Some New Variants of Cauchy's Methods for Solving Nonlinear Equations

Abstract

We present and analyze some variants of Cauchy's methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the well-known existing methods and give better numerical results in many aspects.

1. Introduction

In this paper, we consider iterative methods to find a simple root 𝛼; that is, 𝑓(𝛼)=0 and 𝑓^′(𝛼)≠0, of a nonlinear equation 𝑓(𝑥)=0, (1.1) where 𝑓∶𝐼⊂𝑅→𝑅 for an open interval 𝐼 is a scalar function.

Finding the simple roots of the nonlinear equation (1.1) is one of the most important problems in numerical analysis of science and engineering, and iterative methods are usually used to approximate a solution of these equations. We know that Newton’s method is an important and basic approach for solving nonlinear equations [1, 2], and its formulation is given by 𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛; (1.2) this method converges quadratically.

The classical Cauchy’s methods [2] are expressed as 𝑥_𝑛+1=𝑥_𝑛−21+1−2𝐿_𝑓𝑥_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (1.3) where 𝐿_𝑓𝑥_𝑛=𝑓^′′𝑥_𝑛𝑓𝑥_𝑛𝑓^′2𝑥_𝑛. (1.4) This family of methods given by (1.3) is a well-known third-order method. However, the methods depend on the second derivatives in computing process, and therefore their practical applications are restricted rigorously. In the recent years, several methods with free second derivatives have been developed; see [3–9] and references therein.

In this paper, we will improve the family defined by (1.3) and obtain a three-parameter family of second-derivative-free variants of Cauchy’s methods. The rest of the paper is organized as follows. In Section 2, we describe new variants of Cauchy’s methods and analyze the order of convergence. 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 approximating the equation 𝑓(𝑥)=0 around the point (𝑥_𝑛,𝑓(𝑥_𝑛)) by the quadratic equation in 𝑥 and 𝑦 in the following form: 𝑎𝑥²+𝑏𝑦²+𝑐𝑥𝑦+𝑑𝑥+𝑔𝑦+ℎ=0. (2.1) We impose the tangency conditions 𝑦𝑥_𝑛=𝑓𝑥_𝑛,𝑦^′𝑥_𝑛=𝑓^′𝑥_𝑛,𝑦𝑤_𝑛=𝑓𝑤_𝑛, (2.2) where 𝑥_𝑛 is 𝑛th iterate and 𝑤_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (2.3) By using the tangency conditions from (2.2), we obtain the value of 𝑔 is determined in terms of 𝑎,𝑏,𝑐 in the following: 𝑔=−𝑎𝑓²𝑥_𝑛+𝑏𝑓²𝑤_𝑛𝑓^′2𝑥_𝑛+𝑏𝑓²𝑥_𝑛𝑓^′2𝑥_𝑛𝑓^′2𝑥_𝑛𝑓𝑤_𝑛−𝑐𝑥_𝑛𝑓𝑤_𝑛𝑓^′2𝑥_𝑛+𝑐𝑓²𝑥_𝑛𝑓^′𝑥_𝑛−𝑐𝑓𝑥_𝑛𝑓𝑤_𝑛𝑓^′𝑥_𝑛𝑓^′2𝑥_𝑛𝑓𝑤_𝑛. (2.4) From (2.1), we have 𝑓^′′𝑥_𝑛≈𝑦^′′𝑥_𝑛=−2𝑎+2𝑏𝑓^′2𝑥_𝑛+2𝑐𝑓^′𝑥_𝑛2𝑏𝑓𝑥_𝑛+𝑐𝑥_𝑛.+𝑔 (2.5) Substituting (2.4) into (2.5) yields 𝑓^′′𝑥_𝑛≈2𝑎+2𝑏𝑓^′2𝑥_𝑛+2𝑐𝑓^′𝑥_𝑛𝑓^′2𝑥_𝑛𝑓𝑤_𝑛𝑎𝑓²𝑥_𝑛+𝑏𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑐𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛. (2.6) Using (2.6) we can approximate 𝐿_𝑓𝑥_𝑛=𝑓^′′𝑥_𝑛𝑓𝑥_𝑛𝑓^′2𝑥_𝑛≈2𝑎+2𝑏𝑓^′2𝑥_𝑛+2𝑐𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛𝑎𝑓²𝑥_𝑛+𝑏𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑐𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛. (2.7) We define 𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛=2𝑎+2𝑏𝑓^′2𝑥_𝑛+2𝑐𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛𝑎𝑓²𝑥_𝑛+𝑏𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑐𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛. (2.8) Using 𝐾_{𝑎,𝑏,𝑐}(𝑥_𝑛,𝑤_𝑛) instead of 𝐿_𝑓(𝑥_𝑛), we obtain a new three-parameter family of methods free from second derivative 𝑥_𝑛+1=𝑥_𝑛−21+1−2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (2.9) where 𝑎,𝑏,𝑐∈𝑅. Similar to the classical Cauchy’s method, a square root is required in (2.9). However, this may cost expensively, even fail in the case 1−2𝐾_{𝑎,𝑏,𝑐}(𝑥_𝑛,𝑤_𝑛)<0. In order to avoid the calculation of the square roots, we will derive some forms free from square roots by the Taylor approximation [10].

It is easy to know that the Taylor approximation of 1−2𝐾_{𝑎,𝑏,𝑐}(𝑥_𝑛,𝑤_𝑛) is 1−2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛=_𝑘≥012𝑘−2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘. (2.10) Using (2.10) in (2.9), we can obtain the following form: 𝑥_𝑛+1=𝑥_𝑛−21+∑^𝑚_𝑘≥0𝑘^1/2−2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (2.11) where 𝑎,𝑏,𝑐,𝛽∈𝑅, 𝑚≥0.

On the other hand, it is clear that 21+1−2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛=1−1−2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛=_𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘. (2.12) Then, Using (2.12) in (2.9), we also can construct a new family of iterative methods as follows: 𝑥_𝑛+1=𝑥_𝑛−^𝑚_𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (2.13) where 𝑎,𝑏,𝑐∈𝑅, 𝑚≥0.

We have the convergence analysis of the methods by (2.13).

Theorem 2.1. Let 𝛼∈𝐼 be a simple zero of sufficiently differentiable function 𝑓∶𝐼⊂𝑅→𝑅 for an open interval 𝐼. If 𝑥₀ is sufficiently close to 𝛼, for 𝑚≥1, the methods defined by (2.13) are at least cubically convergent; as particular cases, if 𝑚=1,𝑎=𝑐=0 or 𝑚≥2,𝑏=𝑐=0 the methods have convergence order four.

Proof. Let 𝑒_𝑛=𝑥_𝑛−𝛼; we use the following Taylor expansions: 𝑓𝑥_𝑛=𝑓^′(𝛼)𝑒_𝑛+𝑐₂𝑒²_𝑛+𝑐₃𝑒³_𝑛+𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛, (2.14) where 𝑐_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. (2.15) Dividing (2.14) by (2.15), 𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝑒_𝑛−𝑐₂𝑒²_𝑛+2𝑐²₂−𝑐₃𝑒³_𝑛+7𝑐₂𝑐₃−4𝑐³₂−3𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.16) From (2.16), we get 𝑤_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝛼+𝑐₂𝑒²_𝑛−2𝑐²₂−𝑐₃𝑒³_𝑛−7𝑐₂𝑐₃−4𝑐³₂−3𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.17) Expanding 𝑓(𝑤_𝑛), 𝑓^′(𝑤_𝑛) in Taylor’s series about 𝛼 and using (2.17), 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,𝑓^′𝑤_𝑛=𝑓^′(𝛼)1+2𝑐₂𝑤_𝑛−𝛼+3𝑐₃𝑤_𝑛−𝛼²+4𝑐₄𝑤_𝑛−𝛼³+⋯=𝑓^′(𝛼)1+2𝑐²₂𝑒²_𝑛+4𝑐₂𝑐₃−𝑐³₂𝑒³_𝑛+8𝑐⁴₂+6𝑐₂𝑐₄−11𝑐²₂𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.18)(2.19) From (2.14) and (2.18), we have 𝑓𝑥_𝑛𝑓𝑤_𝑛=𝑓^′2(𝛼)𝑐₂𝑒³_𝑛+2𝑐₃−𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.20) Because of (2.15), we obtain 𝑓^′2𝑥_𝑛=𝑓^′2(𝛼)1+4𝑐₂𝑒_𝑛+6𝑐₃+4𝑐²₂𝑒²_𝑛+8𝑐₄+12𝑐₂𝑐₃𝑒³_𝑛+10𝑐₅+16𝑐₂𝑐₄+9𝑐²₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.21) From (2.20) and (2.21), we have 𝑓^′2𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛=𝑓^′4(𝛼)𝑐₂𝑒³_𝑛+2𝑐₃+3𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.22) Because of (2.15) and (2.20), we get 𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛=𝑓^′3(𝛼)𝑐₂𝑒³_𝑛+2𝑐₃+𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.23) From (2.20), (2.22), and (2.23), we also easily obtain 2𝑎+2𝑏𝑓^′2𝑥_𝑛+2𝑐𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛=2𝑓^′2(𝛼)𝑎𝑐₂+𝑏𝑐₂𝑓^′2(𝛼)+𝑐𝑐₂𝑓^′(𝛼)𝑒³_𝑛+2𝑎𝑐₃−𝑎𝑐²₂+3𝑏𝑐²₂𝑓^′2(𝛼)+2𝑏𝑐₃𝑓^′2(𝛼)+2𝑐𝑐₃𝑓^′(𝛼)+𝑐𝑐²₂𝑓^′(𝛼)𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.24) Because of (2.14) and (2.18), we get 𝑓²𝑥_𝑛=𝑓^′2(𝛼)𝑒²_𝑛+2𝑐₂𝑒³_𝑛+2𝑐₃+𝑐²₂𝑒⁴_𝑛+𝑂𝑒⁵_𝑛,𝑓𝑥_𝑛−𝑓𝑤_𝑛=𝑓^′(𝛼)𝑒_𝑛+2𝑐²₂−𝑐₃𝑒³_𝑛+7𝑐₂𝑐₃−5𝑐³₂−2𝑐₄𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.25)(2.26) Furthermore, we have 𝑓𝑥_𝑛−𝑓𝑤_𝑛²=𝑓^′2(𝛼)𝑒²_𝑛+22𝑐²₂−𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.27) Because of (2.21) and (2.27), we have 𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²=𝑓^′4(𝛼)𝑒²_𝑛+4𝑐₂𝑒³_𝑛+8𝑐²₂+4𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.28) From (2.14) and (2.15), we also easily have 𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝑓^′2(𝛼)𝑒_𝑛+3𝑐₂𝑒²_𝑛+2𝑐²₂+4𝑐₃𝑒³_𝑛+5𝑐₄+5𝑐₂𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.29) By a simple manipulation with (2.26) and (2.29), we obtain 𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛=𝑓^′3(𝛼)𝑒²_𝑛+3𝑐₂𝑒³_𝑛+4𝑐²₂+3𝑐₃𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.30) Substituting (2.25), (2.29), and (2.30) in the denominator of 𝐾_{𝑎,𝑏,𝑐}(𝑥_𝑛), we obtain 𝑎𝑓²𝑥_𝑛+𝑏𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑐𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛=𝑓^′2(𝛼)𝑎+𝑏𝑓^′2(𝛼)+𝑐𝑓^′(𝛼)𝑒²_𝑛+2𝑎𝑐₂+4𝑏𝑐₂𝑓^′2(𝛼)+3𝑐𝑐₂𝑓^′(𝛼)𝑒³_𝑛+2𝑎𝑐₃+𝑎𝑐²₂+8𝑏𝑐²₂𝑓^′2(𝛼)+4𝑏𝑐₃𝑓^′2(𝛼)+4𝑐𝑐²₂𝑓^′(𝛼)+3𝑐𝑐₃𝑓^′(𝛼)𝑒⁴_𝑛+𝑂𝑒⁵_𝑛. (2.31) Dividing (2.24) by (2.31) gives us 𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛=2𝑎+2𝑏𝑓^′2𝑥_𝑛+2𝑐𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛𝑎𝑓²𝑥_𝑛+𝑏𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑐𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛=2𝑐₂𝑒_𝑛−𝚆(𝑎,𝑏,𝑐)−2𝑐₃𝑒²_𝑛+𝑂𝑒³_𝑛, (2.32) where 𝚆(𝑎,𝑏,𝑐)=3𝑎+𝑏𝑓^′2(𝛼)+2𝑐𝑓^′(𝛼)𝑎+𝑏𝑓^′2(𝛼)+𝑐𝑓^′(𝛼)𝑐²₂. (2.33) Since _𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘1=1+2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛+12𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛²+58𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛³+78𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛⁴+⋯. (2.34) If we consider 𝑚=1, from (2.12) we obtain _1𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘1=1+2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛=1+𝑐₂𝑒_𝑛−𝚆(𝑎,𝑏,𝑐)−2𝑐₃𝑒²_𝑛. (2.35) Because of (2.13), we have 𝑥_𝑛+1=𝑥_𝑛−¹_𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝑥_𝑛−𝑒_𝑛−𝑐²₂−𝚆(𝑎,𝑏,𝑐)𝑒³_𝑛+𝑂𝑒⁴_𝑛. (2.36) From 𝑒_𝑛+1=𝑥_𝑛+1−𝛼, we have 𝑒_𝑛+1=−𝑐²₂−𝚆(𝑎,𝑏,𝑐)𝑒³_𝑛+𝑂𝑒⁴_𝑛. (2.37) This means that if 𝑚=1, the methods defined by (2.13) are at least of order three for any 𝑎,𝑏,𝑐∈𝑅. Furthermore, we consider that if 𝑐²₂=𝚆(𝑎,𝑏,𝑐), (2.38) then the methods defined by (2.9) are shown to converge of the order four. From (2.37) and (2.38), it is obvious that the methods defined by (2.13) are of order four by taking 𝑎=𝑐=0.
If considering 𝑚≥2, we from (2.32) have _𝑚𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘1=1+2𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛+12𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛²+𝑂𝑒³_𝑛=1+𝑐₂𝑒_𝑛−𝚆(𝑎,𝑏,𝑐)−2𝑐₃−2𝑐²₂𝑒²_𝑛+𝑂𝑒³_𝑛. (2.39) From (2.13), we can obtain 𝑥_𝑛+1=𝑥_𝑛−^𝑚_𝑘≥012()𝑘+1−1^𝑘2^𝑘+1𝐾_{𝑎,𝑏,𝑐}𝑥_𝑛,𝑤_𝑛^𝑘𝑓𝑥_𝑛𝑓^′𝑥_𝑛=𝑥_𝑛−𝑒_𝑛−3𝑐²₂−𝚆(𝑎,𝑏,𝑐)𝑒³_𝑛+𝑂𝑒⁴_𝑛. (2.40) From (2.40) and 𝑒_𝑛+1=𝑥_𝑛+1−𝛼, we have 𝑒_𝑛+1=−3𝑐²₂−𝚆(𝑎,𝑏,𝑐)𝑒³_𝑛+𝑂𝑒⁴_𝑛. (2.41)
This means that the methods defined by (2.13) are at least of order three for any 𝑎,𝑏,𝑐∈𝑅. Furthermore, we consider that if 3𝑐²₂=𝚆(𝑎,𝑏,𝑐), (2.42) then the methods defined by (2.13) are shown to converge of the order four. From (2.41) and (2.42), it is obvious that the methods defined by (2.13) are of order four by taking 𝑏=𝑐=0.

Similar to the proof of Theorem 2.1, we can prove that for 𝑚≥1, the methods defined by (2.9) and (2.11) are at least cubically convergent; as particular cases, if 𝑚=1, 𝑎=𝑐=0, or 𝑚≥2, 𝑏=𝑐=0, the methods have convergence order four.

3. Some Special Cases

(1⁰) If 𝑎=𝑏=𝑐=1, from (2.8) we obtain 𝐾_1,1,1𝑥_𝑛=2+2𝑓^′2𝑥_𝑛+2𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑓𝑤_𝑛𝑓²𝑥_𝑛+𝑓^′2𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛²+𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛. (3.1) For 𝑚=1, we obtain from (2.13) a third-order method (LM1) 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_1,1,1𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.2) For 𝑚=2, we obtain from (2.13) a third-order method (LM2) 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_1,1,1𝑥_𝑛,𝑤_𝑛+12𝐾_1,1,1𝑥_𝑛,𝑤_𝑛²𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.3) (2⁰) If 𝑎=0, 𝑐=0, from (2.8) we obtain 𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛=2𝑓𝑥_𝑛𝑓𝑤_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛². (3.4) For 𝑚=1, we obtain from (2.13) a fourth-order method [11] 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.5) For 𝑚=2, we obtain from (2.13) a third-order method (LM3) 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛+12𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛²𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.6) (3⁰) If 𝑏=0,𝑐=0, from (2.8) we obtain 𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛=2𝑓𝑤_𝑛𝑓𝑥_𝑛. (3.7) For 𝑚=1, we obtain from (2.13) a third-order method (LM4) 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.8) For 𝑚=2, we also obtain the fourth-order method 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛+12𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛²𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (3.9) which was obtained in [10].

For 𝑚=3, we obtain a fourth-order method as follows [11]: 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛+12𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛²+58𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛³𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.10) For 𝑚=4, we obtain the new fourth-order method 𝑥_𝑛+1=𝑥_𝑛−11+2𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛+12𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛²+58𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛³+78𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛⁴𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.11) (4⁰) If 𝑎=0, 𝑐=0, for 𝑚=1, we obtain a fourth-order method from (2.11) and (3.4) 𝑥_𝑛+1=𝑥_𝑛−22−𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.12) For 𝑚=2, we obtain from (2.11) a third-order method 𝑥_𝑛+1=𝑥_𝑛−44−2𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛−𝐾_0,𝑏,0𝑥_𝑛,𝑤_𝑛²𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.13) (5⁰) If 𝑏=0, 𝑐=0, from (2.11) we obtain some iterative methods as follows:

For 𝑚=1, we obtain a third-order method (LM5) 𝑥_𝑛+1=𝑥_𝑛−22−𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (3.14) where 𝐾_𝑎,0,0(𝑥_𝑛,𝑤_𝑛) is defined by (3.7).

For 𝑚=2, we obtain a fourth-order method (LM6) 𝑥_𝑛+1=𝑥_𝑛−44−2𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛−𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛²𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.15) For 𝑚=3, we obtain the fourth-order method as follows [10]: 𝑥_𝑛+1=𝑥_𝑛−44−2𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛−𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛²−𝐾_𝑎,0,0𝑥_𝑛,𝑤_𝑛³𝑓𝑥_𝑛𝑓^′𝑥_𝑛. (3.16)

4. Numerical Examples

In this section, firstly, we present some numerical test results about the number of iterations (𝑛) for some cubically convergent iterative methods in Table 1. The following methods were compared: Newton’s method (NM), the method of Weerakoon and Fernando [12] (WF), Halley’s method (HM), Chebyshev’s method (CHM), Super-Halley’s method (SHM), and our new methods (3.2) (LM1), (3.3) (LM2), (3.6) (LM3), (3.8) (LM4), and (3.14) (LM5).

Table 1 

Comparison of various third-order methods and newton's method.

Secondly, we employ our new fourth-order methods defined by (3.15) (LM6) and the super cubic convergence method by (3.2) (LM1), to solve some nonlinear equations and compare them with Newton’s method (NM), Newton-secant method [13] (NSM), and Ostrowski’s method [14] (OM). Displayed in Table 2 are the number of iterations (𝑛) and the number of function evaluations (NFEs) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative.

Table 2 

Comparison of the methods and Newton's method.

All computations were done using Matlab7.1. We accept an approximate solution rather than the exact root, depending on the precision 𝜖 of the computer. We use the following stopping criteria for computer programs: |𝑓(𝑥_𝑛+1)|<𝜖, we used the fixed stopping criterion 𝜖=10⁻¹⁵. In table, “−” is divergence.

We used the following test functions and display the computed approximate zero 𝑥^∗ [15]: 𝑓₁(𝑥)=𝑥³+4𝑥²−10,𝑥^∗𝑓=1.3652300134140969,₂(𝑥)=𝑥²−𝑒^𝑥−3𝑥+2,𝑥^∗𝑓=0.25753028543986076,₃(𝑥)=𝑥³−sin²𝑥+3cos𝑥+5,𝑥^∗𝑓=−1.582687045752069901,₄(𝑥)=sin(𝑥)𝑒^𝑥+ln1+𝑥²,𝑥^∗𝑓=0,₅(𝑥)=(𝑥−1)³−1,𝑥^∗𝑓=2,₆(𝑥)=cos𝑥−𝑥,𝑥^∗𝑓=0.73908513321516067,₇(𝑥)=sin²𝑥−𝑥²+1,𝑥^∗𝑓=1.4044916482153411,₈(𝑥)=𝑒^{𝑥2+7𝑥−30}−1,𝑥^∗=3. (4.1)

5. Conclusions

In this paper, we presented some variants of Cauchy’s methods free from second derivative for solving nonlinear equations. Per iteration the methods require two-function and one first-derivative evaluations. These methods are at least three-order convergence, if 𝑚=1, 𝑎=𝑐=0, or 𝑚≥2, 𝑏=𝑐=0, the methods have convergence order four, respectively, and if 𝑚=1, 𝑎=𝑏=𝑐=1, the method has super cubic convergence. We observed from numerical examples that the proposed methods are efficient and demonstrate equal or better performance as compared with other well-known methods.