Iterative Methods for Solving Nonlinear Equations and Systems Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Juan R. Torregrosa, Alicia Cordero and Fazlollah Soleymani Edited by Iterative Methods for Solving Nonlinear Equations and Systems Iterative Methods for Solving Nonlinear Equations and Systems Special Issue Editors Juan R. Torregrosa Alicia Cordero Fazlollah Soleymani MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Juan R. Torregrosa Polytechnic University of Valencia Spain Alicia Cordero Polytechnic University of Valencia Spain Fazlollah Soleymani Institute for Advanced Studies in Basic Sciences (IASBS) Iran Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) from 2018 to 2019 (available at: https://www.mdpi.com/journal/ mathematics/special issues/Iterative Methods Solving Nonlinear Equations Systems). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year , Article Number , Page Range. ISBN 978-3-03921-940-7 (Pbk) ISBN 978-3-03921-941-4 (PDF) c © 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to ”Iterative Methods for Solving Nonlinear Equations and Systems” . . . . . . . . . . xi Shengfeng Li, Xiaobin Liu and Xiaofang Zhang A Few Iterative Methods by Using [1 , n ] -Order Pad ́ e Approximation of Function and the Improvements Reprinted from: Mathematics 2019 , 7 , 55, doi:10.3390/math7010055 . . . . . . . . . . . . . . . . . . 1 Fuad W. Khdhr, Rostam K. Saeed and Fazlollah Soleymani Improving the Computational Efficiency of a Variant of Steffensen’s Method for Nonlinear Equations Reprinted from: Mathematics 2019 , 7 , 306, doi:10.3390/math7030306 . . . . . . . . . . . . . . . . . 15 Yanlin Tao and Kalyanasundaram Madhu Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application Reprinted from: Mathematics 2019 , 7 , 322, doi:10.3390/math7040322 . . . . . . . . . . . . . . . . . 24 Prem B. Chand, Francisco I. Chicharro, Neus Garrido and Pankaj Jain Design and Complex Dynamics of Potra–Pt ́ ak-Type Optimal Methods for Solving Nonlinear Equations and Its Applications Reprinted from: Mathematics 2019 , 7 , 942, doi:10.3390/math7100942 . . . . . . . . . . . . . . . . . 46 Jian Li, Xiaomeng Wang and Kalyanasundaram Madhu Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction Reprinted from: Mathematics 2019 , 7 , 1052, doi:10.3390/math7111052 . . . . . . . . . . . . . . . . 67 Min-Young Lee, Young Ik Kim and Beny Neta A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics Underlying Purely Imaginary Extraneous Fixed Points Reprinted from: Mathematics 2019 , 7 , 562, doi:10.3390/math7060562 . . . . . . . . . . . . . . . . . 82 Fiza Zafar, Alicia Cordero and Juan R. Torregrosa An Efficient Family of Optimal Eighth-Order Multiple Root Finders Reprinted from: Mathematics 2018 , 6 , 310, doi:10.3390/math6120310 . . . . . . . . . . . . . . . . . 108 Ramandeep Behl, Eulalia Mart ́ ınez, Fabricio Cevallos and Diego Alarc ́ on A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots Reprinted from: Mathematics 2019 , 7 , 55, doi:10.3390/math7040339 . . . . . . . . . . . . . . . . . . 124 Saima Akram, Fiza Zafar and Nusrat Yasmin An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots Reprinted from: Mathematics 2019 , 7 , 672, doi:10.3390/math7080672 . . . . . . . . . . . . . . . . . 135 Auwal Bala Abubakar, Poom Kumam, Hassan Mohammad and Aliyu Muhammed Awwal An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications Reprinted from: Mathematics 2019 , 7 , 767, doi:10.3390/math7090767 . . . . . . . . . . . . . . . . . 149 v Ioannis K. Argyros, ́ A. Alberto Magre ̃ n ́ an, Lara Orcos and ́ I ̃ nigo Sarr ́ ıa Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications Reprinted from: Mathematics 2019 , 7 , 299, doi:10.3390/math7030299 . . . . . . . . . . . . . . . . . 174 D. R. Sahu, Ravi P. Agarwal and Vipin Kumar Singh A Third Order Newton-Like Method and Its Applications Reprinted from: Mathematics 2019 , 7 , 31, doi:10.3390/math7010031 . . . . . . . . . . . . . . . . . . 186 Ioannis K. Argyros and Ramandeep Behl Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak Conditions Reprinted from: Mathematics 2019 , 7 , 89, doi:10.3390/math7010089 . . . . . . . . . . . . . . . . . . 208 Ioannis K. Argyros and Stepan Shakhno Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions Reprinted from: Mathematics 2019 , 7 , 207, doi:10.3390/math7020207 . . . . . . . . . . . . . . . . . 222 Cristina Amor ́ os, Ioannis K. Argyros, Ruben Gonz ́ alez Study of a High Order Family: Local Convergence and Dynamics Reprinted from: Mathematics 2019 , 7 , 225, doi:10.3390/math7030225 . . . . . . . . . . . . . . . . . 234 Zhang Yong, Neha Gupta, J. P. Jaiswal and Kalyanasundaram Madhu On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case Reprinted from: Mathematics 2019 , 7 , 540, doi:10.3390/math7060540 . . . . . . . . . . . . . . . . . 248 Pawicha Phairatchatniyom, Poom Kumam, Yeol Je Cho, Wachirapong Jirakitpuwapat and Kanokwan Sitthithakerngkiet The Modified Inertial Iterative Algorithm for Solving Split Variational Inclusion Problem for Multi-Valued Quasi Nonexpansive Mappings with Some Applications Reprinted from: Mathematics 2019 , 7 , 560, doi:10.3390/math7060560 . . . . . . . . . . . . . . . . . 262 Mozafar Rostami, Taher Lotfi and Ali Brahmand A Fast Derivative-Free Iteration Scheme for Nonlinear Systems and Integral Equations Reprinted from: Mathematics 2019 , 7 , 637, doi:10.3390/math7070637 . . . . . . . . . . . . . . . . . 284 Sergio Amat, Ioannis Argyros, Sonia Busquier, Miguel ́ Angel Hern ́ andez-Ver ́ on and Mar ́ ıa Jes ́ us Rubio A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators Reprinted from: Mathematics 2019 , 7 , 701, doi:10.3390/math7080701 . . . . . . . . . . . . . . . . . 295 Auwal Bala Abubakar, Poom Kumam, Hassan Mohammad, Aliyu Muhammed Awwal and Kanokwan Sitthithakerngkiet A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications Reprinted from: Mathematics 2019 , 7 , 745, doi:10.3390/math7080745 . . . . . . . . . . . . . . . . . 307 Alicia Cordero, Cristina Jord ́ an, Esther Sanabria and Juan R. Torregrosa A New Class of Iterative Processes for Solving Nonlinear Systems by Using One Divided Differences Operator Reprinted from: Mathematics 2019 , 7 , 776, doi:10.3390/math7090776 . . . . . . . . . . . . . . . . . 332 vi Abdolreza Amiri, Mohammad Taghi Darvishi, Alicia Cordero and Juan Ram ́ on Torregrosa An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly Nonlinear Systems Reprinted from: Mathematics 2019 , 7 , 815, doi:10.3390/math7090815 . . . . . . . . . . . . . . . . . 344 Hessah Faihan Alqahtani, Ramandeep Behl, Munish Kansal Higher-Order Iteration Schemes for Solving Nonlinear Systems of Equations Reprinted from: Mathematics 2019 , 7 , 937, doi:10.3390/math7100937 . . . . . . . . . . . . . . . . . 361 Dilan Ahmed, Mudhafar Hama, Karwan Hama Faraj Jwamer and Stanford Shateyi A Seventh-Order Scheme for Computing the Generalized Drazin Inverse Reprinted from: Mathematics 2019 , 7 , 622, doi:10.3390/math7070622 . . . . . . . . . . . . . . . . . 375 Haifa Bin Jebreen Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme Reprinted from: Mathematics 2019 , 7 , 731, doi:10.3390/math7080731 . . . . . . . . . . . . . . . . . 385 Zhinan Wu and Xiaowu Li An Improved Curvature Circle Algorithm for Orthogonal Projection onto a Planar Algebraic Curve Reprinted from: Mathematics 2019 , 7 , 912, doi:10.3390/math7100912 . . . . . . . . . . . . . . . . . 396 Anantachai Padcharoen and Pakeeta Sukprasert Nonlinear Operators as Concerns Convex Programming and Applied to Signal Processing Reprinted from: Mathematics 2019 , 7 , 866, doi:10.3390/math7090866 . . . . . . . . . . . . . . . . . 420 Juan Liang, Linke Hou, Xiaowu Li, Feng Pan, Taixia Cheng and Lin Wang Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n -Dimensional Euclidean Space Reprinted from: Mathematics 2018 , 6 , 306, doi:10.3390/math6120306 . . . . . . . . . . . . . . . . . 435 Jos ́ e Antonio Ezquerro and Miguel ́ Angel Hern ́ andez–Ver ́ on How to Obtain Global Convergence Domains via Newton’s Method for Nonlinear Integral Equations Reprinted from: Mathematics 2018 , 6 , 553, doi:10.3390/math7060553 . . . . . . . . . . . . . . . . . 458 Malik Zaka Ullah Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Order FD Scheme Reprinted from: Mathematics 2019 , 7 , 704, doi:10.3390/math7080704 . . . . . . . . . . . . . . . . . 467 vii About the Special Issue Editors Juan Ram ́ on Torregrosa , Dr., has a Bachelor in Mathematical Sciences (Universitat de Val` encia) and obtained his PhD (1990, Universitat de Val` encia) defending his thesis “Algunas propiedades geom ́ etricas uniformes y no uniformes de un espacio de Banach.” He is Full Professor of Applied Mathematics in the Institute for Multidisciplinary Mathematics of the Polytechnical University of Val` encia. He published several papers about locally convex spaces and Banach spaces in the 1990s. Afterwards, he launched new research projects in linear algebra, matrix analysis, and combinatorics. He has supervised several PhD theses on these topics. He also has published a significant number of papers in related journals: Linear Algebra and Its Applications , Applied Mathematics Letters , and SIAM Journal Matrix Analysis His current research is in numerical analysis. He focuses on different problems related to the solution of nonlinear equations and systems, matrix equations, and dynamical analysis of rational functions involved in iterative methods. He has published more than 200 papers in JCR journals and he also has presented numerous communications in international conferences. Alicia Cordero , Dr., has a Bachelor in Mathematic Sciences (1995, Universitat de Val` encia). She obtained her PhD in Mathematics (2003, Universitat Jaume I) defending her PhD Thesis “Cadenas de ́ orbitas peri ́ odicas en la variedad S2xS1,“ which was supervised by Jos ́ e Mart ́ ınez and Pura Vindel. Through the years, she has published many papers about the decomposition into round loops of three-dimensional varieties, links of periodic orbits of non-singular Morse-Smale fluxes, and their applications to celestial mechanics. She is Full Professor of Applied Mathematics in the Institute for Multidisciplinary Mathematics of the Polytechnical University of Val` encia. Her current research is focused on dynamical systems and numerical analysis, highlighting the iterative methods for solving nonlinear equations and systems as well as the dynamical study of the rational functions involved in these processes. She has published more than 150 papers in JCR Journals. She also has presented numerous communications in international conferences. Fazlollah Soleymani , Dr., acquired his Ph.D. degree in numerical analysis at Ferdowsi University of Mashhad in Iran. He also accomplished his postdoctoral fellowship at the Polytechnic University of Valencia in Spain. Soleymani’s main research interests are in the areas of computational mathematics. Recently, he has been working on a number of different problems, which fall under semi-discretized and (localized) RBF(–(H)FD) meshfree schemes for financial partial (integro–) differential equations, high-order iterative methods for nonlinear systems, numerical solution of stochastic differential equations, and iteration methods for generalized inverses. ix Preface to ”Iterative Methods for Solving Nonlinear Equations and Systems” Solving nonlinear equations in any Banach space (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others) is a non-trivial task that involves many areas of science and technology. Usually the solution is not directly affordable and requires an approach utilizing iterative algorithms. This is an area of research that has grown exponentially over the last few years. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new iterative schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots, either with or without derivatives, iterative methods for approximating different generalized inverses, and real or complex dynamics associated with the rational functions resulting from the application of an iterative method on a polynomial function. Additionally, the analysis of the convergence of the proposed methods has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, partial differential equations, or convex programming reveal the connection between iterative methods and other branches of science and engineering. Juan R. Torregrosa, Alicia Cordero, Fazlollah Soleymani Special Issue Editors xi mathematics Article A Few Iterative Methods by Using [ 1, n ] -Order Padé Approximation of Function and the Improvements Shengfeng Li 1, * , Xiaobin Liu 2 and Xiaofang Zhang 1 1 Institute of Applied Mathematics, Bengbu University, Bengbu 233030, China; zhangxiaofangah@126.com 2 School of Computer Engineering, Bengbu University, Bengbu 233030, China; 18807925677@139.com * Correspondence: lsf@bbc.edu.cn; Tel.: +86-552-317-5158 Received: 15 November 2018; Accepted: 30 December 2018; Published: 7 January 2019 Abstract: In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying [ 1, n ] -order Padé approximation of function for finding the roots of nonlinear equations at first. In order to avoid the operation of high-order derivatives of function, we modify the presented methods with fourth-order convergence by using the approximants of the second derivative and third derivative, respectively. Thus, several modified two-step iterative methods are obtained for solving nonlinear equations, and the convergence of the variants is then analyzed that they are of the fourth-order convergence. Finally, numerical experiments are given to illustrate the practicability of the suggested variants. Henceforth, the variants with fourth-order convergence have been considered as the imperative improvements to find the roots of nonlinear equations. Keywords: nonlinear equations; Padé approximation; iterative method; order of convergence; numerical experiment 1. Introduction It is well known that a variety of problems in different fields of science and engineering require to find the solution of the nonlinear equation f ( x ) = 0 where f : I → D , for an interval I ⊆ R and D ⊆ R , is a scalar function. In general, iterative methods, such as Newton’s method, Halley’s method, Cauchy’s method, and so on, are the most used techniques. Hence, iterative algorithms based on these iterative methods for finding the roots of nonlinear equations are becoming one of the most important aspects in current researches. We can see the works, for example, [ 1 – 22 ] and references therein. In the last few years, some iterative methods with high-order convergence have been introduced to solve a single nonlinear equation. By using various techniques, such as Taylor series, quadrature formulae, decomposition techniques, continued fraction, Padé approximation, homotopy methods, Hermite interpolation, and clipping techniques, these iterative methods can be constructed. For instance, there are many ways of introducing Newton’s method. Among these ways, using Taylor polynomials to derive Newton’s method is probably the most widely known technique [ 1 , 2 ]. By considering different quadrature formulae for the computation of the integral, Weerakoon and Fernando derive an implicit iterative scheme with cubic convergence by the trapezoidal quadrature formulae [ 4 ], while Cordero and Torregrosa develope some variants of Newton’s method based in rules of quadrature of fifth order [ 5 ]. In 2005, Chun [ 6 ] have presented a sequence of iterative methods improving Newton’s method for solving nonlinear equations by applying the Adomian decomposition method. Based on Thiele’s continued fraction of the function, Li et al. [ 7 ] give a fourth-order convergent iterative method. Using Padé approximation of the function, Li et al. [ 8 ] rederive the Halley’s method and by the divided differences to approximate the derivatives, they arrive at some modifications with third-order convergence. In [ 9 ], Abbasbandy et al. present an efficient numerical algorithm for Mathematics 2019 , 7 , 55; doi:10.3390/math7010055 www.mdpi.com/journal/mathematics 1 Mathematics 2019 , 7 , 55 solving nonlinear algebraic equations based on Newton–Raphson method and homotopy analysis method. Noor and Khan suggest and analyze a new class of iterative methods by using the homotopy perturbation method in [ 10 ]. In 2015, Wang et al. [ 11 ] deduce a general family of n -point Newton type iterative methods for solving nonlinear equations by using direct Hermite interpolation. Moreover, for a particular class of functions, for instance, if f is a polynomial, there exist some efficient univariate root-finding algorithms to compute all solutions of the polynomial equation (see [ 12 , 13 ]). In the literature [ 13 ], Barto ̆ n et al. present an algorithm for computing all roots of univariate polynomial based on degree reduction, which has the higher convergence rate than Newton’s method. In this article, we will mainly solve more general nonlinear algebraic equations. Newton’s method is probably the best known and most widely used iterative algorithm for root-finding problems. By applying Taylor’s formula for the function f ( x ) , let us recall briefly how to derive Newton iterative method. Suppose that f ( x ) ∈ C n [ I ] , n = 1, 2, 3, . . . , and η ∈ I is a single root of the nonlinear equation f ( x ) = 0. For a given guess value x 0 ∈ I and a δ ∈ R , assume that f ′ ( x ) = 0 for each x belongs to the neighborhood ( x 0 − δ , x 0 + δ ) . For any x ∈ ( x 0 − δ , x 0 + δ ) , we expand f ( x ) into the following Taylor’s formula about x 0 : f ( x ) = f ( x 0 ) + f ′ ( x 0 )( x − x 0 ) + 1 2! f ′′ ( x 0 )( x − x 0 ) 2 + · · · + 1 k ! ( x − x 0 ) k f ( k ) ( x 0 ) + · · · , where k = 0, 1, 2, · · · . Let | η − x 0 | be sufficiently small. Then the terms involving ( η − x 0 ) k , k = 2, 3, . . . , are much smaller. Hence, we think the fact that the first Taylor polynomial is a good approximation to the function near the point x 0 and give that f ( x 0 ) + f ′ ( x 0 )( η − x 0 ) ≈ 0. Notice the fact f ′ ( x 0 ) = 0, and solving the above equation for η yields η ≈ x 0 − f ( x 0 ) f ′ ( x 0 ) , which follows that we can construct the Newton iterative scheme as below x k + 1 = x k − f ( x k ) f ′ ( x k ) , k = 0, 1, 2, . . . . It has been known that Newton iterative method is a celebrated one-step iterative method. The order of convergence of Newton’s method is quadratic for a simple zero and linear for multiple root. Motivated by the idea of the above technique, in this paper, we start with using Padé approximation of a function to construct a few one-step iterative schemes which includes classical Newton’s method and Halley’s method to find roots of nonlinear equations. In order to avoid calculating the high-order derivatives of the function, then we employ the approximants of the higher derivatives to improve the presented iterative method. As a result, we build several two-step iterative formulae, and some of them do not require the operation of high-order derivatives. Furthermore, it is shown that these modified iterative methods are all fouth-order convergent for a simple root of the equation. Finally, we give some numerical experiments and comparison to illustrate the efficiency and performance of the presented methods. The rest of this paper is organized as follows. we introduce some basic preliminaries about Padé approximation and iteration theory for root-finding problem in Section 2. In Section 3, we firstly construct several one-step iterative schemes based on Padé approximation. Then, we modify the presented iterative method to obtain a few iterative formulae without calculating the high-order derivatives. In Section 4, we show that the modified methods have fourth-order convergence at least for a simple root of the equation. In Section 5 we give numerical examples to show the performance of 2 Mathematics 2019 , 7 , 55 the presented methods and compare them with other high-order methods. At last, we draw conclusions from the experiment results in Section 6. 2. Preliminaries In this section, we briefly review some basic definitions and results for Padé approximation of function and iteration theory for root-finding problem. Some surveys and complete literatures about iteration theory and Padé approximation could be found in Alfio [ 1 ], Burden et al. [ 2 ], Wuytack [ 23 ], and Xu et al. [24]. Definition 1. Assume that f ( x ) is a function whose ( n + 1 ) -st derivative f ( n + 1 ) ( x ) , n = 0, 1, 2, . . . , exists for any x in an interval I. Then for each x ∈ I, we have f ( x ) = f ( x 0 ) + f ′ ( x 0 )( x − x 0 ) + f ′′ ( x 0 ) 2! ( x − x 0 ) 2 + · · · + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + o [( x − x 0 ) n ] , (1) which is called the Taylor’s formula with Peano remainder term of order n based at x 0 , and the error o [( x − x 0 ) n ] is called the Peano remainder term or the Peano truncation error. Definition 2. If P ( x ) is a polynomial, the accurate degree of the polynomial is ∂ ( P ) , and the order of the polynomial is ω ( P ) , which is the degree of the first non-zero term of the polynomial. Definition 3. If it can be found two ploynomials P ( x ) = m ∑ i = 0 a i ( x − x 0 ) i and Q ( x ) = n ∑ i = 0 b i ( x − x 0 ) i such that ∂ ( P ( x )) ≤ m , ∂ ( Q ( x )) ≤ n , ω ( f ( x ) Q ( x ) − P ( x )) ≥ m + n + 1, then we have the following incommensurable form of the rational fraction P ( x ) Q ( x ) : R m , n ( x ) = P 0 ( x ) Q 0 ( x ) = P ( x ) Q ( x ) , which is called [ m , n ] -order Padé approximation of function f ( x ) We give the computational formula of Padé approximation of function f ( x ) by the use of determinant, as shown in the following lemma [23,24]. Lemma 1. Assume that R m , n ( x ) = P 0 ( x ) Q 0 ( x ) is Padé approximation of function f ( x ) . If the matrix A m , n = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ a m a m − 1 · · · a m + 1 − n a m + 1 a m · · · a m + 2 − n . . . a m + n − 1 a m + n − 2 · · · a m ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ is nonsingular, that is the determinant | A m , n | = d = 0 , then P 0 ( x ) , Q 0 ( x ) can be written by the following determinants 3 Mathematics 2019 , 7 , 55 P 0 ( x ) = 1 d ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ T m ( x ) ( x − x 0 ) T m − 1 ( x ) · · · ( x − x 0 ) n T m + 1 − n ( x ) a m + 1 a m · · · a m + 1 − n . . . a m + n a m + n − 1 · · · a m ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ and Q 0 ( x ) = 1 d ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 ( x − x 0 ) · · · ( x − x 0 ) n a m + 1 a m · · · a m + 1 − n . . . a m + n a m + n − 1 · · · a m ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ , where a n = f ( n ) ( x 0 ) n ! , n = 0, 1, 2, . . . , and we appoint that T k ( x ) = ⎧ ⎨ ⎩ k ∑ i = 0 a i ( x − x 0 ) i , f or k ≥ 0, 0, f or k < 0. Next, we recall the speed of convergence of an iterative scheme. Thus, we give the following definition and lemma. Definition 4. Assume that a sequence { x i } ∞ i = 0 converges to η , with x i = η for all i , i = 0, 1, 2, . . . . Let the error be e i = x i − η . If there exist two positive constants α and β such that lim i → ∞ | e i + 1 | | e i | α = β , then { x i } ∞ i = 0 converges to the constant η of order α . When α = 1 , the sequence { x i } ∞ i = 0 is linearly convergent. When α > 1 , the sequence { x i } ∞ i = 0 is said to be of higher-order convergence. For a single-step iterative method, sometimes it is convenient to use the following lemma to judge the order of convergence of the iterative method. Lemma 2. Assume that the equation f ( x ) = 0, x ∈ I , can be rewritten as x = φ ( x ) , where f ( x ) ∈ C [ I ] and φ ( x ) ∈ C γ [ I ] , γ ∈ N + . Let η be a root of the equation f ( x ) = 0 . If the iterative function φ ( x ) satisfies φ ( j ) ( η ) = 0, j = 1, 2, . . . , γ − 1, φ ( γ ) ( η ) = 0, then the order of convergence of the iterative scheme x i + 1 = φ ( x i ) , i = 0, 1, 2, . . . , is γ 3. Some Iterative Methods Let η be a simple real root of the equation f ( x ) = 0, where f : I → D , I ⊆ R , D ⊆ R . Suppose that x 0 ∈ I is an initial guess value sufficiently close to η , and the function f ( x ) has n -th derivative f ( n ) ( x ) , n = 1, 2, 3, . . . , in the interval I . According to Lemma 1, [ m , n ] -order Padé approximation of function f ( x ) is denoted by the following rational fraction: 4 Mathematics 2019 , 7 , 55 f ( x ) ≈ R m , n ( x ) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ T m ( x ) ( x − x 0 ) T m − 1 ( x ) · · · ( x − x 0 ) n T m + 1 − n ( x ) a m + 1 a m · · · a m + 1 − n . . . a m + n a m + n − 1 · · · a m ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 ( x − x 0 ) · · · ( x − x 0 ) n a m + 1 a m · · · a m + 1 − n . . . a m + n a m + n − 1 · · · a m ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (2) Recall Newton iterative method derived by Taylor’s series in Section 1. The first Taylor polynomial is regarded as a good approximation to the function f ( x ) near the point x 0 . Solving the linear equation denoted by f ( x 0 ) + f ′ ( x 0 )( η − x 0 ) ≈ 0 for η gives us the stage for Newton’s method. Then, we think whether or not a novel or better linear function is selected to approximate the function f ( x ) near the point x 0 . Maybe Padé approximation can solve this question. In the process of obtaining new iterative methods based on Padé approximation of function, on the one hand, we consider that the degree of the numerator of Equation (2) is always taken as 1, which guarantees to obtain the different linear function. On the other hand, we discuss the equations are mainly nonlinear algebraic equations, which differ rational equations and have not the poles. Clearly, as n grows, the poles of the denominator of Equation (2) do not affect the linear functions that we need. These novel linear functions may be able to set the stage for new methods. Next, let us start to introduce a few iterative methods by using [ 1, n ] -order Padé approximation of function. 3.1. Iterative Method Based on [ 1, 0 ] -Order Padé Approximation Firstly, when m = 1, n = 0, we consider [ 1, 0 ] -order Padé approximation of function f ( x ) It follows from the expression (2) that f ( x ) ≈ R 1,0 ( x ) = T 1 ( x ) = a 0 + a 1 ( x − x 0 ) Let R 1,0 ( x ) = 0, then we have a 0 + a 1 ( x − x 0 ) = 0. (3) Due to the determinant | A 1,0 | = 0, i.e., f ′ ( x 0 ) = 0, we obtain the following equation from Equation (3). x = x 0 − a 0 a 1 In view of a 0 = f ( x 0 ) , a 1 = f ′ ( x 0 ) , we reconstruct the Newton iterative method as below. Method 1. Assume that the function f : I → D has its first derivative at the point x 0 ∈ I . Then we obtain the following iterative method based on [ 1, 0 ] -order Padé approximation of function f ( x ) : x k + 1 = x k − f ( x k ) f ′ ( x k ) , k = 0, 1, 2, . . . . (4) Starting with an initial approximation x 0 that is sufficiently close to the root η and using the above scheme (4), we can get the iterative sequence { x i } ∞ i = 0 Remark 1. Method 1 is the well-known Newton’s method for solving nonlinear equation [1,2]. 5 Mathematics 2019 , 7 , 55 3.2. Iterative Method Based on [ 1, 1 ] -Order Padé Approximation Secondly, when m = 1, n = 1, we think about [ 1, 1 ] -order Padé approximation of function f ( x ) Similarly, it follows from the expression (2) that f ( x ) ≈ R 1,1 ( x ) = ∣ ∣ ∣ ∣ ∣ T 1 ( x ) ( x − x 0 ) T 0 ( x ) a 2 a 1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 ( x − x 0 ) a 2 a 1 ∣ ∣ ∣ ∣ ∣ Let R 1,1 ( x ) = 0, then we get a 0 a 1 + a 2 1 ( x − x 0 ) − a 0 a 2 ( x − x 0 ) = 0. (5) Due to the determinant | A 1,1 | = 0, that is, ∣ ∣ ∣ ∣ ∣ a 1 a 0 a 2 a 1 ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ f ′ ( x 0 ) f ( x 0 ) f ′′ ( x 0 ) 2 f ′ ( x 0 ) ∣ ∣ ∣ ∣ ∣ = f ′ 2 ( x 0 ) − f ( x 0 ) f ′′ ( x 0 ) 2 = 0. Thus, we obtain the following equality from Equation (5): x = x 0 − a 0 a 1 a 2 1 − a 0 a 2 Combining a 0 = f ( x 0 ) , a 1 = f ′ ( x 0 ) , a 2 = 1 2 f ′′ ( x 0 ) , gives Halley iterative method as follows. Method 2. Assume that the function f : I → D has its second derivative at the point x 0 ∈ I . Then we obtain the following iterative method based on [ 1, 1 ] -order Padé approximation of function f ( x ) : x k + 1 = x k − 2 f ( x k ) f ′ ( x k ) 2 f ′ 2 ( x k ) − f ( x k ) f ′′ ( x k ) , k = 0, 1, 2, . . . . (6) Starting with an initial approximation x 0 that is sufficiently close to the root η and applying the above scheme (6), we can obtain the iterative sequence { x i } ∞ i = 0 Remark 2. Method 2 is the classical Halley’s method for finding roots of nonlinear equation [ 1 , 2 ], which converges cubically. 3.3. Iterative Method Based on [ 1, 2 ] -Order Padé Approximation Thirdly, when m = 1, n = 2, we take into account [ 1, 2 ] -order Padé approximation of function f ( x ) . By the same manner, it follows from the expression (2) that f ( x ) ≈ R 1,2 ( x ) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ T 1 ( x ) ( x − x 0 ) T 0 ( x ) 0 a 2 a 1 a 0 a 3 a 2 a 1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 x − x 0 ( x − x 0 ) 2 a 2 a 1 a 0 a 3 a 2 a 1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ Let R 1,2 ( x ) = 0, then one has a 0 a 2 1 + a 2 0 a 2 + ( a 3 1 − 2 a 0 a 1 a 2 + a 2 0 a 3 )( x − x 0 ) = 0. (7) 6 Mathematics 2019 , 7 , 55 Due to the determinant | A 1,2 | = 0, that is, ∣ ∣ ∣ ∣ ∣ ∣ ∣ a 1 a 0 0 a 2 a 1 a 0 a 3 a 2 a 1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ f ′ ( x 0 ) f ( x 0 ) 0 f ′′ ( x 0 ) 2 f ′ ( x 0 ) f ( x 0 ) f ′′′ ( x 0 ) 6 f ′′ ( x 0 ) 2 f ′ ( x 0 ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ = f ′ 3 ( x 0 ) − f ( x 0 ) f ′ ( x 0 ) f ′′ ( x 0 ) + f 2 ( x 0 ) f ′′′ ( x 0 ) 6 = 0. Thus, we gain the following equality from Equation (7): x = x 0 − a 0 a 2 1 − a 2 0 a 2 a 3 1 − 2 a 0 a 1 a 2 + a 2 0 a 3 Substituting a 0 = f ( x 0 ) , a 1 = f ′ ( x 0 ) , a 2 = 1 2 f ′′ ( x 0 ) , and a 3 = 1 6 f ′′′ ( x 0 ) into the above equation gives a single-step iterative method as follows. Method 3. Assume that the function f : I → D has its third derivative at the point x 0 ∈ I . Then we obtain the following iterative method based on [ 1, 2 ] -order Padé approximation of function f ( x ) : x k + 1 = x k − 3 f ( x k ) ∣ 2 f ′ 2 ( x k ) − f ( x k ) f ′′ ( x k ) ) 6 f ′ 3 ( x k ) − 6 f ( x k ) f ′ ( x k ) f ′′ ( x k ) + f 2 ( x k ) f ′′′ ( x k ) , k = 0, 1, 2, . . . . (8) Starting with an initial approximation x 0 that is sufficiently close to the root η and applying the above scheme (8), we can receive the iterative sequence { x i } ∞ i = 0 Remark 3. Method 3 could be used to find roots of nonlinear equation. Clearly, for the sake of applying this iterative method, we must compute the second derivative and the third derivative of the function f ( x ) , which may generate inconvenience. In order to overcome the drawback, we suggest approximants of the second derivative and the third derivative, which is a very important idea and plays a significant part in developing some iterative methods free from calculating the higher derivatives. 3.4. Modified Iterative Method Based on Approximant of the Third Derivative In fact, we let z k = x k − f ( x k ) f ′ ( x k ) . Then expanding f ( z k ) into third Taylor’s series about the point x k yields f ( z k ) ≈ f ( x k ) + f ′ ( x k )( z k − x k ) + 1 2! f ′′ ( x k )( z k − x k ) 2 + 1 3! f ′′′ ( x k )( z k − x k ) 3 , which follows that f ′′′ ( x k ) ≈ 3 f 2 ( x k ) f ′ ( x k ) f ′′ ( x k ) − 6 f ( z k ) f ′ 3 ( x k ) f 3 ( x k ) (9) Substituting (9) into (8), we can have the following iterative method. Method 4. Assume that the function f : I → D has its second derivative about the point x 0 ∈ I . Then we possess a modified iterative method as below: ⎧ ⎨ ⎩ z k = x k − f ( x k ) f ′ ( x k ) , x k + 1 = x k − x k − z k 1 + 2 f ( z k ) f ′ 2 ( x k ) L − 1 ( x k ) , k = 0, 1, 2, . . . , (10) where L ( x k ) = f ( x k ) ∣ f ( x k ) f ′′ ( x k ) − 2 f ′ 2 ( x k ) ) . Starting with an initial approximation x 0 that is sufficiently close to the root η and using the above scheme (10), we can have the iterative sequence { x i } ∞ i = 0 Remark 4. Methods 4 is a two-step iterative method free from third derivative of the function. 7