Convergence Analysis and Numerical Study of a Fixed-Point Iterative Method for Solving Systems of Nonlinear Equations

Abstract

We present a fixed-point iterative method for solving systems of nonlinear equations. The convergence theorem of the proposed method is proved under suitable conditions. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.

1. Introduction

One of the basic problems in mathematics is how to solve nonlinear equations 𝑓(𝑥)=0. (1) In order to solve these equations, we can use iterative methods such as Newton’s method and its variants. Recently, there has been some progress on iterative methods with higher order of convergence using decomposition techniques; see [1–15] and the reference therein.

The zeros of a nonlinear equation cannot in general be expressed in closed form; thus we have to use approximate methods. Nowadays, we often use iterative methods to get the approximate solution of the system (1); the best known method is the classical Newton’s method. Recently, there has been some progress on solving the system (1), which allows us to get the iterative formula by using essentially Taylor’s polynomial (see [16, 17]), quadrature formulas (see [7, 9–12]), homotopy perturbation method (see [8]), and so on.

In this paper, we will present a new fixed point iterative method for solving the system (1) and prove that the method is cubic convergent under suitable conditions.

This paper is organized as follows. In Section 2, we introduce new iterative methods to solve (1). In Section 3 we extend these methods to solve systems of nonlinear equations, and we also prove convergence of the proposed method. Some numerical results are reported in Section 4, while the paper is concluded in the last section.

2. Iterative Methods and Convergence Analysis

We now consider the following nonlinear equation: 𝑓(𝑥)=0,𝑥∈[𝑎,𝑏]. (2) Assume that 𝛼^∗ is a simple root of (2); that is, 𝑓(𝛼^∗)=0. For 𝛼,𝛼_𝑘∈[𝑎,𝑏], using Taylor’s formula, we have 𝑓(𝛼)=𝑓𝛼_𝑘+𝑓^′𝛼_𝑘𝛼−𝛼_𝑘+1𝑓2!^′′𝛼_𝑘𝛼−𝛼_𝑘²+1𝑓3!^′′′𝛼_𝑘𝛼−𝛼_𝑘³1+⋯+()!𝑓𝑟−1^(𝑟−1)𝛼_𝑘𝛼−𝛼_𝑘^𝑟−1+¹₀(1−𝑡)^𝑟−1()!𝑓𝑟−1^(𝑟)𝛼_𝑘+𝑡𝛼−𝛼_𝑘𝛼−𝛼_𝑘^𝑟d𝑡. (3) Taking 𝑟=1 in the above equality, we get 𝑓(𝛼)=𝑓𝛼_𝑘+¹₀𝑓^′𝛼_𝑘+𝑡𝛼−𝛼_𝑘𝛼−𝛼_𝑘d𝑡. (4) If the value of 𝑓^′(𝛼_𝑘+𝑡(𝛼−𝛼_𝑘)) in the interval [0,1] is replaced with its value in 𝑡=0, that is, with 𝑓^′(𝛼_𝑘), then we have ¹₀𝑓^′𝛼_𝑘+𝑡𝛼−𝛼_𝑘𝛼−𝛼_𝑘d𝑡≅𝑓^′𝛼_𝑘𝛼−𝛼_𝑘. (5) By using (5) in (4), we have 𝑓(𝛼)≅𝑓𝛼_𝑘+𝑓^′𝛼_𝑘𝛼−𝛼_𝑘. (6) We can get an iterative method from (6) to solve the system (2); it is the famous Newton’s formula 𝛼_𝑘+1=𝛼_𝑘−𝑓𝛼_𝑘𝑓^′𝛼_𝑘. (7) The formula (7) has already been proved to be quadratically convergent. Now we begin to deduce a higher order iterative method. In fact, if we estimate 𝑓^′(𝛼_𝑘+𝑡(𝛼−𝛼_𝑘)) in the interval [0,1] by its value in 𝑡=1, that is, by 𝑓^′(𝛼), then we have ¹₀𝑓^′𝛼_𝑘+𝑡𝛼−𝛼_𝑘𝛼−𝛼_𝑘d𝑡=𝑓^′(𝛼)𝛼−𝛼_𝑘+𝑅𝑓(𝛼). (8) By using (8) in (4), and cutting off the error 𝑅[𝑓(𝛼)] (still use “=”), we have 𝑓(𝛼)=𝑓𝛼_𝑘+𝑓^′(𝛼)𝛼−𝛼_𝑘. (9) Let 𝛼_𝑘+1 be the solution of (9); we can obtain a new iterative method 𝛼_𝑘+1=𝛼_𝑘−𝑓𝛼_𝑘−𝑓𝛼_𝑘+1𝑓^′𝛼_𝑘+1. (10) Since the iterative method (10) is implicit-type method, we use classical Newton’s formula (7) as predictor and then use the above scheme (10) as corrector; in this way, we can get a workable iterative method.

Algorithm 1. Step 0 (initialization). Choose the initial value 𝛼₀∈𝑁(𝛼^∗,𝛿), where 𝛼^∗ is certain real zero of nonlinear mapping 𝑓(𝛼) and 𝛿>0 is a sufficiently small constant. Take the stopping criteria 𝜖₁,𝜖₂>0. Set 𝑘∶=0.
Step 1 (the predictor step). Compute the predictor 𝛽_𝑘=𝛼_𝑘−𝑓𝛼_𝑘𝑓^′𝛼_𝑘. (11)
Step 2 (the corrector step). Computing the corrector 𝛼_𝑘+1=𝛼_𝑘−𝑓𝛼_𝑘−𝑓𝛽_𝑘𝑓^′𝛽_𝑘. (12)
Step 3. If |𝑓(𝛼_𝑘+1)|≤𝜖₁ or |𝛼_𝑘+1−𝛼_𝑘|≤𝜖₂ then stop; otherwise, set 𝛼_𝑘∶=𝛼_𝑘+1, 𝑘∶=𝑘+1; go to Step 1.

In this section, we consider the convergence and convergent rate of Algorithm 1. We obtain a convergence theorem as follows.

Theorem 2. Let 𝛼^∗∈𝐼 be a simple zero of sufficiently differentiable function 𝑓∶[𝑎,𝑏]⊆𝑅→𝑅. If 𝛼₀ is sufficiently close to 𝛼^∗, then the two-step iterative method defined by (11)-(12) has three-order convergence and satisfies the following error equation: 𝑒_𝑘+1=𝑐₁𝑒³_𝑘+𝑂𝑒⁴_𝑘, (13) where 𝑒_𝑘=𝛼_𝑘−𝛼^∗ and 𝑐₁=𝑓^′′′(𝛼^∗)/6𝑓^′(𝛼^∗).

Proof. By (12) we get 𝑒_𝑘+1=𝑒_𝑘−𝑓𝛽_𝑘−𝑓𝛼_𝑘𝑓^′𝛽_𝑘. (14) Multiplying the above equation by 𝑓^′(𝛽_𝑘), we can get 𝑓^′𝛽_𝑘𝑒_𝑘+1=𝑓^′𝛽_𝑘𝑒_𝑘+𝑓𝛽_𝑘−𝑓𝛼_𝑘. (15) Let 𝛼=𝛼^∗ in (3) and 𝑓(𝛼^∗)=0; we have 0=𝑓𝛼_𝑘+𝑓^′𝛼_𝑘𝛼^∗−𝛼_𝑘+1𝑓2!^′′𝛼_𝑘𝛼^∗−𝛼_𝑘²+1𝑓3!^′′′𝛼_𝑘𝛼^∗−𝛼_𝑘³+𝑂𝛼^∗−𝛼_𝑘⁴; (16) thus, 𝑓𝛼_𝑘=𝑓^′𝛼_𝑘𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑒²_𝑘+16𝑓^′′′𝛼_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘. (17) Dividing both sides of the above equation by 𝑓^′(𝛼_𝑘), we can get 𝑓𝛼_𝑘𝑓^′𝛼_𝑘=𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒²_𝑘+16𝑓^′′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘=𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘. (18) Furthermore, let 𝛼=𝛽_𝑘 in (3); we have 𝑓𝛽_𝑘=𝑓𝛼_𝑘+𝑓^′𝛼_𝑘𝛽_𝑘−𝛼_𝑘+1𝑓2!^′′𝛼_𝑘𝛽_𝑘−𝛼_𝑘²+1𝑓3!^′′′𝛼_𝑘𝛽_𝑘−𝛼_𝑘³+𝑂𝛽_𝑘−𝛼_𝑘⁴. (19) From (11) we get 𝛽_𝑘−𝛼_𝑘𝑓𝛼=−^𝑘𝑓^′𝛼_𝑘. (20) It follows from the above equation that 𝑓𝛽_𝑘=𝑓𝛼_𝑘−𝑓^′𝛼_𝑘𝑓𝛼_𝑘𝑓^′𝛼_𝑘+12𝑓^′′𝛼_𝑘𝑓𝛼_𝑘𝑓^′𝛼_𝑘²−16𝑓^′′′𝛼_𝑘𝑓𝛼_𝑘𝑓^′𝛼_𝑘³𝑓𝛼+𝑂^𝑘𝑓^′𝛼_𝑘⁴=12𝑓^′′𝛼_𝑘𝑓𝛼_𝑘𝑓^′𝛼_𝑘²−16𝑓^′′′𝛼_𝑘𝑓𝛼_𝑘𝑓^′𝛼_𝑘³𝑓𝛼+𝑂^𝑘𝑓^′𝛼_𝑘⁴. (21) By applying (18) we have 𝑓𝛽_𝑘=12𝑓^′′𝛼_𝑘𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘²−16𝑓^′′′𝛼_𝑘𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘³+𝑂𝑒⁴_𝑘. (22) After some manipulations we obtain 𝑓𝛽_𝑘=12𝑓^′′𝛼_𝑘𝑒²_𝑘−12𝑓^′′𝛼_𝑘²𝑓^′𝛼_𝑘𝑒³_𝑘−16𝑓^′′′𝛼_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘. (23) Now, applying Taylor’s expansion for 𝑓^′(𝛽_𝑘) at the point 𝛼_𝑘, we have 𝑓^′𝛽_𝑘=𝑓^′𝛼_𝑘+𝑓^′′𝛼_𝑘𝛽_𝑘−𝛼_𝑘+1𝑓2!^′′′𝛼_𝑘𝛽_𝑘−𝛼_𝑘²+𝑂𝛽_𝑘−𝛼_𝑘³. (24) From (11), (18), and (24), we obtain 𝑓^′𝛽_𝑘=𝑓^′𝛼_𝑘−𝑓^′′𝛼_𝑘𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘+12𝑓^′′′𝛼_𝑘𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑓^′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘²+𝑂𝑒³_𝑘. (25) And after some manipulations we can get 𝑓^′𝛽_𝑘=𝑓^′𝛼_𝑘−𝑓^′′𝛼_𝑘𝑒_𝑘+12𝑓^′′𝛼_𝑘²𝑓^′𝛼_𝑘𝑒²_𝑘+12𝑓^′′′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘. (26) By substituting (17), (23), and (26) into (15) we have 𝑓^′𝛽_𝑘𝑒_𝑘+1=𝑓^′𝛽_𝑘𝑒_𝑘+𝑓𝛽_𝑘−𝑓𝛼_𝑘=𝑓^′𝛼_𝑘−𝑓^′′𝛼_𝑘𝑒_𝑘+12𝑓^′′𝛼_𝑘²𝑓^′𝛼_𝑘𝑒²_𝑘+12𝑓^′′′𝛼_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘𝑒_𝑘+12𝑓^′′𝛼_𝑘𝑒²_𝑘−12𝑓^′′𝛼_𝑘²𝑓^′𝛼_𝑘𝑒³_𝑘−16𝑓^′′′𝛼_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘−𝑓^′𝛼_𝑘𝑒_𝑘−12𝑓^′′𝛼_𝑘𝑒²_𝑘+16𝑓^′′′𝛼_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘=16𝑓^′′′𝛼_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘. (27) Note that 𝑓^′𝛽_𝑘=𝑓^′𝛼^∗+𝑂𝑒_𝑘,𝑓^′′′𝛼_𝑘=𝑓^′′′𝛼^∗+𝑂𝑒_𝑘. (28) Then, it follows from (27) that 𝑒_𝑘+1=16𝑓^′′′𝛼^∗+𝑂𝑒_𝑘𝑓^′(𝛼^∗)+𝑂𝑒_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘=16𝑓^′′′𝛼^∗𝑓^′(𝛼^∗)+𝑂𝑒_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘=16𝑓^′′′𝛼^∗𝑓^′(𝛼^∗)𝑒³_𝑘+𝑂𝑒⁴_𝑘. (29) This proves the conclusion of the theorem and the proof is completed.

3. The n-Dimensional Case

In this section, we consider the 𝑛-dimensional case of the method, and we also study these iterative methods’ order of convergence. Consider the system of nonlinear equations 𝑓₁𝑥₁,𝑥₂,…,𝑥_𝑛𝑓=0,₂𝑥₁,𝑥₂,…,𝑥_𝑛⋮𝑓=0,_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛=0, (30) where each function 𝑓_𝑖(𝑥₁,𝑥₂,…,𝑥_𝑛), 𝑖=1,2,…,𝑛, maps a vector 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 of the 𝑛-dimensional space 𝑅^𝑛 into the real line 𝑅. The system (30) of 𝑛 nonlinear equations in 𝑛 unknowns can also be represented by defining a function 𝐹 mapping 𝑅^𝑛 into 𝑅^𝑛 as 𝐹(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑛(𝑥))^𝑇. Thus, the system (30) can be written in the form 𝐹(𝑥)=0. (31) Let 𝐹∶𝐷⊆𝑅^𝑛→𝑅^𝑛 be a sufficiently differentiable function on a convex set 𝐷⊆𝑅^𝑛 and let 𝑥^∗ be a real zero of the nonlinear mapping 𝐹(𝑥); that is, 𝐹(𝑥^∗)=0. For any 𝑥,𝑥_𝑘∈𝐷, we may write Taylor’s expansion for 𝐹 as follows (see [17]): 𝐹(𝑥)=𝐹𝑥_𝑘+𝐹^′𝑥_𝑘𝑥−𝑥_𝑘+1𝐹2!^′′𝑥_𝑘𝑥−𝑥_𝑘²+1𝐹3!^′′′𝑥_𝑘𝑥−𝑥_𝑘³1+⋯+()!𝐹𝑟−1^(𝑟−1)×𝑥_𝑘𝑥−𝑥_𝑘^𝑟−1+¹₀(1−𝑡)^𝑟−1()!𝐹𝑟−1^(𝑟)𝑥_𝑘+𝑡𝑥−𝑥_𝑘𝑥−𝑥_𝑘^𝑟d𝑡; (32) for 𝑟=1, we have 𝐹(𝑥)=𝐹𝑥_𝑘+¹₀𝐹^′𝑥_𝑘+𝑡𝑥−𝑥_𝑘𝑥−𝑥_𝑘d𝑡. (33) If we estimate 𝐹^′(𝑥_𝑘+𝑡(𝑥−𝑥_𝑘)) in the interval [0,1] by its value in 𝑡=0, that is, by 𝐹^′(𝑥_𝑘), then we have ¹₀𝐹^′𝑥_𝑘+𝑡𝑥−𝑥_𝑘𝑥−𝑥_𝑘d𝑡≅𝐹^′𝑥_𝑘𝑥−𝑥_𝑘. (34) By using (34) in (33), we have 𝐹(𝑥)≅𝐹𝑥_𝑘+𝐹^′𝑥_𝑘𝑥−𝑥_𝑘. (35) We can get an iterative method from (35) to solve the system (31); it is known as Newton’s method 𝑥_𝑘+1=𝑥_𝑘−𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘. (36) The method (36) has already been proved that it has quadratic convergence. Now we begin to deduce a higher order iterative method. If we estimate 𝐹^′(𝑥_𝑘+𝑡(𝑥−𝑥_𝑘)) in the interval [0,1] by its value in 𝑡=1, that is, by 𝐹^′(𝑥), then we have ¹₀𝐹^′𝑥_𝑘+𝑡𝑥−𝑥_𝑘𝑥−𝑥_𝑘d𝑡=𝐹^′(𝑥)𝑥−𝑥_𝑘+𝑅[𝐹(𝑥)]. (37) By using (37) in (33), and cutting off the error 𝑅[𝐹(𝑥)] (still use “=”), we have 𝐹(𝑥)=𝐹𝑥_𝑘+𝐹^′(𝑥)𝑥−𝑥_𝑘. (38) Let 𝑥_𝑘+1 be the solution of (38); we can obtain a new iterative method 𝑥_𝑘+1=𝑥_𝑘−𝐹^′𝑥_𝑘+1⁻¹𝐹𝑥_𝑘−𝐹𝑥_𝑘+1. (39) On the other hand, from the easy Newton’s method (see [4]), we obtain 𝑥_𝑘+1=𝑥_𝑘−𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘+𝐹𝑥_𝑘+1. (40)

Now we consider the convex combination of (39) and (40). Let 𝛼≥0, 𝛽≥0, and 𝛼+𝛽=1; then we can deduce from (39) and (40) that the following iterative formula holds: 𝑥_𝑘+1=𝑥_𝑘−𝛼𝐹^′𝑥_𝑘+1⁻¹+𝛽𝐹^′𝑥_𝑘⁻¹×𝐹𝑥_𝑘+𝐹𝑥_𝑘+1+2𝛼𝐹^′𝑥_𝑘+1⁻¹𝐹𝑥_𝑘+1. (41) Since the iterative method (41) is implicit-type method, we use Newton’s method as predictor and then use the new method (41) as corrector; in this way, we can get a workable iterative method.

Algorithm 3. Step 0 (initialization). Choose the initial value 𝑥₀∈𝑁(𝑥^∗,𝛿), 𝛼∈[0,1] and 𝛽=1−𝛼, where 𝑥^∗ is certain real zero of nonlinear mapping 𝐹(𝑥) and 𝛿>0 is a sufficiently small constant. Take the stopping criterions 𝜖₁,𝜖₂>0. Set 𝑘∶=0.
Step 1 (the predictor step). Compute the predictor 𝑦_𝑘=𝑥_𝑘−𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘. (42)
Step 2 (the corrector step). Computing the corrector 𝑥_𝑘+1=𝑥_𝑘−𝛼𝐹^′𝛽_𝑘⁻¹+𝛽𝐹^′𝑥_𝑘⁻¹×𝐹𝛽_𝑘+𝐹𝑥_𝑘+2𝛼𝐹^′𝛽_𝑘⁻¹𝐹𝛽_𝑘. (43)
Step 3. If ‖𝐹(𝑥_𝑘+1)‖≤𝜖₁ or ‖𝑥_𝑘+1−𝑥_𝑘‖≤𝜖₂ then stop; otherwise, set 𝑘∶=𝑘+1; go to Step 1.

Remark 4. If we take 𝛽=0,𝛼=1, our algorithms (42) and (43) can be written in the following form: 𝑦_𝑘=𝑥_𝑘−𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘,𝑥_𝑘+1=𝑥_𝑘−𝐹^′𝑦_𝑘⁻¹𝐹𝑥_𝑘−𝐹𝑦_𝑘. (44) Notice that, at each iteration, the number of functional evaluations is 2𝑛²+2𝑛.

In this section, we consider the convergence and convergent rate of Algorithm 1. We obtain a convergence theorem as follows.

Theorem 5. Let 𝐹∶𝐷⊆𝑅^𝑛→𝑅^𝑛 be a sufficiently differentiable function on a convex set 𝐷⊆𝑅^𝑛 containing a root 𝑥^∗ of the nonlinear system (31). The iterative method (42)-(43) has cubic convergence and satisfies the error equation 𝑒_𝑘+1=𝐹^′𝑥^∗−1×16𝛼𝐹^′′′𝑥^∗+12𝛽𝐹^′′𝑥^∗𝐹^′𝑥^∗−1×𝐹^′′𝑥^∗𝑒³_𝑘+𝑂𝑒⁴_𝑘. (45)

Proof. Defining 𝑒_𝑘=𝑥_𝑘−𝑥^∗, from (42) and (43), we have 𝑒_𝑘+1=𝑒_𝑘−𝛼𝐹^′𝑦_𝑘⁻¹+𝛽𝐹^′𝑥_𝑘⁻¹×𝐹𝑦_𝑘+𝐹𝑥_𝑘+2𝛼𝐹^′𝑦_𝑘⁻¹𝐹𝑦_𝑘. (46) Premultiplying the above equation by 𝐹^′(𝑦_𝑘), we can get 𝐹^′𝑦_𝑘𝑒_𝑘+1=𝐹^′𝑦_𝑘𝑒_𝑘−𝛼𝐼+𝛽𝐹^′𝑦_𝑘𝐹^′𝑥_𝑘⁻¹×𝐹𝑦_𝑘+𝐹𝑥_𝑘+2𝛼𝐹𝑦_𝑘. (47) Let 𝑥=𝑥^∗ in (32) and 𝐹(𝑥^∗)=0; we have 0=𝐹𝑥_𝑘+𝐹^′𝑥_𝑘𝑥^∗−𝑥_𝑘+1𝐹2!^′′𝑥_𝑘𝑥^∗−𝑥_𝑘²+1𝐹3!^′′′𝑥_𝑘𝑥^∗−𝑥_𝑘³+𝑂𝑥^∗−𝑥_𝑘⁴; (48) thus, 𝐹𝑥_𝑘=𝐹^′𝑥_𝑘𝑒_𝑘−12𝐹^′′𝑥_𝑘𝑒²_𝑘+16𝐹^′′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘. (49) Multiplying 𝐹^′(𝑥_𝑘)⁻¹ to the two sides of the above equation, we obtain 𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘=𝑒_𝑘−12𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+16𝐹^′𝑥_𝑘⁻¹𝐹^′′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘=𝑒_𝑘−12𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘. (50) On the other hand, let 𝑥=𝑦_𝑘 in (32); we have 𝐹𝑦_𝑘=𝐹𝑥_𝑘+𝐹^′𝑥_𝑘𝑦_𝑘−𝑥_𝑘+1𝐹2!^′′𝑥_𝑘𝑦_𝑘−𝑥_𝑘²+1𝐹3!^′′′𝑥_𝑘𝑦_𝑘−𝑥_𝑘³+𝑂𝑦_𝑘−𝑥_𝑘⁴. (51) From (42) we get 𝑦_𝑘−𝑥_𝑘=−𝐹^′(𝑥_𝑘)⁻¹𝐹(𝑥_𝑘). It follows from the above equation that 𝐹𝑦_𝑘=𝐹𝑥_𝑘−𝐹^′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘+12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘²−16𝐹^′′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘³+𝑂𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘⁴=12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘²−16𝐹^′′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘³+𝑂𝐹^′𝑥_𝑘⁻¹𝐹𝑥_𝑘⁴. (52) By applying (50) we have 𝐹𝑦_𝑘=12𝐹^′′𝑥_𝑘×𝑒_𝑘−12𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘²−16𝐹^′′′𝑥_𝑘×𝑒_𝑘−12𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘³+𝑂𝑒⁴_𝑘. (53) After some manipulations we obtain 𝐹𝑦_𝑘=12𝐹^′′𝑥_𝑘𝑒²_𝑘−12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒³_𝑘−16𝐹^′′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘. (54) Now, applying Taylor’s formula for 𝐹^′(𝑦_𝑘) at the point 𝑥_𝑘, we have 𝐹^′𝑦_𝑘=𝐹^′𝑥_𝑘+𝐹^′′𝑥_𝑘𝑦_𝑘−𝑥_𝑘+1𝐹2!^′′′𝑥_𝑘𝑦_𝑘−𝑥_𝑘²+𝑂𝑦_𝑘−𝑥_𝑘³. (55) From (42), (50), and (55), we obtain 𝐹^′𝑦_𝑘=𝐹^′𝑥_𝑘−𝐹^′′𝑥_𝑘×𝑒_𝑘−12𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘+12𝐹^′′′𝑥_𝑘×𝑒_𝑘−12𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘²+𝑂𝑒³_𝑘. (56) And after some manipulations we can get 𝐹^′𝑦_𝑘=𝐹^′𝑥_𝑘−𝐹^′′𝑥_𝑘𝑒_𝑘+12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘+12𝐹^′′′𝑥_𝑘𝑒²_𝑘+𝑂𝑒³_𝑘. (57) Therefore, we have 𝛼𝐼+𝛽𝐹^′𝑦_𝑘𝐹^′𝑥_𝑘⁻¹=𝐼−𝛽𝐹^′′𝑥_𝑘𝑒_𝑘−12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹×𝐹^′′𝑥_𝑘𝑒²_𝑘−12𝐹^′′′𝑥_𝑘𝑒²_𝑘𝐹^′𝑥_𝑘⁻¹+𝑂𝑒³_𝑘. (58) From (49) and (54), we get 𝐹𝑥_𝑘+𝐹𝑦_𝑘=𝐹^′𝑥_𝑘𝑒_𝑘−12𝐹^′′𝑥_𝑘×𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘. (59) By substituting (57), (58), (59), and (54) into (47) we have 𝐹^′𝑦_𝑘𝑒_𝑘+1=𝐹^′𝑦_𝑘𝑒_𝑘−𝛼𝐼+𝛽𝐹^′𝑦_𝑘𝐹^′𝑥_𝑘⁻¹×𝐹𝑦_𝑘+𝐹𝑥_𝑘+2𝛼𝐹𝑦_𝑘=𝐹^′𝑥_𝑘𝑒_𝑘−𝐹^′′𝑥_𝑘𝑒²_𝑘+12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒³_𝑘+12𝐹^′′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘−𝐼−𝛽𝐹^′′𝑥_𝑘𝑒_𝑘−12𝐹^′′𝑥_𝑘×𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒²_𝑘−12𝐹^′′′𝑥_𝑘𝑒²_𝑘×𝐹^′𝑥_𝑘⁻¹+𝑂𝑒³_𝑘×𝐹^′𝑥_𝑘𝑒_𝑘−12𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹×𝐹^′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘1+2𝛼2𝐹^′′𝑥_𝑘𝑒²_𝑘−12𝐹^′′𝑥_𝑘×𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒³_𝑘−16𝐹^′′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘=16𝛼𝐹^′′′𝑥_𝑘+12𝛽𝐹^′′𝑥_𝑘×𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘𝑒³_𝑘+𝑂𝑒⁴_𝑘; (60) that is, 𝑒_𝑘+1=𝐹^′𝑦_𝑘⁻¹×16𝛼𝐹^′′′𝑥_𝑘+12𝛽𝐹^′′𝑥_𝑘𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘×𝑒³_𝑘+𝑂𝑒⁴_𝑘. (61) Notice that 𝐹^′𝑦_𝑘=𝐹^′𝑥^∗+𝑂𝑒_𝑘𝐹^(𝑘)𝑥_𝑘=𝐹^(𝑘)𝑥^∗+𝑂𝑒_𝑘,𝑘=1,2,3. (62) Equation (61) can be written as follows: 𝑒_𝑘+1=𝐹^′𝑥^∗−1×16𝛼𝐹^′′′𝑥^∗+12𝛽𝐹^′′𝑥^∗𝐹^′𝑥^∗−1𝐹^′′𝑥^∗×𝑒³_𝑘+𝑂𝑒⁴_𝑘, (63) which proves the conclusion of our theorem. The proof is completed.

4. Numerical Examples

In this section we present some examples to illustrate the efficiency and the performance of the newly developed method (42)-(43) (present study HM). This new method was compared with Newton’s method (NM), the method of Aslam Noor and Waseem [12] (NR1), the method of Cordero et al. [13] (NAd1), the method of Darvishi and Barati [4] (DV), and the method of Cordero and Torregrosa [10] (CT) in the number of iterations, CPU time, error, and convergence order. All computations were done using the PC with Pentium(R) Dual-Core CPU T4400 @2.20 GHz. All the programming is implemented in MATLAB 7.9. The convergence order 𝑝 is computed approximately by the following formula: 𝑥𝑝≅ln_𝑘+1−𝑥_𝑘∞/𝑥_𝑘−𝑥_𝑘−1∞ln𝑥_𝑘−𝑥_𝑘−1∞/𝑥_𝑘−1−𝑥_𝑘−2∞,if𝑘≥3;+∞,otherwise. (64)

As the iterative formula (43) contains parameters 𝛼 and 𝛽, we make the numerical examples based on 𝛼=1 and 𝛽=0.

Example 1. The test function is as follows (see [10]): 𝐹(𝑥)=(𝑥−1)⁶−1. (65) This problem has a solution 𝑥^∗=2. We test this problem by using 𝑥₀=4 as a starting point. The test results are listed in Table 1.

Table 1 

Numerical results of Example 1 with initial value 𝑥₀=4.

Example 2. The test function is as follows (see [9, 10, 13, 14]): 𝐹𝑥₁,𝑥₂=𝑥₁+𝑒^𝑥2−cos𝑥₂3𝑥₁−𝑥₂−sin𝑥₂. (66) This problem has a solution 𝑥^∗=(0,0)^𝑇. We test this problem by using initial value 𝑥₀=(0.3,−0.3)^𝑇 as a starting point. The test results are listed in Table 2.

Table 2 

Numerical results of Example 2 with initial value 𝑥₀=(0.3,−0.3)^𝑇.

Example 3. The test function is as follows (see [4]): 𝑥²₁+𝑥²₂+𝑥²₃−1=0,2𝑥²₁+𝑥²₂−4𝑥₃=0,3𝑥²₁−4𝑥²₂+𝑥²₃=0. (67) We test this problem by using 𝑥₀=(0.5,0.5,0.5)^𝑇. The test results are listed in Table 3.

Table 3 

Numerical results of Example 3 with initial value 𝑥₀=(0.5,0.5,0.5)^𝑇.

Example 4. The test function is 𝐹(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),𝑓₃(𝑥),𝑓₄(𝑥))^𝑇, where 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄)^𝑇∈𝑅⁴ and 𝑓_𝑖∶𝑅⁴→𝑅,𝑖=1,2,3,4, such that (see [9, 10, 13]) 𝑓₁(𝑥)=𝑥₂𝑥₃+𝑥₄𝑥₂+𝑥₃,𝑓₂(𝑥)=𝑥₁𝑥₃+𝑥₄𝑥₁+𝑥₃,𝑓₃(𝑥)=𝑥₁𝑥₂+𝑥₄𝑥₁+𝑥₂,𝑓₄(𝑥)=𝑥₁𝑥₂+𝑥₁𝑥₃+𝑥₂𝑥₃−1. (68) This problem has two solutions: they are 𝑥^∗=−1√31,−√31,−√3,12√3^𝑇,𝑥^∗∗=1√3,1√3,1√31,−2√3^𝑇. (69) We test this problem by using 𝑥₀=(−1,−1,−1,−1)^𝑇 (the iterative sequence converges to 𝑥^∗) and 𝑥₀=(1,1,1,−2)^𝑇 (the iterative sequence converges to 𝑥^∗∗) as starting point, respectively. The test results are listed in Tables 4 and 5.

Table 4 

Numerical results of Example 4 with initial value 𝑥₀=(−1,−1,−1,−1)^𝑇.

Table 5 

Numerical results of Example 4 with initial value 𝑥₀=(1,1,1,−2)^𝑇.

Example 5. The test function is as follows (see [10, 13, 14]): 𝐹_𝑖(𝑥)=𝑥_𝑖𝑥_𝑖+1−1,𝑖=1,…,𝑛−1,𝐹_𝑛(𝑥)=𝑥₁𝑥_𝑛−1. (70) This problem has two solutions: they are 𝑥^∗=(1,1,…,1)^𝑇 and 𝑥^∗∗=(−1,−1,…,−1)^𝑇. We test this problem by using 𝑥₀=(2,2,…,2)^𝑇 (the iterative sequence converges to 𝑥^∗) and 𝑥₀=(−2,−2,…,−2)^𝑇 (the iterative sequence converges to 𝑥^∗∗) as starting point, respectively. The test results are listed in Tables 6 and 7, which are obtained for 𝑛=101.

Table 6 

Numerical results of Example 5 with initial value 𝑥₀=(2,2,…,2)^𝑇.

Table 7 

Numerical results of Example 5 with initial value 𝑥₀=(−2,−2,…,−2)^𝑇.

Example 6. Consider the unrestraint optimum problem (see [18]) min_{𝑥∈𝑅𝑛}𝑓(𝑥), (71) where 𝑓∶𝑅^𝑛→𝑅 is defined by 𝑓(𝑥)=^𝑛_𝑖=2𝑥²_𝑖−1+𝑥²_𝑖²−4𝑥_𝑖−1+3. (72) By KKT condition we have 𝐹(𝑥)=14∇𝑓(𝑥)=0, (73) where 𝐹(𝑥)=(𝐹₁(𝑥),𝐹₂(𝑥),…,𝐹_𝑛(𝑥))^𝑇, 𝐹₁(𝑥)=𝑥₁𝑥²₁+𝑥²₂𝐹−1,_𝑖(𝑥)=𝑥_𝑖𝑥²_𝑖−1+2𝑥²_𝑖+𝑥²_𝑖+1𝐹−1,𝑖=2,…,𝑛−1,_𝑛(𝑥)=𝑥_𝑛𝑥²_𝑛−1+𝑥²_𝑛. (74) We test this problem by using 𝑥₀=(1,1,…,1)^𝑇 as starting point. The test results are listed in Table 8, which are obtained for 𝑛=512.

Table 8 

Numerical results of Example 6 for dimensions 𝑛=512.

Example 7. Consider the discrete two-point boundary value problems (see [19]): 𝐹(𝑥)1=𝐴𝑥+(𝑛+1)²𝐺(𝑥)=0, (75) where 𝐴 is an 𝑛×𝑛 tridiagonal matrix defined by 𝐴=2−1−12−1⋱⋱⋱−12, (76)𝐺(𝑥)=(𝐺₁(𝑥),𝐺₂(𝑥),…,𝐺_𝑛(𝑥))^𝑇, and 𝐺_𝑖(𝑥)=sin𝑥_𝑖−1, 𝑖=1,2,…,𝑛. We test this problem by using 𝑥₀=(1,1,…,1)^𝑇 as starting point. The test results are listed in Table 9, which are obtained for 𝑛=1024.

Table 9 

Numerical results of Example 7 for dimensions 𝑛=1024.

5. Conclusion

From the seven examples in Section 4, we can see that the newly developed method (42)-(43) has the advantages of fast convergence speed (we can get from the CPU time), small number of iterations. Especially, the value of convergence order 𝑝 that appears in Tables 2–7 is the highest compared to the other four methods. Although our method’s convergence order is not always higher than the method of Cordero and Martínez and Torregrosa (NAd1), ours is superior in the number of iterations and CPU time to the other four methods. In a word, our method (42)-(43) is quite robust and effective.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This project is supported by National Natural Science Foundation of China (Grant no. 11071041), Fujian Natural Science Foundation (Grant no. 2013J01006), and R & D of Key Instruments and Technologies for Deep Resources Prospecting (the National R & D Projects for Key Scientific Instruments) under Grant no. ZDYZ2012-1-02-04.