dots+alpha_(n)b_(n)]:}\begin{aligned} a_{1} b_{1} & +\alpha_{j_{2}} b_{2}+\ldots \ldots \alpha_{n} b_{n} \leqq \\ & \alpha_{1} a_{2}+\alpha_{2} b_{2}+\ldots+\alpha_{n} b_{n} \end{aligned}a1b1+aj2b2+……anbn≦a1a2+a2b2+…+anbn
For α 1 ≦ … .6 α n α 1 ≦ … .6 α n alpha_(1) <= dots.6alpha_(n)\alpha_{1} \leqq \ldots .6 \alpha_{n}a1≦….6an of b 1 ≦ … . ≦ b n b 1 ≦ … . ≦ b n b_(1) <= dots. <= b_(n)b_{1} \leqq \ldots . \leqq b_{n}b1≦….≦bn ．Si on suppose α 1 ≦ … ⋯ ≤ Δ C n ≜ C n + 1 , D 1 ≦ … ≦ b n ≦ b n + 1 α 1 ≦ … ⋯ ≤ Δ C n ≜ C n + 1 , D 1 ≦ … ≦ b n ≦ b n + 1 alpha_(1) <= dots cdots <= DeltaC_(n)≜C_(n+1),D_(1) <= dots <= b_(n) <= b_(n+1)\alpha_{1} \leqq \ldots \cdots \leq \Delta C_{n} \triangleq C_{n+1}, D_{1} \leqq \ldots \leqq b_{n} \leqq b_{n+1}a1≦…⋯≤DCn≜Cn+1,D1≦…≦bn≦bn+1 ot α j 1 = 1 α j 1 = 1 alpha_(j_(1))=1\alpha_{j_{1}}=1aj1=1，then do （34）dd xymatto1
α 1 b 1 + … + α j n b n + α j n + 1 b n + 1 = b 1 ( α j 1 + … + α j n + 1 ) + + ( b 2 − b 1 ) α j 2 + ( b 3 − b 1 ) α j 3 + … + ( b n + 1 − b 1 ) α j n + 1 ≦ b 1 ( c 1 & … + α n + 1 ) + ( b 2 − b 1 ) α 1 + ( b 3 − b 1 ) α 2 + … … ( b 1 − b 1 ) α 1 + 1 + ( b 1 + 1 − b 1 ) α 1 + 1 + … + ( b n + 1 − b 1 ) α n + 1 ≤ b 1 ( α 1 + … + α n + 1 ) + ( b 2 − b 1 ) α 2 + ( b 3 − b 1 ) α 3 + … … + ( b 1 − b 1 ) α 1 + ( b 1 + 1 − b 1 ) α 1 + 1 + … + ( b n + 1 − b 1 ) α n + 1 = b 1 α 1 + b 2 α 2 + … + b n + 1 α n + 1 , α 1 b 1 + … + α j n b n + α j n + 1 b n + 1 = b 1 α j 1 + … + α j n + 1 + + b 2 − b 1 α j 2 + b 3 − b 1 α j 3 + … + b n + 1 − b 1 α j n + 1 ≦ b 1 c 1 & … + α n + 1 + b 2 − b 1 α 1 + b 3 − b 1 α 2 + … … b 1 − b 1 α 1 + 1 + b 1 + 1 − b 1 α 1 + 1 + … + b n + 1 − b 1 α n + 1 ≤ b 1 α 1 + … + α n + 1 + b 2 − b 1 α 2 + b 3 − b 1 α 3 + … … + b 1 − b 1 α 1 + b 1 + 1 − b 1 α 1 + 1 + … + b n + 1 − b 1 α n + 1 = b 1 α 1 + b 2 α 2 + … + b n + 1 α n + 1 , {:[alpha_(1)b_(1)+dots+alpha_(j_(n))b_(n)+alpha_(j_(n+1))b_(n+1)=b_(1)(alpha_(j_(1))+dots+alpha_(j_(n+1)))+],[+(b_(2)-b_(1))alpha_(j_(2))+(b_(3)-b_(1))alpha_(j_(3))+dots+(b_(n+1)-b_(1))alpha_(j_(n+1)) <= ],[b_(1)(c_(1)&dots+alpha_(n+1))+(b_(2)-b_(1))alpha_(1)+(b_(3)-b_(1))alpha_(2)+dots],[dots(b_(1)-b_(1))alpha_(1+1)+(b_(1+1)-b_(1))alpha_(1+1)+dots+(b_(n+1)-b_(1))alpha_(n+1) <= ],[b_(1)(alpha_(1)+dots+alpha_(n+1))+(b_(2)-b_(1))alpha_(2)+(b_(3)-b_(1))alpha_(3)+dots],[dots+(b_(1)-b_(1))alpha_(1)+(b_(1+1)-b_(1))alpha_(1+1)+dots+(b_(n+1)-b_(1))alpha_(n+1)],[=b_(1)alpha_(1)+b_(2)alpha_(2)+dots+b_(n+1)alpha_(n+1)","]:}\begin{gathered} \alpha_{1} b_{1}+\ldots+\alpha_{j_{n}} b_{n}+\alpha_{j_{n+1}} b_{n+1}=b_{1}\left(\alpha_{j_{1}}+\ldots+\alpha_{j_{n+1}}\right)+ \\ +\left(b_{2}-b_{1}\right) \alpha_{j_{2}}+\left(b_{3}-b_{1}\right) \alpha_{j_{3}}+\ldots+\left(b_{n+1}-b_{1}\right) \alpha_{j_{n+1}} \leqq \\ b_{1}\left(c_{1} \& \ldots+\alpha_{n+1}\right)+\left(b_{2}-b_{1}\right) \alpha_{1}+\left(b_{3}-b_{1}\right) \alpha_{2}+\ldots \\ \ldots\left(b_{1}-b_{1}\right) \alpha_{1+1}+\left(b_{1+1}-b_{1}\right) \alpha_{1+1}+\ldots+\left(b_{n+1}-b_{1}\right) \alpha_{n+1} \leq \\ b_{1}\left(\alpha_{1}+\ldots+\alpha_{n+1}\right)+\left(b_{2}-b_{1}\right) \alpha_{2}+\left(b_{3}-b_{1}\right) \alpha_{3}+\ldots \\ \ldots+\left(b_{1}-b_{1}\right) \alpha_{1}+\left(b_{1+1}-b_{1}\right) \alpha_{1+1}+\ldots+\left(b_{n+1}-b_{1}\right) \alpha_{n+1} \\ =b_{1} \alpha_{1}+b_{2} \alpha_{2}+\ldots+b_{n+1} \alpha_{n+1}, \end{gathered}a1b1+…+ajnbn+ajn+1bn+1=b1(aj1+…+ajn+1)++(b2−b1)aj2+(b3−b1)aj3+…+(bn+1−b1)ajn+1≦b1(c1&…+an+1)+(b2−b1)a1+(b3−b1)a2+……(b1−b1)a1+1+(b1+1−b1)a1+1+…+(bn+1−b1)an+1≤b1(a1+…+an+1)+(b2−b1)a2+(b3−b1)a3+……+(b1−b1)a1+(b1+1−b1)a1+1+…+(bn+1−b1)an+1=b1a1+b2a2+…+bn+1an+1,
co that he was going to demonstrate the inequality (33) of the demonstration of the above volume is known (33, p. 539) From the above considerations and the volume the result
THEORMAL Of touies 10 s ( n + 1 ) 1 10 s ( n + 1 ) 1 10 s(n+1)110 s(n+1) 110s(n+1)1iterative methods of forging (25) (27), 90110 for 18que12e we obtain 1 maximum order of con－
yergency is the method obtained by arranging the numbers. D1 in
decreasing order and the numbers α 1 α 1 alpha_(1)\alpha_{1}a1in ascending order.

BIBLIOGRAPHIF

[1]. C. IANCU, I. PAVALOIU, La nésolution des équations par integ; polation inverse de type Hermite. Seminar of Functional Analysis and Numerical Methods. "Babes-Bolyai" University, Faculty of Mathematics, Research Seminaries, Preprint No. 4 (1981), 7248,
[2]. I. PAVALOIU, Rezlovarea equaitiolor prin interpolation. Dacia Publishing House, Oluj-Napoca, 1981.
[3]. S. POPA, Asupra unei probleme a lui P. Erdös si G. Wezss, Study 6i Cercetari Matematice, 33, 5 (1981), 539-542.
1983

Related Posts