A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations

Abstract

We present two new families of iterative methods for obtaining simple roots of nonlinear equations. The first family is developed by fitting the model 𝑚(𝑥)=𝑒^𝑝𝑥(𝐴𝑥²+𝐵𝑥+𝐶) to the function 𝑓(𝑥) and its derivative 𝑓^′(𝑥), 𝑓^′′(𝑥) at a point 𝑥_𝑛. In order to remove the second derivative of the first methods, we construct the second family of iterative methods by approximating the equation 𝑓(𝑥)=0 around the point (𝑥_𝑛,𝑓(𝑥_𝑛)) by the quadratic equation. Analysis of convergence shows that the new methods have third-order or higher convergence. Numerical experiments show that new iterative methods are effective and comparable to those of the well-known existing methods.

1. Introduction

In this paper, we consider iterative methods to find a simple root𝛼, that is,𝑓(𝛼)=0and𝑓^′(𝛼)≠0, of a nonlinear equation: 𝑓(𝑥)=0, (1) where𝑓∶𝐼⊂𝑅→𝑅for an open interval𝐼is a scalar function.

Many of the complex problems in science and engineering contains the function of nonlinear and transcendental nature in (1), so finding the simple roots of the nonlinear equation is one of the most important problems in numerical analysis. Numerical iterative methods are often used to obtain the approximate solution of such problems because it is not always possible to obtain its exact solution by usual algebraic process. We all know that Newton’s method is an important and basic approach for solving nonlinear equations [1, 2], its formulation is given by 𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (2) and this method converges quadratically. Earlier, many investigations [3–21] have been made to explain the root of nonlinear algebraic and transcendental equations.

The outline of the paper is as follows. In Section 2, we firstly describe a one-parameter family of third-order methods by fitting the model𝑚(𝑥)=𝑒^𝑝𝑥(𝐴𝑥²+𝐵𝑥+𝐶)to the function𝑓(𝑥)and its derivative𝑓^′(𝑥),𝑓^′′(𝑥)at a point𝑥_𝑛, and then we use a quadratic equation for approximating the equation𝑓(𝑥)=0to obtain a four-parameter family of second-derivative-free iterative methods. In Section 3, we obtain some different iterative methods by taking several parameters. In Section 4, different numerical tests confirm the theoretical results, and the new methods are comparable with other known methods and give better results in many cases. Finally, we infer some conclusions.

2. Development of Methods and Convergence Analysis

Consider the exponential model: 𝑚(𝑥)=𝑒^𝑝𝑥𝐴𝑥²+𝐵𝑥+𝐶, (3) where𝐴,𝐵,𝐶, and𝑝are parameters. We construct a new iteration scheme by fitting model (3) to the function𝑓(𝑥) and its derivative𝑓^′(𝑥) and𝑓^′′(𝑥)at a point𝑥_𝑛. Imposing the conditions as follows 𝑚𝑥_𝑛=𝑓𝑥_𝑛,𝑚^′𝑥_𝑛=𝑓^′𝑥_𝑛,𝑚^′′𝑥_𝑛=𝑓^′′𝑥_𝑛 (4) at the point (𝑥_𝑛,𝑚(𝑥_𝑛)) and then solving (3) and (4) for𝐴,𝐵, and 𝐶, we have 1𝐴=2𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑒^{−𝑝𝑥𝑛},𝐵=𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑥_𝑛𝑒^{−𝑝𝑥𝑛},𝐶=𝑓𝑥_𝑛−12𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑥²_𝑛−𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑥_𝑛𝑥_𝑛𝑒^{−𝑝𝑥𝑛}. (5)(6)(7) From (3), (4), and (5), we take the values of𝐴,𝐵, and𝐶into (3) and give 𝑚(𝑥)−𝑓𝑥_𝑛𝑒^{𝑝𝑥−𝑝𝑥𝑛}=12𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑥−𝑥_𝑛²𝑒^{𝑝𝑥−𝑝𝑥𝑛}+𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛𝑥−𝑥_𝑛𝑒^{𝑝𝑥−𝑝𝑥𝑛}. (8) At the root estimate𝑥_𝑛+1, it follows that𝑚(𝑥_𝑛+1)=0. We consider reducing (8) as follows 12𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑥−𝑥_𝑛²+𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛𝑥−𝑥_𝑛+𝑓𝑥_𝑛=0. (9) Thus from (9) we develop the iteration formula: 𝑥_𝑛+1=𝑥_𝑛−𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛1−𝐿_𝑓,𝑝𝑥_𝑛𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛, (10) where 𝐿_𝑓,𝑝𝑥_𝑛=2𝑓^′′𝑥_𝑛𝑓𝑥_𝑛+𝑝²𝑓²𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓^′2𝑥_𝑛. (11) The square root is required in (10); however, this may cost expensively and even fail in the case1−𝐿_𝑓,𝑝(𝑥_𝑛)<0. In order to avoid the calculation of the square roots, we will derive some forms free from square roots by Taylor approximation [5].

It is easy to know that Taylor approximation of1−𝐿_𝑓,𝑝(𝑥_𝑛)is 1−𝐿_𝑓,𝑝𝑥_𝑛=_𝑘≥012𝑘−𝐿_𝑓,𝑝𝑥_𝑛^𝑘. (12)

Using (12) in (10), we can obtain the following form: 𝑥_𝑛+1=𝑥_𝑛−𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛^𝑚_𝑘≥012𝑘−𝐿_𝑓,𝑝𝑥_𝑛^𝑘×𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛⁻¹. (13)

We have the convergence analysis of the methods by (13).

Theorem 1. Let𝛼∈𝐼be a simple zero of sufficiently differentiable function𝑓∶𝐼⊂𝑅→𝑅for an open interval𝐼. If𝑥₀is sufficiently close to𝛼, for𝑚≥1, the methods defined by (13) are cubically convergent.

Proof. Let𝑒_𝑛=𝑥_𝑛−𝛼, and we use the following Taylor expansions: 𝑓𝑥_𝑛=𝑓^′(𝛼)𝑒_𝑛+𝑐₂𝑒²_𝑛+𝑐₃𝑒³_𝑛+𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛, (14) where𝑐_𝑘=(1/𝑘!)(𝑓^(𝑘)(𝛼)/𝑓^′(𝛼)); furthermore, we have 𝑓^′𝑥_𝑛=𝑓^′(𝛼)1+2𝑐₂𝑒_𝑛+3𝑐₃𝑒²_𝑛+4𝑐₄𝑒³_𝑛+5𝑐₅𝑒⁴_𝑛+𝑂𝑒⁵_𝑛,𝑓^′′𝑥_𝑛=𝑓^′(𝛼)2𝑐₂+6𝑐₃𝑒_𝑛+12𝑐₄𝑒²_𝑛+20𝑐₅𝑒³_𝑛+30𝑐₆𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (15)(16) Since (15), we obtain 𝑓^′2𝑥_𝑛=𝑓^′2(𝛼)1+4𝑐₂𝑒_𝑛+6𝑐₃+4𝑐²₂𝑒²_𝑛+8𝑐₄+12𝑐₂𝑐₃𝑒³_𝑛+10𝑐₅+16𝑐₂𝑐₄+9𝑐²₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (17) Since (14), we get 𝑓²𝑥_𝑛=𝑓^′2(𝛼)𝑒²_𝑛+2𝑐₂𝑒³_𝑛+2𝑐₃+𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (18) From (14) and (15), we also easily have 𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝑓^′2(𝛼)𝑒_𝑛+3𝑐₂𝑒²_𝑛+2𝑐²₂+4𝑐₃𝑒³_𝑛+5𝑐₄+5𝑐₂𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (19) We obtain the following expression by taking into account (14) and (16): 𝑓𝑥_𝑛𝑓^′′𝑥_𝑛=𝑓^′2(𝛼)2𝑐₂𝑒_𝑛+2𝑐²₂+6𝑐₃𝑒²_𝑛+8𝑐₂𝑐₃+12𝑐₄𝑒³_𝑛+14𝑐₂𝑐₄+6𝑐²₃+20𝑐₅𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (20) From (18), (19), and (20), we obtain 2𝑓^′′𝑥_𝑛𝑓𝑥_𝑛+𝑝²𝑓²𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑓𝑥_𝑛=𝑓^′2(𝛼)4𝑐₂−2𝑝𝑒_𝑛+4𝑐²₂+12𝑐₃+𝑝²−6𝑝𝑐₂𝑒²_𝑛+16𝑐₂𝑐₃+24𝑐₄+2𝑝²𝑐₂−4𝑝𝑐²₂−8𝑝𝑐₃𝑒³_𝑛+28𝑐₂𝑐₄+12𝑐²₃+40𝑐₅+2𝑝²𝑐₃+𝑝²𝑐²₂−10𝑝𝑐₄−10𝑝𝑐₂𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (21) From (17) and (21), we have 𝐿_𝑓,𝑝𝑥_𝑛=𝑓^′(𝛼)4𝑐₂−2𝑝𝑒_𝑛+12𝑐₃−12𝑐²₂+𝑝²+2𝑝𝑐₂𝑒²_𝑛+24𝑐₄−56𝑐₂𝑐₃−2𝑝²𝑐₂−4𝑝𝑐²₂+4𝑝𝑐₃+32𝑐³₂𝑒³_𝑛+𝑂𝑒⁴_𝑛. (22) From (14) and (15), we have 𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛=𝑓^′(𝛼)1+2𝑐₂−𝑝𝑒_𝑛+3𝑐₃−𝑝𝑐₂𝑒²_𝑛+4𝑐₄−𝑝𝑐₃𝑒³_𝑛+5𝑐₅−𝑝𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (23) Using (14), (15), and (16), we have 𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛=𝑓^′(𝛼)2𝑐₂−2𝑝+6𝑐₃+𝑝²−4𝑝𝑐₂𝑒_𝑛+12𝑐₄+𝑝²𝑐₂−6𝑝𝑐₃𝑒²_𝑛+20𝑐₅+𝑝²𝑐₃−8𝑝𝑐₄𝑒³_𝑛+30𝑐₆+𝑝²𝑐₄−10𝑝𝑐₅𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (24) We know that _𝑚𝑘≥012𝑘−𝐿_𝑓,𝑝𝑥_𝑛^𝑘1=1−2𝐿_𝑓,𝑝𝑥_𝑛−18𝐿_𝑓,𝑝𝑥_𝑛²−1𝐿16^𝑓,𝑝𝑥_𝑛³+⋯=1−2𝑐₂−𝑝𝑒_𝑛+𝑝𝑐₂+4𝑐²₂−6𝑐₃−𝑝²𝑒²_𝑛−2𝑝²𝑐₂−16𝑐₂𝑐₃−4𝑝𝑐₃+12𝑐₄+8𝑐³₂−𝑝³𝑒³_𝑛+𝑂𝑒⁴_𝑛. (25) From (15) and (25), we obtain 𝑓^′𝑥_𝑛^𝑚_𝑘≥012𝑘−𝐿_𝑓,𝑝𝑥_𝑛^𝑘=𝑓^′(𝛼)1+𝑝𝑒_𝑛+3𝑝𝑐₂−𝑝²−3𝑐₃𝑒²_𝑛+7𝑝𝑐₃−4𝑝²𝑐₂−2𝑐₂𝑐₃+𝑝³−8𝑐₄+2𝑝𝑐²₂𝑒³_𝑛+𝑂𝑒⁴_𝑛. (26) From (23) and (26), we have 𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛^𝑚_𝑘≥012𝑘−𝐿_𝑓,𝑝𝑥_𝑛^𝑘=2𝑐₂−𝑝𝑒_𝑛−4𝑝𝑐₂−6𝑐₃−𝑝²𝑒²_𝑛−8𝑝𝑐₃−12𝑐₄−4𝑝²𝑐₂−2𝑐₂𝑐₃+𝑝³+2𝑝𝑐²₂𝑒³_𝑛+𝑂𝑒⁴_𝑛. (27) Using (13), (24), and (27), we have 𝑥_𝑛+1=𝑥_𝑛−𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛^𝑚_𝑘≥012𝑘−𝐿_𝑓,𝑝𝑥_𝑛^𝑘×𝑓^′′𝑥_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛⁻¹=𝑥_𝑛−𝑒_𝑛−12𝑝²+𝑐₃−𝑝𝑐₂𝑒³_𝑛+𝑂𝑒⁴_𝑛. (28) From𝑒_𝑛+1=𝑥_𝑛+1−𝛼, we have 𝑒_𝑛+1=𝑝𝑐₂−12𝑝²−𝑐₃𝑒³_𝑛+𝑂𝑒⁴_𝑛, (29) which completes the proof.

The family methods given by (13) are novel third-order methods, but the methods depend on the second derivatives in computing process, and therefore their practical applications are restricted in some cases. In recent years, several methods with free second derivatives have been developed; see [4–15] and references therein.

In order to avoid the calculation of the second derivatives, we consider approximating the equation 𝑓(𝑥)=0 around the point(𝑥_𝑛,𝑓(𝑥_𝑛))by the quadratic equation in𝑥and𝑦in the following form [8]: 𝑎₁𝑥²+𝑎₂𝑦²+𝑎₃𝑥𝑦+𝑎₄𝑥+𝑎₅𝑦+𝑎₆=0, (30) where𝑎_𝑖∈𝑅, 𝑖=1,2,…,6, are parameters. We impose the tangency conditions 𝑦𝑥_𝑛=𝑓𝑥_𝑛,𝑦^′𝑥_𝑛=𝑓^′𝑥_𝑛,𝑦𝑤_𝑛=𝑓𝑤_𝑛, (31) where𝑥_𝑛is𝑛th iterate and 𝑤_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (32) From (30) and (31), we have 𝑓^′′𝑥_𝑛≈𝑦^′′𝑥_𝑛=𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛=2𝑎₁+2𝑎₂𝑓^′2𝑥_𝑛+2𝑎₃𝑓^′𝑥_𝑛𝑓^′2𝑥_𝑛𝑓𝑤_𝑛×𝑎₁𝑓²𝑥_𝑛+𝑎₂𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑎₃𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛⁻¹, (33) where𝑖=1,2,3.

From (33) we can approximate 𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛=2𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛+𝑝²𝑓²𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓^′2𝑥_𝑛. (34) Using𝐿_{𝑎𝑖,𝑓,𝑝}(𝑥_𝑛,𝑤_𝑛)instead of𝐿_𝑓,𝑝(𝑥_𝑛)(11), we obtain a new four-parameter family of methods free from second derivative: 𝑥_𝑛+1=𝑥_𝑛−𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛^𝑚_𝑘≥012𝑘−𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛^𝑘×𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛⁻¹, (35) where 𝑎_𝑖 (𝑖=1,2,3), 𝑝∈𝑅,𝑚≥0.

We also have the convergence analysis of the methods by (35).

Theorem 2. Let𝛼∈𝐼be a simple zero of sufficiently differentiable function𝑓∶𝐼⊂𝑅→𝑅for an open interval𝐼. If𝑥₀is sufficiently close to𝛼, for𝑚≥1, 𝑎_𝑖 (𝑖=1,2,3), 𝑝∈𝑅, the methods defined by (35) are at least cubically convergent; as particular cases, if𝑚≥2, 𝑎₂=𝑎₃=𝑝=0, 𝑎₁∈𝑅 and the methods have convergence order four.

Proof. Let𝑒_𝑛=𝑥_𝑛−𝛼, and we use the following Taylor expansions: 𝑓𝑥_𝑛=𝑓^′(𝛼)𝑒_𝑛+𝑐₂𝑒²_𝑛+𝑐₃𝑒³_𝑛+𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛, (36) where𝑐_𝑘=(1/𝑘!)(𝑓^(𝑘)(𝛼)/𝑓^′(𝛼)); furthermore, we have 𝑓²𝑥_𝑛=𝑓^′2(𝛼)𝑒²_𝑛+2𝑐₂𝑒³_𝑛+2𝑐₃+𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛,𝑓^′𝑥_𝑛=𝑓^′(𝛼)1+2𝑐₂𝑒_𝑛+3𝑐₃𝑒²_𝑛+4𝑐₄𝑒³_𝑛+5𝑐₅𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (37)(38) Dividing (36) by (38) 𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝑒_𝑛−𝑐₂𝑒²_𝑛+2𝑐²₂−𝑐₃𝑒³_𝑛+7𝑐₂𝑐₃−4𝑐³₂−3𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (39) From (39), we get 𝑤_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝛼+𝑐₂𝑒²_𝑛−2𝑐²₂−𝑐₃𝑒³_𝑛−7𝑐₂𝑐₃−4𝑐³₂−3𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (40) Expanding𝑓(𝑤_𝑛)in Taylor’s Series about𝛼and using (40), we get 𝑓𝑤_𝑛=𝑓^′(𝛼)𝑤_𝑛−𝛼+𝑐₂𝑤_𝑛−𝛼²+𝑐₃𝑤_𝑛−𝛼³+𝑐₄𝑤_𝑛−𝛼⁴+⋯=𝑓^′(𝛼)𝑐₂𝑒²_𝑛+2𝑐₃−𝑐²₂𝑒³_𝑛+5𝑐³₂+3𝑐₄−7𝑐₂𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (41) From (36) and (41), we have 𝑓𝑥_𝑛𝑓𝑤_𝑛=𝑓^′2(𝛼)𝑐₂𝑒³_𝑛+2𝑐₃−𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (42) From (38), we obtain 𝑓^′2𝑥_𝑛=𝑓^′2(𝛼)1+4𝑐₂𝑒_𝑛+6𝑐₃+4𝑐²₂𝑒²_𝑛+8𝑐₄+12𝑐₂𝑐₃𝑒³_𝑛+10𝑐₅+16𝑐₂𝑐₄+9𝑐²₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (43) From (38), (41), and (43) we also easily obtain 2𝑎₁+2𝑎₂𝑓^′2𝑥_𝑛+2𝑎₃𝑓^′𝑥_𝑛𝑓^′2𝑥_𝑛𝑓𝑤_𝑛=𝑓^′3(𝛼)2𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁𝑐₂𝑒²_𝑛+4𝑎₂𝑓^′2(𝛼)𝑐₃+3𝑎₂𝑓^′2(𝛼)𝑐²₂+𝑎₃𝑓^′(𝛼)𝑐₃+2𝑓^′(𝛼)𝑎₃𝑐²₂+𝑎₁𝑐₃+𝑎₁𝑐²₂𝑒³_𝑛+23𝑎₂𝑓^′2(𝛼)𝑐₄+21𝑓^′2(𝛼)𝑎₂𝑐₂𝑐₃+5𝑓^′(𝛼)𝑎₃𝑐³₂+3𝑎₃𝑓^′(𝛼)𝑐₄+14𝑓^′(𝛼)𝑎₃𝑐₂𝑐₃+7𝑎₁𝑐₃𝑐₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (44) Substituting (36), (37), (38), (41), and (43) in the denominator of𝐿_𝑎𝑖(𝑥_𝑛,𝑤_𝑛), we obtain 𝑎₁𝑓²𝑥_𝑛+𝑎₂𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑎₃𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛=𝑓^′2(𝛼)𝑎₁+𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)𝑒²_𝑛+2𝑎₁𝑐₂+4𝑎₂𝑐₂𝑓^′2(𝛼)+3𝑎₃𝑐₂𝑓^′(𝛼)𝑒³_𝑛+2𝑎₁𝑐₃+𝑎₁𝑐²₂+8𝑎₂𝑐²₂𝑓^′2(𝛼)+4𝑎₂𝑐₃𝑓^′2(𝛼)+4𝑎₃𝑐²₂𝑓^′(𝛼)+3𝑎₃𝑐₃𝑓^′(𝛼)𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (45) Using (44) and (45), we have 𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛=𝑓^′(𝛼)2𝑐₂+22𝑎₂𝑓^′2(𝛼)𝑐₃+2𝑎₂𝑓^′2(𝛼)𝑐²₂+2𝑎₃𝑓^′(𝛼)𝑐₃+𝑓^′(𝛼)𝑎₃𝑐²₂+2𝑎₁𝑐₃𝑒_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁−+16𝑎²₂𝑓^′4(𝛼)𝑐³₂−14𝑓^′3(𝛼)𝑎₂𝑐₂𝑐₃𝑎₃−6𝑎₂𝑓^′3(𝛼)𝑐₄𝑎₃−9𝑓^′4(𝛼)𝑎²₂𝑐₂𝑐₃+17𝑓^′3(𝛼)𝑎₃𝑐³₂𝑎₂−5𝑓^′2(𝛼)𝑎²₃𝑐₂𝑐₃−3𝑎₂𝑓^′2(𝛼)𝑐₄𝑎₁+2𝑓^′(𝛼)𝑎₃𝑐³₂𝑎₁−3𝑎₃𝑓^′(𝛼)𝑐₄𝑎₁+𝑎²₁𝑐³₂−3𝑎²₂𝑓^′(𝛼)⁴𝑐₄+2𝑓^′(𝛼)²𝑎²₃𝑐³₂−3𝑎²₃𝑓^′(𝛼)²𝑐₄−𝑎²₁𝑐₃𝑐₂−10𝑓^′2(𝛼)𝑎₂𝑐₂𝑐₃𝑎₁−6𝑓^′(𝛼)𝑎₃𝑐₂𝑐₃𝑎₁+13𝑎₁𝑐³₂𝑎₂𝑓^′2(𝛼)𝑒²_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁²+𝑂𝑒³_𝑛. (46) From (36), (37), and (46), we have 2𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛+𝑝²𝑓²𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑓𝑥_𝑛=𝑓^′2(𝛼)22𝑐₂−𝑝𝑒_𝑛+12𝑎₂𝑓^′2(𝛼)𝑐²₂+8𝑓^′(𝛼)𝑎₃𝑐²₂+4𝑎₁𝑐²₂+𝑝²𝑎₂𝑓^′(𝛼)²+𝑝²𝑎₂𝑓^′(𝛼)+𝑝²𝑎₁+8𝑎₂𝑓^′2(𝛼)𝑐₃+8𝑎₃𝑓^′(𝛼)𝑐₃+8𝑎₁𝑐₃−6𝑝𝑐₂𝑎₂𝑓^′2(𝛼)−6𝑝𝑐₂𝑎₃𝑓^′(𝛼)−6𝑝𝑐₂𝑎₁𝑒²_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁−22𝑝𝑐²₂𝑐²𝑓^′2(𝛼)+2𝑝𝑐²₂𝑎²₂𝑓^′4(𝛼)−6𝑓^′2(𝛼)𝑎²₃𝑐₄−6𝑓^′4(𝛼)𝑎²₂𝑐₄+28𝑓^′4(𝛼)𝑎²₂𝑐³₂+2𝑎²₁𝑐³₂−𝑝²𝑐₂𝑎²₂𝑓^′4(𝛼)−𝑝²𝑐₂𝑎²₃𝑓^′2(𝛼)−2𝑝²𝑐₂𝑎₂𝑓^′3(𝛼)𝑎₃−2𝑝²𝑐₂𝑎₂𝑓^′2(𝛼)𝑎₁−2𝑝²𝑐₂𝑎₃𝑓^′(𝛼)𝑎₁+2𝑝𝑐²₂𝑎²₁+4𝑝𝑐₃𝑎²₁+2𝑓^′(𝛼)²𝑎²₃𝑐³₂−32𝑓^′2(𝛼)𝑎₂𝑐₂𝑐₃𝑎₁−16𝑓^′2(𝛼)𝑐²𝑐₂𝑐₃−40𝑓^′3(𝛼)𝑎₂𝑐₂𝑐₃𝑎₃−6𝑓^′2(𝛼)𝑎₂𝑐₄𝑎₁+2𝑓^′(𝛼)𝑎₃𝑐³₂𝑎₁−4𝑓^′2(𝛼)𝑎₁𝑐³₂𝑎₂−12𝑓^′3(𝛼)𝑎₂𝑐₄𝑎₃−6𝑓^′(𝛼)𝑎₃𝑐₄𝑎₁−6𝑓^′(𝛼)³𝑎₃𝑐³₂𝑎₂−24𝑓^′(𝛼)𝑎₃𝑐₂𝑐₃𝑎₁−8𝑎²₁𝑐₃𝑐₂−24𝑓^′4(𝛼)𝑎²₂𝑐₂𝑐₃+4𝑝𝑐²₂𝑎₂𝑓^′(𝛼)³𝑎₃+4𝑝𝑐²₂𝑎₂𝑓^′2(𝛼)𝑎₁+4𝑝𝑐²₂𝑎₃𝑓^′(𝛼)𝑎₁+8𝑝𝑐₃𝑎₂𝑓^′3(𝛼)𝑎₃+8𝑝𝑐₃𝑎₂𝑓^′2(𝛼)𝑎₁+8𝑝𝑐₃𝑎₃𝑓^′(𝛼)𝑎₁−𝑝²𝑐₂𝑎²₁+4𝑝𝑐₃𝑎²₃𝑓^′2(𝛼)+4𝑝𝑐₃𝑎²₂𝑓^′(𝛼)⁴𝑒²_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁²+𝑂𝑒³_𝑛. (47) Since (43) and (47), we get 𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛=22𝑐₂−𝑝𝑒_𝑛−4𝑎₂𝑓^′2(𝛼)𝑐²₂+8𝑎₃𝑓^′(𝛼)𝑐²₂+12𝑎₁𝑐²₂−𝑝²𝑎₂𝑓^′2(𝛼)−𝑝²𝑎₃𝑓^′(𝛼)−𝑝²𝑎₁−8𝑎₂𝑓^′2(𝛼)𝑐₃−8𝑎₃𝑓^′(𝛼)𝑐₃−8𝑎₁𝑐₃−2𝑝𝑐₂𝑎₂𝑓^′2(𝛼)−2𝑝𝑐₂𝑎₃𝑓^′(𝛼)−2𝑝𝑐₂𝑎₁𝑒²_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁+2−2𝑝𝑐²₂𝑎²₃𝑓^′2(𝛼)−2𝑝𝑐²₂𝑎²₂𝑓^′4(𝛼)+6𝑓^′2(𝛼)𝑎²₃𝑐₄+6𝑓^′4(𝛼)𝑎²₂𝑐₄−28𝑓^′4(𝛼)𝑎²₂𝑐³₂+14𝑎²₁𝑐³₂−𝑝²𝑐₂𝑎²₂𝑓^′4(𝛼)−𝑝²𝑐₂𝑎²₃𝑓^′2(𝛼)−2𝑝²𝑐₂𝑎₂𝑓^′3(𝛼)𝑎₃−2𝑝²𝑐₂𝑎₂𝑓^′2(𝛼)𝑎₁−2𝑝²𝑐₂𝑎₃𝑓^′(𝛼)𝑎₁−2𝑝𝑐²₂𝑎²₁+2𝑝𝑐₃𝑎²₁+6𝑓^′2(𝛼)𝑎²₃𝑐³₂−24𝑓^′2(𝛼)𝑎₂𝑐₂𝑐₃𝑎₁−12𝑓^′2(𝛼)𝑎²₃𝑐₂𝑐₃−16𝑓^′3(𝛼)𝑎₂𝑐₂𝑐₃𝑎₃+6𝑓^′2(𝛼)𝑎₂𝑐₄𝑎₁+22𝑓^′(𝛼)𝑎₃𝑐³₂𝑎₁+20𝑓^′2(𝛼)𝑎₁𝑐³₂𝑎₂+12𝑓^′3(𝛼)𝑎₂𝑐₄𝑎₃+6𝑓^′(𝛼)𝑎₃𝑐₄𝑎₁+14𝑓^′3(𝛼)𝑎₃𝑐³₂𝑎₂−32𝑓^′(𝛼)𝑎₃𝑐₂𝑐₃𝑎₁−20𝑎²₁𝑐₃𝑐₂−4𝑓^′4(𝛼)𝑎²₂𝑐₂𝑐₃−4𝑝𝑐²₂𝑎₂𝑓^′3(𝛼)𝑎₃−4𝑝𝑐²₂𝑎₂𝑓^′2(𝛼)𝑎₁−4𝑝𝑐²₂𝑎₃𝑓^′(𝛼)𝑎₁+4𝑝𝑐₃𝑎₂𝑓^′3(𝛼)𝑎₃+4𝑝𝑐₃𝑎₂𝑓^′2(𝛼)𝑎₁+4𝑝𝑐₃𝑎₃𝑓^′(𝛼)𝑎₁−𝑝²𝑐₂𝑎²₁+2𝑝𝑐₃𝑎²₃𝑓^′2(𝛼)+2𝑝𝑐₃𝑎²₂𝑓^′4(𝛼)𝑒³_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁²+𝑂𝑒⁴_𝑛. (48) Using (48), we write _𝑚𝑘≥012𝑘−𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛^𝑘1=1−2𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛−18𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛²−1𝐿16^{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛³+⋯=1−2𝑐₂−𝑝𝑒_𝑛+124𝑎₂𝑓^′2(𝛼)𝑐²₂+8𝑐𝑓^′(𝛼)𝑐²₂+12𝑎₁𝑐²₂−𝑝²𝑎₂𝑓^′2(𝛼)−𝑝²𝑎₃𝑓^′(𝛼)−𝑝²𝑎₁−8𝑎₂𝑓^′2(𝛼)𝑐₃−8𝑎₃𝑓^′(𝛼)𝑐₃−8𝑎₁𝑐₃−2𝑝𝑐₂𝑎₂𝑓^′2(𝛼)−2𝑝𝑐₂𝑎₃𝑓^′(𝛼)−2𝑝𝑐₂𝑎₁𝑒²_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁+𝑂𝑒³_𝑛. (49) Using (36), (38), and (46), we obtain 𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛+𝑝²𝑓𝑥_𝑛−𝑝𝑓^′𝑥_𝑛=𝑓^′(𝛼)2𝑐₂+−𝑝4𝑎₂𝑓^′2(𝛼)𝑐₃+4𝑎₂𝑓^′2(𝛼)𝑐²₂+4𝑎₃𝑓^′(𝛼)𝑐₃+2𝑎₃𝑓^′(𝛼)𝑐²₂+4𝑎₁𝑐₃+𝑝²𝑎₂𝑓^′2(𝛼)+𝑝²𝑎₃𝑓^′(𝛼)+𝑝²𝑎₁−4𝑝𝑐₂𝑎₂𝑓^′2(𝛼)−4𝑝𝑐₂𝑎₃𝑓^′(𝛼)−4𝑝𝑐₂𝑎₁𝑒_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁−6𝑝𝑐₃𝑎²₁−𝑝²𝑐₂𝑎²₂𝑓^′4(𝛼)−𝑝²𝑐₂𝑎²₁+6𝑝𝑓^′4(𝛼)𝑐₃𝑎²₂+6𝑝𝑓^′2(𝛼)𝑐₃𝑎²₃−2𝑝²𝑐₂𝑎₃𝑓^′(𝛼)𝑎₁−2𝑝²𝑐₂𝑎₂𝑓^′2(𝛼)𝑎₁−2𝑝²𝑐₂𝑎₂𝑓^′3(𝛼)𝑎₃−𝑝²𝑐₂𝑎²₃𝑓^′2(𝛼)+12𝑓^′3(𝛼)𝑝𝑐₃𝑎₂𝑎₃+12𝑝𝑓^′2(𝛼)𝑐₃𝑎₂𝑎₁+12𝑝𝑓^′(𝛼)𝑐₃𝑎₃𝑎₁−18𝑎²₂𝑓^′4(𝛼)𝑐₂𝑐₃+26𝑎₁𝑐³₂𝑎₂𝑓^′2(𝛼)−20𝑐₂𝑎₂𝑓^′2(𝛼)𝑐₃𝑎₁−12𝑐₂𝑎₃𝑓^′(𝛼)𝑐₃𝑎₁−12𝑓^′3(𝛼)𝑎₂𝑐₄𝑎₃−6𝑎²₂𝑓^′4(𝛼)𝑐₄+4𝑎²₃𝑓^′2(𝛼)𝑐³₂−6𝑎²₃𝑓^′2(𝛼)𝑐₄−2𝑎²₁𝑐₃𝑐₂+2𝑎²₁𝑐³₂+32𝑎²₂𝑓^′4(𝛼)𝑐³₂−28𝑐₂𝑎₂𝑓^′3(𝛼)𝑐₃𝑎₃+34𝑓^′3(𝛼)𝑎₃𝑐³₂𝑎₂−10𝑎²₃𝑓^′2(𝛼)𝑐₂𝑐₃−6𝑎₂𝑓^′2(𝛼)𝑐₄𝑎₁+4𝑎₃𝑓^′(𝛼)𝑐³₂𝑎₁−6𝑎₃𝑓^′(𝛼)𝑐₄𝑎₁𝑒²_𝑛/𝑎₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁²+𝑂𝑒³_𝑛. (50) Taking into account (36), (38), (49), and (50), we finally obtain 𝑥_𝑛+1=𝑥_𝑛−𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛×^𝑚_𝑘≥012𝑘−𝐿_{𝑎𝑖,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛^𝑘×𝐿_𝑎𝑖𝑥_𝑛,𝑤_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛⁻¹=𝑥_𝑛−𝑒_𝑛+12−40𝑓^′3(𝛼)𝑎₃𝑐³₂𝑎₂−30𝑎₁𝑐³₂𝑎₂𝑓^′2(𝛼)−2𝑎₃𝑓^′(𝛼)𝑐³₂𝑎₁−2𝑎²₃𝑓^′2(𝛼)𝑐³₂−4𝑎²₂𝑓^′4(𝛼)𝑐³₂+2𝑐²₂𝑝𝑎²₁+6𝑝𝑓^′(𝛼)𝑐²₂𝑎₃𝑎₁+8𝑝𝑓^′2(𝛼)𝑐²₂𝑎₂𝑎₁+4𝑝𝑓^′2(𝛼)𝑐²₂𝑎²₃+10𝑓^′3(𝛼)𝑝𝑐²₂𝑎₂𝑎₃+6𝑝𝑓^′4(𝛼)𝑐²₂𝑎²₂−3𝑝²𝑐₂𝑎²₃𝑓^′2(𝛼)−6𝑝²𝑐₂𝑎₂𝑓^′2(𝛼)𝑎₁−6𝑝²𝑐₂𝑎₃𝑓^′(𝛼)𝑎₁−3𝑝²𝑐₂𝑎²₁−3𝑝²𝑐₂𝑎²₂𝑓^′4(𝛼)−6𝑝²𝑐₂𝑎₂𝑓^′3(𝛼)𝑎₃+2𝑓^′2(𝛼)𝑝³𝑎₁𝑎₂+2𝑓^′3(𝛼)𝑝³𝑎₂𝑎₃+𝑝³𝑎²₁+2𝑓^′(𝛼)𝑝³𝑎₁𝑎₃+𝑓^′4(𝛼)𝑝³𝑎²₂+𝑓^′2(𝛼)𝑝³𝑎²₃𝑒³_𝑛/𝑐₂/𝑎−𝑝₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁²+𝑂𝑒⁴_𝑛. (51) Taking into account the last expression (51) and𝑒_𝑛+1=𝑥_𝑛+1−𝛼, we have 𝑒_𝑛+1=12−40𝑓^′3(𝛼)𝑎₃𝑐³₂𝑎₂−30𝑎₁𝑐³₂𝑎₂𝑓^′2(𝛼)−2𝑎₃𝑓^′(𝛼)𝑐³₂𝑎₁−2𝑎²₃𝑓^′2(𝛼)𝑐³₂−4𝑎²₂𝑓^′4(𝛼)𝑐³₂+2𝑐²₂𝑝𝑎²₁+6𝑝𝑓^′(𝛼)𝑐²₂𝑎₃𝑎₁+8𝑝𝑓^′2(𝛼)𝑐²₂𝑎₂𝑎₁+4𝑝𝑓^′2(𝛼)𝑐²₂𝑎²₃+10𝑓^′3(𝛼)𝑝𝑐²₂𝑎₂𝑎₃+6𝑝𝑓^′4(𝛼)𝑐²₂𝑎²₂−3𝑝²𝑐₂𝑎²₃𝑓^′2(𝛼)−6𝑝²𝑐₂𝑎₂𝑓^′2(𝛼)𝑎₁−6𝑝²𝑐₂𝑎₃𝑓^′(𝛼)𝑎₁−3𝑝²𝑐₂𝑎²₁−3𝑝²𝑐₂𝑎²₂𝑓^′4(𝛼)−6𝑝²𝑐₂𝑎₂𝑓^′3(𝛼)𝑎₃+2𝑓^′2(𝛼)𝑝³𝑎₁𝑎₂+2𝑓^′3(𝛼)𝑝³𝑎₂𝑎₃+𝑝³𝑎²₁+2𝑓^′(𝛼)𝑝³𝑎₁𝑎₃+𝑓^′4(𝛼)𝑝³𝑎²₂+𝑓^′2(𝛼)𝑝³𝑎²₃𝑒³_𝑛/𝑐₂/𝑎−𝑝₂𝑓^′2(𝛼)+𝑎₃𝑓^′(𝛼)+𝑎₁²+𝑂𝑒⁴_𝑛. (52)
This means that the methods defined by (35) are at least of order three for any 𝑎_𝑖 (𝑖=1,2,3),  𝑝∈𝑅. Furthermore, we consider that if 𝑚≥2, 𝑎₂=𝑎₃=𝑝=0, then the methods defined by (35) are shown to converge the order four.

3. Some Special Cases

From (33)–(35), we have 𝐿_{𝑎1,𝑎2,𝑎3}𝑥_𝑛,𝑤_𝑛=2𝑎₁+2𝑎₂𝑓^′2𝑥_𝑛+2𝑎₃𝑓^′𝑥_𝑛𝑓^′2𝑥_𝑛𝑓𝑤_𝑛×𝑎₁𝑓²𝑥_𝑛+𝑎₂𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑎₃𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛⁻¹, (53) where𝑎₁,𝑎₂,𝑎₃∈𝑅. From (33) we can approximate 𝐿_{𝑎1,𝑎2,𝑎3,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛=2𝐿_{𝑎1,𝑎2,𝑎3}𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛+𝑝²𝑓²𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛𝑓𝑥_𝑛×𝑓^′2𝑥_𝑛⁻¹. (54) Let 𝐾_{𝑎1,𝑎2,𝑎3,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛=𝑓^′𝑥_𝑛−𝑝𝑓𝑥_𝑛−𝑓^′𝑥_𝑛^𝑚_𝑘≥012𝑘−𝐿_{𝑎1,𝑎2,𝑎3,𝑓,𝑝}𝑥_𝑛,𝑤_𝑛^𝑘×𝐿_{𝑎1,𝑎2,𝑎3}𝑥_𝑛,𝑤_𝑛+𝑝²𝑓𝑥_𝑛−2𝑝𝑓^′𝑥_𝑛⁻¹, (55) where𝑎₁,𝑎₂,𝑎₃,𝑝∈𝑅,𝑚≥0.1⁰:If𝑎₁=𝑎₂=𝑎₃=𝑝=1, and 𝑚=3, from (55), we obtain a third-order method (LM1): 𝑥_𝑛+1=𝑥_𝑛−𝐾_1,1,1,𝑓,1𝑥_𝑛,𝑤_𝑛. (56)2⁰:If𝑎₁=𝑎₂=0, 𝑎₃=𝑝=1, and 𝑚=3, from (55) we also obtain a third-order method (LM2): 𝑥_𝑛+1=𝑥_𝑛−𝐾_0,0,1,𝑓,1𝑥_𝑛,𝑤_𝑛. (57)3⁰:If𝑎₁=1, 𝑎₂=0, 𝑎₃=𝑝=1, and 𝑚=3, from (55) we obtain a new third-order method (LM3): 𝑥_𝑛+1=𝑥_𝑛−𝐾_1,0,1,𝑓,1𝑥_𝑛,𝑤_𝑛. (58)4⁰:If𝑎₁=−1, 𝑎₂=𝑎₃=0, 𝑝=1, and 𝑚=3, from (55) we obtain a third-order method (LM4): 𝑥_𝑛+1=𝑥_𝑛−𝐾_{−1,0,0,𝑓,1}𝑥_𝑛,𝑤_𝑛. (59)5⁰:If 𝑎₁=𝑎₂=0, 𝑎₃=−1, 𝑝=1, and 𝑚=3, from (55) we obtain a new third-order method (LM5): 𝑥_𝑛+1=𝑥_𝑛−𝐾_{0,0,−1,𝑓,1}𝑥_𝑛,𝑤_𝑛. (60)

4. Numerical Examples

In this section, some numerical examples commonly used in the literature are presented in Table 1 to check the effectiveness of the new methods. The following methods were compared: Newton method (NM), the method of Weerakoon and Fernando [10] (WF), the method of Potra and Pták (PP) [11], Chebyshev’s method (CHM) [12, 13], Halley’s method (HM) [12], and our new methods (56) (LM1), (57) (LM2), (58) (LM3), (59) (LM4), and (60) (LM5). Displayed in Table 1 are the number of iterations (IT), the number of function evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative, the value𝑓(𝑥_𝑛+1), the computing time (TIME, the unit of time is one second), and the distance of two consecutive approximations𝛿=|𝑥_𝑛+1−𝑥_𝑛|. All computations were done using Matlab 7.1 environment with a ADM athlon (tm) II X2 250-3.01 GHz based PC. We accept an approximate solution rather than the exact root, depending on the precision𝜖of the computer. We use the following stopping criteria for computer programs:|𝑓(𝑥_𝑛+1)|<𝜖, we used the fixed stopping criterion𝜖=10⁻¹⁵.

Table 1 

Comparison of various third-order methods and Newton’s method.

We used the following test functions and display the computed approximate zero𝑥^∗ [16]: 𝑓₁(𝑥)=𝑥³+4𝑥²−10,𝑥^∗𝑓=1.3652300134140969,₂(𝑥)=𝑥²−𝑒^𝑥−3𝑥+2,𝑥^∗𝑓=0.25753028543986076,₃(𝑥)=sin(𝑥)𝑒^𝑥+ln1+𝑥²,𝑥^∗𝑓=0,₄(𝑥)=(𝑥−1)³−1,𝑥^∗𝑓=2,₅(𝑥)=cos𝑥−𝑥,𝑥^∗𝑓=0.73908513321516067,₆(𝑥)=sin²𝑥−𝑥²+1,𝑥^∗𝑓=1.4044916482153411,₇(𝑥)=𝑒^{𝑥2+7𝑥−30}−1,𝑥^∗=3. (61)

5. Conclusions

In this paper, we presented two new families of iterative methods for solving nonlinear equations. One is developed by fitting the model𝑚(𝑥)=𝑒^𝑝𝑥(𝐴𝑥²+𝐵𝑥+𝐶)to the function𝑓(𝑥)and its derivative𝑓^′(𝑥),𝑓^′′(𝑥)at a point𝑥_𝑛. The other family of iterative methods was constructed by approximating the equation𝑓(𝑥)=0around the point(𝑥_𝑛,𝑓(𝑥_𝑛))with the quadratic equation to avoid the calculation of the second derivatives. Analysis of convergence shows that the new methods have third-order or higher convergence: if 𝑚≥2, 𝑎₂=𝑎₃=𝑝=0, then the methods defined by (35) are shown to converge the order four. We observed from numerical examples that the proposed methods are efficient and demonstrate equal or better performance as compared with other well-known methods.