The document compares the convergence rates of the bisection, Newton-Raphson, and secant methods for finding roots of functions. It finds the root of the function f(x)=x-cos(x) on the interval [0,1] using each method. The bisection method converges at the 52nd iteration, while Newton-Raphson and secant methods converge to the exact root of 0.739085 with an error of 0 at the 8th and 6th iterations respectively. Therefore, the document concludes that the secant method is the most effective of the three for this problem.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
The best known deterministic polynomial-time algorithm for primality testing right now is due to
Agrawal, Kayal, and Saxena. This algorithm has a time complexity O(log15=2(n)). Although this algorithm is
polynomial, its reliance on the congruence of large polynomials results in enormous computational requirement.
In this paper, we propose a parallelization technique for this algorithm based on message-passing
parallelism together with four workload-distribution strategies. We perform a series of experiments on an
implementation of this algorithm in a high-performance computing system consisting of 15 nodes, each with
4 CPU cores. The experiments indicate that our proposed parallelization technique introduce a significant
speedup on existing implementations. Furthermore, the dynamic workload-distribution strategy performs
better than the others. Overall, the experiments show that the parallelization obtains up to 36 times speedup.
Algorithmic entropy can be seen as a special case of entropy as studied in
statistical mechanics. This viewpoint allows us to apply many techniques
developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing
D. Mladenov - On Integrable Systems in CosmologySEENET-MTP
Lecture by Prof. Dr. Dimitar Mladenov (Theoretical Physics Department, Faculty of Physics, Sofia University, Bulgaria) on December 7, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
The best known deterministic polynomial-time algorithm for primality testing right now is due to
Agrawal, Kayal, and Saxena. This algorithm has a time complexity O(log15=2(n)). Although this algorithm is
polynomial, its reliance on the congruence of large polynomials results in enormous computational requirement.
In this paper, we propose a parallelization technique for this algorithm based on message-passing
parallelism together with four workload-distribution strategies. We perform a series of experiments on an
implementation of this algorithm in a high-performance computing system consisting of 15 nodes, each with
4 CPU cores. The experiments indicate that our proposed parallelization technique introduce a significant
speedup on existing implementations. Furthermore, the dynamic workload-distribution strategy performs
better than the others. Overall, the experiments show that the parallelization obtains up to 36 times speedup.
Algorithmic entropy can be seen as a special case of entropy as studied in
statistical mechanics. This viewpoint allows us to apply many techniques
developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing
D. Mladenov - On Integrable Systems in CosmologySEENET-MTP
Lecture by Prof. Dr. Dimitar Mladenov (Theoretical Physics Department, Faculty of Physics, Sofia University, Bulgaria) on December 7, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
Holographic Dark Energy in LRS Bianchi Type-II Space Timeinventionjournals
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
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
A Non Parametric Estimation Based Underwater Target ClassifierCSCJournals
Underwater noise sources constitute a prominent class of input signal in most underwater signal processing systems. The problem of identification of noise sources in the ocean is of great importance because of its numerous practical applications. In this paper, a methodology is presented for the detection and identification of underwater targets and noise sources based on non parametric indicators. The proposed system utilizes Cepstral coefficient analysis and the Kruskal-Wallis H statistic along with other statistical indicators like F-test statistic for the effective detection and classification of noise sources in the ocean. Simulation results for typical underwater noise data and the set of identified underwater targets are also presented in this paper.
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Parallel hybrid chicken swarm optimization for solving the quadratic assignme...IJECEIAES
In this research, we intend to suggest a new method based on a parallel hybrid chicken swarm optimization (PHCSO) by integrating the constructive procedure of GRASP and an effective modified version of Tabu search. In this vein, the goal of this adaptation is straightforward about the fact of preventing the stagnation of the research. Furthermore, the proposed contribution looks at providing an optimal trade-off between the two key components of bio-inspired metaheuristics: local intensification and global diversification, which affect the efficiency of our proposed algorithm and the choice of the dependent parameters. Moreover, the pragmatic results of exhaustive experiments were promising while applying our algorithm on diverse QAPLIB instances . Finally, we briefly highlight perspectives for further research.
I am Samantha K. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, McGill University, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The Inverse Scattering Series (ISS) is a direct inversion method
for a multidimensional acoustic, elastic and anelastic earth. It
communicates that all inversion processing goals are able to
be achieved directly and without any subsurface information.
This task is reached through a task-specific subseries of the
ISS. Using primaries in the data as subevents of the first-order
internal multiples, the leading-order attenuator can predict the
time of all the first-order internal multiples and is able to attenuate
them.
However, the ISS internal multiple attenuation algorithm can
be a computationally demanding method specially in a complex
earth. By using an approach that is based on two angular
quantities and that was proposed in Terenghi et al. (2012), the
cost of the algorithm can be controlled. The idea is to use the
two angles as key-control parameters, by limiting their variation,
to disregard some calculated contributions of the algorithm
that are negligible. Moreover, the range of integration
can be chosen as a compromise of the required degree of accuracy
and the computational time saving.
This time-saving approach is presented
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Holographic Dark Energy in LRS Bianchi Type-II Space Timeinventionjournals
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
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
A Non Parametric Estimation Based Underwater Target ClassifierCSCJournals
Underwater noise sources constitute a prominent class of input signal in most underwater signal processing systems. The problem of identification of noise sources in the ocean is of great importance because of its numerous practical applications. In this paper, a methodology is presented for the detection and identification of underwater targets and noise sources based on non parametric indicators. The proposed system utilizes Cepstral coefficient analysis and the Kruskal-Wallis H statistic along with other statistical indicators like F-test statistic for the effective detection and classification of noise sources in the ocean. Simulation results for typical underwater noise data and the set of identified underwater targets are also presented in this paper.
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Parallel hybrid chicken swarm optimization for solving the quadratic assignme...IJECEIAES
In this research, we intend to suggest a new method based on a parallel hybrid chicken swarm optimization (PHCSO) by integrating the constructive procedure of GRASP and an effective modified version of Tabu search. In this vein, the goal of this adaptation is straightforward about the fact of preventing the stagnation of the research. Furthermore, the proposed contribution looks at providing an optimal trade-off between the two key components of bio-inspired metaheuristics: local intensification and global diversification, which affect the efficiency of our proposed algorithm and the choice of the dependent parameters. Moreover, the pragmatic results of exhaustive experiments were promising while applying our algorithm on diverse QAPLIB instances . Finally, we briefly highlight perspectives for further research.
I am Samantha K. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, McGill University, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The Inverse Scattering Series (ISS) is a direct inversion method
for a multidimensional acoustic, elastic and anelastic earth. It
communicates that all inversion processing goals are able to
be achieved directly and without any subsurface information.
This task is reached through a task-specific subseries of the
ISS. Using primaries in the data as subevents of the first-order
internal multiples, the leading-order attenuator can predict the
time of all the first-order internal multiples and is able to attenuate
them.
However, the ISS internal multiple attenuation algorithm can
be a computationally demanding method specially in a complex
earth. By using an approach that is based on two angular
quantities and that was proposed in Terenghi et al. (2012), the
cost of the algorithm can be controlled. The idea is to use the
two angles as key-control parameters, by limiting their variation,
to disregard some calculated contributions of the algorithm
that are negligible. Moreover, the range of integration
can be chosen as a compromise of the required degree of accuracy
and the computational time saving.
This time-saving approach is presented
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the method has at most an order of convergence (at least) linear. Keywords: Iteration Function, Order of Convergence, Fractional Derivative.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
In this work, we propose to apply trust region optimization to deep reinforcement
learning using a recently proposed Kronecker-factored approximation to
the curvature. We extend the framework of natural policy gradient and propose
to optimize both the actor and the critic using Kronecker-factored approximate
curvature (K-FAC) with trust region; hence we call our method Actor Critic using
Kronecker-Factored Trust Region (ACKTR). To the best of our knowledge, this
is the first scalable trust region natural gradient method for actor-critic methods.
It is also a method that learns non-trivial tasks in continuous control as well as
discrete control policies directly from raw pixel inputs. We tested our approach
across discrete domains in Atari games as well as continuous domains in the MuJoCo
environment. With the proposed methods, we are able to achieve higher
rewards and a 2- to 3-fold improvement in sample efficiency on average, compared
to previous state-of-the-art on-policy actor-critic methods. Code is available at
https://github.com/openai/baselines.
Computational language have been used in physics research
for many years and there is a plethora of programs and packages on the Web which can be used to solve dierent problems. In this report I trying to use as many of these available solutions as possible and not reinvent the wheel. Some of these packages have been written in C program. As I stated above, physics relies heavily on graphical representations. Usually,the scientist would save the results
from some calculations into a file, which then can be read and used for display by a graphics package like Gnuplot.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear
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.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
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.
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.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
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.
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.
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/
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
The Art of the Pitch: WordPress Relationships and Sales
A04410107
1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 04, Issue 04 (April. 2014), ||V1|| PP 01-07
International organization of Scientific Research 1 | P a g e
Comparative Study of Bisection, Newton-Raphson and Secant
Methods of Root- Finding Problems
Ehiwario, J.C., Aghamie, S.O.
Department of Mathematics, College of Education, Agbor, Delta State.
Abstract: - The study is aimed at comparing the rate of performance, viz-aviz, the rate of convergence of
Bisection method, Newton-Raphson method and the Secant method of root-finding. The software, mathematica
9.0 was used to find the root of the function, f(x)=x-cosx on a close interval [0,1] using the Bisection method,
the Newton’s method and the Secant method and the result compared. It was observed that the Bisection method
converges at the 52 second iteration while Newton and Secant methods converge to the exact root of 0.739085
with error 0.000000 at the 8th
and 6th
iteration respectively. It was then concluded that of the three methods
considered, Secant method is the most effective scheme. This is in line with the result in our Ref.[4].
Keywords: - Convergence, Roots, Algorithm, Iterations, Bisection method, Newton-Raphson method, Secant
method and function
I. INTRODUCTION
Root finding problem is a problem of finding a root of the equation 0xf , where xf is a
function of a single variable, x . Let xf be a function, we are interested in finding x such
that 0f . The number is called the root or zero of xf . xf may be algebraic, trigonometric or
transcendental function.
The root finding problem is one of the most relevant computational problems. It arises in a wide variety of
practical applications in Physics, Chemistry, Biosciences, Engineering, etc. As a matter of fact, the
determination of any unknown appearing implicitly in scientific or engineering formulas, gives rise to root
finding problem [1]. Relevant situations in Physics where such problems are needed to be solved include finding
the equilibrium position of an object, potential surface of a field and quantized energy level of confined
structure [2]. The common root-finding methods include: Bisection, Newton-Raphson, False position, Secant
methods etc. Different methods converge to the root at different rates. That is, some methods are faster in
converging to the root than others. The rate of convergence could be linear, quadratic or otherwise. The higher
the order, the faster the method converges [3]. The study is at comparing the rate of performance (convergence)
of Bisection, Newton-Raphson and Secant as methods of root-finding.
Obviously, Newton-Raphson method may converge faster than any other method but when we compare
performance, it is needful to consider both cost and speed of convergence. An algorithm that converges quickly
but takes a few seconds per iteration may take more time overall than an algorithm that converges more slowly,
but takes only a few milliseconds per iteration [4]. Secant method requires only one function evaluation per
iteration, since the value of 1nxf can be stored from the previous iteration [1,4]. Newton’s method, on the
other hand, requires one function and the derivative evaluation per iteration. It is often difficult to estimate the
cost of evaluating the derivative in general (if it is possible) [1, 4-5]. It seem safe, to assume that in most cases,
evaluating the derivative is at least as costly as evaluating the function [4]. Thus, we can estimate that the
Newton iteration takes about two functions evaluation per iteration. This disparity in cost means that we can run
two iterations of the secant method in the same time it will take to run one iteration of Newton method.
In comparing the rate of convergence of Bisection, Newton and Secant methods,[4] used C++
programming
language to calculate the cube roots of numbers from 1 to 25, using the three methods. They observed that the
rate of convergence is in the following order: Bisection method < Newton method < Secant method. They
concluded that Newton method is 7.678622465 times better than the Bisection method while Secant method is
1.389482397 times better than the Newton method.
II. METHODS
Bisection Method:
2. Comparative Study Of Bisection, Newton-Raphson And Secant Methods Of Root- Finding Problems
International organization of Scientific Research 2 | P a g e
Given a function 0xf , continuous on a closed interval ba, , such that 0bfaf , then, the function
0xf has at least a root or zero in the interval ba, . The method calls for a repeated halving of sub-
intervals of ba, containing the root. The root always converges, though very slow in converging [5].
Algorithm of Bisection Method for Root- Finding:
Inputs: (i) xf – the given function,
(ii) 00 ,ba – the two numbers, such that 0bfaf .
Output: An approximation of the root of 0xf in 00 ,ba , for ,...2,1,0k do until satisfied.
Compute
2
kk
k
ba
C
Test if kC is the desired root. If so, stop.
If kC is not the desired root, test if 0kk afCf . If so, set kk Cb 1 and kk Ca 1
Otherwise, set kkk bbC 1
End
Stopping Criteria for Bisection Method
The following are the stopping criteria as suggested by [1]: Let be the error tolerance, that is we would like to
obtain the root with an error of at most of . Then, accept kCx as a root of 0xf . If any of the
following criteria is satisfied:
(i) )( kCf (ie the functional value is less than or equal to the tolerance).
(ii)
k
kk
C
CC 1
(ie the relative change is less than or equal to the tolerance).
(iii)
k
ab
2
)(
(ie the length of the interval after k iterations is less than or equal to tolerance).
(iv) The number of iterations k is greater than or equal to a predetermined number, say N.
Theorem 1: The number of iterations, N needed in the Bisection method to obtain an accuracy of is given by:
N
2
10
1010
log
)(log)(log
oo ab
Proof: Let the interval length after N iteration be N
oo ab
2
. So to obtain an accuracy of
N
oo ab
2
.That is
)(2 oo
N
ab
)(
2
oo
N
ab
)log(2log10
oo ab
N
)log(2log10
oo ab
N
2log
)(log
10
10
oo ab
N
2log
)(log)(log
10
1010
oo ab
N as required.
Note: Since the number of iterations, N needed to achieve a certain accuracy depends upon the initial length of
the interval containing the root, it is desirable to choose the initial interval 00 ,ba as small as possible.
We solve 0cos xxxf at [0,1] using Bisection method with the aid of the software, Mathematica
9.0.
3. Comparative Study Of Bisection, Newton-Raphson And Secant Methods Of Root- Finding Problems
International organization of Scientific Research 3 | P a g e
III. NEWTON-RAPHSON METHOD
The Newton-Raphson method finds the slope (tangent line) of the function at the current point and uses
the zero of the tangent line as the next reference point. The process is repeated until the root is found [5-7]. The
method is probably the most popular technique for solving nonlinear equation because of its quadratic
convergence rate. But it is sometimes damped if bad initial guesses are used [8-9].It was suggested however,
that Newton’s method should sometimes be started with Picard iteration to improve the initial guess [9]. Newton
Raphson method is much more efficient than the Bisection method. However, it requires the calculation of the
derivative of a function as the reference point which is not always easy or either the derivative does not exist at
all or it cannot be expressed in terms of elementary function [6,7-8]. Furthermore, the tangent line often shoots
wildly and might occasionally be trapped in a loop [6]. The function, 0xf can be expanded in the
neighbourhood of the root ox through the Taylor
expansion: 0)(("
!2
)(
)()()()(
2
'
o
o
oo xf
xx
xfxxxfxf , where x can be seen as a trial value
for the root at the nth step and the approximate value of the next step 1kx can be derived from
0)()()()( '
11 kkkkk xfxxxfxf .
,...2,1,0,
)('
)(
1 k
xf
xf
xx
k
k
kk called the Newton-Raphson method.
Algorithm of the Newton- Raphson Method
Inputs: f(x) –the given function, xo –the initial approximation, -the error tolerance and N –the maximum
number of iteration.
Output: An approximation to the root x or a message of a failure.
Assumption: x is a simple root of
0xf
Compute )(xf and )(' xf
Compute
)('
)(
1
k
k
kk
xf
xf
xx , for k = 0,1,2,… do until convergence or failure.
Test for convergence of failure: If )( kxf or
k
kk
x
xx 1
or k>N, stop.
End.
It was remarked in [1], that if none of the above criteria has been satisfied, within a predetermined, say,
N, iteration, then the method has failed after the prescribed number of iteration. In this case, one could try the
method again with a different xo. Meanwhile, a judicious choice of xo can sometimes be obtained by drawing the
graph of f(x), if possible. However, there does not seems to exist a clear- cut guideline on how to choose a right
starting point, xo that guarantees the convergence of the Newton-Raphson method to a desire root.
We implement the function, 0cos)( xxxf using the Newton-Raphson method with the aid of the
software, Mathematica 9.0.
IV. THE SECANT METHOD
As we have noticed, the main setback of the Newton-Raphson method is the requirement of finding the
value of the derivative of f(x) at each iterations. There are some functions that are either extremely difficult (if
not impossible) or time consuming. The way out out of this, according to [1] is to approximate the derivative by
knowing the values of the function at that and the previous approximation. Knowing
)( 1kxf , we can then
approximate )(' xf as
1
1)()(
)('
kk
kk
k
xx
xfxf
xf *
Putting * into the Newton iterations, we have:
)()(
))((
1
1
1
kk
kkk
kk
xfxf
xxxf
xx **
** is referred to as Secant iteration (method).
4. Comparative Study Of Bisection, Newton-Raphson And Secant Methods Of Root- Finding Problems
International organization of Scientific Research 4 | P a g e
Algorithm of Secant Method
Input: )(xf -the given function, 10 , xx –the two initial approximation of the root,
-the error tolerance and
N –the maximum number of iterations.
Output: An approximation of the exact solution or a message of failure, for k= 1,2,… do until convergence
or otherwise.
Compute )( kxf and )( 1kxf
Compute the next approximation
)()(
))((
1
1
1
kk
kkk
kk
xfxf
xxxf
xx
Test for convergence or maximum number of iterations: If kk xx 1 or k>N, stop.
We implement the function 0cos)( xxxf , using the Secant method with the aid of software,
Mathematica 9.0.
Analysis of Convergence Rates of Bisection, Newton-Raphson and Secant methods.
Definition: Suppose that the sequence kx converges to . Then, the sequence kx is said to converge to
with order of convergence if there exists a positive constant p such that:
k
k
x
x
k 1
lim =
p
e
e
k
k
1
lim .
Thus, if
=1, the convergence is linear. If =2, the convergence is quadratic, and so on. The number is
called the convergence factor. Based on this definition, we show the rate of convergence of Newton and Secant
methods of root-finding. Meanwhile, we may not border to show that of Bisection method, sequel to the fact that
many literatures consulted are in agreement that Bisection method will always converge, and has the least
convergence rate. It was also maintained that it converges linearly [1-7].
Rate of Convergence of the Newton-Raphson Method
To investigate into the convergence of Newton-Raphson method, we need to apply the Taylor’s theorem. Thus:
Theorem 2: Taylor’s Theorem of order n:
Suppose that the function f(x) possesses continuous derivatives of order up to (n+1) in the interval [a,b], and p
is a point in this interval. Then, for every x in this interval, there exist a number, c between p and x such that
)()(
!
)(
...)(
!2
)("
))((')()( 2
xRpn
n
pf
px
pf
pxpfpfxf n
n
n
where, )(xRn , called the remainder after n terms, is given by:
1
)1(
)(
)!1(
)(
)(
n
n
n px
n
cf
xR
Let us choose a small interval around the root x . Then, for any x in this interval, we have by Taylor’s
theorem of order 1, the following expansion of the function g(x):
)("
!2
)(
)(')()()(
2
kg
x
gxgxg
where , k lies between x and . Now for the Newton method, we have seen that .0)(' g
)("
!2
)(
)()( k
k
k g
x
gxg
)("
!2
)(
)()(
2
kg
x
gxg
5. Comparative Study Of Bisection, Newton-Raphson And Secant Methods Of Root- Finding Problems
International organization of Scientific Research 5 | P a g e
)("
!2
)(
)()(
2
k
k
k g
x
gxg
Since 1 kk xxg and
)(g , this gives: )("
!2
)( 2
1 k
k
k g
x
x
That is )("
2
2
1 k
k
k g
x
x
Or
2
)("
2
1 k
k
k g
e
e
Since k lies between x and , for every k, it follows that
kklim
So, we have
)("
2
)("
limlim 2
1
g
g
e
e k
k
k
k
This shows that Newton-Raphson method converges quadratically. By implication, the quadratic convergence
we mean that the accuracy gets doubled at each iteration.
Rate of convergence of Secant Method
The iterates kx of the Secant method converges to a root of f(x), if the initial values 1,0 xx ,are sufficiently close
to the root. The order of convergence is α where
618.1
2
51
is the golden ratio. In particular, the
convergence is superlinear, but not quite quadratic. This result only holds under some technical conditions,
namely that f(x) be twice continuously differentiable and the root in question be simple. That is with multiplicity
1. If the initial values are not close enough to the root, then there is no guarantee that the Secant method
converges [7].
3.0 Result and Discussion
The Bisection, Newton-Raphson and Secant methods were applied to a single-variable function:
xxxf cos)( on [0,1], using the software, Mathematica 9.0. The results are presented in Table 1 to 3.
Table 1: Iteration Data for Bisection Method
Steps A Function values B Function values
0 0 -1 1 0.459698
1 0.5 -0.377583 1 0.459698
2 0.5 -0.377583 0.75 0.0183111
3 0.625 -0.185963 0.75 0.0183111
4 0.6875 -0.0853349 0.75 0.0183111
5 0.71875 -0.0338794 0.75 0.0183111
6 0.734375 -0.00787473 0.75 0.0183111
7 0.734375 -0.00787473 0.7421875 0.00519571
8 0.73828125 -0.00134515 0.7421875 0.00519571
9 0.73828125 -0.00134515 0.740234375 0.00192387
10 0.73828125 -0.00134515 0.7392578125 0.000289009
11 0.73876953125 -0.000528158 0.7392578125 0.000289009
12 0.739013671875 -0.000119597 0.7392578125 0.000289009
13 0.739013671875 -0.000119597 0.7391357421875 0.0000847007
14 0.73907470703125 -0.0000174493 0.7391357421875 0.0000847007
15 0.73907470703125 -0.0000174493 0.739105224609375 0.0000336253
16 0.73907470703125 -0.0000174493 0.7390899658203125 8.08791x10-6
17 0.7390823364257813 -4.68074x10-6
0.7390899658203125 8.08791x10-6
18 0.7390823364257813 -4.68074x10-6
0.7390861511230469 1.70358x10-6
19 0.7390842437744141 -1.48858x10-6
0.7390861511230469 1.70358x10-6
20 0.7390842437744141 -1.48858x10-6
0.7390851974487305 1.07502x10-7
21 0.7390847206115723 -6.90538x10-7
0.7390851974487305 1.07502x10-7
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International organization of Scientific Research 6 | P a g e
22 0.7390849590301514 -2.91518x10-7
0.7390851974487305 1.07502x10-7
23 0.7390850782394409 -9.2008x10-8
0.7390851974487305 1.07502x10-7
24 0.7390850782394409 -9.2008x10-8
0.7390851378440857 7.74702x10-9
25 0.7390851080417633 -4.21305x10-8
0.7390851378440857 7.74702x10-9
26 0.7390851229429245 -1.71917x10-8
0.7390851378440857 7.74702x10-9
27 0.7390851303935051 -4.72236x10-9
0.7390851378440857 7.74702x10-9
28 0.7390851303935051 -4.72236x10-9
0.7390851341187954 1.51233x10-9
29 0.7390851322561502 -1.60501x10-9
0.7390851341187954 1.51233x10-9
30 0.7390851331874728 -4.63387x10-11
0.7390851341187954 1.51233x10-9
31 0.7390851331874728 -4.63387x10-11
0.7390851336531341 7.32998x10-10
32 0.7390851331874728 -4.63387x10-11
0.7390851334203035 3.43329x10-10
33 0.7390851331874728 -4.63387x10-11
0.7390851333038881 1.48495x10-10
34 0.7390851331874728 -4.63387x10-11
0.7390851332456805 5.10784x10-11
35 0.7390851331874728 -4.63387x10-11
0.7390851332165767 2.36988x10-12
36 0.7390851332020247 -2.19844x10-11
0.7390851332165767 2.36988x10-12
37 0.7390851332093007 -9.80727x10-12
0.7390851332165767 2.36988x10-12
38 0.7390851332129387 -3.71869x10-12
0.7390851332165767 2.36988x10-12
39 0.7390851332147577 -6.7446x10-13
0.7390851332165767 2.36988x10-12
40 0.7390851332147577 -6.7446x10-13
0.7390851332156672 8.47655x10-13
41 0.7390851332147577 -6.7446x10-13
0.7390851332152124 8.65974x10-14
42 0.7390851332149850 -2.93876x10-13
0.7390851332152124 8.65974x10-14
43 0.7390851332150987 -1.03584x10-13
0.7390851332152124 8.65974x10-14
44 0.7390851332151556 -8.54872x10-15
0.7390851332152124 8.65974x10-14
45 0.7390851332151556 -8.54872x10-15
0.7390851332151840 3.90799x10-14
46 0.7390851332151556 -8.54872x10-15
0.7390851332151698 1.53211x10-14
47 0.7390851332151556 -8.54872x10-15
0.7390851332151627 3.44169x10-15
48 0.7390851332151591 -2.55351x10-15
0.7390851332151627 3.44169x10-15
49 0.7390851332151591 -2.55351x10-15
0.7390851332151609 4.44089x10-16
50 0.7390851332151600 -1.11022x10-15
0.7390851332151609 4.44089x10-16
51 0.7390851332151605 -3.33067x10-16
0.7390851332151609 4.44089x10-16
52 0.7390851332151607 0 0.7390851332151607 0
Table 1 shows the iteration data obtained for Bisection method with the aid of Mathematica 9.0. It was observed
that in Table 1 that using the Bisection method, the function, f(x) = x-cosx = 0 at the interval [0,1] converges to
0.7390851332151607 at the 52 second iterations with error level of 0.000000.
Table 2: Iteration Data for Newton- Raphson Method with xo = 0.5.
Steps xk f(xk+1)
1 0.5 -9.62771
2 -9.62771 -2.43009
3 -2.43009 2.39002
4 2.39002 0.535581
5 0.535581 0.750361
6 0.750361 0.739113
7 0.739113 0.739085
8 0.739085 0.739085
Table 2 revealed that with xo=0.5, the function converges to 0.739085 the 8th
iteration with error 0.000000.
Table 3: Iteration Data for the Secant Method with [0,2]
xo 0
x1 1
x2 0.685073
x3 0.736299
x4 0.739119
x5 0.739085
x6 0.739085
x7 0.739085
From Table3, we noticed that the function converges to 0.739085 after the 6th
iteration with error 0.000000
7. Comparative Study Of Bisection, Newton-Raphson And Secant Methods Of Root- Finding Problems
International organization of Scientific Research 7 | P a g e
Comparing the results of the three methods under investigation, we observed that the rates of
convergence of the methods are in the following order: Secant method > Newton-Raphson method > Bisection
method. This is in line with the findings of [4]. Comparing the Newton-Raphson method and the Secant method,
we noticed that theoretically, Newton’s method may converge faster than Secant method (order 2 as against
α=1.6 for Secant). However, Newton’s method requires the evaluation of both the function f(x) and its derivative
at every iteration while Secant method only requires the evaluation of f(x). Hence, Secant method may
occasionally be faster in practice as in the case of our study. (see Tables 2 and 3). In Ref. [10-11], it was argued
that if we assume that evaluating f(x) takes as much time as evaluating its derivative, and we neglect all other
costs, we can do two iterations of Secant (decreasing the logarithm of error by factor α2
=2.6) for the same cost
as one iteration of Newton-Raphson method (decreasing the logarithm of error by a factor 2). So, on this
premises also, we can claim that Secant method is faster than the Newton’s method in terms of the rate of
convergence.
V. CONCLUSION
Based on our results and discussions, we now conclude that the Secant method is formally the most
effective of the methods we have considered here in the study. This is sequel to the fact that it has a converging
rate close to that of Newton-Raphson method, but requires only a single function evaluation per iteration. We
also concluded that though the convergence of Bisection is certain, its rate of convergence is too slow and as
such it is quite difficult to extend to use for systems of equations.
REFERENCE
[1] Biswa Nath Datta (2012), Lecture Notes on Numerical Solution of root Finding Problems.
www.math.niu.edu/dattab.
[2] Iwetan, C.N, Fuwape,I.A, Olajide, M.S and Adenodi, R.A(2012), Comparative Study of the Bisection and
Newton Methods in solving for Zero and Extremes of a Single-Variable Function. J. of NAMP
vol.21pp173-176.
[3] Charles A and W.W Cooper (2008), Non-linear Power of Adjacent Extreme Points Methods in Linear
Programming, Econometrica Vol. 25(1) pp132-153.
[4] Srivastava,R.B and Srivastava, S (2011), Comparison of Numerical Rate of Convergence of Bisection,
Newton and Secant Methods. Journal of Chemical, Biological and Physical Sciences. Vol 2(1) pp 472-
479.
[5] Ehiwario, J.C (2013), Lecture Notes on MTH213: Introduction to Numerical Analysis, College of
Education Agbor. An Affiliate of Delta State University, Abraka.
[6] http://www.efunda.com/math/num_rootfinding-cfm February,2014
[7] Wikipedia, the free Encyclopedia
[8] Autar, K.K and Egwu, E (2008), http://www.numericalmethods.eng.usf.edu Retrieved on 20th
February,
2014.
[9] McDonough, J.M (2001), Lecture in Computationl Numerical Analysis . Dept. of Mechanical
Engineering, University of Kentucky.
[10] Allen, M.B and Isaacson E.L (1998), Numerical Analysis for Applied Science. John Wiley and sons.
Pp188-195.