International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
New Two-Step Method with Fifth-Order Convergence for Solving Nonlinear Equationsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
A Fifth-Order Iterative Method for Solving Nonlinear Equationsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
New Two-Step Method with Fifth-Order Convergence for Solving Nonlinear Equationsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
A Fifth-Order Iterative Method for Solving Nonlinear Equationsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Some Mixed Quadrature Rules for Approximate Evaluation of Real Cauchy Princip...IJERD Editor
In this paper some mixed quadrature rules have been constructed for numerical integration of real
Cauchy principal value integrals and their asymptotic error estimates have been derived. The numerical
verification of the rules has been done by considering some standard Cauchy principal value integrals.
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
ABSTRACT: In this article a three point boundary value problem associated with a second order differential equation with integral type boundary conditions is proposed. Then its solution is developed with the help of the Green’s function associated with the homogeneous equation. Using this idea and Iteration method is proposed to solve the corresponding linear problem.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Some Mixed Quadrature Rules for Approximate Evaluation of Real Cauchy Princip...IJERD Editor
In this paper some mixed quadrature rules have been constructed for numerical integration of real
Cauchy principal value integrals and their asymptotic error estimates have been derived. The numerical
verification of the rules has been done by considering some standard Cauchy principal value integrals.
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
ABSTRACT: In this article a three point boundary value problem associated with a second order differential equation with integral type boundary conditions is proposed. Then its solution is developed with the help of the Green’s function associated with the homogeneous equation. Using this idea and Iteration method is proposed to solve the corresponding linear problem.
We give an elementary exposition of a method to obtain the infinitesimal point symmetries of Lagrangians.Besides, we exhibit the Lanczos approach to Noether’s theorem to construct the first integral associated with each symmetry.
MSC 2010:49S05, 58E30, 70H25, 70H33
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Derivation and Application of Six-Point Linear Multistep Numerical Method for...IOSR Journals
A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 6 and collocating the first and second derivative of the
continuous interpolant at 𝑥𝑛+𝑗 , 𝑗 = 0 and 𝑗 = 2, 3, … 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
Nonstationary Relaxed Multisplitting Methods for Solving Linear Complementari...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous multisplitting methods for solving the linear complementarity problems with their coefficient matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency is shown by numerical tests.
We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered.
ABSTRACT: In this paper, we construct new classes of derivative-free of tenth-order iterative methods for solving nonlinear equations. The new methods of tenth-order convergence derived by combining of theSteffensen's method, the Kung and Traub’s of optimal fourth-order and the Al-Subaihi's method. Several examples to compare of other existing methods and the results of new iterative methods are given the encouraging results and have definite practical utility.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
The study of this paper suggests on dependency problem in fuzzy computational method by using the numerical solution of Fuzzy differential equations(FDEs) in Milne’s predictor-corrector method. This method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value problems to illustrate the theory.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 2 Issue 8 || August. 2014 || PP-01-06
www.ijmsi.org 1 | P a g e
Some New Iterative Methods Based on Composite Trapezoidal Rule for Solving Nonlinear Equations Ogbereyivwe Oghovese1, Emunefe O. John2 1((Department of Mathematics and Statistics, Delta State Polytechnic, Ozoro, Nigeria) 2(Department of General Studies, Petroleum Training Institute, Effurun, Delta State, Nigeria) ABSTRACT: In this paper, new two steps family of iterative methods of order two and three constructed based on composite trapezoidal rule and fundamental theorem of calculus, for solving nonlinear equations. Several numerical examples are given to illustrate the efficiency and performance of the iterative methods; the methods are also compared with well known existing iterative method. KEYWORDS: Nonlinear equations, computational order of convergence, Newton’s method
I. INTRODUCTION
Solving nonlinear equations is one of the most predominant problems in numerical analysis. A classical and very popular method for solving nonlinear equations is the Newton’s method. Some historical points on this method can be found in [1]. Recently, some methods have been proposed and analyzed for solving nonlinear equations [2-13]. Some of these methods have been suggested either by using quadrature formulas, homotopy, decomposition or Taylor’s series [2-13]. Motivated by these techniques applied by various authors [2-13] and references therein, in constructing numerous iterative methods for solving nonlinear equations, we suggest a two steps family of iterative method based on composite trapezoidal rule and fundamental theorem of calculus for solving nonlinear equations. We also considered the convergence analysis of these methods. Several examples of functions, some of which are same as in [2-13] were used to illustrate the performance of the methods and comparison with other existing methods.
II. PRELIMINARIES
We use the following definitions: Definition 1. (See Dennis and Schnable [2] ) Let 훼∈ℝ, 푥푛∈ℝ, 푛=0,1,2,… Then, the sequence 푥푛 is said to converge to 훼 if 푙푖푚 푛→∞ 푥푛−훼 =0 (1) If, in addition, there exists a constant 푐≥0, an integer 푥0≥0, and 푝≥0 such that for all 푛≥푥0, 푥푛+1−훼 ≤푐 푥푛−훼 푝 (2) then 푥푛 is said to converge to 훼 with 푞-order at least 푝. If 푝=2, the convergence is said to be of order . Definition 2 (See Grau-Sanchez et al. [14]) The computational local order of convergence, 휌푛 , (CLOC) of a sequence 푥푛 푛≥0 is defined by 휌푛 = 푙표푔 푒푛 푙표푔 푒푛−1 , (3) where 푥푛−1 and 푥푛 are two consecutive iterations near the roots 훼 and 푒푛=푥푛−1−훼 . Notation 1: (See [6]) The notation 푒푛=푥푛−훼 is the error in the nth iteration. The equation 푒푛+1=푐푒푛 푝+푂 푒푛 푝+1 , (4) is called the error equation. By substituting 푒푛=푥푛−훼 for all 푛 in any iterative method and simplifying, we obtain the error equation for that method. The value of 푝 obtained is called the order of this method.
2. Some New Iterative Methods Based on Composite…
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III. DEVELOPMENT OF THE METHODS Consider a nonlinear equation 푓 푥 =0 (5) By the Fundamental Theorem of Calculus, if 푓(푥) is continuous at every point of [푎,푏] and 퐹 is any anti- derivatives of 푓(푥) on [푎,푏], then 푓(푥) 푏 푎 푑푥=퐹 푏 −퐹 푎 (6) Differentiating both side of (6) with respect to 푥, we have; 푓 푥 =푓 푏 −푓 푎 (7) where 푓(푏) and 푓(푎) are derivatives of 퐹(푏) and 퐹(푎) respectively. Recall the Composite Trapezoidal rule given by; 푓(푥) 푏 푎 푑푥= 푏−푎 2푛 푓 푎 +2 푓푖 푛−1 푖=1+푓(푏) (8) If 푛=2 in (8) we have; 푓(푥) 푏 푎 푑푥= 푏−푎 4 푓 푎 +2푓 푎+푏 2 +푓(푏) (9) Differentiating (9) with respect to 푥, we have; 푓 푥 = 푏−푎 4 푓′ 푎 +2푓′ 푎+푏 2 +푓′ 푏 (10) Equating (7) and (10) we have; 푓 푏 −푓 푎 = 푏−푎 4 푓′ 푎 +2푓′ 푎+푏 2 +푓′ 푏 (11) From(5), we have; 푥=푎−4 푓(푎) 푓′ 푎 − 푥−푎 푓′ 푥 푓′ 푎 −2 푥−푎 푓′ 푎+푏 2 푓′ 푎 (12) Using(12), one can suggest the following iterative method for solving the nonlinear equation(5). Algorithm 1: Given an initial approximation 푥0 (close to 훼 the root of (5)), we find the approximate solution 푥푛+1 by the implicit iterative method: 푥푛+1=푥푛−4 푓(푥푛) 푓′ 푥푛 − 푥−푎 푓′ 푥푛+1 푓′ 푥푛 −2 푥−푎 푓′ 푥푛+푥푛+12 푓′ 푥푛 , 푛=0,1,2,… (13) The implicit iterative method in (13) is a predictor-corrector scheme, with Newton’s method as the predictor, and Algorithm 1 as the corrector. The first consequence of (13) is the suggested two-step iterative method for solving (5) stated as follows: Algorithm 2: Given an initial approximation 푥0 (close to 훼 the root of (5)), we find the approximate solution 푥푛+1 by the iterative schemes: 푦푛=푥푛− 푓(푥푛) 푓′ 푥푛 (14푎) 푥푛+1=푥푛−4 푓(푥푛) 푓′ 푥푛 − 푦푛−푥푛 푓′ 푦푛 푓′ 푥푛 −2 푦푛−푥푛 푓′ 푥푛+푦푛 2 푓′ 푥푛 , 푛=0,1,2,… (14푏) From(14푎), we have that; 푦푛−푥푛=− 푓(푥푛) 푓′ 푥푛 (15) Using (15) in (14푏) we suggest another new iterative scheme as follows: Algorithm 3: Given an initial approximation 푥0 (close to 훼 the root of (5), we find the approximate solution 푥푛+1 by the iterative schemes:
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푥푛+1=푥푛−4 푓(푥푛) 푓′ 푥푛 + 푓(푥푛) 푓′ 푥푛 푓′ 푦푛 푓′ 푥푛 −2 푓(푥푛) 푓′ 푥푛 푓′ 푥푛+푦푛 2 푓′ 푥푛 , 푛=0,1,2,… (16) From (5) and (11) we can have the fixed point formulation given by 푥=푎− 4푓(푎) 푓′ 푎 +2푓′ 푎+푥 2 +푓′ 푥 (17 ) The formulation (17 ) enable us to suggest the following iterative method for solving nonlinear equations. Algorithm 4: Given an initial approximation 푥0 (close to 훼 the root of (5), we find the approximate solution 푥푛+1 by the iterative schemes: 푦푛=푥푛− 푓(푥푛) 푓′(푥푛) 푥푛+1=푥푛− 4푓(푥푛) 푓′ 푥푛 +2푓′ 푥푛+푦푛 2 +푓′(푦푛) , 푛=0,1,2,… (18) In the next section, we present the convergence analysis of Algorithm 2 and 4. Similar procedures can be applied to analyze the convergence of Algorithm 3. IV. CONVERGENCE ANALYSIS OF THE METHODS Theorem 1: Let 훼∈퐼 be a simple zero of sufficiently differentiable function 푓:퐼⊆ ℝ →ℝ for an open interval 퐼. If 푥0 is sufficiently close to 훼, then the iterative method defined by (14) is of order two and it satisfies the following error equation: 푒푛+1=훼−3푐2푒푛 2+ 푐3+6푐22− 32 푐2 푒푛 3+푂 푒푛 3 (19 ) where 푐2= 푓′′(훼) 2푓′(훼) (20) Proof Let 훼 be a simple zero of 푓, and 푒푛=푥푛−훼. Using Taylor expansion around 푥=훼 and taking into account 푓 훼 =0, we get 푓 푥푛 =푓′ 훼 푒푛+푐2푒푛 2+푐3푒푛 3+푐4푒푛 4+⋯ , (21) 푓′ 푥푛 =푓′ 훼 1+2푐2푒푛+3푐3푒푛 2+4푐4푒푛 3+5푐5푒푛 4+⋯ (22) where 푐푘= 푓푘(훼) 푘!푓′(훼) , 푘=2,3,4,… (23) Using 21 and 22 , we have; 푓(푥푛) 푓′ 푥푛 = 푒푛−푐2푒푛 2+2(푐22−푐3)푒푛 3+(7푐2푐3−4푐23−3푐4)푒푛 4+⋯ (24)
But 푦푛=푥푛− 푓 푥푛 푓′ 푥푛 25 = 훼+푐2푒푛 2+2(푐22−푐3)푒푛 3−(7푐2푐3−4푐23−3푐4)푒푛 4+⋯ (26) Hence, 푦푛−푥푛=−푒푛+푐2푒푛 2+2(푐22−푐3)푒푛 3−(7푐2푐3−4푐23−3푐4)푒푛 4+⋯ (27) From 26 , we have; 푓′ 푦푛 =푓′ 훼 1+2푐22푒푛 2+4 푐2푐3−푐23 푒푛 3+(−11푐22푐3+6푐2푐4+8푐24)푒푛 4+⋯ (28) Combining (22) and (28), we have; 푓′ 푦푛 푓′ 푥푛 =−2푐2푒푛+ −3푐3+6푐22 푒푛 2+ −16푐23−4푐4+16푐2푐3 푒푛 3+⋯ (29) From (27) and (29) we have; 푦푛−푥푛 푓′ 푦푛 푓′ 푥푛 =−푒푛+3푐2푒푛 2+(5푐3−10푐22)푒푛 3+ −30푐2푐3+30푐23+7푐4 푒푛 4+⋯ (30)
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From the relation; 푥푛+푦푛 2=푥푛− 푓 푥푛 2푓′ 푥푛 =훼+ 12 푒푛+ 12 푐2푒푛 2− 푐22−푐3 푒푛 3− 12 7푐2푐3−4푐23−3푐4 푒푛 4+⋯ (31 ) we have; 푓 푥푛+푦푛 2 =푓 푥푛− 푓 푥푛 2푓′ 푥푛 =푓(훼) 12 푒푛+ 34 푐2푒푛 2+ − 12 푐22+ 98 푐3 푒푛 3+ 54 푐23− 178 푐2푐3+ 2516 푐4 푒푛 4 + −3푐24+ 578 푐3푐22− 94 푐32− 134 푐2푐4+ 6532 푐5 푒푛 5+⋯ (32) 푓′ 푥푛+푦푛 2 =푓(훼) 1+푐2푒푛+ 푐22+ 34 푐3 푒푛 2+ −2푐23+ 72 푐2푐3+ 12 푐4 푒푛 3 + 92 푐2푐4+푐24− 374 푐22푐3+3푐32+ 516 푐5 푒푛 4+⋯ (33) Using (22) and (33) we have; 푓′ 푥푛+푦푛 2 푓′ 푥푛 =1−푐2푒푛+ 3푐22− 34 푐3−3푐3 푒푛 2+⋯ (34) And (27) with (34) gives; 푦푛−푥푛 푓′ 푥푛+푦푛 2 푓′ 푥푛 =−푒푛+2푐2푒푛 2+ 34 푐2+푐3−2푐22 푒푛 3+⋯ (35) Using 29 ,(30) and (35) in 푥푛+1=푥푛−4 푓 푥푛 푓′ 푥푛 − 푦푛−푥푛 푓′ 푦푛 푓′ 푥푛 −2 푦푛−푥푛 푓′ 푥푛+푦푛 2 푓′ 푥푛 =훼−3푐2푒푛 2+ 푐3+6푐22− 32 푐2 푒푛 3+푂 푒푛 4 ∎ (36) Thus, we observe that the Algorithm 2 is second order convergent. Theorem 2: Let 훼∈퐼 be a simple zero of sufficiently differentiable function 푓:퐼⊆ ℝ →ℝ for an open interval 퐼. If 푥0 is sufficiently close to 훼, then the iterative method defined by (18) is of order three and it satisfies the following error equation: 푒푛+1=훼+ 18 푐3+푐22 푒푛 3+푂 푒푛 4 (37 ) where 푐3= 푓′′(훼) 3!푓′(훼) (38) Proof Using 22 ,(33) and (28) we have; 푓′ 푥푛 +2푓′ 푥푛+푦푛 2 +푓′ 푦푛 =푓′ 훼 4+4푐2푒푛+ 92 푐3+4푐22 푒푛 2 + 5푐4+11푐2푐3−8푐23 푒푛 3+(−11푐22푐3+6푐2푐4+8푐24)푒푛 4+⋯ (39) and from (21) we have; 4푓′ 푥푛 =푓′ 훼 4푒푛+4푐2푒푛 2+4푐3푒푛 3+4푐4푒푛 4+⋯ (40) Combining (39) and (40) in (18) gives;
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푒푛+1=푥푛− 4푓 푥푛 푓′ 푥푛 +2푓′ 푥푛+푦푛 2 +푓′ 푦푛 = 훼+ 18 푐3+푐22 푒푛 3+푂 푒푛 4 ∎ (41) This means the method defined by (18) is of third-order. That completes the proof. V. NUMERICAL EXAMPLES In this section, we present some examples to illustrate the efficiency of our developed methods which are given by the Algorithm 1 – 4. We compare the performance of Algorithm 2 (AL2) and Algorithm 4 (AL4) with that of Newton Method (NM). All computations are carried out with double arithmetic precision. Displayed in Table 1 are the number of iterations (NT) required to achieve the desired approximate root 푥푛 and respective Computational Local Order of Convergence (CLOC), 휌푛 . The following stopping criteria were used. 푖. 푥푛+1−푥푛 <휀 푖푖. 푓 푥푛+1 <휀 (37) where 휀=10−15. We used the following functions, some of which are same as in [2-4,6-12,14] 푓1 푥 = 푥−1 3−1 푓2 푥 =cos 푥 −푥 푓3 푥 =푥3−10 푓4(푥)=푥2−푒푥−3푥+2 푓5 푥 = 푥+2 푒푥−1 푓6(푥)=푥3+4푥2−10 푓7 푥 =푙푛푥+ 푥−5 푓8 푥 =푒푥푠푖푛푥+ln(푥2+1) (38) Table 1: Comparison between methods depending on the number of iterations (IT) and Computational Local Order of Convergence.
푓(푥)
푥0
Number of iterations (NT)
Computational Local Order of Convergence (CLOC)
푵푴
푨푳 ퟐ
푨푳 ퟒ
푵푴
푨푳 ퟐ
푨푳 ퟒ
푓1
3.5
7
8
5
1.99999
1.92189
2.99158
푓2
1.7
4
5
3
2.19212
1.89891
3.55514
푓3
1.5
6
34
4
2.05039
1.97063
3.18850
푓4
2
5
6
4
2.17194
2.10516
3.42355
푓5
2
9
34
5
2.03511
1.03401
3.17161
푓6
2
5
6
3
2.06888
1.97365
3.42128
푓7
7
4
5
3
2.38378
2.16311
4.17738
푓8
0.5
6
14
5
1.94071
1.86901
2.00000
The computational results presented in Table 1 shows that the suggested methods are comparable with Newton Method. This means that; the new methods (Algorithm 4 in particular) can be considered as a significant improvement of Newton Method, hence; they can serve as an alternative to other second and third order convergent respectively, methods of solving nonlinear equations. VI. CONCLUSION We derived a two step family of iterative methods based on composite trapezoidal rule and fundamental theorem of calculus, for solving nonlinear equations. Convergence proof is presented in detail for algorithm 2 and 4 and they are of order two and three respectively. Analysis of efficiency showed that these methods can be used as alternative to other existing order two and three iterative methods for zero of nonlinear equations. Finally, we hoped that this study makes a contribution to solve nonlinear equations.
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