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 𝑓𝐷𝐑𝐑, 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=𝑥𝑓(𝑥)𝑓,(𝑥)𝑧=𝑦𝑓(𝑦)𝑓(𝑥)𝑓(𝑥),𝑓(𝑥)2𝑓(𝑦)̂𝑥=𝑧𝑓(𝑧)𝑓,(𝑧)(1.1) where 𝑥 is a current approximation and ̂𝑥 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=𝑓(𝑧)𝑓(𝑦),𝑣=𝑓(𝑧).𝑓(𝑥)(1.3) Now the iterative scheme (1.1) becomes 𝑦=𝑥𝑓(𝑥)𝑓,(𝑥)𝑧=𝑦𝑓(𝑦)𝑓(𝑥)𝑓(𝑥),𝑓(𝑥)2𝑓(𝑦)̂𝑥=𝑧𝑓(𝑧)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 𝑡0, 𝑠0, and 𝑣0 when 𝑥0. We have 𝜙(𝑡)=𝜙(0)+𝜙𝜙(0)𝑡+(0)𝑡2!2+𝜙(0)𝑡3!3+,𝜓(𝑣)=𝜓(0)+𝜓(0)𝑠+,𝜔(𝑤)=𝜔(0)+𝜔(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 e ̂ 2+c3 e ̂ 3+c4 e ̂ 4); f1x=D[fx,e];
ey=e-Series[fx/f1x, { e,0,8 } ];
fy=f1a*(ey+c2 ey ̂ 2+c3 ey ̂ 3+c4 ey ̂ 4);
t=Series[fy/fx, { e,0,8 } ];
ez=ey-Series[1/(1−2t)*fy/f1x, { e,0,8 } ];
fz=f1a*(ez+c2 ez ̂ 2+c3 ez ̂ 3);
s=Series[fz/fy, { e,0,8 } ];
v=Series[fz/fx, { e,0,8 } ];
gt=t0+t10*t+t20*t ̂ 2/2+t30*t ̂ 3/6;
gs=s0+s10*s+s20*s ̂ 2/2;
gv=v0+v10*v+v20*v ̂ 2/2;
e1=ez-Series [fz/f1z, { e,0,8 } ]//Simplify
C1: “Out[a4] = 𝑐 2 ( 𝑐 2 2 𝑐 3 ) (-1+t0 s0 v0)/(t0 s0 v0)
t0=1; s0=1; v0=1; a5=Coefficient[e1,e ̂ 5] //Simplify
C2: “Out [ a5 ] = 𝑐 2 2 ( 𝑐 2 2 𝑐 3 ) (2 + t10)”
t10=−2;  a6=Coefficient[e1,e ̂ 6]//Simplify
C3: “Out [ a6 1 ] = 2 𝑐 2 ( 𝑐 2 2 𝑐 3 ) ( 2 𝑐 3 (1+s10) + 𝑐 2 2 (4 + t20 + 2v10) )
s10=−1;  t20=-2;  a7=Coefficient[e1,e ̂ 7]//Simplify
C4: “Out [ a7 1 ] = 6 𝑐 2 2 ( 𝑐 2 2 𝑐 3 ) ( 6 𝑐 3 ( 2 + v10 ) + 𝑐 2 2 ( t30 + 6 ( 2 + v10 ) ) )
v10=−2;  t30=0;  a8=Coefficient[e1,e ̂ 8]//Simplify
C5: “Out [ a8 1 ] = 2 𝑐 2 ( 𝑐 2 2 𝑐 3 ) ( 2 𝑐 2 𝑐 4 + 𝑐 3 2 𝑣 2 0 + 𝑐 2 4 ( 4 + 𝑠 2 0 ) 2 𝑐 2 2 𝑐 3 ( 4 + 𝑠 2 0 ) ) 𝑒 8 + 𝑂 [ 𝑒 9 ]

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=𝑥𝛼, ey=𝑦𝛼, ez=𝑧𝛼, e1=̂𝑥𝛼, 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 ̂𝜀=̂𝑥𝛼 we observe that ̂𝜀 is of the form ̂𝜀=𝑎4𝜀4+𝑎5𝜀5+𝑎6𝜀6+𝑎7𝜀7+𝑎8𝜀8𝜀+𝑂9.(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=0𝜙(0)=2,C3:𝜓(0)+1=0𝜙(0)+2𝜓(0)+4=0𝜓(0)=1, 𝜙(0)=2,C4:𝜔(0)+2=0𝜙(0)+6(𝜔(0)+2)=0𝜔(0)=2, 𝜙(0)=0.

Comment 2.   C5: substituting the quantities 𝜙(0),𝜙(0),,𝜔(0) in the expression of ̂𝜀, found in the described interactive procedure, we obtain 1̂𝜀=2𝑐2𝑐22𝑐32𝑐2𝑐4+4𝑐428𝑐22𝑐3+𝜓𝑐(0)23+𝑐422𝑐22𝑐3𝑒8𝜀+𝑂9.(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 𝑦𝑘=𝑥𝑘𝑓𝑥𝑘𝑥𝑓𝑘,𝑧𝑘=𝑦𝑘𝑓𝑦𝑘𝑥𝑓𝑘𝑓𝑥𝑘𝑓𝑥𝑘𝑦2𝑓𝑘,𝑥𝑘+1=𝑧𝑘𝑓𝑧𝑘𝑥𝑓𝑘𝜙𝑡𝑘𝜓𝑠𝑘𝜔𝑣𝑘,𝑡𝑘=𝑓𝑦𝑘𝑓𝑥𝑘,𝑠𝑘=𝑓𝑧𝑘𝑓𝑦𝑘,𝑣𝑘=𝑓𝑧𝑘𝑓𝑥𝑘,(2.5) has the order eight if sufficiently times differentiable functions 𝜙, 𝜓, and 𝜔 are chosen so that the following conditions are fulfilled: 𝜙(0)=1,𝜙(0)=2,𝜙(0)=2,𝜙(0)=0,𝜓(0)=1,𝜓||𝜓(0)=1,||(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, 𝑦𝑘=𝑥𝑘𝑓𝑥𝑘𝑓𝑥𝑘,𝑧𝑘=𝑦𝑘𝑓𝑦𝑘𝑓𝑥𝑘𝑓𝑥𝑘𝑓𝑥𝑘𝑦2𝑓𝑘,𝑥𝑘+1=𝑧𝑘𝑓𝑧𝑘𝑓𝑥𝑘𝑦12𝑓𝑘𝑥/𝑓𝑘𝑓𝑦𝑘𝑥/𝑓𝑘2𝑧1𝑓𝑘𝑦/𝑓𝑘𝑧12𝑓𝑘𝑥/𝑓𝑘.(2.7) Using the approximation 1/(1𝑎𝑞)1+𝑎𝑞 for sufficiently small |𝑞|, the last iterative formula may be modified to the form 𝑦𝑘=𝑥𝑘𝑓𝑥𝑘𝑓𝑥𝑘,𝑧𝑘=𝑦𝑘𝑓𝑦𝑘𝑓𝑥𝑘𝑓𝑥𝑘𝑓𝑥𝑘𝑦2𝑓𝑘,𝑥𝑘+1=𝑧𝑘𝑓𝑧𝑘𝑧1+𝑓𝑘𝑦/𝑓𝑘𝑧1+2𝑓𝑘𝑥/𝑓𝑘𝑓𝑥𝑘𝑦12𝑓𝑘𝑥/𝑓𝑘𝑓𝑦𝑘𝑥/𝑓𝑘2.(2.8)

Some other simple forms of functions 𝜙, 𝜓, and 𝜔 are 𝜙(𝑡)=14𝑡2(1𝑡)(1+𝑡)2,𝜙(𝑡)=512𝑡52𝑡+𝑡2,𝑠𝜓(𝑠)=1𝑛𝑛𝑠,preferable𝜓(𝑠)=122for𝑛=2,𝜓(𝑠)=1+𝑎𝑠11+(𝑎+1)𝑠,𝑎𝐑,preferable𝜓(𝑠)=11+𝑠for𝑎=0,𝜓(𝑠)=1+𝑠+𝑐𝑠2,𝑐𝐑,𝜔(𝑣)=12𝑣𝑛𝑛,preferable𝜔(𝑣)=(1𝑣)2for𝑛=2,𝜔(𝑣)=1+𝑏𝑣11+(𝑏+2)𝑣,𝑏𝐑,preferable𝜔(𝑣)=11+2𝑣for𝑏=0,𝜔(𝑣)=1+2𝑣+𝑑𝑣2,𝑑𝐑.(2.9) It is interesting to note that functions 𝜓(𝑠)=𝑒𝑠 and 𝜔(𝑣)=𝑒2𝑣 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]
𝑦𝑘=𝑥𝑘𝑓𝑥𝑘𝑓𝑥𝑘,𝑧𝑘=𝑦𝑘𝜇𝑘𝑓𝑦𝑘𝑓𝑥𝑘,𝑥𝑘+1=𝑧𝑘𝑓𝑥𝑘𝑧+𝛽𝑓𝑘𝑓𝑥𝑘𝑧+(𝛽2)𝑓𝑘𝑓𝑧𝑘𝑓𝑧𝑘,𝑦𝑘𝑧+𝑓𝑘,𝑥𝑘,𝑥𝑘𝑧𝑘𝑦𝑘(𝛽𝐑),(3.1) where 𝜇𝑘=𝑓(𝑦𝑘)/𝑓(𝑥𝑘),(𝑡) is a real-valued function and 𝑓[]=𝑧,𝑦𝑓(𝑧)𝑓(𝑦)[]=𝑓[]𝑧𝑦,𝑓𝑧,𝑥,𝑥𝑧,𝑥𝑓(𝑥).𝑧𝑥(3.2) We have tested two methods belonging to the family (3.1), obtained by choosing two different forms of the weight function in the same way as in [4] (see Tables 1 and 2).

Kung-Traub's Method
Version 1 without derivatives [5]: 𝑦𝑘=𝑥𝑘𝑥𝛾𝑓𝑘2𝑓𝑥𝑘𝑥+𝛾𝑓𝑘𝑥𝑓𝑘,𝑧𝑘=𝑦𝑘𝑓𝑦𝑘𝑓𝑥𝑘𝑥+𝛾𝑓𝑘𝑓𝑥𝑘𝑥+𝛾𝑓𝑘𝑦𝑓𝑘𝑓𝑥𝑘,𝑦𝑘𝑥,(𝑘=0,1,),𝑘+1=𝑧𝑘𝑓𝑦𝑘𝑓𝑥𝑘𝑥+𝛾𝑓𝑘𝑦𝑘𝑥𝑘𝑥+𝑓𝑘𝑥/𝑓𝑘,𝑧𝑘𝑓𝑦𝑘𝑧𝑓𝑘𝑓𝑥𝑘𝑥+𝛾𝑓𝑘𝑧𝑓𝑘+𝑓𝑦𝑘𝑓𝑦𝑘,𝑧𝑘,(3.3) where 𝛾 is a real parameter.

Kung-Traub's Method
Version 2 with derivative [5]: 𝑦𝑘=𝑥𝑘𝑓𝑥𝑘𝑓𝑥𝑘,𝑧𝑘=𝑦𝑘𝑓𝑥𝑘2𝑓𝑦𝑘𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑘2𝑥,(𝑘=0,1,),𝑘+1=𝑧𝑘𝑓𝑥𝑘2𝑓𝑦𝑘Δ(𝑘)𝑦𝑧1Δ(𝑘)𝑥𝑧𝑥𝑘𝑧𝑘Δ(𝑘)𝑥𝑧1𝑓𝑥𝑘𝑓𝑦𝑘𝑓𝑥𝑘Δ(𝑘)𝑥𝑦2,(3.4) where, for example, Δ(𝑘)𝑥𝑧𝑥=𝑓𝑘𝑧𝑓𝑘.(3.5)

Liu-Wang's Method [6]

Remark 3.1. There are other three-point methods with optimal order eight, see, for instance, [3, 715]. 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: 𝑥𝑓(𝑥)=log2+1+𝑒𝑥sin𝑥,𝑥0𝑓=0.3,𝛼=0,(𝑥)=1+𝑒𝑥3𝑥cos1𝑥2+𝑥3,𝑥0=1.65,𝛼=1.(3.7) The errors |𝑥𝑘𝛼| of approximations to the zeros are given in Tables 1 and 2, where 𝐴() denotes 𝐴×10. These tables include the values of the computational order of convergence 𝑟𝑐 calculated by the formula 𝑟𝑐=||𝑓𝑥log𝑘𝑥/𝑓𝑘1||||𝑓𝑥log𝑘1𝑥/𝑓𝑘2||,(3.8) taking into consideration the last three approximations in the iterative process.


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