A Family of Three-Point Methods of Ostrowski's Type for Solving Nonlinear Equations

Abstract

A class of three-point methods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two-point Ostrowski's fourth-order methods and a modified Newton's method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three-step methods have order eight costing only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Two numerical examples for various weight functions are given to demonstrate very fast convergence and high computational efficiency of the proposed multipoint methods.

1. Introduction

Multipoint methods for solving nonlinear equations 𝑓(𝑥)=0, where 𝑓∶𝐷⊂𝐑→𝐑, possess an important advantage since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. More details may be found in the book [1] and many papers published in the first decade of the 21st century. In this paper we present a new family of three-point methods which employs Ostrowski's method in the first two steps and suitably chosen weight functions in the third step. The order of this family is eight requiring four function evaluations.

We start with a three-step scheme (omitting iteration index for simplicity) 𝑦=𝑥−𝑓(𝑥)𝑧=𝑦−𝑓(𝑦)𝑓^(𝑥),𝑥𝑓̂=^(𝑥𝑧)−⋅𝑓𝑓((𝑥𝑧)𝑓)−(𝑥2)𝑓(𝑦),𝑓^(𝑧),(1.1) where 𝑥 is a current approximation and 𝑥̂ is a new approximation to a simple real zero 𝛼 of 𝑓. Note that the first two steps form Ostrowski's two-point method [2] of order four.

The iterative method (1.1) has order eight but it requires five function evaluations, which is expensive from the computational point of view. To decrease this cost from 5 to 4 function evaluations, we want to approximate 𝑓′(𝑧) in the third step of (1.1) using available data 𝑓(𝑥), 𝑓′(𝑥), 𝑓(𝑦), 𝑓(𝑧). We are seeking this approximation in the form 𝑓^(𝑧)≈𝑓^(𝑥)𝜙(𝑡)𝜓(𝑠)𝜔(𝑣),(1.2) where 𝜙, 𝜓, and 𝜔 are sufficiently differentiable real-valued functions with the arguments 𝑡=𝑓(𝑦)𝑓(𝑥),𝑠=𝑓𝑓((𝑧𝑦)),𝑣=𝑓𝑓((𝑥𝑧)).(1.3) Now the iterative scheme (1.1) becomes 𝑦=𝑥−𝑓(𝑥)𝑧=𝑦−𝑓(𝑦)𝑓^(𝑥),𝑥̂=𝑧−𝑓^(𝑥)⋅𝑓𝑓((𝑥𝑧)𝑓)−(𝑥2)𝑓(𝑦),𝑓^(𝑥)𝜙(𝑡)𝜓(𝑠)𝜔(𝑣).(1.4)

Functions 𝜙, 𝜓, and 𝜔 should be determined in such a way that the iterative method (1.4) attains the order eight. Such procedure will be presented in Section 2.

2. Construction and Convergence of New Three-Point Root Solvers

To find the weight functions 𝜙, 𝜓, and 𝜔 in (1.4) providing order eight, we will use the method of undetermined coefficients and Taylor's series about 0 since 𝑡→0, 𝑠→0, and 𝑣→0 when 𝑥→0. We have 𝜙(𝑡)=𝜙(0)+𝜙^(0)𝑡+𝜙^(0)2!𝑡²+𝜙^(0)3!𝑡³𝜓(𝑣)=𝜓(0)+𝜓_+⋯,𝜔(𝑤)=𝜔(0)+𝜔_(0)𝑠+⋯,(0)𝑣+⋯.(2.1) The simplest method for finding the coefficients of the above Taylor expansions is the use of symbolic computation by a computer algebra system and an interactive procedure (comments C1–C4 in Algorithm  1), as already carried out for some of the previously developed methods, see, for example, [3]. The corresponding program can always display any desired formula or expression, although these expressions are cumbersome and only of academic interest.

fx=f1a*(e+c2 ê‌2+c3 ê‌3+c4 ê‌4); f1x=D[fx,e]; ey=e-Series[fx/f1x, { e,0,8 } ]; fy=f1a*(ey+c2 eŷ‌2+c3 eŷ‌3+c4 eŷ‌4); t=Series[fy/fx, { e,0,8 } ]; ez=ey-Series[1/(1−2t)*fy/f1x, { e,0,8 } ]; fz=f1a*(ez+c2 eẑ‌2+c3 eẑ‌3); s=Series[fz/fy, { e,0,8 } ]; v=Series[fz/fx, { e,0,8 } ]; gt=t0+t10*t+t20*t̂‌2/2+t30*t̂‌3/6; gs=s0+s10*s+s20*ŝ‌2/2; gv=v0+v10*v+v20*v̂‌2/2; f1z=f1x*gt*gs*gv; e1=ez-Series [fz/f1z, { e,0,8 } ]//Simplify C1: “Out[a4]=𝑐2(𝑐22−𝑐3)(-1+t0 s0 v0)/(t0 s0 v0)” t0=1; s0=1; v0=1; a5=Coefficient[e1,ê‌5] //Simplify C2: “Out[a5]=𝑐22(𝑐22−𝑐3)(2 + t10)” t10=−2;  a6=Coefficient[e1,ê‌6]//Simplify C3: “Out[a6]=12𝑐2(𝑐22−𝑐3)(−2𝑐3 (1+s10) + 𝑐22(4 + t20 + 2v10))” s10=−1;  t20=-2;  a7=Coefficient[e1,ê‌7]//Simplify C4: “Out[a7]=16𝑐22(𝑐22−𝑐3)(−6𝑐3(2+ v10)+𝑐22(t30+6(2+v10)))” v10=−2;  t30=0;  a8=Coefficient[e1,ê‌8]//Simplify C5: “Out[a8]=12𝑐2(𝑐22−𝑐3)(2𝑐2𝑐4+𝑐32𝑣20+𝑐24(4+𝑠20)−2𝑐22𝑐3(4+𝑠20))𝑒8+𝑂[𝑒9]”

Algorithm 1 

Program (written in Mathematica).

We introduce the following abbreviations:ck=𝑓^(𝑘)(𝛼)/(𝑘!𝑓′(𝛼)), fx=𝑓(𝑥), f1x=𝑓′(𝑥), f1a=𝑓′(𝛼), fy=𝑓(𝑦), fz=𝑓(𝑧),e=𝑥−𝛼, ey=𝑦−𝛼, ez=𝑧−𝛼, e1=𝑥̂−𝛼, t0=𝜓(0), t10=𝜓′(0), t20=𝑝^(0), t30=𝜓^(0), s0=𝜓(0), s10=𝜓′(0),v0=𝜔(0), v10=𝜔′(0).

Comment 1. C1: from the expression of the error 𝜀̂=𝑥̂−𝛼 we observe that 𝜀̂ is of the form 𝜀̂=𝑎₄𝜀⁴+𝑎₅𝜀⁵+𝑎₆𝜀⁶+𝑎₇𝜀⁷+𝑎₈𝜀⁸+𝑂𝜀⁹.(2.2) The iterative three-point method (1.4) will have the order of convergence equal to eight if we determine the coefficients of the developments appearing in (2.1) in such way that 𝑎₄, 𝑎₅, 𝑎₆, 𝑎₇ (in (2.2)) all vanish. We find these coefficients equating shaded expressions in boxed formulas to 0. First, from Out[a4] we have −1+𝜙(0)𝜓(0)𝜔(0)=0.(2.3) Without the loss of generality, we can take 𝜙(0)=𝜓(0)=𝜔(0)=1 with the benefit that the term 𝜙(0)𝜓(0)𝜔(0) becomes 1 simplifying subsequent expressions.
In what follows, equating coefficient 𝑎₅, 𝑎₆, 𝑎₇ to 0, one obtainsC2:𝜙^(0)+2=0⇒𝜙^(0)=−2,C3:𝜓^(0)+1=0∧𝜙^(0)+2𝜓^(0)+4=0⇒𝜓^(0)=−1, 𝜙^(0)=−2,C4:𝜔^(0)+2=0∧𝜙^(0)+6(𝜔^(0)+2)=0⇒𝜔^(0)=−2, 𝜙^(0)=0.

Comment 2.   C5: substituting the quantities 𝜙(0),𝜙^(0),…,𝜔′(0) in the expression of 𝜀̂, found in the described interactive procedure, we obtain 𝜀̂=12𝑐²𝑐²²−𝑐₃2𝑐₂𝑐₄+4𝑐⁴²−8𝑐²²𝑐₃+𝜓^(0)𝑐²³+𝑐⁴²−2𝑐²²𝑐₃𝑒⁸+𝑂𝜀⁹.(2.4) Observe from (2.4) that 𝜓^(0) must be bounded.
According to the above analysis we can state the following theorem.

Theorem 2.1   1. If 𝑥₀ is a sufficiently close approximation to a zero 𝛼 of 𝑓, then the family of three-point methods 𝑦_𝑘=𝑥_𝑘−𝑓𝑥^𝑘𝑧_𝑘𝑓′𝑥_𝑘,=𝑦_𝑘−𝑓𝑦^𝑘𝑓′𝑥_𝑘⋅𝑓𝑥^𝑘𝑥_𝑘+1𝑓𝑥_𝑘−2𝑓𝑦_𝑘,=𝑧_𝑘−𝑓𝑧^𝑘𝑓′𝑥_𝑘𝜙𝑡_𝑘𝜓𝑠_𝑘𝜔𝑣_𝑘,𝑡^𝑘=𝑓𝑦^𝑘𝑠_𝑘𝑓𝑥_𝑘,=𝑓𝑧^𝑘𝑓𝑦_𝑘,𝑣^𝑘=𝑓𝑧^𝑘𝑓𝑥_𝑘,(2.5) has the order eight if sufficiently times differentiable functions 𝜙, 𝜓, and 𝜔 are chosen so that the following conditions are fulfilled: 𝜓(0)=𝜙1,(0)𝜓=_1,𝜙^(0)=2,𝜙^(0)=−2,𝜙^(0)=0,(0)=−1,||𝜓^(0)||<∞,𝜔(0)=1,𝜔′(0)=−2.(2.6) Values of higher order derivatives of 𝜙, 𝜓, and 𝜔, not explicitly given in (2.6), can be arbitrary at the point 0.

Weight functions 𝜙, 𝜓, and 𝜔 should be chosen as simple as possible. One of the simplest forms is that obtained by using the Taylor polynomials of these functions according to (2.6), that is, 𝑦_𝑘=𝑥_𝑘−𝑓𝑥^𝑘𝑧_𝑘𝑓^𝑥_𝑘,=𝑦_𝑘−𝑓𝑦^𝑘𝑓^𝑥_𝑘⋅𝑓𝑥^𝑘𝑥_𝑘+1𝑓𝑥_𝑘−2𝑓𝑦_𝑘,=𝑧_𝑘−𝑓𝑧^𝑘𝑓^𝑥_𝑘1−2𝑓𝑦_𝑘/𝑓𝑥_𝑘−𝑓𝑦_𝑘/𝑓𝑥_𝑘²1−𝑓𝑧^𝑘/𝑓𝑦_𝑘1−2𝑓𝑧_𝑘/𝑓𝑥_𝑘(2..7) Using the approximation 1/(1−𝑎𝑞)≈1+𝑎𝑞 for sufficiently small |𝑞|, the last iterative formula may be modified to the form 𝑦_𝑘=𝑥_𝑘−𝑓𝑥^𝑘𝑧_𝑘𝑓^𝑥_𝑘,=𝑦_𝑘−𝑓𝑦^𝑘𝑓^𝑥_𝑘⋅𝑓𝑥^𝑘𝑥_𝑘+1𝑓𝑥_𝑘−2𝑓𝑦_𝑘,=𝑧_𝑘−𝑓𝑧^𝑘1+𝑓𝑧_𝑘/𝑓𝑦_𝑘1+2𝑓𝑧_𝑘/𝑓𝑥_𝑘𝑓^𝑥_𝑘1−2𝑓𝑦_𝑘/𝑓𝑥_𝑘−𝑓𝑦_𝑘/𝑓𝑥_𝑘².(2.8)

Some other simple forms of functions 𝜙, 𝜓, and 𝜔 are 𝜙(𝑡)=1−4𝑡²(1−𝑡)(1+𝑡)²𝜙(𝑡)=5−12𝑡,5−2𝑡+𝑡²𝜓(𝑠)=1−𝑠,𝑛^𝑛,preferable𝜓(𝑠)=1−𝑠2²𝜓(𝑠)=1+𝑎𝑠for𝑛=2,𝜓(𝑠)=1+(𝑎1+1)𝑠,𝑎∈𝐑,preferable𝜓(𝑠)=1+1𝑠for𝑎=0,1+𝑠+𝑐𝑠²𝜔(𝑣)=1−2𝑣,𝑐∈𝐑,𝑛^𝑛𝜔(𝑣)=1+𝑏𝑣,preferable𝜔(𝑣)=(1−𝑣)²for𝑛=2,𝜔(𝑣)=1+(𝑏1+2)𝑣,𝑏∈𝐑,preferable𝜔(𝑣)=1+12𝑣for𝑏=0,1+2𝑣+𝑑𝑣²,𝑑∈𝐑.(2.9) It is interesting to note that functions 𝜓(𝑠)=𝑒^−𝑠 and 𝜔(𝑣)=𝑒^−2𝑣 do satisfy the requested conditions (2.6), but the calculation of exponential function increases computational cost, so such choice is not acceptable.

3. Numerical Results

The family of three-point methods (2.5) has been tested on numerous nonlinear equations along with some other methods of the same convergence rate. The programming package Mathematica with multiprecision arithmetic (800 significant decimal digits) was employed to provide very high accuracy. For comparison purposes, we have also tested the three-point methods of optimal order eight given below.

Bi-Wu-Ren's Family [4]
𝑦_𝑘=𝑥_𝑘−𝑓𝑥^𝑘𝑧_𝑘𝑓^𝑥_𝑘,=𝑦_𝑘−ℎ𝜇_𝑘𝑓𝑦^𝑘𝑥_𝑘+1𝑓^𝑥_𝑘,=𝑧_𝑘−𝑓𝑥^𝑘+𝛽𝑓𝑧_𝑘𝑓𝑥_𝑘+(𝛽−2)𝑓𝑧_𝑘⋅𝑓𝑧^𝑘𝑓𝑧_𝑘,𝑦_𝑘+𝑓𝑧_𝑘,𝑥_𝑘,𝑥_𝑘𝑧_𝑘−𝑦_𝑘(𝛽∈𝐑),(3.1) where 𝜇_𝑘=𝑓(𝑦_𝑘)/𝑓(𝑥_𝑘),ℎ(𝑡) is a real-valued function and 𝑓[𝑧,𝑦]=𝑓(𝑧)−𝑓(𝑦)𝑧−𝑦,𝑓[𝑧,𝑥,𝑥]=𝑓[𝑧,𝑥]−𝑓^(𝑥)𝑧−𝑥.(3.2) We have tested two methods belonging to the family (3.1), obtained by choosing two different forms of the weight function ℎ in the same way as in [4] (see Tables 1 and 2).

Table 1 

Table 2 

Kung-Traub's Method
Version 1 without derivatives [5]: 𝑦_𝑘=𝑥_𝑘−𝛾𝑓𝑥^𝑘²𝑧_𝑘𝑓𝑥_𝑘+𝛾𝑓𝑥_𝑘−𝑓𝑥_𝑘,=𝑦_𝑘−𝑓𝑦^𝑘𝑓𝑥_𝑘+𝛾𝑓𝑥_𝑘𝑥𝑓𝑥_𝑘+𝛾𝑓𝑥_𝑘−𝑓𝑦_𝑘𝑓𝑥_𝑘,𝑦_𝑘,(𝑘=0,1,…),_𝑘+1=𝑧_𝑘−𝑓𝑦^𝑘𝑓𝑥_𝑘+𝛾𝑓𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝑓𝑥_𝑘/𝑓𝑥_𝑘,𝑧_𝑘𝑓𝑦_𝑘−𝑓𝑧_𝑘𝑓𝑥_𝑘+𝛾𝑓𝑥_𝑘−𝑓𝑧_𝑘+𝑓𝑦^𝑘𝑓𝑦_𝑘,𝑧_𝑘,(3.3) where 𝛾 is a real parameter.

Kung-Traub's Method
Version 2 with derivative [5]: 𝑦_𝑘=𝑥_𝑘−𝑓𝑥^𝑘𝑧_𝑘𝑓^𝑥_𝑘,=𝑦_𝑘−𝑓𝑥^𝑘²𝑓𝑦_𝑘𝑓^𝑥_𝑘𝑓𝑥_𝑘−𝑓𝑦_𝑘²𝑥,(𝑘=0,1,…),_𝑘+1=𝑧_𝑘−𝑓𝑥^𝑘²𝑓𝑦_𝑘Δ^{(𝑘)𝑦𝑧}⎡⎢⎢⎢⎣1Δ^{(𝑘)𝑥𝑧}𝑥^𝑘−𝑧_𝑘Δ^{(𝑘)𝑥𝑧}−1𝑓^𝑥_𝑘−𝑓𝑦^𝑘𝑓^𝑥_𝑘Δ^{(𝑘)𝑥𝑦}²⎤⎥⎥⎥⎦,(3.4) where, for example, Δ^{(𝑘)𝑥𝑧}=𝑓𝑥_𝑘−𝑓𝑧_𝑘.(3.5)

Liu-Wang's Method [6]
𝑦_𝑘=𝑥_𝑘−𝑓𝑥^𝑘𝑧_𝑘𝑓^𝑥_𝑘,=𝑥_𝑘−𝑓𝑦^𝑘𝑓^𝑥_𝑘𝐻𝑓𝑦^𝑘𝑥_𝑘+1𝑓𝑥_𝑘,=𝑧_𝑘−𝑓𝑧^𝑘𝑓^𝑥_𝑘𝑈𝑓𝑦^𝑘𝑓𝑥_𝑘+𝑉𝑓𝑧^𝑘𝑓𝑦_𝑘+𝑊𝑓𝑧^𝑘𝐻(0)=1,𝐻_𝑓𝑥_𝑘(𝑎∈𝐑),𝑈(0)=2,𝑈(0)=1−𝑉(0)−𝑊(0),𝑈^(0)=2,_(0)=2+𝐻^(0),𝑈^(0)=−24+6𝐻^(0)+𝐻^(0),𝑉^(0)=1,𝑊^(0)=4.(3.6)

Remark 3.1. There are other three-point methods with optimal order eight, see, for instance, [3, 7–15]. However, these methods produce results of approximately same quality so that did not display them in Tables 1 and 2.
For demonstration, among many numerical experiments, we have selected the following two functions: 𝑓(𝑥)=𝑓(1𝑥+)=𝑒log_𝑥3−𝑥𝑥²+1+𝑒^𝑥sin𝑥,𝑥₀=0.3,𝛼=0,−cos1−𝑥²+𝑥³,𝑥₀=−1.65,𝛼=−1.(3.7) The errors |𝑥_𝑘−𝛼| of approximations to the zeros are given in Tables 1 and 2, where 𝐴(−ℎ) denotes 𝐴×10^−ℎ. These tables include the values of the computational order of convergence 𝑟_𝑐 calculated by the formula 𝑟_𝑐=log||𝑓𝑥^𝑘/𝑓𝑥_𝑘−1||log||𝑓𝑥_𝑘−1/𝑓𝑥_𝑘−2||,(3.8) taking into consideration the last three approximations in the iterative process.

Acknowledgment

This work was supported by the Serbian Ministry of Science under Grant no. 174022.